Mathematical Models of Cell-Based Morphogenesis: Passive and Active Remodeling (Theoretical Biology) 9811929157, 9789811929151

Table of contents :
Contents
Chapter 1: Introduction
References
Chapter 2: Cell Center Model
2.1 Cell Properties (Honda 1978, 1983)
2.2 Mathematics of the Cell Center Model (Honda 1983)
2.3 Dirichlet Approximation (Honda 1978)
2.4 Deviation from Dirichlet Domains (Honda 1978)1
2.5 Approximation of Actual Cell Patterns Using the Cell Center Model (Honda 1978)
2.6 Cell Division
2.7 Cell Disappearance1
2.8 Summary
References
Chapter 3: Applications of the Cell Center Model
3.1 Neat Arrangement of Cells in the Mammalian Epidermis (Honda et al. 1996)
3.2 Cell Patterns Consisting of Large and Small Cells (Honda et al. 2000)
3.3 Formation of Pore Patterns Using the Dirichlet Geometry (Honda and Yoshizato 1994a, b, c)
3.4 From Polygonal Patterns to Branching Patterns
3.4.1 Formation of Blood Vessel Branching In Vivo
3.4.2 Computer Simulation of the Branching Formation of Blood Vessels
3.5 Other Applications of the Geometry of Dirichlet Domains and Voronoi Polyhedra
3.6 Summary
References
Chapter 4: Vertex Model
4.1 Cell Boundaries and Tricellular Contact
4.2 Geometry of a Trijunction of Three Lines (Honda 1983)
4.3 Boundary Shortening (BS) Procedure (Honda and Eguchi 1980)3
4.4 Discrimination of Normal Epithelia from Nonepithelial Tissues Using the BS Procedure (Honda 1983)
4.5 Tissue Transformation from an Aggregate of Packed Cells to a Surface-Contracting Cell Sheet
4.6 Reconnection of Paired Vertices of an Edge
4.7 Vertex Dynamics (Nagai and Honda 2001)
4.8 Summary
References
Chapter 5: Applications of 2D Cell Models
5.1 Wound Closure in Epithelial Tissues (Nagai and Honda 2009)
5.2 Cell Death Leads to Ordered Cell Patterns
5.3 Passively Elongated Epithelial Tubes (Honda et al. 2009)
5.4 Cell Patterns Composed of Heterogeneous Cells
5.4.1 Differential Cell Adhesion Theory
5.4.2 Two Types of Cells in the Oviduct Epithelium
5.4.3 The Kagome Pattern (Star Pattern)6
5.4.4 A Balanced State between Boundary Contraction and Differential Adhesion6
5.4.5 Quantitative Estimation of the Difference in Intercellular Adhesions6
5.4.6 Computer Simulations of the Kagome-Checkerboard Pattern Transformation6
5.4.7 Molecular Basis of Cell Adhesion between Heterogenic Cell Types
5.5 Summary
References
Chapter 6: 3D Vertex Model
6.1 Expression of an Aggregation of Polyhedral Cells
6.2 Equation of Motion in 3D Vertex Dynamics (Honda et al. 2004)7
6.3 Reconnection of Neighboring Vertices7
6.4 Structure of Packed Polyhedra in 3D Space7
6.5 Formation of a Spherical Cell Aggregate
6.6 Flattening of a Cell Aggregate by Centrifugal Force7
6.7 Viscoelastic Properties of Cell Aggregates7
6.8 Modification of 3D Vertex Dynamics
6.9 Transition of a Cell Aggregate to a Vesicle (Honda et al. 2008)
6.10 Fundamentals of Vertex Dynamics
6.11 Summary
References
Chapter 7: The World of Epithelial Sheets
7.1 Morphogenesis of Multicellular Animals Is Deformation of a Closed Epithelial Envelope
7.2 Classification of Epithelial Cells
7.3 Crucial Roles of Vacuolar Apical Compartments (VACs) in Epithelization
7.4 Evolutionary Merit of Enclosure of the Animal Body Within an Epithelial Cell Sheet
7.5 Signaling Molecules for Formation of the Apicobasal Polarity of Epithelial Cells
7.6 Summary
References
Chapter 8: Cells Themselves Produce Force for Active Remodeling
8.1 Cell Rearrangement Involving Cell Intercalation
8.2 Convergent Extension (CE) Mediated by an Anisotropic Contractile Force in 3D Space
8.3 CE Mediated by an Anisotropic Contractile Force on a Cylindrical Surface10
8.4 Global Dynamics of a Cylindrical Surface10
8.5 CE Mediated by Anisotropic Cell Stiffness on a Cylindrical Surface10
8.6 Planar Cell Polarity Signaling Links CE Orientations of Tissues
8.7 Mathematical Modeling of Supracellular Actomyosin Cable Formation11
8.8 Observation of PCP Signaling Proteins in the Neural Plate11
8.9 Invagination of Epithelial Sheets
8.10 Summary
References
Chapter 9: Expansion of Shape-Dimension
9.1 Twisting of a Strip of Sheet: A Narrow Rectangular Sheet (Honda et al. 2019)
9.2 Twisting of the Heart Tube
9.2.1 Helical Looping in Computer Simulations
9.2.2 Mechanism of Determination of Output of the Handedness of Helical Heart Tubes
9.2.2.1 Anisotropic Convergent Extension (CE)
9.2.2.2 Peculiar Remodeling of an Artificial Tube via CE of Constituent Cells
9.2.3 Computer Simulations of the Initial Heart Tube13
9.2.4 Position-Specific Deformation of Cell Colonies in the Process of Helix Loop Formation13
9.2.5 Mechano-physical Mechanism That Determines the Handedness of the Helical Loop13
9.2.5.1 Distinctive Feature of the Cell-Based Vertex Dynamics Model13
9.2.5.2 Intrinsic and Extrinsic Factors Causing Left-Handed Helical Looping13
9.2.5.3 Consideration of CE of Collective Cells Across Different Animal Species13
9.2.6 Conclusion Regarding the Mechanism Underlying Left-Handed Helical Looping of the Heart Tube
9.3 Summary
References
Chapter 10: Mathematical Cell Models and Morphogenesis
10.1 Mathematical Cell Models Are a Bridge to Link Shape Formation with Genes
10.2 Self-Construction of Shapes Recapitulated via Mathematical Models
10.3 Successive Self-Construction
10.4 Summary
Reference

Citation preview

Theoretical Biology

Hisao Honda Tatsuzo Nagai

Mathematical Models of Cell-Based Morphogenesis Passive and Active Remodeling

Theoretical Biology Series Editor Yoh Iwasa, Kyushu University, Fukuoka, Japan

The “Theoretical Biology” series publishes volumes on all aspects of life sciences research for which a mathematical or computational approach can offer the appropriate methods to deepen our knowledge and insight. Topics covered include: cell and molecular biology, genetics, developmental biology, evolutionary biology, behavior sciences, species diversity, population ecology, chronobiology, bioinformatics, immunology, neuroscience, agricultural science, and medicine. The main focus of the series is on the biological phenomena whereas mathematics or informatics contribute the adequate tools. Target audience is researchers and graduate students in biology and other related areas who are interested in using mathematical techniques or computer simulations to understand biological processes and mathematicians who want to learn what are the questions biologists like to know using diverse mathematical tools.

Hisao Honda • Tatsuzo Nagai

Mathematical Models of Cell-Based Morphogenesis Passive and Active Remodeling

Hisao Honda Graduate School of Medicine Kobe University Kobe, Hyogo, Japan

Tatsuzo Nagai Kyushu Kyoritsu University Kitakyushu, Fukuoka, Japan

ISSN 2522-0438 ISSN 2522-0446 (electronic) Theoretical Biology ISBN 978-981-19-2915-1 ISBN 978-981-19-2916-8 (eBook) https://doi.org/10.1007/978-981-19-2916-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2

Cell Center Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Cell Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematics of the Cell Center Model . . . . . . . . . . . . . . . . . . 2.3 Dirichlet Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Deviation from Dirichlet Domains . . . . . . . . . . . . . . . . . . . . . 2.5 Approximation of Actual Cell Patterns Using the Cell Center Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Cell Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Cell Disappearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

5 5 6 8 10

. . . . .

10 12 14 14 15

Applications of the Cell Center Model . . . . . . . . . . . . . . . . . . . . . . 3.1 Neat Arrangement of Cells in the Mammalian Epidermis . . . . . 3.2 Cell Patterns Consisting of Large and Small Cells . . . . . . . . . . 3.3 Formation of Pore Patterns Using the Dirichlet Geometry . . . . 3.4 From Polygonal Patterns to Branching Patterns . . . . . . . . . . . . 3.4.1 Formation of Blood Vessel Branching In Vivo . . . . . . . 3.4.2 Computer Simulation of the Branching Formation of Blood Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Other Applications of the Geometry of Dirichlet Domains and Voronoi Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

19 19 24 25 29 29

.

30

. . .

35 36 36

Vertex Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Cell Boundaries and Tricellular Contact . . . . . . . . . . . . . . . . . . 4.2 Geometry of a Trijunction of Three Lines . . . . . . . . . . . . . . . . .

39 39 41

3

4

v

vi

Contents

4.3 4.4

Boundary Shortening (BS) Procedure . . . . . . . . . . . . . . . . . . . Discrimination of Normal Epithelia from Nonepithelial Tissues Using the BS Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Tissue Transformation from an Aggregate of Packed Cells to a Surface-Contracting Cell Sheet . . . . . . . . . . . . . . . . . . . . 4.6 Reconnection of Paired Vertices of an Edge . . . . . . . . . . . . . . 4.7 Vertex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

.

42

.

45

. . . . .

47 49 51 54 55

Applications of 2D Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Wound Closure in Epithelial Tissues . . . . . . . . . . . . . . . . . . . . 5.2 Cell Death Leads to Ordered Cell Patterns . . . . . . . . . . . . . . . . 5.3 Passively Elongated Epithelial Tubes . . . . . . . . . . . . . . . . . . . . 5.4 Cell Patterns Composed of Heterogeneous Cells . . . . . . . . . . . . 5.4.1 Differential Cell Adhesion Theory . . . . . . . . . . . . . . . . . 5.4.2 Two Types of Cells in the Oviduct Epithelium . . . . . . . . 5.4.3 The Kagome Pattern (Star Pattern) . . . . . . . . . . . . . . . . 5.4.4 A Balanced State between Boundary Contraction and Differential Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Quantitative Estimation of the Difference in Intercellular Adhesions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Computer Simulations of the Kagome–Checkerboard Pattern Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.7 Molecular Basis of Cell Adhesion between Heterogenic Cell Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 62 64 69 69 72 73 74 75 76 77 78 80

6

3D Vertex Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Expression of an Aggregation of Polyhedral Cells . . . . . . . . . . 6.2 Equation of Motion in 3D Vertex Dynamics . . . . . . . . . . . . . . 6.3 Reconnection of Neighboring Vertices . . . . . . . . . . . . . . . . . . 6.4 Structure of Packed Polyhedra in 3D Space . . . . . . . . . . . . . . 6.5 Formation of a Spherical Cell Aggregate . . . . . . . . . . . . . . . . 6.6 Flattening of a Cell Aggregate by Centrifugal Force . . . . . . . . 6.7 Viscoelastic Properties of Cell Aggregates . . . . . . . . . . . . . . . 6.8 Modification of 3D Vertex Dynamics . . . . . . . . . . . . . . . . . . . 6.9 Transition of a Cell Aggregate to a Vesicle . . . . . . . . . . . . . . . 6.10 Fundamentals of Vertex Dynamics . . . . . . . . . . . . . . . . . . . . . 6.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 83 . 83 . 85 . 86 . 86 . 88 . 89 . 94 . 97 . 98 . 104 . 108 . 109

7

The World of Epithelial Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1 Morphogenesis of Multicellular Animals Is Deformation of a Closed Epithelial Envelope . . . . . . . . . . . . . . . . . . . . . . . . 113

Contents

vii

7.2 7.3

. 119

Classification of Epithelial Cells . . . . . . . . . . . . . . . . . . . . . . . Crucial Roles of Vacuolar Apical Compartments (VACs) in Epithelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Evolutionary Merit of Enclosure of the Animal Body Within an Epithelial Cell Sheet . . . . . . . . . . . . . . . . . . . . . . . 7.5 Signaling Molecules for Formation of the Apicobasal Polarity of Epithelial Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9

. 122 . 124 . 124 . 126 . 127

Cells Themselves Produce Force for Active Remodeling . . . . . . . . . 8.1 Cell Rearrangement Involving Cell Intercalation . . . . . . . . . . . . 8.2 Convergent Extension (CE) Mediated by an Anisotropic Contractile Force in 3D Space . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 CE Mediated by an Anisotropic Contractile Force on a Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Global Dynamics of a Cylindrical Surface . . . . . . . . . . . . . . . . 8.5 CE Mediated by Anisotropic Cell Stiffness on a Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Planar Cell Polarity Signaling Links CE Orientations of Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Mathematical Modeling of Supracellular Actomyosin Cable Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Observation of PCP Signaling Proteins in the Neural Plate . . . . . 8.9 Invagination of Epithelial Sheets . . . . . . . . . . . . . . . . . . . . . . . 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion of Shape–Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Twisting of a Strip of Sheet: A Narrow Rectangular Sheet . . . . 9.2 Twisting of the Heart Tube . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Helical Looping in Computer Simulations . . . . . . . . . . 9.2.2 Mechanism of Determination of Output of the Handedness of Helical Heart Tubes . . . . . . . . . . . . . . . 9.2.3 Computer Simulations of the Initial Heart Tube . . . . . . 9.2.4 Position-Specific Deformation of Cell Colonies in the Process of Helix Loop Formation . . . . . . . . . . . . . . . . 9.2.5 Mechano-physical Mechanism That Determines the Handedness of the Helical Loop . . . . . . . . . . . . . . . . . 9.2.6 Conclusion Regarding the Mechanism Underlying Left-Handed Helical Looping of the Heart Tube . . . . . . 9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

129 130 130 133 136 138 141 142 143 146 150 150 153 153 160 161

. 167 . 170 . 171 . 175 . 183 . 183 . 185

viii

10

Contents

Mathematical Cell Models and Morphogenesis . . . . . . . . . . . . . . . 10.1 Mathematical Cell Models Are a Bridge to Link Shape Formation with Genes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Self-Construction of Shapes Recapitulated via Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Successive Self-Construction . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 189 . 189 . . . .

190 190 191 192

Chapter 1

Introduction

Genes are deeply related to the shape of living organisms, and elucidation of a pathway of shape formation from genes is one of the fundamental problems in biology. Information on genes inherited from ancestors defines proteins as having several functions, i.e., enzymes synthesizing and degrading materials, signaling molecules providing signal transduction and regulation, and cytoskeleton-related molecules producing force and supporting configuration for morphogenesis. Recently, much knowledge related to the shape formation of biological bodies has accumulated. For example, membrane ruffles (lamellipodia), cell protrusions (filopodia), and focal adhesion with stress fibers are organized with the actin skeleton and are regulated by members of the Rho family of small GTPases (Rac, Rho, and Cdc42). Rac regulates the accumulation of actin filaments at the plasma membrane to produce lamellipodia and membrane ruffles, Rho controls the assembly of actin stress fibers and focal adhesion complexes, and Cdc42 stimulates the formation of filopodia (Olson et al. 1995). In addition, we have knowledge of planar cell polarity (PCP) on the apical surface of epithelial cells (Adler 2002; Shimada et al. 2006; Harumoto et al. 2010), the mechanism underlying the formation of apical–basal (AB) polarity within epithelial cells (Suzuki and Ohno 2006), directional proliferation of cells, and anisotropic contraction of cells with polarized contracting molecules (Segalen and Bellaiche 2009). However, this accumulated knowledge does not automatically lead us to an understanding of shape formation during developmental processes. We do not have methods in hand to integrate the knowledge related to the shape formation into an understanding of the formation of shapes, especially largescale shapes. We need mathematical models for this integration. When we view the field of physics, there are various shapes, e.g., a spherical drop of water, a polyhedral crystal of quartz, the hexagonal pattern of a snowflake, and a wind-wrought pattern on sand, which consist of constituent elements: atoms, molecules, or particles. These shapes are self-constructed in assemblages of their constituent elements, that is, spontaneously constructed under certain conditions without outside aids. These self-constructions have been described using mathematical models involving equations of motion. On the other hand, multicellular © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_1

1

2

1 Introduction

organisms consist of cells. Cells are constituent elements. We would like to consider that the shape formation of multicellular organisms is also a self-construction of cells. We attempt to describe this shape formation using mathematical models. In this book, we establish mathematical cell models to elucidate the mechanisms underlying the formation of biological shapes. We would like to show the possibility that the formation of some biological shapes is the result of self-construction. We hope that many would agree with us that mathematical cell models in addition to conventional tools in biological sciences are indispensable to understanding shape formation. We would like to use “self-construction” in this book instead of “selforganization” that includes pattern formations via chemical reactions. We are interested in cells that construct a physico-mechanic structure via physical force. Here, we will briefly introduce the chapters in this book. We describe two mathematical cell models to which the authors have contributed. The first is a cell center model, where there is a one-to-one correspondence between cells and points (cell centers). Cell proliferation and cell disappearance (cell loss and apoptosis) were incorporated into the cell center model (Chap. 2). The cell center model was applied to elucidate the formation of neat cell arrangements in the epidermis, cell patterns consisting of heterogeneous-sized cells, capillary networks, and the branching patterns of blood vessels (Chap. 3). The second mathematical cell model is a vertex model. The history of its completion in biological cells is described (Chap. 4). The vertex model was applied to elucidate the mechanisms of wound healing of the epithelium and ordered pattern formation involving apoptosis. Pattern formation with differential cell adhesion is also described (Chap. 5). Next, the vertex model is extended from a two-dimensional (2D) to a three-dimensional (3D) model. The origin of the viscoelastic properties of a cell aggregate is discussed. To elucidate the development of the mammalian blastocyst or the formation of an epithelial vesicle, a cell aggregate involving a large cavity is described (Chap. 6). Then, the epithelial tissues are comprehensively reviewed from the perspective of the generalized points, and the process of polarity formation of the epithelium is explained (Chap. 7). The vertex model has been ascertained to recapitulate active remodeling of tissues, during which the cells themselves produce force for active remodeling. Therefore, convergent extension (CE) of collective cells takes place, resulting in elongation of bodies along the body axis (Chap. 8). Furthermore, the vertex model describes twisting of tissue, in which chiral properties emerge. A flat strip twists three-dimensionally, and a straight tube becomes a helical loop. Helical looping of a mathematical artificial tube contributes to understanding cardiac loop formation of the embryonic heart tube (Chap. 9). Finally, in the last chapter, we conclude that mathematical cell models are indispensable tools to understand the shape formation of living organisms. Successive iterations of self-constructions will lead to an understanding of the remarkable and mysterious morphogenesis that occurs during the development of living organisms (Chap. 10). Acknowledgments We would like to thank Professor Masaharu Tanemura (Institute of Statistical Mathematics, Tokyo) for cooperation in modeling research, Professor Yasuhiro Minami (Kobe

References

3

University Graduate School of Medicine, Kobe) and Professor Shigeo Hayashi (RIKEN Rikagaku Kenkyusho, Kobe) for encouraging us to perform our research, Dr. Hideru Togashi (Kobe University Graduate School of Medicine, Kobe) and Dr. Tetsuya Hiraiwa (Mechanobiology Institute, National University of Singapore, Singapore) for information in preparing this book, and Professor Yoh Iwasa (Kyushu University, Fukuoka) for advice in preparing this book. We thank the RIKEN Integrated Cluster of Clusters (Wako, Saitama) and the Institute of Statistical Mathematics (Tachikawa, Tokyo) for their supercomputation facilities. This work was funded by grants from Japan Society for the Promotion of Science KAKENHI (Kagaku-Kenkyuhi).

References Adler, P.N.: Planar Signaling and Morphogenesis in Drosophila. Dev. Cell. 2, 525–535 (2002) Harumoto, T., et al.: Atypical cadherins Dachsous and Fat control dynamics of noncentrosomal microtubules in planar cell polarity. Dev. Cell. 19, 389–401 (2010) Olson, M.F., et al.: An essential role for Rho, Rac, and Cdc42 GTPases in cell cycle progression through G1. Science. 269, 1270–1272 (1995) Segalen, M., Bellaiche, Y.: Cell division orientation and planar cell polarity pathways. Semin. Cell Dev. Biol. 20, 972–977 (2009) Shimada, Y., et al.: Polarized transport of Frizzled along the planar microtubule arrays in Drosophila wing epithelium. Dev. Cell. 10, 209–222 (2006) Suzuki, A., Ohno, S.: The PAR-aPKC system: lessons in polarity. J. Cell. Sci. 119, 979–987 (2006)

Chapter 2

Cell Center Model

Outline We describe two mathematical cell models in this book. Here, the first cell model, a cell center model, is described. In the cell center model, there is a one-toone correspondence between cells and cell centers. Actual cell patterns are approximated by the cell center model, in which the discrepancy between the model and real patterns is quantitatively estimated. Cell proliferation and cell disappearance behaviors (cell loss and apoptosis) are described by this model.

2.1

Cell Properties (Honda 1978, 1983)

When there are many flexible spheres of cells in a limited space and they are packed tighter, they show a polygonal cellular pattern (Fig. 2.1). Generally, an isolated cell that is suspended in a medium is spherical in shape. This is true of animal cells and of plant cells (e.g., protoplasts, when plant cells remove their cell walls via specific enzymes; Nagata and Takebe 1970). When a cell attaches to the substratum, its thin cytoplasm (filopodia with webbing or lamellipodia) often spreads radially in a “fried egg” shape. Figure 2.2 shows the spreading process of a rat hepatocyte attached to a serum-pretreated glass surface (Seglen and Gjessing 1978). Similar processes have been observed in other cell types, e.g., a human cell line (normal human diploid WI-38 cells, Rajaraman et al. 1974), mouse embryo fibroblasts (Figs. 2–4 in Ivanova et al. 1976), sea urchin coelomocytes (Fig. la in Edds 1980), and pig kidney epithelial cells (PK15 in Connolly et al. 1981). When spreading cells collide with each other, they fail to move over one another’s upper surfaces (DiPasquale and Bell 1974) and eventually arrange themselves as a confluent monolayer (cells contacted by neighbors on all sides). In a culture of dissociated hepatocytes, spherical cells attach to and spread on the bottom surface, forming a monolayered cell sheet with a polygonal pattern (Seglen and Gjessing 1978; Rubin et al. 1978). A schematic drawing is shown in Fig. 2.2. In a culture of dissociated epithelial cells from embryonic chick pigmented retina, a monolayered sheet shows a polygonal pattern. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_2

5

6

2 Cell Center Model

Fig. 2.1 Spherical cells packed in a plane (starfish eggs). Bar, 500 μm

Fig. 2.2 Drawing of a spherical cell spreading on the surface of a culture dish

Here, stable contacts with junctional complexes were found between cells (Crawford et al. 1972; Crawford 1975; Newsome et al. 1974; Middleton and Pegrum 1976). Radial spreading and collision of cells have also been observed in vivo. The spreading of epithelial cells during wound closure was studied in the transparent fin of Xenopus laevis tadpoles (Radice 1980). The marginal cells around the wound extended broad lamellipodia in a radial fashion. When the lamellipodia met near the center of the wound, they adhered to each other and stopped advancing (see Fig. 2.9 and Figs. 5.1 and 5.2 in Chap. 5). After artificially wounding the epithelium by removing a small number of cells, Yoji Ogita and Shoichi Higuchi observed that in the tissue of living mammals (the corneal endothelium of cats) cells began to flatten and migrate to heal the wounded epithelium (Honda et al. 1982). On the basis of these observations, we will initially consider cells having certain properties, such as spreading and formation of boundaries between cells.

2.2

Mathematics of the Cell Center Model (Honda 1983)

In addition to biological cells, spheres of deformable plastic substances (e.g., fat clay or plasticine) show similar behaviors. When many uniform spheres of fat clay are distributed in a plane and pressed with a transparent glass plate, they are deformed into a polygonal pattern (Fig. 2.3). These pattern formations can be recapitulated in a simple geometry, as shown in Fig. 2.4. Initially, there are many points that are fixed in a plane (Fig. 2.4a), and the points become growing circles. Then, the circles are

2.2 Mathematics of the Cell Center Model (Honda 1983)

7

Fig. 2.3 Spheres of fat clay in the array pressed with a transparent glass plate [Reproduced from Fig. 5a and c in Honda 1983 with permission of Elsevier]

Fig. 2.4 Point distribution defines a polygonal pattern. Points (a) become centers of circles (b), and the circles grow. The circles overlap with each other and define boundaries (c). As the circles grow, the number of boundaries increases, the lengths of boundaries elongate, and the boundaries connect with each other (d~f). Finally, the boundaries form a polygonal pattern (g). Polygons in the polygonal pattern correspond one to one with the initial points (h and a) [Original]

packed tighter. The circles touch each other. When the circles become larger, they overlap with each other and cover the plane without leaving any gaps. However, in reality, a pair of neighboring circles defines a boundary as the intersecting lines of two circles. The straight boundary is a bisector of the connecting line of two neighboring circles and is perpendicular to the connecting line (Fig. 2.4c). When

8

2 Cell Center Model

Fig. 2.5 Method to obtain the approximated Dirichlet center. The line AiPi is the direction to the Dirichlet center, where ∠Ai-1AiPi ¼ 180
– ∠BiAiAi+1. When all AiPi do not intersect at a single point, a reasonably approximated center is defined as described in the text. P is the approximated Dirichlet center of the domain [Original]

three straight boundaries among three neighboring cells connect, a point is defined as a polygon corner. Then, the plane is partitioned into domains, which are convex polygons (i.e., having all interior angles less than 180
) that cover the plane without leaving any gaps or overlaps (Fig. 2.4h). Such a domain is mathematically defined as a polygon whose interior consists of all the points in the plane that are nearer to a particular point than to any other point. This is called a Dirichlet domain (Dirichlet 1850; Coxeter 1969; Loeb 1976), which is the same concept as the two-dimensional case of a Voronoi polyhedron (Voronoi 1908; Rogers 1964). Such a domain is often called another name, a Maijering cell (see Gilbert 1962) or a Wigner–Seitz cell (Wigner and Seitz 1933). The geometry of Voronoi polygons was used to analyze space division patterns via packing of hard disks (Ogawa and Tanemura 1974) and the territories of individuals in behavioral biology (Hasegawa and Tanemura 1976; Tanemura and Hasegawa 1980). We also constructed a computer program of Dirichlet domains (Voronoi polygons) and used the domains as a cell center model (Honda 1978). This model is one of the cell models applied to study collective biological cells (Honda 1983).

2.3

Dirichlet Approximation (Honda 1978)1

Cellular polygonal patterns look like Dirichlet domains. We developed a method to make a Dirichlet domain that is as similar as possible to an actual cellular pattern. Centers of the circles (Dirichlet centers) in a plane correspond one-to-one to Dirichlet polygonal domains. Dirichlet domains uniquely determine Dirichlet centers as follows: There is a method to determine the direction to a Dirichlet center from the vertex (corner) of a domain, at which three sides meet (Chap. 13 in Loeb 1976). In Fig. 2.5, Ai is a corner of the domain. The line AiPi is the direction to the

1

Reproduced from excerpts on pages 193–196 and 205 in Honda 1983 with permission of Elsevier.

2.3 Dirichlet Approximation (Honda 1978)

9

Fig. 2.6 Examples of polygonal patterns with large and small Δ values. (a) Δ ¼ 1  102. (b) Δ ¼ 0. Thin lines in each polygon are directions to the Dirichlet center. Solid circles in (b) are Dirichlet centers. Dotted lines in (a) show the approximated Dirichlet pattern based on the approximated Dirichlet centers (solid circles in a) [Reproduced from Fig. 3 in Honda et al. 1983 with permission of Elsevier (Copyright 1984)]

Dirichlet center, where ∠Ai–1AiPi ¼ 180
– ∠BiAiAi+1. When the directions to the Dirichlet center from every vertex of a particular domain are determined, they all intersect at a single point P, which is the Dirichlet center (Fig. 2.6b). However, when the domain is a real cellular pattern, i.e., a general polygon that is not necessarily a Dirichlet polygon, the lines of the direction do not intersect at a single point (Fig. 2.6a). Therefore, the reasonably approximated center P is defined so that the value of Σili2 becomes minimal, (Σili2)min where li is the distance from P to line AAiPi, Σi is the summation from i ¼ 1 to n, and n is the number of vertices in the domain (Fig. 2.5). After we calculate P of all domains, we can make a polygonal pattern based on these P, which is the approximated Dirichlet domain (dotted lines in Fig. 2.6a).

10

2.4

2 Cell Center Model

Deviation from Dirichlet Domains (Honda 1978)1

To quantitatively determine how closely a polygonal pattern is approximated by Dirichlet domains, a deviation value (the Δ value) is defined as an average value of (Σili2)min/nJ over all domains in a pattern, where nJ is the number of vertices of the Jth domain and the scale of a pattern is determined so that the averaged area of the polygons becomes unity. For practical calculation, Δ¼




 
 ΣJ¼1 N ΔJ =N = ΣJ¼1 N SJ =N

ð2:1Þ

where N is the number of all domains in a pattern, SJ is the area of the J-th domain, and ΔJ ¼ Σi¼1 nJ li 2


min

=nJ

ð2:2Þ

Δ is zero when the pattern consists of exact Dirichlet domains, while it becomes larger as the pattern diverges away from the Dirichlet domains.

2.5

Approximation of Actual Cell Patterns Using the Cell Center Model (Honda 1978)

There are one-to-one correspondences between polygons and centers in Dirichlet domain patterns. In computer simulations, the description of the behaviors of polygonal patterns using centers of polygons is easier than using the polygons themselves. We will refer to a mathematical model using Dirichlet domains as a cell center model. Some polygonal patterns (Fig. 2.7) were approximated by cell center models (broken line) and compared with each actual pattern (solid line) with deviation values: polygonal patterns of pressed fat clay spheres (A) (Δ ¼ 0.324  102), closed strings of stretched rubber (B) (Δ ¼ 1.06  102), soap bubbles on a plane (C) (Δ ¼ 1.55), a cultured monolayer of retinal pigment epithelial cells (D) (Δ ¼ 1.62  102), a cultured monolayer of lung epithelial cells (E) (Δ ¼ 2.79  102), and a cultured monolayer of chondrocytes (F) (Δ ¼ 2.18  102). It should be noted that the Δ value of pressed fat clay spheres is remarkably small. The pressed fat clay spheres seem to be precisely recapitulated by the cell center model.

2.5 Approximation of Actual Cell Patterns Using the Cell Center Model (Honda 1978)

11

Fig. 2.7 Photographs of polygonal patterns and their analyses with the Dirichlet approximation and the BS procedure. Solid line, trace pattern after photo; Broken line, pattern approximated by the Dirichlet domain; Dotted line, pattern after the BS procedure. (a) Pressed fat clay spheres (Δ ¼ 0.324  102, s ¼ 1.033) [Reproduced from Fig. 5b; Fig. 5d; Fig. 6 in Honda 1983 with permission of Elsevier]. (b) Closed strings of stretched rubber (Δ ¼ 1.06  102, s ¼ 0.072) [Reproduced from Fig. 25e; Fig. 25c; Fig. 31 in Honda 1983 with permission of Elsevier]. (c) Soap bubbles on a plane (Δ ¼ 1.55, s ¼ 0.065) [Reproduced from Fig. 25d in Honda 1983 with permission of Elsevier; Original; Reproduced from Fig. 4a in Honda and Eguchi 1980 with permission of Elsevier]. (d) Cultured monolayer of retinal pigment epithelial cells, courtesy of Kunio Yasuda (Δ ¼ 1.62  102, s ¼ 0.687). Bar, 50 μm [Reproduced from Plate Ia; Fig. 6 in

12

2.6

2 Cell Center Model

Cell Division

A cell divides into two cells. We observed cell division of the epithelial cell in the blastular wall of the starfish (Honda et al. 1984). Epithelial cells first rounded up, then adopted a “peanut” shape due to constriction of the cleavage furrow (Fig. 4d in Honda et al. 1984; Guillot and Lecuit 2013), and finally restored their polygonal shape. Recently, the division process in Drosophila embryos with cell invagination by a contractile actomyosin ring was observed in detail (Guillot and Lecuit 2013). According to the cell center model, cell division can be simulated by replacement of one cell center by two cell centers (daughter cell centers). For the purpose of simulating cell division, we must know the orientation of the line on which the two points reside and the distance between the two points. Based on an actual cell pattern, the position of a cell center is obtained by the Dirichlet approximation of actual cell patterns. For orientation of two centers in the simulation, we assume that the cell center splits into two centers along the long axis of the initial polygonal cell. The long axis of the polygon is obtained by approximation of a polygon using an ellipse of inertia. The long axis of the polygon is considered the long axis of the ellipse. We confirmed this assumption by observing cell divisions in the blastular wall of the starfish (Honda et al. 1984). This assumption was described by Hofmeister (1863), which is quoted in Korn and Spalding (1973) and broadly known as Hertwig’s rule (1884). The mechanism of cell division along the long axis was recently elucidated (Bosveld et al. 2016). Next, we must define the distance between two points. To determine the cell pattern after cell division, we used the Dirichlet center–gravity center method, which has previously been used in the field of ecology by Tanemura and Hasegawa (1980). The procedure can be explained using an actual observation (Fig. 2.8a). First, we obtained the gravity center of an actual cell polygon (Fig. 2.8c). The gravity center was replaced by two points, whose orientation was along the long axis of the actual cell, and the distance was temporarily small (Fig. 2.8d left and center). Based on the two points, we obtained two Dirichlet domains. Next, we obtained two gravity centers of the two Dirichlet domains (Fig. 2.8d center). Based on the two gravity centers, we reconstructed a new Dirichlet polygonal pattern. Based on the new Dirichlet polygonal pattern, we again obtained gravity centers. We repeated the procedure to identify the gravity center–Dirichlet center until a final polygonal pattern was reached (Fig. 2.8d center and right). The distance between the two centers was asymptotically settled. The final pattern was automatically obtained and was a theoretical pattern that the

Fig. 2.7 (continued) Honda 1978; Fig. 4b in Honda and Eguchi 1980 with permission of Elsevier]. (e) Cultured monolayer of lung epithelial cells, courtesy of Kunio Yasuda (Δ ¼ 2.79  102, s ¼ 0.598). Bar, 50 μm [Reproduced from Plate Ib; Fig. 7 in Honda 1978; Fig. 2 in Honda and Eguchi 1980 with permission of Elsevier]. (f) Cultured monolayer of chondrocytes, courtesy of Kazuo Watanabe (Δ ¼ 2.18  102, s ¼ 1.84). Bar, 50 μm [Reproduced from Fig. 5g; Fig. 10; Fig. 41 in Honda 1983 with permission of Elsevier]

2.6 Cell Division

13

Fig. 2.8 Prediction of a pattern of divided cells from a cell pattern before cell division. (a) Observation of cell division in the blastula wall of the starfish. The right photograph was obtained 1 h 25 min after the left photograph. A cell (white dot in left) divided into two daughter cells (two white dots in right) [Reproduced from Fig. 4a in Honda et al. 1984 with permission of Elsevier]. (b) Comparison between (a) and (c). Pattern C is superimposed on pattern A [Reproduced from Fig. 4d in Honda et al. 1984 with permission of Elsevier]. (c) The observed cell pattern (left) and the patterns predicted by the cell models (right). Left, thin lines are drawn after photograph (a). Thick lines represent the Dirichlet approximation of the thin lines. Right, thin lines present the final Dirichlet pattern, which is the same as the pattern in (d) (right). The thick lines were obtained after the BS procedure was applied to the pattern of thin lines [Reproduced from Fig. 5 in Honda et al.

14

2 Cell Center Model

mathematical model predicted (Fig. 2.8d right). The theoretical pattern was compared with the actual cell pattern in Fig. 2.8b. The cell center model seems to work well.

2.7

Cell Disappearance1

Cells in tissues sometimes disappear during wounding, cell apoptosis, and other processes. According to the cell center model, cell disappearance corresponds to a loss of center points, which can be simulated simply with the cell center model using computers. There is an experiment to remove a single cell from an epithelium and observe the healing process (Hudspeth 1975). When a single cell is artificially removed from the mudpuppy gallbladder epithelium, contiguous cells deform to fill the defect. The actual cellular patterns before and after cell removal are shown in Fig. 2.9a. The cell to be removed is indicated by the arrow. Dirichlet centers corresponding to these polygonal cells were obtained and are plotted using solid circles in Fig. 2.9b (left), where polygons are approximated Dirichlet domains. The Dirichlet center, which corresponds to a cell to be removed, is indicated by an arrow (Fig. 2.9b left), and the cell was removed to simulate the cell removal experiment. The Dirichlet domains after the removal of the particular Dirichlet center are shown in Fig. 2.9b (right), which should be compared with the actual pattern presented in Fig. 2.9a (right). The two patterns closely resemble each other (Honda 1978, 1983). An experiment similar to that of Hudspeth (1975) was performed in rabbit corneal endothelium (Sherrard 1976). The corneal endothelium shows a polygonal, almost hexagonal cellular pattern. In a damage experiment involving glycerin drops, a single cell that fell away was replaced by the spreading of neighboring cells during the healing process, and a rosette pattern eventually formed (Sherrard 1976).

2.8

Summary

A biological tissue consists of cells. For the study of tissues using computer facilities, we constructed two mathematical models. One is the cell center model, which is based on Dirichlet or Voronoi geometry. There is a distribution of points in 2D or 3D space. At each point (center), a circle appears and grows. Growing circles

Fig. 2.8 (continued) 1984 with permission of Elsevier]. (d) Simulation of cell division via repetition of the gravity center–Dirichlet center procedure. Left, orientation of cell division was determined by the long axis of the initial cell. Center, repetition of the gravity center–Dirichlet center procedure. The smallest solid circles are initial centers, and the largest are the final centers. Right, the final Dirichlet pattern is shown [Reproduced from Fig. 5 in Honda et al. 1984 with permission of Elsevier]

References

15

Fig. 2.9 Computer simulation of cell disappearance using the cell center model. (a) After a single cell was experimentally removed from the mudpuppy gallbladder epithelium, contiguous cells deformed to fill the defect, and the epithelial sheet was repaired. (b) Computer simulation of cell disappearance. Left, the actual cell pattern was approximated by the Dirichlet domains. Dots are approximated Dirichlet centers from pattern (a). After the dot with the arrow was removed, the change in the cell pattern was predicted by the Dirichlet model (right), which is close to the actual pattern (a, right) [Reproduced from Fig. 12 in Honda 1978 with permission of Elsevier]

from neighboring points intersect with each other and define the boundaries between neighboring points. Finally, the boundaries form a polygonal pattern. These patterns can be used to approximate real tissues consisting of cells. We succeeded in quantifying how closely the models approximate real tissues using the Δ value (deviation value). We then established a method to describe cell patterns using points (centers), where cells and centers correspond one to one. Using the cell center model, we performed computer simulations of cell division and cell disappearance.

References Bosveld, F., et al.: Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis. Nature. 530, 495–498 (2016) Connolly, J.A., et al.: Microtubules and microfilaments during cell spreading and colony formation in PK 15 epithelial cells. Proc. Natl. Acad. Sci. USA. 78, 6922–6926 (1981) Coxeter, H.S.M.: Introduction to Geometry. Wiley, New York (1969) Crawford, B., et al.: Cloned pigmented retinal cells; the affects of cytochalasin B on ultrastructure and behavior. Z. Zellforsch. 130, 135–151 (1972) Crawford, B.J.: The structure and development of the pigmented retinal clone. Can. J. Zool. 53, 560–570 (1975)

16

2 Cell Center Model

DiPasquale, A., Bell, P.B.: The upper cell surface: its inability to support active cell movement in culture. J. Cell Biol. 62, 198–214 (1974) Dirichlet, G.L.: Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angewandte Mathematik. 40, 209–227 (1850). Quoted in Coxeter 1969. Edds, K.T.: Coelomocyte cytoskeletons. Exp. Cell Res. 130, 371–376 (1980) Gilbert, E.N.: Random subdivisions of space into crystals. Ann. Math. Stat. 33, 958–972 (1962) Guillot, C., Lecuit, T.: Adhesion disengagement uncouples intrinsic and extrinsic forces to drive cytokinesis in epithelial tissues. Dev. Cell. 24, 227–241 (2013) Hasegawa, M., Tanemura, M.: On the pattern of space division by territories. Ann. Inst. Statist. Math. 28 B, 509–519 (1976) Hertwig, O.: Das Problem der Befruchtung und der Isotropie des Eies, eine Theory der Vererbung. Jenaische Zeitschrift fuer Naturwissenschaft (1884). Quoted in Bosveld et al. 2016. Hofmeister, W.: Zusaetze und Berichtigungen zu den 1851 veroeffentlichen Untersuchungengen der Entwicklung hoeherer Kryptogamen. Jb. wiss. Bot. 3, 259 (1863). Quoted in Korn and Spalding 1973. Honda, H.: Description of cellular patterns by Dirichlet domains: the two-dimensional case. J. Theor. Biol. 72, 523–543 (1978) Honda, H.: Geometrical models for cells in tissues. Int. Rev. Cytol. 81, 191–248 (1983) Honda, H., Eguchi, G.: How much does the cell boundary contract in a monolayered cell sheet? J. Theor. Biol. 84, 575–588 (1980) Honda, H., et al.: Geometrical analysis of cells becoming organized into a tensile sheet, the blastular wall, in the starfish. Differentiation. 25, 16–22 (1983) Honda, H., et al.: Cell movements in a living mammalian tissue: long-term observation of individual cells in wounded corneal endothelia of cats. J. Morphol. 174, 25–39 (1982) Honda, H., Yamanaka, H., Dan-Sohkawa, M.: A computer simulation of geometrical configurations during cell division. J. Theor. Biol. 106, 423–435 (1984) Hudspeth: Establishment of tight junctions between epithelial cells. Proc. Nat. Acad. Sci. USA. 72, 2711–2713 (1975) Ivanova, O.Y., et al.: Effect of colcemid on the spreading of fibroblasts in culture. Exp. Cell Res. 101, 207–219 (1976) Korn, R.W., Spalding, R.M.: The geometry of plant epidermal cells. Plant Phytol. 72, 1357–1365 (1973) Loeb, A.L.: Space Structures – Their Harmony and Counterpoint. Addison-Wesley, London (1976) Middleton, C.A., Pegrum, S.M.: Contacts between pigmented retina epithelial cells in culture. J. Cell Sci. 22, 371–383 (1976) Nagata, T., Takebe, I.: Cell wall regeneration and cell division in isolated tobacco mesophyll protoplasts. Planta. 92, 301–308 (1970) Newsome, D.A., et al.: Effects of cyclic amp and sephadex fractions of chick embryo extract on cloned retinal pigmented epithelium in tissue culture. J. Cell Biol. 61, 369–382 (1974) Ogawa, T., Tanemura, M.: Geometrical considerations on hard core problems. Prog. Theoret. Phys. 51, 399–416 (1974) Radice, G.P.: The spreading of epithelial cells during wound closure in xenopus larvae. Dev. Biol. 76, 26–46 (1980) Rajaraman, R., et al.: A scanning electron microscope study of cell adhesion and spreading in vitro. Exp. Cell Res. 88, 327–339 (1974) Rogers, C.A.: Packing and Covering. Cambridge University Press, Cambridge (1964) Rubin, K., et al.: Adhesion of rat hepatocytes to collagen. Exp. Cell Res. 117, 165–177 (1978) Seglen, P.O., Gjessing, R.: Effect of temperature and divalent cations on the substratum attachment of rat hepatocytes in vitro. J. Cell Sci. 34, 117–131 (1978)

References

17

Sherrard, E.S.: The corneal endothelium in vivo: its response to mild trauma. Exp. Eye Res. 22, 347–357 (1976) Tanemura, M., Hasegawa, M.: Geometrical models of territory I. Models for synchronous and asynchronous settlement of territories. J. Theor. Biol. 82, 477–496 (1980) Voronoi, G.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. J. Reine Angewandte Mathematik. 134, 198–287 (1908). Quoted in Rogers 1964. Wigner, E., Seitz, F.: On the constitution of metallic sodium. Phys. Rev. 43, 804–810 (1933)

Chapter 3

Applications of the Cell Center Model

Outline The cell center model was applied to elucidate the formation of neat cell arrangements in the mammalian epidermis, cell patterns consisting of heterogeneous-sized cells in the pupal wing epidermis of a butterfly, and capillary networks and branching patterns of blood vessels in quail embryos.

3.1

Neat Arrangement of Cells in the Mammalian Epidermis (Honda et al. 1996)

Mammalian epidermal cells, particularly cells in the stratum corneum of the epidermis, have a striking architectural organization (Mackenjie 1969; Christophers 1971, 1972; Yokouchi et al. 2016), as shown in Fig. 3.1. The cells are precisely stacked in columns in a honeycomb fashion. The cells in the ordered structure of the epidermis are vertically compressed and flattened at the skin surface. The stacked organization of cells is closely approximated by polyhedra in a 3D geometry, specifically, a flattened Kelvin’s tetrakaidecahedron, as shown in Fig. 3.2a (Menton 1976). This pattern is compatible with cell renewal, where epidermal cells are supplied from the basal layer, migrate to the surface, and are removed at the surface as flakes of skin (Allen and Potten 1974, 1976; Potten and Allen 1975; Honda and Oshibe 1984). According to the cell center model, the neat arrangement of cells is expressed by an ordered arrangement of points. Toshiteru Morita and Akira Tanabe experimentally examined mouse skin, and we assumed that points migrating upward occupy less crowded regions (Honda et al. 1979). The idea is schematically presented in Fig. 3.3a. Cells (black circles) are randomly arranged in the first layer (the uppermost layer in Fig. 3.3a). In the second layer, a cell migrating upward from the basal layer settles at a position that is just under the middle point of neighboring cells in the first layer (see M in Fig. 3.3a). Repetition of the process makes an ordered point arrangement, as shown in the lower layers (gray circles). Cells connected with © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_3

19

20

3 Applications of the Cell Center Model

Fig. 3.1 Schematic drawing of a vertical section of columns of stratified flattened cells

Fig. 3.2 Aggregates of tetrakaidecahedron-like polyhedra. (a) A cluster of flattened tetrakaidecahedra without interstices, which closely approximates stacked epidermal cells. The flattened tetrakaidecahedron consists of a pair of hexagonal faces, six equal and opposite hexalateral faces, and six equal and opposite quadrilateral faces. The edges of polyhedra in a column interdigitate with those of six surrounding columns. (b) A cell stack modified from that shown in Fig. A after removal of interdigitation between neighboring columns [Reproduced from Fig. 1B and C in Honda et al. 1996a with permission of Elsevier]

lines are vertically arranged, forming columns. During stacking of layers, the uppermost layer peels off sequentially as a flake of skin, which is not drawn in Fig. 3.3a. In a computer simulation, we considered point arrangements in stacked flat layers. First, we performed a simplified computer simulation under a limited condition in which each cell has just six neighbors in its layer. Cells migrate and settle at locally less crowded positions. We succeeded in the formation of a neat cell arrangement (Honda et al. 1979; Honda 1983; Honda and Oshibe 1984). The results of this computer simulation were mathematically confirmed (Saito 1982). Later, we introduced a completely random cell arrangement to the initial condition; in other words, the cells do not necessarily have six neighbors (Honda et al. 1996). We considered a network of triangles (Delaunay triangles) that consists of lines connecting neighboring points in a layer (Tanemura et al. 1991; Tanemura et al.

3.1 Neat Arrangement of Cells in the Mammalian Epidermis (Honda et al. 1996)

21

Fig. 3.3 Proposed mechanism of formation of neat cell arrangement. (a) Explanation of the mechanism of columnar array formation of cells in two-dimensional space. The uppermost layer is the first cell array, in which cells are supplied from the bottom. Cells (points) in the first cell array are randomly distributed. Depending on the first cell array, the next cell array forms as follows. A point (thin arrow) on the lower line is settled at a position that is just under the middle point (M) of neighboring cells in the upper layer. The irregular distribution of points on the upper line becomes more uniform in the lower layer. As a result, the points on every other layer are approximately arranged on a vertical line (arrows). During stacking of layers, the uppermost layer peels off sequentially as flakes of skin (not drawn in a). A periodic (circular) boundary condition is used for the points next to the terminals, where one side of the line segment continues to the opposite side of the neighboring line segment [Reproduced from Fig. 2 and an excerpt of its legend in Honda et al. 1979 with permission of Elsevier]. (b) Triangle k has three neighboring triangles 1~3. If two or three triangles are empty (i.e., not occupied by cells), triangle k can accept a cell. (c) View from below the piled layers of cells, which are represented by spheres. The positions of cells in layer t are determined by the cell positions in the two preceding layers t – 1 and t – 2. A network of Delaunay triangles is constructed by connecting neighboring points in layer t – 1. A cell in layer t occupies the central region of a triangle (the inner center of the triangle) if no cell in layer t – 2 interferes (solid arrow). Open arrow, the position in layer t with interference by a cell in layer t – 2 [Reproduced from Fig. 3 in Honda et al. 1996a with permission of Elsevier]

1983). Each cell in the next layer (lower layer) occupies one of the triangles. We constructed three algorithms for computer simulations. 1. Lateral interference: Points on layer t settle just below the central regions of selected triangles of the network of layer t – 1 (upper layer of layer t). The selection of triangles was performed as follows. The larger the triangle is, the greater the chance of occupying a point. Therefore, the trial of point occupation is performed sequentially from the larger empty triangle in the computer simulation. We have empty and occupied triangles in the occupation process. Triangle k has the three neighboring triangles 1~3, as shown in Fig. 3.3b. If two or three triangles among them are empty, triangle k accepts a point; otherwise, triangle k does not have a point. Furthermore, an additional condition should be satisfied in which a new point is not closer than the distance d to neighboring points. d is a critical distance; at distances less than d, points cannot get close to each other. 2. Vertical interference: Cells in layer t are arranged at positions just below the triangles of layer t – 1. A cell settles at the central region of a triangle in layer t – 1, but only if no cell in layer t – 2 interferes (solid arrow in Fig. 3.3c). A cell cannot

22

3 Applications of the Cell Center Model

Fig. 3.4 Initiation of cell stacking. (a) Random distribution of cells in layer 1 as the initial condition for the computer simulation. (b) The cell distribution in layer 2 is designated by a solid circle, which is determined by that in layer 1 [Reproduced from Fig. 4 in Honda et al. 1996a with permission of Elsevier]

occupy the central region of the triangle indicated by the open arrow in Fig. 3.3c because the cell in layer t – 2 interferes with the occupation. 3. Lateral adjustment: Each point in layer t is surrounded by neighboring points. Neighboring points compose a polygon. Each point position is adjusted by force so that each point becomes the central region of its polygon. These algorithms are useful to perform a process in 3D space in computer simulations. A cell produced in a basal layer migrates upward, and its position is governed by three cells in the upper plane, since the central regions of the triangles comprising the three cells are considered to be a less crowded region. A square (100
100) was used under the periodic (circular) boundary condition, where the square area satisfies two-dimensional tiling in which one side of the square continues to the opposite side of the neighboring square, as shown in Fig. 3.4a. Points were randomly distributed on the square under the condition that the neighboring points were not closer to one another than the distance d (d ¼ 6 in the simulation). The method of distribution using nonoverlapping disks (radius: d/2), instead of points, meets this condition, as described elsewhere (Tanemura et al. 1991). One hundred seventy points on layer 1 are distributed in Fig. 3.4a (the packing density, which is defined as the ratio of the total disk area to the square area, was 0.481). A network of triangles (Delaunay triangles) was formed by connecting neighboring points in layer 1, as described elsewhere (Tanemura and Hasegawa 1980, 1983, 1991). We made a point distribution in layer 2 according to (1) lateral interference. After all triangles that are allowed to have a point have accepted the point, (3) lateral adjustment is made among the points within layer 2 (solid circles in Fig. 3.4b). The procedure proceeds in the next layer. In layer 3 and beyond, a similar method is used with the

3.1 Neat Arrangement of Cells in the Mammalian Epidermis (Honda et al. 1996)

23

Fig. 3.5 Computer simulation of cell stacking. A point distribution pattern in every layer is sequentially calculated until layer 200. To display the surface view of stacked cells, we used the deformation method to remove the interdigitation between columns (see Fig. 3.2b). Polygonal patterns in layers 2, 19, 79, 109, and 159 (from left to right) [Reproduced from Fig. 5 in Honda et al. 1996a with permission of Elsevier]

additional consideration of (2) vertical interference, as shown in Fig. 3.3c. The procedure continued until layer 200. Finally, a structure similar to a face-centered lattice is obtained. The vertical distance between two layers was determined so small that the stacking structure is vertically flattened (Honda et al. 1979). The point distribution in a pile of layers forms a polyhedral pattern; the pattern comprises 3D flattened Voronoi polyhedra (Fig. 3.2a) and is so complicated that it is displayed on a 2D paper. To clearly display the results of computer simulations, we deformed the pattern by the method shown in Fig. 3.2b. The interdigitation between neighboring columns in Fig. 3.2a is removed. The deformation provides a simple surface view of the polygonal pattern of stacked cells. To construct 2D domains in layer t, the deformation of polyhedra was performed using the Dirichlet model (2D Voronoi model) based on the points at which the points of layers t – 1, t, and t + 1 were superposed on layer t. The results are shown in Fig. 3.5 (layers 2, 19, 79, 109, and 159). The pattern in the final layer (Fig. 3.5, rightmost) presents an almost regular honeycomb appearance and becomes steady. The present study demonstrates that the local rule that cells occupy less crowded regions leads to an entirely ordered arrangement of tissue. That is, an ordered arrangement is spontaneously organized, although the cells are supplied at random. While the neat arrangement of cells is curious and mysterious, our computer simulation suggests a lack of a supervising control system in the organization of the epidermal architecture.

24

3.2

3 Applications of the Cell Center Model

Cell Patterns Consisting of Large and Small Cells (Honda et al. 2000)

We often observe cell patterns consisting of cells of nonequivalent size. For example, the pupal wing epidermis of the butterfly Pieris rapae shows a transition from an equivalent-size cell pattern to a pattern involving large cells (Fig. 3.6). The transition process was analyzed using the method of weighted Voronoi tessellation, which is useful for treatment of polygons of nonequivalent size (Tanemura 1992). Figure 3.7 explains the weighted Voronoi tessellation. A straight boundary line is defined between two different-size disks (Fig. 3.7a). Two neighboring disks are drawn whose centers are x1 and x2 and whose radii are r1 and r2, respectively. Line P1P2 is tangential to discs x1 and x2, and point M is the middle of line P1P2. The line that passes point M and is perpendicular to line x1x2 is the boundary of the weighted Voronoi tessellation between disks x1 and x2. Point x on the boundary line satisfies the condition of |x – x1|2 – r12 ¼ |x – x2|2 – r22 (Aurenhammer 1987). When an assemblage of disks of different sizes is given, weighted Voronoi tessellation is performed. Next, each disk moves within its polygon via the compression method (Tanemura 1992). Based on the moved disks, weighted Voronoi tessellation is again performed. After the iterative movement of disks, we obtain a polygonal pattern (Fig. 3.7b). In the early stage of pupal wing development in butterflies, the wing surface is made up of a monolayered epithelial cell sheet consisting of equivalent-size polygonal cells (Fig. 3.6 inset), and during the developmental process, some of the cells

Fig. 3.6 Cell pattern of nonequivalent size cells. The pupal wing epidermis of the butterfly Pieris rapae. The dorsal surface of the forewing was observed under a scanning electron microscope and photographed for approximately 35 h after pupation. Bar, 50 μm. Inset, the pattern at approximately 25 h after pupation. The scale is the same as in the figure [Reproduced from Fig. 2 in Honda et al. 2000 with permission of Springer Nature]

3.3 Formation of Pore Patterns Using the Dirichlet Geometry (Honda. . .

25

Fig. 3.7 Weighted Voronoi tessellation. (a) Line xM is the boundary between discs x1 and x2 and perpendicular to line x1x2. M is the middle point of line P1P2 [Original]. (b) A polygonal cell pattern of the weighted Voronoi tessellation after iterative movement of the disks [Reproduced from Fig. 1b in Honda et al. 2000 with permission of Springer Nature]

become large (Yoshida 1990). The cells seem to decide their fate under lateral inhibition (Honda et al. 1990), and some cells that are to be differentiated become large. According to the lateral inhibition system, initially equivalent cells are all components that can differentiate, but once a cell has differentiated, the cell inhibits its immediate neighbors from following this pathway. Such differentiation repeats until all noninhibited cells have differentiated. The populations of ordinary (small) and differentiated (large) cells are restricted in a different manner from a system consisting of homogeneously sized cells because more small cells are needed to enclose a large cell. We performed computer simulations under the lateral inhibition rule using the method of weighted Voronoi tessellation (Tanemura 1992), as shown in Fig. 3.8, and quantitatively analyzed the system, including large polygonal cells (Honda et al. 2000).

3.3

Formation of Pore Patterns Using the Dirichlet Geometry (Honda and Yoshizato 1994a, b, c)

A capillary network sometimes looks like a sheet in which many pores are packed (Fig. 3.9). We observed angiogenesis on the yolk sac surface of quail embryos (Honda and Yoshizato 1997). Based on the observation, we reproduced the process of capillary network formation using a cell automaton model and analyzed the resultant patterns using the Dirichlet geometry.

26

3 Applications of the Cell Center Model

Fig. 3.8 Computer simulation of the pattern transition from an equivalent-size cell pattern to a pattern involving large cells. The lateral inhibition rule and the method of weighted Voronoi tessellation were used. NS/NL ¼ infinity (a), 19.8 (b), 6.94 (c), and 4.15 (d), where NS and NL are the numbers of small and large cells, respectively [Reproduced from Fig. 5a-d in Honda et al. 2000 with permission of Springer Nature] Fig. 3.9 Schematic presentation of a capillary network. Inset, the network has a cavity inside [Reproduced from Zu 3-55e in Honda 2010]

A blood island that is a mass of cells derived from the mesoderm becomes a vesicle consisting of flattened cells enclosing the plasma. Blood islands expand and anastomose with each other, forming a vascular network. The process was simulated using a cell automaton model (Fig. 3.10). We distributed dots (small squares) on grids at random (Figs. 3.10a and 3.11a). The dots randomly proliferated on neighboring grids and became dot aggregates (Figs. 3.10b and 3.11b). Next, growing dot aggregates sprouted to produce a short stem (Figs. 3.10c, d and 3.11c). The growing dot aggregates anastomose with each other. The processes were repeated to form a network (Fig. 3.11c, d). We performed three additional computer simulations with different random numbers (Fig. 3.11e~g). Because the process involved randomness of proliferation and sprouting, the final network patterns were diverse. These diverse patterns were superposed into one pattern (Fig. 3.11h). On the other hand, we can obtain theoretically predicted network patterns from the initial dot distribution patterns using the Dirichlet geometry as follows. Figure 3.12a is a Dirichlet pattern based on the dot distribution shown in Fig. 3.11a. Figure 3.12b is a distribution pattern of the vertices of the Dirichlet pattern in Fig. 3.12a. The positions of vertices are sites apart from dots and are considered to be less crowded sites of dots. An

3.3 Formation of Pore Patterns Using the Dirichlet Geometry (Honda. . .

27

Fig. 3.10 Procedure of dot pattern formation. (a) Random distribution of dots (small squares). (b) Random proliferation of dots along the periphery of dot aggregates. (c, d) Sprouting of short stems (sprout length is three dots) from the periphery of dot aggregates. When a stem contacts any dot, it becomes fixed; otherwise, it retracts [Original]

inversion between pattern and ground took place in the Dirichlet pattern. The positions of vertices are candidates for sites of pores in a final network. For selection of candidates, we picked up vertex j, measured distances from vertex j to all dots in the pattern, and defined the nearest dot, which is referred to as vertex–dot distance of vertex j, as shown in Fig. 3.12c (vertex j is designated by a triangle). We performed this procedure on all vertices. Then, every vertex has its vertex–dot distance. Vertices with a small vertex–dot distance were discarded. We selected a certain number of vertices with a larger vertex–dot distance. That is, we selected larger distances among the minimum distances. Based on the selected vertices, we constructed Dirichlet domains, as shown in Fig. 3.12d, which are superposed on the dot-aggregate pattern in Fig. 3.11e~h for comparison. Figure 3.11h is the superposed pattern of four patterns (Fig. 3.11c and e~g) on the Dirichlet pattern shown in Fig. 3.12d. The Dirichlet pattern corresponds closely to the superposed dot-aggregate pattern. The Dirichlet geometry is also useful for making pore patterns. In this section, another usage of the Dirichlet geometry was shown, which was used to find not substantial sites but hollow sites. Our study of network formation is a geometrical approach using Dirichlet domains. On the other hand, mathematical studies of the formation of vascular networks have advanced using nonlinear dynamics (Manoussaki et al. 1996; Herrero and Velazquez 1996; Merks et al. 2006; Herrero and Kohn 2009).

28

3 Applications of the Cell Center Model

Fig. 3.11 Computer simulation of the formation of a capillary network. (a) Radom distribution of dots in a lattice (100
90). (b) Proliferation of dots on the periphery of dot aggregates with probability ( p ¼ 0.04). (c, d) Sprouting of short stems from the periphery of dot aggregates. (e~g) Superposition of the Dirichlet pattern (Fig. 3.12d) on dot patterns with different random numbers. (h) Superposition of (c) and (e~g) on the Dirichlet pattern in Fig. 3.12d [Reproduced from Fig. 1 in Honda and Yoshizato 1994a, b, c]

Fig. 3.12 Method to find sites where dots are less crowded. (a) Dirichlet pattern based on the distribution of the initial dots in Fig. 3.11a. (b) Distribution pattern of the vertices of the Dirichlet pattern in (a). (c) Determination of the vertex-dot distance of vertex j (designated by a triangle). The three nearest initial dots from vertex j are indicated by small arrows. Initial dots are designated by circles. See text. (d) Dirichlet pattern based on the selected vertices (small gray circles). A certain number of vertices with a larger vertex-dot distance were selected [Reproduced from Fig. 1 in Honda and Yoshizato 1994a, b, c]

3.4 From Polygonal Patterns to Branching Patterns

3.4

29

From Polygonal Patterns to Branching Patterns2

The present study was performed to provide data that support the idea that branching of blood vessels occurs due to the selection of capillaries in the network of polygonal patterns. This idea was previously believed but not experimentally verified. In an attempt to understand the mechanism underlying the formation of blood vessel branching structures, we recorded the transformation of a capillary network to a branching system using a series of photographs of the wall of the quail yolk sac. Then, a computer simulation was carried out to recapitulate the process of in vivo vascularization that had been recorded in the photographs. The simulation suggested that a positive feedback system participated in the formation of a branching structure. In other words, vessels that had been much used were enlarged, whereas less used vessels were reduced in size and finally extinguished (Honda and Yoshizato 1997). The enlarged vessels became major components of the branching system. The observation and computer simulation are described in detail as follows.

3.4.1

Formation of Blood Vessel Branching In Vivo

We will describe the results of the observation of the formation of capillary vessel networks and the branching pattern of blood vessels in the wall of the quail yolk sac. A window into the eggs was opened by cutting the shell at 45 h of incubation. An area vasculosa of the windowed eggs was observed at time points from 47 to 80 h (Fig. 3.13). We recorded the vascular structure developmental process from the phase of blood islands connecting and anastomosing with one another (Fig. 3.13a) to the phase of branched blood vessels (Fig. 3.13b) by taking photographs every 2–4 h. The developmental process in a local area (indicated by a triangle in Fig. 3.13) is shown in a series of photographs (Fig. 3.14). The blood islands connected and anastomosed to form a network (Fig. 3.14, 47 h–56 h). The network consisted of capillary endothelial tubes containing red blood cells and plasma. The blood volume amount in the capillary tubes was unevenly allocated (Fig. 3.14, 56 h). At 60–64 h, the network pattern was distinctly established (Fig. 3.14, 60 h and 64 h), and then a branching system was formed at 68 h (Fig. 3.14, 68 h). The transformation from the network to the branching system in the 4 h is shown in Fig. 3.15. Vessels in a network (indicated by white dots in Fig. 3.15b) became thick and formed branched chains along which continuous blood flows were observed (Fig. 3.14, 68 h and Fig. 3.15c). Branched blood vessels became prominent, whereas the polygonal networks among them became slender (Fig. 3.14, 76 h and 80 h). The branching vessels were dynamically changing. A transverse vessel at 60 h (vessel 1 in Fig. 3.14, 60 h) became thin at 64, 68, and 72 h (vessel 1 in Fig. 3.14, 64 h, 68 h

2

Reproduced from excerpts in Honda and Yoshizato 1997 with permission of John Wiley and Sons.

30

3 Applications of the Cell Center Model

Fig. 3.13 Development of an area vasculosa in the wall of the quail yolk sac. (a) 47-h embryo. (b) 80-h embryo. Triangles in (a) and (b) indicate the regions shown in Fig. 3.14. Bar, 2 mm [Reproduced from Fig. 1 in Honda and Yoshizato 1997 with permission of John Wily and Sons]

and 72 h), and in contrast, longitudinal vessels at 60 h became thick at 72 h (vessels 2 and 3 in Fig. 3.14, 72 h). The hearts of approximately 50-h quail embryos were observed to exhibit periodic contractions. The vascular area had already acquired a definite peripheral boundary by forming the sinus terminalis at its margin, a terminal vein that was connected to the heart by two anterior vitelline veins. Therefore, the capillary network shown in Fig. 3.14 at 56 h establishes blood circulation from the heart to the sinus terminalis.

3.4.2

Computer Simulation of the Branching Formation of Blood Vessels

The transformation of capillary networks into branched blood vessels is considered as follows. The individual capillary vessels contain blood. Once circulation is established, the law of flow begins to operate, leading the most often used capillary vessels to enlarge. At the same time, the less used capillary vessels collapse and are ultimately eliminated (Hamilton 1965). This process can be described mathematically. One of such trials is introduced below.

3.4 From Polygonal Patterns to Branching Patterns

31

Fig. 3.14 Development of blood vessels at a site shown by the triangle in Fig. 3.13. Photographs were taken successively at the incubation times (37  C) indicated in the figure. Three identified points are linked by lines. Numerals are for location identification: Numeral 1 is plotted in Figures 60 h, 64 h, 68 h, and 72 h. Numerals 2 and 3 are plotted in Figure 72 h. Bar, 1 mm [Reproduced from Fig. 2 in Honda and Yoshizato 1997 with permission of John Wily and Sons]

The blood flow (current) in vessels was formulated by assuming that (1) currents along vessels of a polygonal capillary network can be calculated using Kirchhoff’s law of electrical circuits. The position of the heart (Fig. 3.16a) and the sinus terminalis were arbitrarily defined in the polygonal pattern. When the pressure difference between the exit of the heart and the sinus terminalis is given and when the resistance value of every vessel in the network is given, the current along every vessel can be calculated according to Kirchhoff’s law. (2) The resistance value of every vessel is variable and can be determined by the current of each vessel in the previous stage, utilizing an equation (Eq. 3.1).

32

3 Applications of the Cell Center Model

Fig. 3.15 Transformation of polygonal patterns into a branching pattern. (a) and (b) Polygonal patterns. Photographs from Fig. 3.14 (64 h). (c) Branching pattern. Photograph from Fig. 3.14 (68 h). Dots in (b) indicate vessels that become thick in (c). Bar, 1 mm [Reproduced from Fig. 3 in Honda and Yoshizato 1997 with permission of John Wily and Sons]

ΔRi ¼ a ΔAi

ð3:1Þ

where ΔRi is an increase in the resistance of vessel i, ΔAi is an increase in the current in vessel i, and a is a positive constant. A vessel through which much blood flows becomes thick, and its resistance decreases, whereas a vessel through which little blood flows becomes reduced in diameter, increasing the resistance and resulting in slower blood flow. In this case, vessels gradually degrade and ultimately disappear. The formula (Eq. 3.1) provides a positive feedback system representing the relationship between blood flow and the diameter of blood vessels. (3) There should be threshold values of resistance: Rmax and Rmin. When a resistance value is larger than the upper critical value (Rmax), the resistance value becomes almost infinite (R ¼ 107; blood cannot flow in the vessel, and therefore, the vessel degrades). When the resistance value is smaller than the lower critical value (Rmin), the resistance value is fixed at almost zero (R ¼ 1012; almost no resistance). With these assumptions, a computer simulation of the process of branching of blood vessels was performed. We explain the computer simulation process in Fig. 3.16, which illustrates a partial pattern of one of the computer simulations. The position of the heart is indicated by an arrowhead in Fig. 3.16a. The sinus terminalis is far from the right side of the figure. In Fig. 3.16a, for simplicity, the individual vessels were assumed to have the same resistance value, and the pressure difference between the exit from the heart and the sinus terminalis was fixed. The blood flowed toward the right. The currents of the vessels were calculated and are shown in Fig. 3.16b, in which line thickness is proportional to the current value. The results met the fact that the heart produces much of the flow. We determined new resistance values according to the formula (Eq. 3.1) for the calculation of currents in

3.4 From Polygonal Patterns to Branching Patterns

33

Fig. 3.16 Explanation of computer simulations. (a) Initial polygonal pattern; (b–d) Sequential patterns in the stages of the computer simulation. The current amounts are represented by the line thickness. a ¼ 0.2; Rmax ¼ 0.888; Rmin ¼ 104. Arrowheads show the position of the heart [Reproduced from Fig. 6 in Honda and Yoshizato 1997 with permission of John Wily and Sons]

the next stage. The current calculation results are shown in Fig. 3.16c. The procedure was repeated, and we obtained the pattern presented in Fig. 3.16d. At the beginning, the current differed little among the vessels on the right side in Fig. 3.16b. The differences in currents among the vessels became distinct during the repetition of calculations (Fig. 3.16c, d). Notably, our trial resulted in a remarkable branching pattern, although we assumed that all the resistances of the initial vessels were the same. A whole view of one of the computer simulations is shown in Fig. 3.17. Figure 3.17a shows an initial condition. The position of the heart is indicated by the solid circle, and the sinus terminalis is indicated by a chain of thick solid lines on the right side of the figure. Initially, all the vessels were assumed to have the same resistance value (R ¼ 1.0), and the pressure difference between the exit from the heart and the sinus terminalis was fixed at 100. The blood flowed from left to right in

34

3 Applications of the Cell Center Model

Fig. 3.17 Branching pattern made by a computer simulation. (a) Initial condition. A chain of solid thick lines indicates the position of the sinus terminalis. The pressure difference between the heart and the sinus terminalis was 100, and the resistance of all vessels was 1.0. (b–f) Snapshots of the currents in vessels in stages 1–5 of the computer simulation. A solid circle represents the position of the heart. A periodic boundary condition was used between the top and bottom borders in each figure. a ¼ 0.1; Rmax ¼ 0.94; Rmin ¼ 104 [Reproduced from Fig. 7 in Honda and Yoshizato 1997 with permission of John Wily and Sons]

gross. The currents of the vessels were calculated and are shown in Fig. 3.17b~f, in which line thickness is proportional to the current value. The results met the fact that the heart produces much of the flow. We determined new resistance values according to the formula (Eq. 3.1) for the calculation of currents in the next stage.

3.5 Other Applications of the Geometry of Dirichlet Domains and. . .

35

A branching pattern gradually became obvious. Figure 3.17f presents the final pattern that did not exhibit any further changes. Vessels 1 and 2 in Fig. 3.17e became prominent, but their currents decreased to the average level in the next stage (Fig. 3.17f). Currents increased in some vessels but decreased in others during branching pattern formation. Thus, we demonstrated that capillaries in the network are successively selected by a positive feedback mechanism and form blood vessels. The enlarged vessels became major components of the branching system. As the body of an embryo grew, polygonal capillary networks enlarged, which led each polygon in the network to divide into a few finer polygons (Honda and Yoshizato 1997). Then, some of the capillary vessels were again selected and formed a branching system. This process was repeated during body growth, indicating that the vascular system developed adaptively to body growth. A region in which the growth was fast received much blood flow, producing finer capillary networks. This positive feedback system of blood flow may be related to the formation of bypass blood vessels in vascular infarction.

3.5

Other Applications of the Geometry of Dirichlet Domains and Voronoi Polyhedra

The geometry of Dirichlet domains was applied to biology by assuming a polygon as an animal territory (Hasegawa and Tanemura 1976; Tanemura and Hasegawa 1980). When male mouthbreeder fishes were kept in a large pool with a sand bottom, each fish excavated a breeding pit by spitting sand away from the pit center toward his neighbors, forming remarkable polygonal territories (Barlow 1974). Tanemura and Hasegawa (1980) analyzed the distribution of bird territories of pectoral sandpipers on Arctic tundra and examined how territories became stationary. The study of territory is not restricted to animal biology. Patterns of tree crowns in a forest have been examined (Hasegawa et al. 1981). The study of territory is related to the problem of packing of disks in a restricted area (Tanemura 1979). Voronoi polygons on the surface of a large sphere, not a flat surface, have also been investigated (Tanemura 1998a, b). The tessellation theory was expanded to 3D Voronoi domains (Tanemura et al. 1983). The Dirichlet geometry was applied to cells that are not polygonal but dendritic (or spoke-like) (Numahara et al. 2001). Langerhans cells are dendritic cells situated in the mammalian epidermis and are set in the best formation for their immunosurveillance activity. Their distribution is completely regular and demonstrates repulsive interactions among cells. The pattern of territories of each cell fits the random packing model of Dirichlet domains. In addition, a regular distribution of microglia that are resident immune cells in the retina was investigated using a mathematical model, in which direct cell–cell contact is elucidated to be important (Endo et al. 2021).

36

3.6

3 Applications of the Cell Center Model

Summary

The cell center model was used to investigate neat cell arrangement in the mammalian epidermis. Next, the model was modified to be useful for examination of cell patterns involving large cells that have differentiated from normal-sized cells. On the other hand, a network of capillary vessels was considered to be a sheet involving many pores, and the distribution of pores in a sheet was simulated by the cell center model. This was an inversion between pattern and ground in the Dirichlet pattern. Although the branching pattern seems to be fundamentally different from the network pattern, the branching pattern of blood vessels is unexpectedly shown to form from the capillary network, in which a positive feedback system of blood flow is involved.

References Allen, T.D., Potten, C.S.: Fine-structural identification and organization of the epidermal proliferative unit. J. Cell Sci. 15, 291–319 (1974) Allen, T.D., Potten, C.S.: Significance of cell shape in tissue architecture. Nature. 265, 545–547 (1976) Aurenhammer, F.: Power diagrams: properties, algorithms and applications. Siam J. Comput. 16, 78–96 (1987) Barlow, G.: Hexagolal territories. Anim. Behav. 22, 876–878 (1974) Christophers, E.: Cellular architecture of the stratum corneum. J. Investigative Dermatol. 56, 165–169 (1971) Christophers, E.: Correlation between column formation, thickness and rate of new cell production in guinea pig epidermis. Virchows Arch. B Cell Pathol. 10, 286–292 (1972) Endo, Y., Asanuma, D., Namiki, S., Sugihara, Hirose, K., Uemura, A., Kubota, Y., Miura, T.: Quantitative modeling of regular retinal microglia distribution. Scientific Rep. 11, 22671 (2021) Hamilton, H.L.: Lille’s development of the chick, 3rd edn. Holt, Reinhart & Wilson, New York (1965) Hasegawa, M., Tanemura, M.: On the pattern of space division by territories. Ann. Inst. Statist. Math. 28B, 509–519 (1976) Hasegawa, M., et al.: Spatial patterns in ecology. Proceedings of International Roundtable Congress, The 50th Anniversary of the Japan Statistical Society: 146–161 (1981) Herrero, M.A., Kohn, A.: Modelling vascular morphogenesis: current views on blood vessels development. Math. Models Methods Appl. Sci. 19(Suppl), 1483–1537 (2009) Herrero, M.A., Velazquez, J.J.L.: Chemotactic collapse for the Keller-Segel model. J. Math. Biol. 35, 177–194 (1996) Honda, H.: Geometrical models for cells in tissues. Int. Rev. Cytol. 81, 191–248 (1983) Honda, H.: “Katachi no Seibutugaku” (in Japanese). NHK Shuppan, Tokyo (2010) Honda, H., et al.: Establishment of epidermal cell columns in mammalian skin: computer simulation. J. Theor. Biol. 81, 745–759 (1979) Honda, H., Oshibe, S.: A computer simulation of cell stacking for even thickness in mammalian epidermis. J. Theor. Biol. 111, 625–633 (1984) Honda, H., et al.: Spontaneous architectural organization of mammalian epidermis from random cell packing. J. Invest. Dermatol. 106, 312–315 (1996)

References

37

Honda, H., Tanemura, M., Yoshida, A.: Estimation of neuroblast number in insect neurogenesis using the lateral inhibition hypothesis of cell differentiation. Development. 110, 1349–1352 (1990) Honda, H., et al.: Differentiation of wing epidermal scale cells in a butterfly under the lateral inhibition model—appearance of large cells in a polygonal pattern. Acta Biotheoretica. 48, 121–136 (2000) Honda, H., Yoshizato, K.: Angiogenesis – Blood islands to a capillary network. The Japanese Society of Developmental Biologists (1994.5/25-27 Sendai) The 27th Annual Meeting: p. 132 (131A1200) (1994a) Honda, H., Yoshizato, K.: Morphogenesis of capillary network. The 32nd Annual Meeting of the Biophysical Society of Japan (1994.9/28-30 Yokohama): 3I1130 (1994b) Honda, H., Yoshizato, K.: A computer simulation of vascularization from blood islands. The International KATACHI SYMMETRY Symposium (University of Tsukuba): 18–20 (1994c) Honda, H., Yoshizato, K.: Formation of the branching pattern of blood vessels in the wall of the avian yolk sac studied by a computer simulation. Dev. Growth Differ. 39, 581–589 (1997) Mackenjie, J.C.: Ordered structure of the stratum corneum of mammalian skin. Nature. 222, 881–882 (1969) Manoussaki, D., et al.: A mechanical model for the formation of vascular networks in vitro. Acta Biotheoretica. 44, 271–282 (1996) Menton, D.N.: A minimum-surface mechanism to account for the organization of cells into columns in the mammalian epidermis. Am. J. Anat. 145, 1–22 (1976) Merks, R.M., et al.: Cell elongation is key to in silico replication of in vitro vasculogenesis and subsequent remodeling. Dev. Biol. 289, 44–54 (2006) Numahara, T., et al.: Spatial data analysis by epidermal Langerhans cells reveals an elegant system. J. Dermatol. Sci. 25, 219–228 (2001) Potten, C.S., Allen, T.D.: Control of epidermal proliferative units (EPUs). Differentiation. 3, 161–165 (1975) Saito, N.: Asymptotic regular pattern of epidermal cells in mammalian skin. J. Theor. Biol. 95, 591–599 (1982) Tanemura, M.: On random complete packing by discs. Ann. Inst. Statist. Math. 31B, 351–365 (1979) Tanemura, M.: Models and simulations of random structure of particles. Acta Stereol. 11(Suppl I), 41–52 (1992) Tanemura, M.: Random packing and tessellation network on the sphere. Forma. 13, 99–121 (1998a) Tanemura, M.: Problems of optimal configuration of points on the sphere (in Japanese). Proc. Instit. Statis. Math. 46, 359–381 (1998b) Tanemura, M., Hasegawa, M.: Geometrical models of territory I. Models for synchronous and asynchronous settlement of territories. J. Theor. Biol. 82, 477–496 (1980) Tanemura, M., et al.: Distribution of differentiated cells in a cell sheet under the lateral inhibition rule of differentiation. J. Theor. Biol. 153, 287–300 (1991) Tanemura, M., et al.: A new algorithm for three-dimensional Voronoi tessellation. J. Comput. Phys. 51, 191–207 (1983) Yokouchi, M., et al.: Epidermal cell turnover across tight junctions based on Kelvin’s tetrakaidecahedron cell shape. Elife. 5, e19593 (2016) Yoshida, A.: Cellular pattern development in the pupal wing of a butterfly. Forma. 5, 65–72 (1990)

Chapter 4

Vertex Model

Outline The second mathematical cell model is a vertex model, which is constructed based on cell vertices and cell boundaries. We have two vertex models, the boundary shortening (BS) model and its refinement, the vertex dynamics model. Explanation begins with the BS model, which is an ancestor of vertex dynamics. The vertex dynamics is a dynamical system of differential equations accompanying an elemental process of reconnection of vertices. Vertices and edges of polygons in the cell center model are not real but deduced from positions of cell centers by the geometry of Dirichlet domains. On the other hand, we actually observe cell boundaries and points of trijunctions of boundaries through experiments. We sought to construct cell models based on actual observations and improve the reliability of mathematical cell models. Then, we constructed and used vertex dynamics. Vertex dynamics models of cell populations have been reviewed in detail (Fletcher et al. 2013; 2014; 2017; Osborne et al. 2017; Alt et al. 2017; Fuji et al. 2022).

4.1

Cell Boundaries and Tricellular Contact3

Boundaries or junctions between cells consist of two cell membranes of neighboring cells. Each cell membrane is lined with associated molecules. In particular, an epithelial cell is columnar, and at its apical level, bundles of actomyosin filaments, lie along the cell membrane. The bundles of actomyosin filaments are restricted to a circumferential region just inside the lateral membranes in the apical part of epithelial cells (Crawford et al. 1972; Perry 1975; Isenberg et al. 1976; Middleton and Pegrum 1976; Eguchi 1977; Crawford 1979; Owaribe et al. 1979; Lehtonen and Badley 1980; Kibbelaar et al. 1980; Kreis and Birchmeier 1980; Yamanaka 1990; 3

Reproduced from excerpts on pages 217, 223 and 240 in Honda 1983 with permission of Elsevier.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_4

39

40

4 Vertex Model

Fig. 4.1 A toy model composed of closed rubber strings and small metal rings

Yamanaka and Honda 1990). Boundaries of the apical regions have contractile properties, which have been experimentally confirmed as follows. When stress fibers, which consist of actomyosin filaments, were severed via laser microirradiation, the target fiber was immediately severed, followed by separation of the cut ends (Strahs and Berns 1979). Another experiment was performed with epithelium that was treated with a glycerol solution and transferred to an ATP solution. Each cell constituting the epithelium began to contract. The epithelium was cleaved into many cell groups as a result of the contraction of each cell (Owaribe et al. 1981). These experiments indicate that microfilaments (actomyosin filaments) within cells are in a tensile state. These investigations show that, even in nonmuscle cells, the microfilament has an explicit or latent ability to act as an intracellular contractile system. Based on observation of contractile filaments along polygonal cell boundaries, a toy model of the network was made composed of many polygons whose circumference was a closed string of stretched rubber, as shown in Fig. 4.1 (Honda 1983). The edges of the network contract, and the polygonal sheet is contractile as a whole. In addition, two cells between which a junction is running do not separate from each other. To make the toy model, joints (small metal rings) were necessary at vertices of polygons that connect three rubber strings. The small ring supports the tensile force of actomyosin along edges. We long wondered what actual substance in cells might correspond to the small joint. Generally, corners of three epithelial cells meet and form a tricellular contact. Tricellular contacts can be considered to be points that support the tensile force of actomyosin along cell–cell junctions (Cavey and Lecuit 2009; Yonemura 2011). As molecular constituents of tricellular contacts, tricellulin and angulin family proteins were identified in mouse epithelial cells (Ikenouchi et al. 2005; Masuda et al. 2011; Higashi et al. 2013). Recently, among these molecules, angulin-1 was distinguished from tricellulin. It was elucidated that angulin-1 is responsible for the plasma membrane seal at tricellular contacts independent of tricellulin (Sugawara et al. 2021). Tricellulin is required for the connection of tight junction (TJ) strands to the central sealing elements but not for epithelial barrier function. Tricellulin was shown to regulate actin filament organization through Cdc42 (Oda et al. 2014).

4.2 Geometry of a Trijunction of Three Lines (Honda 1983)

41

Angulin-1 proteins with extracellular and cytoplasmic domains are stacked along the apicobasal axis among three epithelial cells. The cytoplasmic domain of angulin-1 recruits tricellulin to the tricellular contact of each cell (Sugawara et al. 2021). Tricellulin draws TJ strands and induces a meshwork pattern for tricellular TJ formation (Furuse et al. 2014; Oda et al. 2020). Apical junctional complexes involving the TJ network associate with the actin cytoskeleton (Hartsock and Nelson 2008). The small ring supporting three rubber strings in the toy model shown in Fig. 4.1 may correspond to the large molecular complex of angulin-1 and tricellulin that, via TJ meshwork and adhesion junctional complex, is linked to actomyosin bundles. In invertebrates, tricellular junctions are also formed at tricellular contacts. Gliotactin, which is a transmembrane protein and a septate tricellular junction marker, is highly concentrated in tricellular junctions (Furuse et al. 2014). The dynein-associated protein Mud (NuMA in vertebrates) is also known to be localized and enriched at tricellular junctions in Drosophila epithelia (Bosveld et al. 2016). Thinking back, we anticipated the existence of something at the vertices that integrate contractile forces along three cell boundaries, and we constructed a toy model of rubber strings (Honda 1983). Since then, we continued the theoretical investigation of cell edge contraction (Honda and Eguchi 1980) and succeeded in elucidating cell pattern formation via the mechanical properties of epithelia as described in this book. The mechanical properties of vertices were theoretically established. Now, it is time to make a connection between vertex properties theoretically established and the substances discovered experimentally within the vertex.

4.2

Geometry of a Trijunction of Three Lines (Honda 1983)

As described in the preceding section, we can expect some cellular polygonal patterns to contract. There is a geometrical method to arrange line segments so that they have minimal or shorter edge lengths. We tried to use such a geometrical method for analysis of cellular polygonal patterns. First, we considered the geometry of a trijunction with three lines (Fig. 4.2). There is a geometrical theorem for a pattern of three line segments meeting at a vertex (Fig. 4.2a). When vertex P is movable while three vertices (A, B, and C) are fixed in a plane, the total segment length AP + BP + CP becomes minimum when P is fixed at P0 , where the three lines meet at 120 (i.e., ∠AP0 B ¼ ∠BP0 C ¼ ∠CP0 A ¼ 120 ) if such an arrangement (120 trijunction) is possible. The arrangement is approximated by rubber strings with an elaborated connection, where the string is homogeneously stretched. Unfortunately, when one of the corners of the triangle ABC has an angle of 120 or more (Fig. 4.2b), the trijunction in the minimum length arrangement degenerates, and P0 merges with one of the corners, A, B, or C. Figure 4.2b shows a case in which P merges with A. In such a case, we cannot use this geometrical method. We again tried other new P selected at random. Using the geometrical theorem of the minimum length arrangement of 120 trijunctions, we aimed to transform a given polygonal pattern into a pattern whose

42

4 Vertex Model

Fig. 4.2 Consideration of the total length of a trijunction of three line segments. Points A, B, and C are fixed in a plane, and point P is movable. (a) The total segment length AP + BP + CP becomes minimal when P goes to P0 (if such an arrangement is possible), where the three lines meet at 120 , i.e., ∠AP0 B ¼ ∠BP0 C ¼ ∠CP0 A ¼ 120 . (b) When point A is a vertex with an angle of 120 or more, we cannot obtain the proper position of point P because the trijunction degenerates and P merges with A. We skip this adjustment of point P and again choose another point at random [Original]

total segment length was minimal. If we succeeded in such a transformation, we could obtain information on whether a given cellular sheet is operating under the contractile system. However, as shown in Fig. 4.3, if the polygons were not hexagonal, the size of the polygons continued either to be expanded (stippled polygon) or to be reduced (e.g., gray polygon). Reducing polygons ultimately disappear, and we cannot obtain useful information. It was impossible to apply the transformation to know whether a given cellular sheet is operating under a contractile system (details of the disappearance of a polygon are described in the legend of Fig. 4.3).

4.3

Boundary Shortening (BS) Procedure (Honda and Eguchi 1980)3

Next, we tried a method of two moving vertices (Fig. 4.4). In a 2D cellular pattern in which convex polygons cover a plane without leaving any gaps or overlaps, Ryuji Takaki (1978) described that two arbitrary vertices (P and Q in Fig. 4.4) linked by an

4.3 Boundary Shortening (BS) Procedure (Honda and Eguchi 1980)3

43

Fig. 4.3 Computer simulation using the boundary shortening procedure based on the geometry of the trijunction of line segments at 120 (Fig. 4.2a). To transform a polygonal pattern into another pattern whose total length is shorter than that of the original, a vertex (P) was selected at random in a pattern. Three vertices that are directly connected with P are referred to as A, B, and C and fixed tentatively. P was moved to form a trijunction whose internal angles were all 120 . The same procedure was repeated using a series of random numbers. The results show that heptagons, octagons, etc. expand in size (e.g., dotted area) and pentagons, quadrangles, etc. are reduced in size and eventually disappear (e.g., shaded area) during the repetition. Sequential patterns are shown at 0 (a), 121 (b), 240 (c), and 267 (d) steps. Detailed explanation of polygon disappearance: via the reconnection of vertices (see Fig. 4.11c), the number of vertices of a polygon changes. A pentagon (shaded) in B becomes a triangle through two vertex reconnections (insets b0 and c0 ). The triangle becomes small (c) and ultimately disappears (d). The triangle is replaced by a dot [Reproduced from Fig. 27 and an excerpt of the legend of Fig. 27 in Honda 1983 with permission of Elsevier]

edge and related to four neighboring polygons can be moved to maintain a constant area for each polygon. The length AP0 + BP0 + P0 Q0 + Q0 C + Q0 D can be calculated from the position of P0 (which moves along a line parallel to line AB and contains P) and the position of Q0 (which moves along a line parallel to line CD and contains Q). When P0 is moved from P, Q0 moves so that the area of polygon AP0 Q0 C ¼ the area of polygon APQC. When P0 is sequentially moved from P by a small distance, the local minimum length (AP0 + BP0 + P0 Q0 + Q0 C + Q0 D) is obtained, and the positions of P0 and Q0 settle. When a pattern is given in which polygons are packed in a space without gaps or overlaps, we can move an edge between two vertices according to the movement shown in Fig. 4.4 so that the total length of the five edges decreases. Then, we choose another edge at random, and the two-vertex movement is repeated. The procedure is

44

4 Vertex Model

Fig. 4.4 An elemental step in the boundary shortening (BS) procedure. Vertices P and Q linked by an edge are moved while keeping a constant area for each domain so that the total length of five sides (AP0 + BP0 + P0 Q0 + Q0 C + Q0 D) becomes minimal. PP0 and QQ0 are parallel to AB and CD, respectively. When P moves to P0 , Q is forced to move to Q0 to keep a constant area for two respective polygons (polygon APQC and polygon BPQD) [Reproduced from Fig. 3 in Honda et al. 1986 with permission of The Company of Biologists]

performed one-by-one. After many repetitions of the edge movement, the polygonal pattern becomes almost settled. Then, the polygonal pattern was reformed. The total decreased edge length of the final polygonal pattern was estimated. This is called the boundary shortening (BS) procedure. This procedure allows us to obtain a decreased total edge length under the condition that each cell keeps its original area (Honda and Eguchi 1980). Therefore, when a polygonal cell pattern is given, we can estimate how much the cells contract using the BS procedure. The ratio of decrease of the total edge length is represented by the s-value as a percentage. If the decrease in edge length is small (small s-value), the edges of the pattern are considered to be already contracted. Therefore, we can conclude that the given cell pattern for the cell sheet may be epithelial. If the decrease in edge length is large, the cell sheet is not epithelial. An example is shown in Fig. 4.5. Figure 4.5a presents an initial pattern, and Fig. 4.5b shows the pattern after the BS procedure, in which the difference between the initial and boundary shortened patterns is clear. Figure 4.6 shows an actual process of the BS procedure to examine the cell pattern of the cultured lung epithelial cells shown in Fig. 2.7e in Chap. 2 (peripheral points are fixed, designated by sold circles in the bottom figure). An edge was selected at every step using a series of random numbers, and the selections were repeated until the 2500th step. The decrease in the boundary length at every step is shown in the top figure in Fig. 4.6. Decreases in the edge length are large in the initial early steps and became small after the approximately 2000th step. The percentage of the total decrease (svalue) during the BS procedure is shown at the bottom of Fig. 4.6. Patterns obtained

4.4 Discrimination of Normal Epithelia from Nonepithelial Tissues Using the. . .

45

Fig. 4.5 Comparison of polygonal patterns before and after the BS procedure. A polygonal pattern is given as shown in (a). After the BS procedure, the pattern was remodeled, as shown in (b) [Reproduced from Zu 114 in Honda 1991]

using another two different series of random numbers are superimposed on the bottom of Fig. 4.6. The s-values at the 2500th step were 0.601, 0.602, and 0.608. These three patterns after the BS procedure are similar (dotted lines in Fig. 2.7e (bottom) in Chap. 2). The fixed points that are located almost at the periphery of the pattern may also cause the s-value to vary. When the vertices with open ellipses or triangles in Fig. 4.5 were fixed in addition to those represented by the solid circles, the s-values observed at the 2500th step are shown by the arrows in Fig. 4.6 (s ¼ 0.606 and 0.570). The patterns after the BS procedure were quite similar to the dotted lines in Fig. 2.7e (bottom). The BS procedure was reported in 1980 (Honda and Eguchi 1980). Another computational method for the measurement of cell shape in epithelia was presented by detailed observational analysis of cell–cell interface curvature due to cell surface tension (Stein and Gordon 1982).

4.4

Discrimination of Normal Epithelia from Nonepithelial Tissues Using the BS Procedure (Honda 1983)

Polygonal patterns were analyzed using the cell center models described in the previous section (Fig. 2.7). Here, these polygonal patterns were also analyzed using the BS procedure, and the results were compared with the results of the cell center model, as shown in Fig. 2.7. The sheets of retinal pigment epithelial cells and lung epithelial cells show relatively small s-values (s ¼ 0.687 and 0.598) in comparison with the sheet of chondrocytes (s ¼ 1.84). The difference is remarkable, whereas the Δ values (degree of deviation from Dirichlet domains) of these cell sheets are similar (Δ ¼ ca. 102). Epithelial cell sheets (retinal pigment and lung) are concluded to strongly contract their boundary length. This result is consistent with

46

4 Vertex Model

Fig. 4.6 Process of the BS procedure applied to the pattern of cultured lung epithelial cells in Fig. 2.7e (bottom). Abscissa, steps for the BS procedure. Ordinate (top), change in the boundary length in one calculation step. Ordinate (bottom), s-value, which is the ratio (percentage) of the total boundary length to the initial total boundary length. The results in the bottom figure are presented in a graph obtained using three different series of random numbers. Arrows with an open ellipse and a triangle show the levels at which s-values converged when the points with an open ellipse or triangle in Fig. 2.7e (bottom) were fixed in addition to the solid circles during the BS procedure [Reproduced from Fig. 3 and an excerpt of its legend in Honda and Eguchi 1980 with permission of Elsevier]

the difference between the two artificial polygonal patterns, where the s-value of the stretched rubber polygons (s ¼ 0.072) is significantly smaller than that of the pressed clay spheres (s ¼ 1.033). These results suggested the possibility that we can discriminate normal epithelia from nonepithelial tissues using the BS model. Then, we performed the BS procedure on many polygonal cellular patterns. Our data are summarized in Fig. 4.7 (Honda and Eguchi 1988). Sheets of epithelial cells from pigment retina epithelium, lung epithelium, corneal endothelium, and skin epidermis showed small s-values (less than 0.75), whereas the monolayer cell sheet of cartilage cells (nonepithelial cells) showed a large s-value. The s-value of starfish blastula varies during developmental stages, as described in the next section. FL cells, KB cells, and B16 cells are monolayer cultured cell lines. The FL cell line is a human amnion cell strain that shows a relatively large s-value. The KB cell line and B16 cell line are cancer cell strains and show large s-values. Sheets of these cells in culture do not contract. Therefore, the BS procedure seems to be useful in image analysis for tissue classification. We have described a method to determine the mechanical properties of a given cell pattern as a whole. In contrast, a method to estimate the tension of each edge and the pressure of each cell from the observed geometry of the cells has been developed, in which specific statistics, the Bayesian statistics, was used (Ishihara and Sugimura 2012).

4.5 Tissue Transformation from an Aggregate of Packed Cells to. . .

47

Fig. 4.7 Classification of cell sheets using the BS procedure. The total edge lengths are shortened (small s-values) in epithelial cell sheets (retinal pigment cells, lung epithelial cells in culture, corneal endothelium, mammalian skin epidermis, and starfish blastula) in comparison with nonepithelial cell sheets (cultured cartilage cells, FL cells, KB cancer cells, and B16 cancer cells). X, cells before the blastula stage. Open circle, cell sheet in situ. Gray circle, cell sheet in culture [Reproduced from Zu 9 in Honda and Eguchi 1988 with permission of NewScience-sha, Tokyo]

4.5

Tissue Transformation from an Aggregate of Packed Cells to a Surface-Contracting Cell Sheet4

We generated two cell models, the cell center model and the vertex cell model, both of which are useful. The cell center model applies to tightly packed cells, and the vertex cell model applies to contracting cell sheets, such as epithelia. However, cell properties change during the developmental process. For example, blastomeres of the starfish embryo are spherical, are initially packed forming an embryo surface, and are converted into epithelial cells forming a blastula. We analyzed this cell pattern change quantitatively during the developmental process. Photographs of the developmental process of the starfish Asterina pectinifera were taken of the constituent cells of developing embryos up to the time just before rotation (Honda et al. 1983). All cells in an embryo at every stage were similar in volume because they divide evenly and synchronously. An embryo develops into a hollow sphere consisting of a single layer of cells (Fig. 4.8 left), which eventually become tightly packed (Fig. 4.8 right). After cutting off peripheral regions of the photograph, the cell boundaries in photographs were traced and are represented by solid lines. Patterns of the tightly packed cells were analyzed using the two cell models. The deviation value (Δ) of the pattern derived from Dirichlet domains is very small (Δ ¼ 0.16  102) but increases rapidly during successive stages, as shown in Fig. 4.9 (solid circles), i.e., the patterns increasingly deviate from the Dirichlet

4

Reproduced from an excerpt on page 20 in Honda et al. 1983 with permission of Elsevier (Copyright 1984).

48

4 Vertex Model

Fig. 4.8 Schematic drawings of the transformation of the cell pattern in starfish blastula [Reproduced from Zu 11c and d in Honda 1991]

Fig. 4.9 Quantitative analysis of the process of starfish blastula formation. Deviation from the Dirichlet pattern (Δ) increases along with the developmental stage, whereas the total edge length shortens (s-value decreases). The cell pattern changes from the cell center model to an edgecontracting pattern in the vertex cell model. Abscissa, developmental stages of the starfish expressed by cell numbers [Reproduced from Fig. 8 in Honda et al. 1983 with permission of Elsevier (Copyright 1984)]

domains. To estimate the degree of contraction of cell boundaries, s values were obtained from the patterns, which were observed to decrease as development proceeded (Fig. 4.9 open circles). The transformation is schematically presented in Fig. 4.10a. Independent spherical cells were packed into polygonal shapes, the boundaries of polygons contracted, and the cells eventually organized into a tensile sheet. Initially, a polygonal pattern was made under the Dirichlet principle, and the arrays of polygons were then modified by the principle of boundary contraction. Figure 4.10b shows a schematic drawing of the transformation from an aggregate of packed cells to a surfacecontracting cell sheet.

4.6 Reconnection of Paired Vertices of an Edge

49

Fig. 4.10 Transformation of cell types from packed cells to contracting cells. (a) Transformation of packed cells in the cell center model to contracting cells in the vertex cell model. Spherical cells packed in a restricted space form a Dirichlet polygonal pattern (left to middle). Edges of the polygonal pattern contract, and a tensile polygonal pattern forms (right). The dotted line shows the pattern in the previous stage (left side figure) [Reproduced in from Fig. 43 in Honda 1983 with permission of Elsevier]. (b) Schematic 3D views of the transformation of packed cells to contracting cells [Reproduced from Zu 11 in Honda and Eguchi 1988 with permission of NewScience-sha, Tokyo]

4.6

Reconnection of Paired Vertices of an Edge

Vertices are connected by edges, forming networks in a polygonal pattern. Changes in a polygonal pattern were induced by reconnection of edges among vertices. In particular, reconnection of paired vertices forming an edge (Fig. 4.11c) is an elemental process in changing the polygonal patterns. Sometimes, during the BS procedure in cellular polygonal patterns, an edge shortens extremely and forms an “X” pattern, as shown in Fig. 4.11d. Then, if we perform a reconnection of the short edge, we have a chance that the BS procedure continues shortening the total edge length. The reconnection of vertices causes shape changes in polygons. For example, a hexagon becomes a heptagon, and the neighboring hexagon becomes a pentagon. Therefore, the BS procedure accompanying the vertex reconnection alters the shape of polygons and is useful for computer simulation of pattern formations. A typical example is shown as follows. The top figure in Fig. 4.12a shows an artificially horizontally elongated polygonal pattern. The BS procedure was performed in the polygonal pattern. We often found short edges of an “X” pattern. At each short edge, we applied vertex reconnection (Fig. 4.11c). Finally, we obtained a polygonal

50

4 Vertex Model

Fig. 4.11 Reconnection of paired vertices of an edge. (a) Photographs of corneal endothelium on Day 0 and Day 1, where a small number of cells were removed on Day 0 and the corneal epithelium began to repair (Honda et al. 1982). Cells are marked with small and large white circles for identification. Bar, 50 μm. (b, c) Tracing patterns in (a). As cells rearranged, two contiguous cells (marked with small white circles in a, gray in b) separated from each other. Thus, the edge numbers for these four cells changed, decreasing by one or increasing by one: 6 to 5, 6 to 5, 7 to 8, and 6 to 7. Nevertheless, the total outline of the four cells was not changed. Rearrangement took place independently of the edge numbers of surrounding contiguous cells [Reproduced from Fig. 5b and Fig. 5b’ in Honda et al. 1982 with permission of John Wily and Sons]. (d) “X” pattern in the corneal endothelium. This photograph is a live image of the corneal endothelium taken with a specular ophthalmologic microscope using a 20 water immersion objective, as described in Honda et al. (1982). Bar, 50 μm. Courtesy of Yoji Ogita and Shoichi Higuchi

pattern which consisted of nonelongated polygons whose areas are the same as those of the previous polygons. A series of edge reconnections is shown in Fig. 4.12b. After five reconnections of vertices, two vertically arranged polygons became horizontally arranged polygons. We established a computer simulation method for edge reconnection (Honda et al. 1982). Edge reconnection is not only hypothetical but was confirmed by actual observations in 1982 (Honda et al. 1982). Figure 4.11a shows an actual process of edge reconnection of cells of corneal endothelium. Four cells (marked by white discs) rearranged with each other, and a vertex reconnection took place. Figure 4.11d shows some “X” patterns during edge reconnections. The mechanical basis of cell rearrangement was examined in detail through observation of epithelial morphogenesis during Fundulus epiboly (Weliky and Oster 1990). In their model, the forces acting on each vertex are given explicitly.

4.7 Vertex Dynamics (Nagai and Honda 2001)

51

Fig. 4.12 Change in cell patterns through reconnection of vertices. (a) Cells in an elongated polygonal pattern become nonelongated after many reconnections of vertices [Reproduced from Fig. 7b-d Honda et al. 1982 with permission of John Wiley and Sons]. (b) Two stacked cells became two horizontally arranged cells after sequential reconnections of vertices. Snapshots are at steps 0, 1090, 2400, 4044, and 12,000 [Reproduced from Fig. 9 in Honda et al. 1982 with permission of John Wiley and Sons]

4.7

Vertex Dynamics (Nagai and Honda 2001)

The BS procedure was refined by introducing differential equations, and a dynamical system—vertex dynamics for assemblage of biological cells—was established (Nagai and Honda 2001; Honda and Nagai 2015). This system was constructed by Tatsuzo Nagai based on vertex dynamics in physics (Nagai et al. 1988), which modeled the evolution of soap froths and grain aggregates (Fullman 1952). An assemblage of biological cells is considered to be a monolayered sheet consisting of Nc cells, where each cell is prism shaped. Assigning the base area of cell α to be Sα

52

4 Vertex Model

and assuming the height of cells to be uniform, we can describe the system in terms of a two-dimensional polygonal pattern. When initial vertex positions and the relationships of vertex pairs linked by edges are given, we can track vertex positions with time. Since a polygonal cell pattern is defined by constituent polygons, we can investigate cell pattern formation or the morphogenesis of polygonal cell patterns using vertex dynamics. A flat sheet composed of multiple cells is paved with polygons without gaps or overlaps. The edge (boundary between two cells) and area of polygons (corresponding to cell volume) are expressed by the x- and y-coordinates of the vertices. Spatial relationships between neighboring vertices are defined by surrounding polygons. The vertices obey an equation of motion similar to an equation of motion in which massless particles move under potential and frictional forces: η dri =dt ¼ ∇i U

ði ¼ 1 . . . nv Þ,

ð4:1Þ

where ri is a positional vector of vertex i, ∇i is the Nabla differential operator with respect to ri, and nv is the total vertex number. The left side of Eq. (4.1) represents a viscous drag force proportional to the vertex velocity dri/dt, with the positive constant η (an analog of the coefficient of viscosity). Vertices do not have mass (inertia), and thus, the motion of the vertices and cells is completely damped. Equation (4.1) contains U, referred to as potential. Differentiation of U with respect to time t gives the following inequality using Eq. (4.1), dU=dt ¼ Σi ð∇i UÞ  ðdri =dtÞ ¼ η Σi ðdri =dtÞ2  0

ð4:2Þ

The inequality indicates that vertices move to decrease U (strictly, not to increase U ). We can obtain a stable shape described by vertices using Eq. (4.1). The right side of Eq. (4.1), minus the gradient of the potential U represents a potential force (driving force). The potential U includes two main terms related to the edges and surface areas of polygons, which are all expressed by vertex positions. Hence, U is a function of the vertex coordinates, ri ¼ (xi, yi). The potential U contains terms of interfacial energy or edge energy (UL) and elastic surface energy (UES): U ¼ U L þ U ES

ð4:3Þ

The potential UL denotes the total edge energy of the cells: U L ¼ σ L Σ Lij,

ð4:4Þ

where i and j are neighboring vertices forming the edge ij and Lij is the length of edge ij. Summations are performed over all edges. σ L is the edge energy density. UL works such that the edge length of the circumference of each polygon becomes short

4.7 Vertex Dynamics (Nagai and Honda 2001)

53

and the shape of the polygon becomes as round as possible. The potential UES denotes the total elastic energy of the polygon area: U ES ¼ κS Σn a ðSa  So Þ2 ,

ð4:5Þ

where Sa and So are the polygon area at time t and the polygon area in the relaxed state, respectively. κS is the elastic energy density of the polygon area. n is the cell number. UES works so that the area of each polygon becomes an area in the relaxed state. Thus, Eq. (4.1) takes the form: h i η dri =dt ¼ — i σ L Σ Lij þ κS Σn a ðSa  So Þ2 :

ð4:6Þ

There are two ways to use the equilibrium value of the cell area, So: for the global equilibrium value of the cell area, a value is given to So; for local equilibrium dynamics, So in Eq. (4.6) is replaced by   So α ¼ Sα þ Σnα β¼1 Sβ =ðnα þ 1Þ

ð4:7Þ

where nα denotes the edge number of cell α and the β sum is taken over the immediate neighbor cells of cell α (Nagai and Honda 2001). The vertices move according to the equation of motion so that the potential energy U decreases. During calculation, we sometimes encounter a minute edge. To continue calculation, we have to introduce an elementary process, reconnection of paired vertices to the system of differential equations (Nagai and Honda 2001; Honda et al. 2004). When an edge length becomes less than the critical length δ, a reconnection of paired vertices takes place, as previously described in Reconnection of paired vertices (Fig. 4.11c). The reconnection of edges causes shape changes in polygons, e.g., a hexagon becomes a heptagon, and the neighboring hexagon becomes a pentagon. Thus, the reconnection process brings plastic properties to the cell models. We can obtain a stable state of the cell sheet using computer simulations. Numerical calculations of the differential equations were performed using the Runge–Kutta method. We applied the vertex dynamics to a Dirichlet pattern, as shown in Fig. 4.13. A Dirichlet pattern was made with 200 centers that were scattered at random (Fig. 4.13 left), to which the vertex dynamics were applied. Figure 4.13 (middle and right) shows the patterns after t ¼ 0.3 and 200, respectively. Reconnections of paired vertices took place frequently during the process, and the polygonal pattern became regular. A, B, and C in each figure are for cell identification.

54

4 Vertex Model

Fig. 4.13 Computer simulation of the Dirichlet pattern determined by vertex dynamics. Left, initial polygonal pattern (t ¼ 0). A Dirichlet pattern was constructed based on 200 randomly scattered centers in a rectangle. A periodic boundary condition is imposed on the system, where the rectangular area satisfies two-dimensional tiling in which one side of the rectangle continues to the opposite side of the neighboring rectangle. κ S ¼ 5.5; σ L ¼ 1.0; So ¼ 1.0; δ ¼ 0.2. Center and right, snapshots at t ¼ 0.3 and 200, respectively. A, B, and C are for polygon identification [Reproduced from Fig. 5 in Nagai and Honda 2001 with permission of Taylor & Francis]

4.8

Summary

Another mathematical cell model, the vertex model, is described in this chapter. Real cell aggregates consist of cell boundaries and vertices (trijunctions of edges), which have been actually observed, not a mere mathematical concept. We constructed collective polygons consisting of vertices, edges, and faces. Edges and faces in the aggregate are expected to be contractile due to the contractility of the molecules with which the edges and faces are lined. We devised a mathematical procedure that reduces the total edge length of a polygonal pattern. The procedure is referred to as the boundary shortening procedure (BS procedure) and is useful for discriminating whether a given pattern is contractile. Contractile patterns may reflect patterns of epithelial tissue. Nonepithelial tissues may belong to the mesenchyme, tumor tissues, or other tissues. Two types of cell patterns have been observed, one that fits the cell center model (small Δ value) and one that fits the BS shortening pattern (small svalue). These two types of patterns are transformable during the developmental process. We showed the example of the blastula stage of the starfish embryo in which cells convert to epithelial cells. Finally, we introduced vertex dynamics, which involves an equation of the motion of vertices (Eq. 4.1). The equation has a potential U and drives the motion of vertices so that U decreases. During calculation, any edge becomes minute. Here, we introduced an elemental process of pattern formation, i.e., reconnection of a pair of vertices linked by an edge among four polygons (Figs. 4.11b and 4.14a). A short edge, which is a boundary between two polygons and links the other two polygons, is displaced and reconnects vertices. Finally, two polygons intercalate between the other two polygons; in other words, the other two polygons are intercalated. We applied the elemental process to minute edges. After reconnections, the shapes of polygons (edge number of polygons) change, and we continue calculation of the vertex dynamics toward a shorter total edge of the polygonal pattern. Finally, we would like to note that, in our studies, we

References

55

Fig. 4.14 Elementary processes of topological changes in wound closure. δ, short length. (a) Vertex reconnection process (T1): changes in neighbors of vertices i and j. (b) Disappearance process (T2): disappearance of a triangular wound area (w). Process T2 was previously presented in Fig. 4.3d. (c) Adhesion process (T3): left and center, vertex i touches another wound edge kk0 ; right, three cells adhere to one another, and two new vertices l and l0 are created where the distances between wound edges defined by the fine dotted lines (fine dotted line ¼ δ) [Reproduced from Fig. 2 in Nagai and Honda 2009 with permission of American Physical Society]

have observed that experiments follow the results of theoretical work; in this case, the theoretical work predicts the experimental result. Based on observation of actual vertices in cell patterns, we hypothetically assigned mechanical properties to vertices (a small ring supporting three rubber strings) and constructed a useful cell model. After these theoretical investigations, real molecules, such as angulin-1 and tricellulin, were discovered in the vertex.

References Alt, S., Ganguly, P., Salbreux, G.: Vertex models: from cell mechanics to tissue morphogenesis. Philos. Trans. R. Soc. B: Biol. Sci. 327, 2015052 (2017) Bosveld, F.O., Markova, B., Guirao, C., Martin, Z., Wang, A., Pierre, M., Balakireva, I., Gaugue, A., Ainslie, N., Christophorou, D.K., Lubensky, N.M., Bellaiche, Y.: Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis. Nature. 530, 495–498 (2016) Cavey, M., Lecuit, T.: Molecular bases of cell-cell junctions stability and dynamics. Cold Spring Harbor Perspect. Biol. 1, a002998–a002998 (2009) Crawford, B.: Cloned pigmented retinal epithelium–The role of microfilaments in the differentiation of cell shape. J. Cell Biol. 81, 301–315 (1979) Crawford, B.R.A., Cloney, Cahn, R.D.: Cloned pigmented retinal cells; the affects of cytochalasin B on ultrastructure and behavior. Z. Zellforsch. 130, 135–151 (1972) Eguchi, G.: Cell shape change and establishment of tissue structure. Saiensu (Japanese edition of Scientific America). 7, 66–77 (1977) Fletcher, A.G., Osborne, J.M., Maini, P.K., Gavaghan, D.J.: Implementing vertex dynamics models of cell populations in biology within a consistent computational framework. Prog. Biophys. Mol. Biol. 113, 299–326 (2013) Fletcher, A.G., Osterfield, M., Baker, R.E., Shvartsman, S.Y.: Vertex models of epithelial morphogenesis. Biophys. J. 106, 2291–2304 (2014). https://doi.org/10.1016/j.bpj.2013.11.4498

56

4 Vertex Model

Fletcher, A.G., Cooper, F., Baker, R.E.: Mechanocellular models of epithelial morphogenesis. Phil. Trans. R. Soc. B. 372, 20150519 (2017) Fullman, R.L.: Metal interfaces. In: Proceedings of ASM Seminar, Cleveland, OH, USA, 13 October 1951 (1952). Furuse, M.Y., Izumi, Y., Oda, T.H., Iwamoto, N.: Molecular organization of tricellular tight junctions. Tissue Barriers. 2, e28960 (2014) Hartsock, A., Nelson, W.J.: Adherens and tight junctions: structure, function and connections to the actin cytoskeleton. Biochim. Biophys. Acta (BBA) - Biomembranes. 1778, 660–669 (2008) Higashi, T.S., Tokuda, S., Kitajiri, I., Masuda, S., Nakamura, H., Oda, Y., Furuse, M.: Analysis of the ‘angulin’ proteins LSR, ILDR1 and ILDR2 – tricellulin recruitment, epithelial barrier function and implication in deafness pathogenesis. J. Cell Sci. 126, 3797–3797 (2013) Fuji, K., Tanida, S., Sano, M., Nonomura, M., Riveline, D., Honda, H., Hiraiwa, T.: Computational approaches for simulating luminogenesis. (in press) (2022) Honda, H.: Geometrical models for cells in tissues. Int. Rev. Cytol. 81, 191–248 (1983) Honda, H.: “Shiitokarano Karada-zukuri” in Japanese. Chuokoron-sha, 1–230 (1991) Honda, H., Eguchi, G.: How much does the cell boundary contract in a monolayered cell sheet? J. Theor. Biol. 84, 575–588 (1980) Honda, H., Eguchi, G.: Cell sheets on culture dishes (2). Sosiki Baiyou (in Japanese). 14, 420–425 (1988) Honda, H., Nagai, T.: Cell models lead to understanding of multi-cellular morphogenesis consisting of successive self-construction of cells. J. Biochem. 157, 129–136 (2015) Honda, H., Ogita, Y., Higuchi, S., Kani, K.: Cell movements in a living mammalian tissue: longterm observation of individual cells in wounded corneal endothelia of cats. J. Morphol. 174, 25–39 (1982) Honda, H.M., Dan-Sohkawa, Watanabe, K.: Geometrical analysis of cells becoming organized into a tensile sheet, the blastular wall, in the starfish. Differentiation. 25, 16–22 (1983) Honda, H.H., Yamanaka, Eguchi, G.: Transformation of a polygonal cellular pattern during sexual maturation of the avian oviduct epithelium: computer simulation. J. Embryol. Exp. Morphol. 98, 1–19 (1986) Honda, H., Tanemura, M., Nagai, T.: A three-dimensional vertex dynamics cell model of spacefilling polyhedra simulating cell behavior in a cell aggregate. J. Theor. Biol. 226, 439–453 (2004) Ikenouchi, J., Furuse, M., Furuse, K., Sasaki, H., Tsukita, S., Tsukita, S.: Tricellulin constitutes a novel barrier at tricellular contacts of epithelial cells. J. Cell Biol. 171, 939–945 (2005) Isenberg, G., Rathke, P.C., Hülsmann, N., Franke, W.W., Wohlfarth-Bottermann, K.E.: Cytoplasmic actomyosin fibrils in tissue culture cells: direct proof of contractility by visualization of ATP-induced contraction in fibrils isolated by laser micro-beam dissection. Cell Tissue Res. 166, 427–443 (1976) Ishihara, S., Sugimura, K.: Bayesian inference of force dynamics during morphogenesis. J. Theor. Biol. 313, 201–211 (2012) Kibbelaar, M.A., Ramaekers, F.C., Ringens, P.J., Selten-Versteegen, A.M., Poels, L.G., Jap, P.H., von Rossum, A.L., Feltkamp, T.E., Bloemendal, H.: Is actin in eye lens a possible factor in visual accommodation? Nature. 285, 506–508 (1980) Kreis, T.E., Birchmeier, W.: Stress fiber sarcomeres of fibroblasts are contractile. Cell. 22, 555–561 (1980) Lehtonen, E., Badley, R.A.: Localization of cytoskeletal proteins in preimplantation mouse embryos. J. Embryol. Exp. Morphol. 55, 211–225 (1980) Masuda, S., Oda, Y., Sasaki, H., Ikenouchi, J., Higashi, T., Akashi, M., Nishi, E., Furuse, M.: LSR defines cell corners for tricellular tight junction formation in epithelial cells. J. Cell Sci. 124, 548–555 (2011) Middleton, C.A., Pegrum, S.M.: Contacts between pigmented retina epithelial cells in culture. J. Cell Sci. 22, 371–383 (1976)

References

57

Nagai, T., Honda, H.J.: A dynamic cell model for the formation of epithelial tissue. Philos. Mag. B. 81, 699–719 (2001) Nagai, T., Honda, H.: Computer simulation of wound closure in epithelial tissues: Cell–basallamina adhesion. Phys. Rev. E. 80(6), 061903 (2009) Nagai, T., Kawasaki, K., Nakamura, K.: Vertex dynamics of two-dimensional cellular patterns. J. Phys. Soc. Jpn. 57, 2221–2224 (1988) Oda, Y., Otani, T., Ikenouchi, J., Furuse, M.: Tricellulin regulates junctional tension of epithelial cells at tricellular contacts through Cdc42. J. Cell Sci. 127, 4201–4212 (2014) Oda, Y., Sugawara, T., Fukata, Y., Izumi, Y., Otani, T., Higashi, T., Fukata, M., Furuse, M.: The extracellular domain of angulin-1 and palmitoylation of its cytoplasmic region are required for angulin-1 assembly at tricellular contacts. J. Biol. Chem. 295, 4289–4302 (2020) Osborne, J.M., Fletcher, A.G., Pitt-Francis, J.M., Maini, P.K., Gavaghan, D.J.: Comparing individual-based approaches to modelling the self-organization of multicellular tissues. PLOS Comput. Biol. 13, e1005387 (2017) Owaribe, K., Araki, M., Hatano, S., Eguchi, G.: Cell shape and filaments: 491-500. In: Hatano, S., Ishikawa, H., Sato, H. (eds.) Cell motility: Molecules and organization. University of Tokyo Press, Tokyo (1979) Owaribe, K., Kodama, R., Eguchi, G.: Demonstration of contractility of circumferential actin bundles and its morphogenetic significance in pigmented epithelium in vitro and in vivo. J. Cell Biol. 90, 507–514 (1981) Perry, M.M.: Microfilaments in the external surface layer of the early amphibian embryo. J. Embryol. Exp. Morph. 33, 127–146 (1975) Stein, M.B., Gordon, R.: Epithelia as bubble rafts: a new method for analysis of cell shape and intercellular adhesion in embryonic and other epithelia. J. Theor. Biol. 97, 623–639 (1982) Strahs, K.R., Berns, M.W.: Laser microirradiation of stress fibers and intermediate filaments in non-muscle cells from cultured rat heart. Exp. Cell Res. 119, 31–45 (1979) Sugawara, T., Furuse, K., Otani, T., Wakayama, T., Furuse, M.: Angulin-1 seals tricellular contacts independently of tricellulin and claudins. J. Cell Biol. 220, e20200562 (2021) Takaki, R.: Katachi no Tankyuu (In Japanese) Daiamondo-Sha. Figure 17b on page 19 (1978) Weliky, M., Oster, G.: The mechanical basis of cell rearrangement I. Epithelial morphogenesis during Fundulus epiboly. Development. 109, 373–386 (1990) Yamanaka, H.: Pattern formation in the epithelium of the oviduct of Japanese quail. Int. J. Dev. Biol. 34, 385–390 (1990) Yamanaka, H.I., Honda, H.: A checkerboard pattern manifested by the oviduct epithelium of the Japanese quail. Int. J. Dev. Biol. 34, 377–383 (1990) Yonemura, S.: Cadherin-actin interactions at adherens junctions. Curr. Opin. Cell Biol. 23(5), 515–522 (2011)

Chapter 5

Applications of 2D Cell Models

Outline The 2D vertex model was applied to elucidate the mechanism of wound healing of the epithelium through extension of surrounding epithelial cells and the mechanism of zigzag ordered pattern formation involving apoptosis in the Drosophila wing margin. Pattern formation via differential cell adhesion is also discussed considering cell–cell adhesion molecules.

5.1

Wound Closure in Epithelial Tissues (Nagai and Honda 2009)

Cell disappearance in the cell sheet was previously described by the cell center model (Chap. 2). The cell disappearance is a recovery process after the cell loss. However, we did not study the course of the recovery process of the cell sheet. Wound healing involving cell proliferation and biochemical regulations has been studied using mathematical models (Sherratt and Murray 1990). However, there are recovery processes without cell proliferation (e.g., cat cornea endothelium in Chap. 2). Here, we studied wound healing without cell proliferation. We examined the behavior of cell boundaries after cell disappearance using vertex dynamics involving cell adhesion. A wound is an area in which some cells have been removed from the cell monolayer sheets. The system encompassing the wound area consists of the cells surrounding the wound, basal lamina touching the cellular bases, and an environmental medium. The basal lamina is a soft, thin mat on which epithelial cells rest and separates epithelial cells from connective tissue. We introduced interfacial energy between the cell and basal lamina into the vertex dynamics cell model as the primary driving force of wound closure and cell–wound boundary energy as the secondary force (Nagai and Honda 2009). While the latter gives the contractility by actin filaments to the cell boundary, the former gives the new mechanism of adhesion between a cell and the basal lamina, which describes the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_5

59

60

5 Applications of 2D Cell Models

spreading of an isolated rat hepatocyte on a serum-pretreated glass surface, as mentioned previously (Fig. 2.2). It should be noted that in this section, we focus on the closure of small wounds via cell spreading and cell rearrangement. The mitotic process is not included in the simulations. Then, the potential U in Eq. (4.3) has additional terms as follows. U ¼ U L CC þ U L CW þ U ES þ U W :

ð5:1Þ

U L CC ¼ σ L ΣCC Lij ,

ð5:2Þ

U L CW ¼ 1=2 σ L ΣCW Lij ,

ð5:3Þ

where the potential ULCC denotes the total edge energy of cell–cell boundaries and ULCW denotes the total edge energy of cell–wound boundaries. i and j are neighboring vertices forming the edge ij, and Lij is the length of edge ij. Summations ΣCC and ΣCW were performed over all edges of cell–cell boundaries and cell–wound boundaries, respectively. σ L is the edge energy density of cell–cell boundaries. Note that the edge energy of cell–wound boundaries is 1/2 that of cell–cell boundaries. UES is described by Eq. (4.5). The last term UW denotes the exposure energy of the wound given by U W ¼ σ W Σγ S W γ ,

ð5:4Þ

where σ W is the surface energy of the wound and SWγ is the area of wound γ. Using the potential U, we perform computer simulations. When the length of any edge becomes shorter than a critical length δ, we use the elementary process of reconnection of vertices. In the wound healing process, in addition to the elementary process (Figs. 4.11c or 4.14a), two additional elementary processes are necessary, as shown in Fig. 4.14b, c. Figure 4.14b shows the process of wound disappearance. The wound size deceases, and the wound becomes a triangle. When the length of the minimum edge of the triangle becomes less than δ, the triangle of the wound converges to a vertex. The elemental process was introduced to describe the disappearance of soap bubbles or polycrystal grains (Nagai et al. 1988). Figure 4.14c shows collision of a vertex on the wound boundary with another wound boundary. Three cells that contact the vertex remodel their boundaries, as shown in Fig. 4.14c. These procedures in the elemental processes are based on the coarse-grained model, which is an approximation of a real cell system. Elements in our model have a finite size (critical length δ), below which one cannot discuss within the model. For the coarse-grained model, see Fundamentals of vertex dynamics in Chap. 6. A result of the computer simulations is shown in Fig. 5.1. Here, all physical quantities are dimensionless and have been scaled by their characteristic quantities in the system (see in detail Fundamentals of vertex dynamics in Chap. 6). The gray area is the wound, while the surrounding polygons are normal cells. The initial wound in Fig. 5.1 (t ¼ 0) that was created by removing 20 cells was relatively round. As seen in subsequent snapshots, the wound contracts approximately retaining its initial round shape. During wound contraction, the surrounding cells elongate toward the

5.1 Wound Closure in Epithelial Tissues (Nagai and Honda 2009)

61

Fig. 5.1 Snapshots from the computer simulation of wound closure with a round wound. The wound (gray area) disappears at t ¼ 3.95. Right, contraction process of the wound. The wound contours at t ¼ 0, 0.5, 1, 2, 3, 3.5, and 3.9 are superposed [Reduced from Fig. 3 and Fig. 6a in Nagai & Honda 2009 with permission of American Physical Society]

wound, and the closer to the wound they are, the greater they deform. The wound disappears at t ¼ 3.95, and then, the surrounding cells move to decrease the entire length of the cell–cell boundaries. The relaxation movements of the cells mostly finish at t ¼ 50.0. The wound contours at each time t are superposed in Fig. 5.1 (right). Figure 5.2 shows the results of other computer simulations. The initial wound (Fig. 5.2 t ¼ 0) was long and narrow, which was created from the same original normal pattern shown in Fig. 5.1 (t ¼ 0). The wound contracts and approximately retains its initial shape until a vertex on one long side of the wound margin touches the other side. Here, the third elementary process of the edge adhesion process was performed. The third elementary process was constructed by Tatsuzo Nagai, as shown in Fig. 4.14c. A vertex touches another wound edge and becomes two new vertices; then, the wound splits into two parts. The two wound parts continue to decrease in area. The wound contours at each time point are superposed in Fig. 5.2 (right). On the other hand, the spreading of real epithelial cells at the wound margin was recorded during wound closure in the fin of Xenopus laevis tadpoles. Extending borders were drawn in superposition from the recorded photographs (see Fig. 2 in Honda 1983). The actual observation is remarkably similar to the results of the computer simulation of wound closure (Figs. 5.1, right, and 5.2, right). In conclusion, the driving force of wound closure was understood as follows. The primary driving force is a force that attempts to decrease the total potential energy between the cell surface and basal lamina. Cell–wound boundary tension is the secondary driving force of wound closure, which assists the primary force in closing the wound. Wound closure proceeds as the driving forces overcome resistance due to

62

5 Applications of 2D Cell Models

Fig. 5.2 Snapshots from a computer simulation of wound closure with a flattened wound. The wound (gray area) splits at t ¼ 4.23. The left wound disappears at t ¼ 7.50, while the right wound disappears at t ¼ 9.54. After wound closure, the surrounding cells relax into the final state at t ¼ 50.0. Right, the wound contraction process. The wound contours at t ¼ 0–7 (interval time ¼ 0.5), 8, and 9 were superposed [Reproduced from Fig. 4 and Fig. 6b in Nagai & Honda 2009 with permission of American Physical Society]

the two fundamental forces of morphogenesis, cell–cell boundary potential forces, and cell elastic forces. Later, a mathematical model of cell shrinkage and disappearance in 3D tissues was proposed (Okuda et al. 2016).

5.2

Cell Death Leads to Ordered Cell Patterns5

We applied the cell models to two cell disappearance events, the cell center model to artificial cell removal (Chap. 2) and the vertex model to wound closure in the previous section. Here, another cell loss, programmed cell death or apoptosis, is considered. Apoptosis is a type of cell disappearance. Although apoptosis is a relatively recent introduction in the long history of embryology (Saunders and Fallon 1966), its indispensable role in the developmental process has become broadly recognized (Alberts et al. 2002). We examined apoptosis-induced patterning using vertex dynamics (Nagai et al. 2018). Morphogenesis of the Drosophila wing involves apoptosis (Takemura and Adachi-Yamada 2011, 2013), and the Drosophila wing exhibits a well-ordered cell pattern, where hair cells are arranged in a zigzag pattern. 5 Reproduced from excerpts on pages 958, 959 and 961 in Nagai et al. 2018 with permission of Elsevier.

5.2 Cell Death Leads to Ordered Cell Patterns

63

Fig. 5.3 Zigzag pattern formation of wing margin hairs. (a) Adult Drosophila wing. (b) Higher magnification view of the posterior wing margin (the rectangular region in a). (c, d) Lateral views of the Drosophila pupal wing margin at 20 h and 30 h after puparium formation (APF). Cell types are symbolized for clarity. White * indicates the hair cell; o indicates the interhair cell; + indicates the tooth cell; and blank areas indicate preapoptotic or apoptotic cells. The region between the two lines of hair cells (*) is the lateral portion; the upper region (white cells, out of focus) is the dorsal part; and the lower region (white cells, out of focus) is the ventral part. To see this figure in color, go online (Nagai et al. 2018) [Reproduced from Fig. 1 in Nagai et al. 2018 with permission of Elsevier]

The posterior portion of the Drosophila wing margin is covered with long hairs sourced from hair cells arranged in a zigzag pattern in the lateral view (Fig. 5.3a, b). Figure 5.3c, d show photomicrographs of the formation of this zigzag pattern (Takemura and Adachi-Yamada 2011, 2013). Figure 5.3d shows that the dorsal and ventral hair cells along the wing margin between the dorsal and ventral surfaces interlock with each other like a fastener (zipper). According to their observations during pupal wing formation, this zigzag pattern is created by selective apoptosis of wing margin cells that are spatially separated from hair cells. Hair cells send signals to their immediately neighboring cells, preventing their apoptosis. Hair cells rearrange the remaining wing margin cells in a well-ordered manner to create the zigzag pattern. To simulate the zigzag pattern, we inserted additional information on the Drosophila wing. The Drosophila wing blade continues extending along its proximal– distal axis, resulting in oriented elongation of the epithelial wing cells. The reason why the wing blade elongates is that strong contraction of the basal part of the wing (wing hinge) causes elongation of the remaining wing part (wing blade), as described in Aigouy et al. (2010), Ray et al. (2015), and Diaz de la Loza and Thompson (2017). Based on the experimental understanding of elongation of the wing blade, we artificially elongated the outer frame of the polygonal pattern, which induced passive elongation of polygonal cells. Computer simulations were performed using the cell-based model of vertex dynamics for tissues. The model describes the epithelial tissue as a monolayer cell sheet of polyhedral cells, which are approximated by prisms with an equal height.

64

5 Applications of 2D Cell Models

Therefore, our system is expressed by an assembly of polygonal cells on a 2D plane. Vertices of the polygonal cells move according to equations of motion, minimizing the sum total of the interfacial and elastic energies of cells. The interfacial energy densities between cells are introduced consistently with an ideal zigzag cell pattern, which is extracted from the experimental results (Nagai et al. 2018). Based on the experimental findings mentioned above, cells that undergo apoptosis were modeled by gradually reducing their equilibrium areas with a given lifetime as follows. First, we initially specified preapoptotic cells that had no or small contact with hair cells and then gave their apoptosis starting times tAμ at random. Thus, the equilibrium area of apoptotic cells μ is assumed to decrease exponentially at time t after tAμ:

SAμ 0 ðt Þ ¼ ST 0 exp  t  t Aμ =τA for t > t Aμ ,

ð5:5Þ

where ST0 is the initial area and τA is the lifetime. Exponential decay was inferred from the results of experiments showing wound closure in epithelial tissues and an exponential decrease in the wound area (Nagai and Honda 2009). In Eq. (5.5), each preapoptotic cell is indistinguishable from a cell that closely contacts the hair cells before its apoptosis starting time and enters the apoptotic process after that time. The cell decreases its equilibrium area from its initial value ST0 and vanishes at ~τA. When the cell size decreased to less than a critical size in our simulation, the cell disappeared according to the elemental process of cell disappearance (Fig. 4.14b, where wound w should be replaced by an apoptotic cell). The elongation of the wing blade mentioned above (passive elongation of cells) was modeled as elongation of the horizontal outer frame of the cell system under a constant area of the system (Fig. 5.4). Starting with an initial cell pattern similar to the micrograph experimentally obtained just before apoptosis (Fig. 5.4 top), we carried out simulations and successfully reproduced the ideal zigzag cell pattern (Fig. 5.4, top to bottom). The computer simulation theoretically elucidates a physical mechanism of patterning triggered by cell apoptosis and exemplifies a new framework to study apoptosisinduced patterning. We conclude that the zigzag cell pattern is formed by an autonomous communicative process among the participant cells.

5.3

Passively Elongated Epithelial Tubes (Honda et al. 2009)

The morphogenesis of tubes (e.g., capillary blood vessels, insect tracheas) in developing embryos was studied. Tubes are 3D objects. However, when we consider that tubes are mathematically constructed of a monolayer sheet wrapped around a lumen, we can study them using 2D vertex dynamics (Fig. 5.5). The method is referred to as the method of rectangle–cylinder conversion. Here, we use a periodic boundary condition in which the rectangular area satisfies “tiling”: the right side of the rectangle continues to the left side of the neighboring rectangle. In the case of a

5.3 Passively Elongated Epithelial Tubes (Honda et al. 2009)

65

Fig. 5.4 Snapshots of the simulations of the vertex dynamics cell model (top to bottom). NA and λ are the total numbers of preapoptotic and apoptotic cells and the aspect ratio of the system at time t, respectively. Cell types are differentiated by symbols: black (*) represents the hair cell; o, gray indicates the interhair cell; +, white indicates the tooth cell; and blank areas indicate the preapoptotic or apoptotic cells. Numerals in the hair and interhair cells are cell numbers, which were fixed throughout this run. The small letters “v” and “d” denote cells on the ventral and dorsal surfaces, respectively. Cells undergoing apoptosis are enclosed in circles (t ¼ 5 and 18). To see this figure in

66

5 Applications of 2D Cell Models

tube with a single-cell-size circumference, one side of the rectangle continues to the opposite side of itself. Then, the tube has a continuous cell pattern (seamless tubular cell pattern) on the tube side. The Drosophila trachea is a tube structure. Housei Wada, Kagayaki Kato, and Shigeo Hayashi (Honda et al. 2009) experimentally investigated in detail the morphogenesis of fine tubes at high temporal and spatial resolution and confirmed the observation of Ribeiro et al. (2004). The circumference of a multicellular tube comprises two or more cells, whereas the circumference of a fine tube comprises one cell. A cell in the fine tube forms the contact of a cell with itself—that is, the formation of an autocellular junction. A multicellular tube consisting of two neighboring cells that are wrapped around the luminal space (a double-cell-size tube) and a fine tube (a single-cell-size tube) are drawn in Fig. 5.6. Using computer simulations, we investigated the formation mechanism of the fine tube and the transformation from a double-cell-size to a single-cell-size tube. Such a transformation is observed during elongation of the tracheal tube, when a growing body of the embryo pulls the trachea within the developing embryo. This is a passive elongation of the tube. In vertex dynamics, we simulated tube elongation by artificial remodeling of the rectangle of the boundary condition. The rectangle was gradually elongated longitudinally and narrowed horizontally while keeping its area constant. When decreasing the width of the rectangular boundary, the shape of the double-cell-size tube changed, as shown in Fig. 5.7a (top inset). The change in the potential of the tube is plotted in Fig. 5.7a (solid line). The potential increased monotonically with decreasing rectangle width. Potential changes and shape changes of the single-cellsize tube are also shown in Fig. 5.7a (gray line and bottom inset, respectively). When the tube width was decreased to less than a certain size (~1.5), the potential of the single-cell-size tube became smaller than that of the double-cell-size tube. Next, we introduced a perturbation to the tube structure. Assumption of the perturbation is reasonable because actual cells in cell assemblages (e.g., cell sheets in culture dishes and endothelial cells) are observed to be generally fluctuating or motile (Honda et al. 1984; Sugihara et al. 2015; Ozawa et al. 2020). We added a perturbation term into Eq. (4.3). U ¼ σ L Σ Lij þ κS Σn α ðSα  So Þ2 þ Σn α ðW α  W o Þ2 ,

ð5:6Þ

where Wα is the width of cell α. Wo is the width in the relaxed state (Wα/Wo ¼ 0.625). Hexagons on the tube were assumed to be deformed along one axis among three. The axis orientation varied at random, as shown in Fig. 5.8a. Perturbations of orientation were applied periodically and led to a vibrating potential, as shown in Fig. 5.7b (thick solid line). The perturbed tube recovered its original shape in the early stages, but when the rectangle width was decreased less than a certain size

Fig. 5.4 (continued) color, go online (Nagai et al. 2018) [Reproduced from Fig. 6 in Nagai et al. 2018 with permission of Elsevier]

5.3 Passively Elongated Epithelial Tubes (Honda et al. 2009) Fig. 5.5 Transformation of a tube (cylinder) to a flat sheet. Explanation of the rectangle–cylinder conversion method

Fig. 5.6 Transformation from a double-cell-size tube to a single-cell-size tube. (a) 3D view. (b) Unfolded patterns. In a single-cell-size tube, a cell contacts itself, an autocellular junction [Original]

67

68

5 Applications of 2D Cell Models

Fig. 5.7 Transformation from a double-cell-size to a single-cell-size tube. (a) Potential changes in tube structures during deformation of the rectangle of the boundary. The rectangular boundary is longitudinally elongated and horizontally narrowed while its area remains constant. Potential changes in a multicellular tube (double-cell-size circumference) and a fine tube (single-cell-size circumference) are shown as solid and gray lines, respectively. Top and bottom insets, deformation of multicellular and fine tubes, respectively. (b) Potential change in the computer simulation of tube transformation (thick solid line). Perturbed orientation of the cell axis was applied [Original]

Fig. 5.8 Methods of perturbation to hexagons. (a) Random selection of an axis of a hexagon, along which the hexagon is elongated. The width ratio was Wα/Wo ¼ 0.625 in Eq. (5.4). (b) Random selection of an edge of a hexagon, which strongly contracts. σ L ¼ 2.6 for strong contraction instead of 1.0 in Eq. (4.4)

(~1.1), the double-cell-size tube was suddenly transformed to a single-cell-size tube (arrow in Fig. 5.7b). A pattern change is shown in Fig. 5.9, where 2D patterns (top) and 3D views (bottom) are shown. Some of the reconnections of paired vertices took place during the transition. One of the reconnections of paired vertices is shown in Fig. 5.10, where an autocellular contact is zipped down, replacing intercellular

5.4 Cell Patterns Composed of Heterogeneous Cells

69

Fig. 5.9 Details of transformation of a double-cell-size tube to a single-cell-size tube. (a) and (b) Presentations of unfolded figures and 3D views, respectively. Left to right, 5500, 5575, 5600, 5625, and 5800 steps in the computer simulation [Original]

contact with autocellular contact (arrow). We introduced another perturbation of the tube structure, that is, edges of polygons to be strongly contracted were selected at random as shown in Fig. 5.8b (σ L ¼ 2.6 instead of 1.0 in Eq. 4.4). The perturbation also gave us a similar transition of a double-cell-size tube to a single-cell-size tube.

5.4 5.4.1

Cell Patterns Composed of Heterogeneous Cells Differential Cell Adhesion Theory

We have described the study of cell patterns consisting of homogeneous cell species. Here, the study is extended to cell patterns composed of two cell types. When cells of different types are mixed, they can sort themselves into a variety of patterns (Moscona and Moscona 1955). Pattern formation in an aggregate of two cell types of cells was systematically described in terms of differential adhesion between cells in the so-called differential cell adhesion theory (Steinberg 1962a, 1962b, 1962c, 1963). Steinberg considered a system in which there are two types of cells that fluctuate in an aggregate. If the intercellular adhesion between heterotypic (unlike)

70

5 Applications of 2D Cell Models

Fig. 5.10 Formation of autocellular contact. A reconnection of paired vertices (from small circle to large circle) is followed by zipping down, where the intercellular contact is replaced with autocellular contact (open arrow). Left to right, 6070, 6090, 6100, 6110, and 6120 steps in the computer simulation [Original]

cells is weaker than either of the cohesions between homotypic (like) cells, the two cell types sort to form two separate aggregates (Fig. 5.11, top). Furthermore, a model was proposed of deformation of an epithelial sheet consisting of blocks of cells with different adhesive affinities, e.g., deformation of a planar bullseye into a banded tube (Mittenthal and Mazo 1983). Mathematical studies for cell sorting pattern formation were performed with computer simulations using a two-dimensional grid whose squares represent cells (Goel et al. 1970; Goel and Leith 1970; Rogers and Goel 1978). Another mathematical study was performed using a two-dimensional lattice (Graner and Glazier 1992, 1993). Garner and Glazier used the large-Q Potts model, which describes a collection of N cells by defining N degenerate spins, σ(i, j) ¼ 1, 2, . . . N, where i, j identifies a two-dimensional lattice site. A cell σ consists of all sites in the lattice with spin σ. Another mathematical model used for cell sorting is a lattice-structured model of continuous-time Markovian transition (Mochizuki et al. 1996), which was applied to cell–cell adhesion in limb formation (Mochizuki et al. 1998). On the other hand, the cell center model (the Dirichlet cell model) was successfully used to simulate the cell sorting (Graner and Sawada 1993). The differential cell adhesion theory involves an assumption of strong and weak adhesions between two cells. The molecular basis of discrimination between strong and weak cell adhesions was experimentally investigated. Cadherin protein molecules were known as cell adhesion molecules (Takeichi 1977). The idea that

5.4 Cell Patterns Composed of Heterogeneous Cells

71

Fig. 5.11 Schematic presentation of the cell adhesion theory. Adhesion strengths between like cells (white–white or black–black) and unlike cells (white–black) are compared. If the strength of like cells is stronger than that of unlike cells, cell sorting takes place; otherwise, the two cell types intermingle with each other [Original]

cadherin protein molecules work for cell sorting was examined and confirmed by an experiment (Steinberg and Takeichi 1994). The researchers constructed different types of L-cells via transfection with different amounts of P-cadherin cDNA and performed experiments with the L-cells. According to the differential adhesion theory of cell aggregates, the two types of cells intermix in an aggregate if the adhesion between unlike cells is stronger than that between like cells (Fig. 5.11, bottom). However, the intermixing pattern had only been theoretical: no actual case in which two types of cells mix with each other was known until Hachiro Yamanaka reported kagome and checkerboard-like patterns (Yamanaka 1990).

72

5 Applications of 2D Cell Models

Fig. 5.12 Luminal surfaces of quail oviduct epithelium. (a) A luminal surface of an immature oviduct epithelium just before sexual maturation (quail, 41 days after hatching). (b) A luminal surface of a mature oviduct epithelium. A and B were treated with silver nitrate to stain cell boundaries via silver impregnation and photographed through an optical microscope. Bar, 10 μm [Reproduced from Fig. 1 A and B in Honda et al. 1986 with permission of The Company of Biologists]

5.4.2

Two Types of Cells in the Oviduct Epithelium6

The oviduct epithelium of birds (e.g., chickens and quails) consists of two types of cells. The albumen-secreting region of the tissue is a monolayered epithelium consisting of ciliated cells (C-cells) and goblet-type gland cells (G-cells) (Romanoff and Romanoff 1949; Wrenn 1971; Hodges 1974). C-cells were identified by their numerous cilia. G-cells appeared darker and were distinguished from C-cells by metachromasia of toluidine blue (Yamanaka and Honda 1990). Yamanaka examined the cellular patterns of the oviduct epithelium of Japanese quail during maturation (Yamanaka 1990). The epithelium of a juvenile bird showed a jigsaw puzzle pattern consisting of a single, undifferentiated cell type. At the start of maturation, cells were rearranged into a pattern in which G-cells surround large C-cells, as shown in Fig. 5.12a (kagome pattern, see the next section in detail). The photograph shows the oviduct epithelium stained by silver impregnation for observation of boundaries. During the maturation process, cells gradually rearranged themselves into a checkerboard-like pattern (or “ichimatsu mon” in Japanese) through an increase in the population of C-cells and enlargement of the G-cells, as shown in Fig. 5.12b (Honda et al. 1986).

6

Reproduced from excerpts in Honda et al. 1986 with permission of The Company of Biologists.

5.4 Cell Patterns Composed of Heterogeneous Cells

73

Fig. 5.13 Kagome pattern (star pattern). A woven pattern appearing in traditional Japanese bamboo ware [Photograph by the author]

Fig. 5.14 Change in an artificial pattern from honeycomb (left) to kagome (right). The middle is a modified kagome pattern that corresponds to the typical cellular pattern of the avian oviduct just before sexual maturation [Reproduced from Fig. 8 in Honda et al. 1986 with permission of The Company of Biologists]

5.4.3

The Kagome Pattern (Star Pattern)6

An immature oviduct epithelium just before sexual maturation (approximately 40–45 days after hatching) shows a peculiar pattern (Yamanaka 1990). The oviduct epithelium at this stage is a monolayer consisting of columnar C- and G-cells. However, the luminal surface shows a pattern in which large C-cells are surrounded by small G-cells, as shown in Fig. 5.12a. Most of the C-cells do not lie next to each other, whereas all G-cells form chains that enclose isolated C-cells. The arrangement of these cells looks like a “kagome” pattern (star pattern), one of the woven patterns appearing in traditional Japanese bamboo ware (Fig. 5.13). Figure 5.14 shows the theoretical continuous change of a honeycomb pattern to a kagome pattern. The

74

5 Applications of 2D Cell Models

actual immature oviduct epithelium corresponds to the middle part of Fig. 5.14 (we will call it a modified kagome pattern), in which small cells link together to surround a large cell. The ratio of small cells to large cells in the artificial pattern shown in Fig. 5.14 is 2.0, which is close to the cell ratio in an actual immature oviduct epithelium (Honda et al. 1986).

5.4.4

A Balanced State between Boundary Contraction and Differential Adhesion6

First, we used the mechanism of differential cell adhesion to understand the formation of the kagome and checkerboard-like patterns. Assuming that unlike cells adhere to each other more strongly than like cells cohere, we can explain the formation of these peculiar patterns. Indeed, most of the cell boundaries in the pattern are edges along which C- and G-cells meet. The GG and CC boundaries are small or few in number. Here, we introduced an additional factor, boundary contraction. We speculated that the actual oviduct epithelium is simultaneously governed by two factors: boundary contraction and differential cell adhesion. Boundary contraction of cells tends to form a honeycomb pattern, as shown in Fig. 5.15a, because the boundary length of the honeycomb pattern is minimal. On the other hand, the strong intercellular adhesion between unlike cells tends to form a checkerboard pattern in the mature oviduct epithelium, as shown in Fig. 5.15c, because all cell boundaries are between different cell types. Neither pattern simultaneously meets the demands of both boundary contraction and differential adhesion. Actual oviduct epithelia show a pattern similar, but not identical, to a checkerboard, as shown in Fig. 5.15c. Most of the polygons look like squares, but in reality, some of them are octagons that include four short sides (Fig. 5.15b). That is, there are

Fig. 5.15 Hexagonal and checkerboard-like polygonal patterns. An actual cellular pattern of the oviduct epithelium (b) is intermediate between two typical artificial patterns, a honeycomb pattern (a) and a checkerboard pattern (c) [Reproduced from Fig. 4 in Honda et al. 1986 with permission of The Company of Biologists]

5.4 Cell Patterns Composed of Heterogeneous Cells

75

minor (short) GG or CC boundaries in addition to main (long) CG boundaries in the pattern. The deviation from the checkerboard pattern is considered to be caused by the boundary contraction. The actual cellular pattern of the oviduct epithelium is in a balanced state between the two factors, differential adhesion and boundary contraction.

5.4.5

Quantitative Estimation of the Difference in Intercellular Adhesions6

The actual cellular pattern of the oviduct epithelium is considered to be a deformation from a honeycomb pattern on account of the stronger intercellular adhesion between unlike cells than between like cells. To introduce the effect of differential adhesion into the boundary shortening (BS) model of cells, we modified the previous BS model (Fig. 4.4). In the previous BS model, we assumed that the total length of the five edges AP + BP + PQ + QC + QD becomes minimal; here, we used numerical values of weighting factors (σ AP, σ BP, σ PQ, σ QC, and σ QD) for the total lengths of edges (σ APAP+ σ BPBP+ σ PQPQ+ σ QCQC+ σ QDQD). We assumed that σ CG > σ GG and σ CC (σ GG ¼ 1.0), where σ CG, σ GG, and σ CC are the values of weighting factors of boundaries between cells C-G, G-G, and C-C, respectively. An example of BS in a simple pattern is shown in Fig. 5.16. Polygon APQC is a C-cell, and all the other polygons are G-cells. Therefore, we use the value of σ CG for σ AP, σ PQ, and σ QC and the value of σ GG for σ BP and σ QD. We perform an elemental step of boundary length shortening by using σ CG ¼ 1.0. A resultant pattern is shown in Fig. 5.16 (left); this is the same pattern as that obtained with the usual model shown in Fig. 4.4. On the other hand, when using σ CG ¼ 0.4, the pattern is deformed after the elemental step, as shown in Fig. 5.16 (right). The lengths of the CG

Fig. 5.16 Four polygons around edge PQ. Elemental steps in the procedure of shortening nonweighted (left) or weighted (right) boundary length. P and Q are movable vertices. One C-cell is designated by stippling. The other three cells are G-cells. Parameter values of weighting factors are as follows: σ CG ¼ σ GG ¼ 1.0 in A; σ CG ¼ 0.4 and σ GG ¼ 1.0 in (right) [Reproduced from Fig. 5 in Honda et al. 1986 with permission of The Company of Biologists]

76

5 Applications of 2D Cell Models

boundaries (AP0 , P0 Q0 ) are greatly elongated, and the length of the GG boundary (BP0 ) is shortened. This is an effect of the strong adhesion between unlike cells along the CG boundaries, which is produced using σ CG ¼ 0.4 (note that σ CG is less than σ GG). We next examined real oviduct epithelial patterns. When a polygonal pattern of the oviduct epithelium was given, we obtained various patterns of the minimum total weighted boundary length by employing the BS procedure using various values of σ CG and σ CC (σ GG ¼ 1.0). Among them, we looked for a pattern that is scarcely deformed from the given pattern. We found that the pattern with σ CG ¼ 0.43 and σ CC ¼ 1.8 was the most similar to the given pattern (see Honda et al. 1986 in detail). Using this method, we obtained quantitative results on differential cell adhesion.

5.4.6

Computer Simulations of the Kagome–Checkerboard Pattern Transformation6

A computer simulation of the pattern transformation started with the modified kagome pattern shown in the middle part of Fig. 5.14, where the area of C-cells (large regular hexagons) was assumed to be twice that of G-cells (small hexagons) in a luminal surface and the cleavage planes for all C-cells were parallel to each other. The modified boundary shortening procedure described in Fig. 5.16 (right) was performed on the pattern shown in Fig. 5.17 (top left) using σ CG ¼ 0.43, σ GG ¼ 1.0, and σ CC ¼ 1.8. A rectangular cyclic boundary condition was used. The resultant patterns after 2000, 9000, and 40,000 steps are shown in Fig. 5.17. Figure 5.17 (bottom right) shows a pattern resembling a checkerboard; most of the boundaries are edges along which C- and G-cells meet. The inset in Fig. 5.17 is a superposed figure showing cell behavior from the initial pattern to the final pattern. G-cells intercalated between divided C-cells. We used the vertex model (the BS procedure) to investigate cellular patterns of heterogeneous cells (Honda et al. 1986). Other models have also been developed. The checkerboard-like pattern was studied using the large-Q Potts model (Glazier and Garner 1993). On the other hand, the lattice-structured model of continuous-time Markovian transition was modified to explain various cell pattern formations based on repulsive interactions between cells contacting directly (Honda and Mochizuki 2002). The modified lattice-structured model involving repulsive interactions systematically described not only checkerboard-like and kagome-like patterns but also various other cell patterns: patterns of graded cell arrangement of various cell types, cell sorting patterns of two cell types, and formation and maintenance of segmental domains along a body axis (Honda and Mochizuki 2002). The cell center model has also been used to examine the checkerboard-like pattern (Graner and Sawada 1993). Cellular patterns of heterogeneous cells were investigated by a finite-element method (Brodland and Chen 2000; Perrone et al. 2015).

5.4 Cell Patterns Composed of Heterogeneous Cells

77

Fig. 5.17 Computer simulation of the transformation starting with a modified kagome pattern. Top left, A modified kagome pattern in which C-cells (stippled polygons) divide in parallel with each other. All polygons are the same in area. The procedure for shortening the weighted boundary length is performed on this pattern, putting σ CG ¼ 0.43, σ GG ¼ 1.0, and σ CC ¼ 1.8. A rectangular cyclic boundary condition is used. Top right, After 2000 steps. Bottom left, After 9000 steps. Bottom right, After 40,000 steps. The pattern comes to resemble a checkerboard. Inset, correspondence of initial polygons (top) to the final polygons (bottom) in a computer simulation of the pattern formation. Polygons do not shift significantly during the simulation. The resultant polygons are not squares but have been deformed into rhomboids. These are limitation of the computer simulation [Reproduced from Fig. 9 and Fig, 14 in Honda et al. 1986 with permission of The Company of Biologists]

5.4.7

Molecular Basis of Cell Adhesion between Heterogenic Cell Types

We will consider the molecular basis of cell adhesions. The sorting of homotypic cells is considered to be governed by homophilic cell adhesion molecules, such as cadherin molecules. Indeed, as previously mentioned, a study in which cells were transfected with cadherin cDNA supported the idea that cell adhesion via cadherin molecules leads to cell sorting (Steinberg and Takeichi 1994). In contrast to homophilic cell adhesion, the mechanism underlying adhesion between heterotypic cells has been uncertain, although there is a possibility that heterophilic cell adhesion molecules work in the checkerboard cell pattern. A few decades later, Hideru Togashi et al. noticed that the auditory epithelium shows a checkerboard-like cell pattern (Fig. 1, bottom row, middle; from Togashi et al. 2011). They examined the mouse cochlear auditory epithelium consisting of two different cell types, hair cells and supporting cells, which express nectin-1 and -3, respectively (nectins are immunoglobulin-like adhesion molecules). The interaction between these molecules mediates the heterophilic adhesion between these two

78

5 Applications of 2D Cell Models

cell types (Togashi et al. 2011). In other words, heterophilic trans-interaction between nectin-1 on hair cells and nectin-3 on supporting cells preferentially recruits cadherin to heterotypic junctions. Heterotypic junctions between nectin-1 cells and nectin-3 cells accumulate more cadherin molecules than homotypic junctions. The differential distribution of cadherin molecules between junctions promotes cellular intercalations, resulting in the formation of a checkerboard-like cell pattern. Experimental results show strong adhesion between heterotypic cells, as shown in Fig. 5.18. Boundaries between colonies of cells expressing various types of nectin were experimentally constructed, and variation of the boundary patterns was observed (Katsunuma et al. 2016). The initial boundary between the left and right colonies was straight (the figure is not shown). The boundary between cells expressing different types of nectins became more intermingled (Fig. 5.18b, c, rightmost panel) than the boundaries between cells expressing the same type of nectin (Fig. 5.18b, c, left and center panels). These results indicated that the homophilic trans-interactions between cadherins alone could not explain mosaic cellular patterning. Evidence of a heterophilic cell adhesion system between various nectin types has been established. In the past, when we investigated the formation of the checkerboard-like pattern of the oviduct epithelium (Honda et al. 1986), we anticipated any heterophilic cell adhesion molecules and hypothetically introduced them to our model. Since we succeeded in forming a checkerboard-like pattern in the computer simulation (Fig. 5.17), the heterophilic cell adhesion system was theoretically confirmed. Later, Togashi et al. established that the nectin system in the mouse auditory epithelium facilitates formation of a checkerboard-like cell pattern (Togashi et al. 2011). This encouraged us to perform research in which theory cooperates with experiments. Currently, whether nectin molecules or other similar molecules function in the formation of a checkerboard-like pattern in the oviduct epithelium is still uncertain.

5.5

Summary

In this chapter, we present various examples of applications of vertex dynamics. When cells on the basal lamina in a cell sheet are partially lost, an open space is filled through migration of surrounding cells. Wound closure was formulated using a driving force that attempts to decrease the total potential energy between the cell surface and the basal lamina. In another case, some cells are lost through apoptosis in the Drosophila wing blade, and the remaining cell assemblage consisting of a few cell types forms a regular zigzag pattern. When a thin tracheal tube in Drosophila, with two cells comprising the circumference of the tube, is pulled along its long axis, the tube automatically becomes a fine tube, the circumference of which is composed of one cell. The process of tube size transformation was recapitulated by vertex dynamics using the rectangle–cylinder conversion method. We also investigated cell

5.5 Summary

79

Fig. 5.18 Mosaic-forming assay for examination of cell–cell affinity. In the mosaic-forming assay, two types of cells were placed on the left and right sides of a culture dish. The initial boundary between colonies of the two cell types is straight and simple. When the affinity between the two cell types is strong, the two types of cells intermingle, and the boundary becomes interdigitated; otherwise, the boundary maintains its simple shape during cell culture. HEK293 cells were doubly transfected with nectin and EGFP (green) or mCherry (red). In the mosaic-forming assay, cells expressing EGFP (green) and mCherry (red) were placed on the left and right sides, respectively. Three combinations of cells were examined, and the results are shown: n2–293  n2–293 (left panel), n3–293  n3–293 (center panel), and n2–293  n3–293 (right panel), where n2–293 and n3–293 indicate HEK293 cells transfected with nectin-2 and nectin-3, respectively. (a) Results shown in grayscale. The original photo in A was in color, showing a green area (EGFP) and a red area (mCherry). To see this figure in color, go online (Fig. 7B in Katsunuma et al. 2016). (b) Red is replaced by black, which designates cells that were initially placed on the right side. The boundary between n2–293 cells and n3–293 cells (right panel) is strongly interdigitated compared with that of other combinations (left and middle panels). (c) Green is replaced by white, which designates cells that were initially placed on the left side. (c) It also shows that the boundary between n2–293 cells and n3–293 cells (right panel) is strongly interdigitated compared with that of other combinations (left and middle panels). Scale, 50 μm [Reproduced from Fig. 7B in Katsunuma et al. 2016 with permission of Rockefeller University Press]

80

5 Applications of 2D Cell Models

patterns consisting of two different cell types. Based on actual observation of the epithelium of the bird oviduct, we confirmed two types of cellular patterns and further transformed the kagome pattern (star pattern) to a checkerboard pattern (ichimatsu) during cell division of one cell type (ciliate cells) in a kagome pattern. According to the differential cell adhesion theory, we understood that the strength of cell adhesion between unlike cells is stronger than that between like cells. Computer simulations recapitulated the transformation of the kagome pattern to the checkerboard pattern. Finally, a molecular system of nectins that induce heterophilic cell adhesion was described. In other words, unlike molecules (nectin 1–nectin 3) provide stronger affinity than like molecules (nectin 1–nectin 1 or nectin 3–nectin 3). This is a rare case in biology in which substances with certain properties were theoretically expected first, and the actual substances were confirmed later.

References Aigouy, B., Farhadifar, R., Staple, D.B., Sagner, A., Roper, J.C., Julicher, F., Eaton, S.: Cell flow reorients the axis of planar polarity in the wing epithelium of Drosophila. Cell. 142, 773–786 (2010) Alberts, B., Johnson, A., Walter, P.: Molecular Biology of the Cell, 4th edn. Garland Science, New York (2002) Brodland, G.W., Chen, H.H.: The mechanics of heterotypic cell aggregates: insights from computer simulations. J. Biomech. Eng. 122, 402–407 (2000) Diaz de la Loza, M.C., Thompson, B.J.: Forces shaping the Drosophila wing. Mech. Dev. 144, 23–32 (2017) Glazier, J., Graner, F.: Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E. 47, 2128–2154 (1993) Goel, N.S., Leith, A.G.: Self-sorting of anisotropic cells. J. Theor. Biol. 28, 469–482 (1970) Goel, N.S., Campbell, R.D., Gordon, R., Rosen, R., Martinez, H., Yeas, M.: Self-sorting of isotropic cells. J. Theor. Biol. 28, 423–468 (1970) Graner, F., Glazier, J.: Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69, 2013–2016 (1992) Graner, F.A., Sawada, Y.: Can surface adhesion drive cell rearrangement? Part II: a geometrical model. J. Theor. Biol. 164, 477–506 (1993) Hodges, R.D.: The histology of the fowl. Academic Press, London (1974) Honda, H.: Geometrical models for cells in tissues. Int. Rev. Cytol. 81, 191–248 (1983) Honda, H., Mochizuki, A.: Formation and maintenance of distinctive cell patterns by coexpression of membrane-bound ligands and their receptors. Dev. Dyn. 223, 180–192 (2002) Honda, H., Kodama, R., Takeuchi, T., Yamanaka, H., Watanabe, K., Eguchi, G.: Cell behaviour in a polygonal cell sheet. J. Embryol. Exp. Morphol. 83(Suppl), 313–327 (1984) Honda, H., Yamanaka, H., Eguchi, G.: Transformation of a polygonal cellular pattern during sexual maturation of the avian oviduct epithelium: computer simulation. J. Embryol. Exp. Morpho. 98, 1–19 (1986) Honda, H., Nagai, T., Wada, H., Kato, K., Hayashi, S.: Formation ol an epithehal tube of single cellsize circumference. Seibutsu Butsuri, 2009 S62; The 47th Annual Meeting of the Biophysical Society of Japan 3TP4-02 (2009). https://www.jstage.jst.go.jp/article/biophys/49/supplement/4 9_KJ00006865860/_pdf

References

81

Katsunuma, S., Honda, H., Shinoda, T., Ishimoto, Y., Miyata, T., Kiyonari, H., Abe, T., Nibu, K., Takai, Y., Togashi, H.: Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium. J. Cell Biol. 212, 561–575 (2016) Mittenthal, J.E., Mazo, R.M.: A model for shape generation by strain and cell-cell adhesion in the epithelium of an arthropod leg segment. J. Theor. Biol. 100, 443–483 (1983) Mochizuki, A., Iwas, Y., Takeda, Y.: A stochastic model for cell sorting and measuring cell–cell adhesion. J. Theor. Biol. 179, 129–146 (1996) Mochizuki, A., et al.: Cell-cell adhesion in limb-formation, estimated from photographs of cell sorting experiments based on a spatial stochastic model. Dev. Dyn. 211, 204–214 (1998) Moscona, A., Moscona, H.: The dissociation and aggregation of cells from organ rudiments of the early chick embryo. J. Anat. 86, 287–301 (1955) Nagai, T., Kawasaki, K., Nakamura, K.: Vertex dynamics of two-dimensional cellular patterns. J. Phys. Soc. Jpn. 57, 2221–2224 (1988) Nagai, T., Honda, H.: Computer simulation of wound closure in epithelial tissues: cell–basal-lamina adhesion. Phys. Rev. E. 80, 061903 (2009) Nagai, T., Honda, H., Takemura, M.: Simulation of cell patterning triggered by cell death and differential adhesion in Drosophila wing. Biophys. J. 114, 958–967 (2018) Okuda, S., Inoue, Y., Eiraku, M., Adachi, T., Sasai, Y.: Modeling cell apoptosis for simulating three-dimensional multicellular morphogenesis based on a reversible network reconnection framework. Biomech. Model. Mechanobiol. 15, 805–816 (2016) Ozawa, M., Hiver, S., Yamamoto, T., Shibata, T., Upadhyayula, S., Mimori-Kiyosue, Y., Takeichi, M.: Adherens junction regulates cryptic lamellipodia formation for epithelial cell migration. J. Cell Biol. 219, e202006196 (2020) Perrone, M.C., Veldhuis, J.H., Brodland, G.W.: Non-straight cell edges are important to invasion and engulfment as demonstrated by cell mechanics model. Biomech. Model. Mechanobiol. 15, 405–418 (2015) Ray, R.P., Matamoro-Vidal, A., Ribeiro, P.S., Tapon, N., Houle, D., Salazar-Ciudad, I., Thompson, B.J.: Patterned anchorage to the apical extracellular matrix defines tissue shape in the developing appendages of Drosophila. Dev. Cell. 34, 310–322 (2015) Ribeiro, C., Neumann, M., Affolter, M.: Genetic control of cell intercalation during tracheal morphogenesis in Drosophila. Curr. Biol. 14, 2197–2207 (2004) Rogers, G., Goel, N.S.: Computer simulation of cellular movements: cell-sorting, cellular migration through a mass of cells and contact inhibition. J. Theor. Biol. 71, 141–166 (1978) Romanoff, A.F., Romanoff, A.J.: The avian egg. Wiley, New York (1949) Saunders Jr., J.W., Fallon, J.F.: Cell death in morphogenesis. In: Locke, M. (ed.) Major problems in developmental biology, pp. 289–314. Academic Press, New York (1966) Sherratt, J.A., Murray, J.D.: Models of epidermal wound healing. Proc. R. Soc. Lond. B. 241, 29–36 (1990) Steinberg, M.S.: On the mechanism of tissue construction by dissociated cells. I. Population kinetics, differential adhesiveness, and the absence of directed migration. Proc. NAS. 48, 1577–1582 (1962a) Steinberg, M.S.: Mechanism of tissue reconstruction by dissociated cells, II: time-course of events. Science. 137(3532), 762–763 (1962b) Steinberg, M.S.: On the mechanism of tissue construction by dissociated cells. III. Free energy relations and the reorganization of fused heteronomic tissue fragments. Proc. Natl. Acad. Sci. USA. 48, 1769–1776 (1962c) Steinberg, M.S.: Reconstruction of tissues by dissociated cells. Some morphogenetic tissue movements and the sorting out of embryonic cells may have a common explanation. Science. 141, 401–408 (1963) Steinberg, M.S., Takeichi, M.: Experimental specification of cell sorting, tissue spreading, and specific spatial patterning by quantitative differences in cadherin expression. Proc. Natl. Acad. Sci. USA. 91, 206–209 (1994)

82

5 Applications of 2D Cell Models

Sugihara, K., Nishiyama, K., Fukuhara, S., Uemura, A., Arima, S., Kobayashi, R., Köhn-Luque, A., Mochizuki, N., Suda, T., Ogawa, H., Kurihara, H.: Autonomy and non-autonomy of angiogenic cell movements revealed by experiment-driven mathematical modeling. Cell Rep. 13, 1814–1827 (2015) Takeichi, M.: Functional correlation between cell adhesive properties and some cell surface proteins. J. Cell Biol. 75, 464–474 (1977) Takemura, M., Adachi-Yam, T.: Apoptosis—Chapter 6. In: Apoptosis During Cellular Pattern Formation, pp. 113–124. Intech (2013). https://doi.org/10.5772/52166. https://www. intechopen.com/chapters/44694 Takemura, M., Adachi-Yamada, T.: Cell death and selective adhesion reorganize the dorsoventral boundary for zigzag patterning of Drosophila wing margin hairs. Dev. Biol. 357, 336–346 (2011) Togashi, H., Kominami, K., Waseda, M., Komura, H., Miyoshi, J., Takeichi, M., Takai, Y.: Nectins establish a checkerboard-like cellular pattern in the auditory epithelium. Science. 333, 1144–1147 (2011) Wrenn, J.T.: An analysis of tubular gland morphogenesis in chick oviduct. Dev. Biol. 26, 400–415 (1971) Yamanaka, H.: Pattern formation in the epithelium of the oviduct of Japanese quail. Int. J. Dev. Biol. 34, 385–390 (1990) Yamanaka, H.I., Honda, H.: A checkerboard pattern manifested by the oviduct epithelium of the Japanese quail. Int. J. Dev. Biol. 34, 377–383 (1990)

Chapter 6

3D Vertex Model

Outline The vertex model is extended from 2D to 3D space. A method of illustrating the polyhedral pattern in 3D space is described, and 3D vertex dynamics is performed. A cell aggregate becomes spherical due to the surface tension of the cell. The viscoelastic properties of a cell aggregate are discussed. A computer simulation of the development of the mammalian blastocyst from a cell aggregate is performed, which is one case of formation of an epithelial vesicle, a cell aggregate involving a large cavity.

6.1

Expression of an Aggregation of Polyhedral Cells7

For extension of 2D vertex dynamics to 3D vertex dynamics, we described methods to express vertex, edge, face, and body using 3D coordinates. The method describing polyhedral patterns was primarily developed by Masaharu Tanemura (Tanemura et al. 1983). An aggregate of polyhedra is geometrically defined by the positions of vertices and neighboring relationships between two vertices forming edges. We consider a tissue (cell aggregate) consisting of many polyhedra to be a 3D space tessellation, which consists of convex polyhedra without gaps or overlaps. When we consider a random cell aggregate that is topologically stable, a vertex generally connects to four neighboring vertices via four edges in 3D space (Weaire and Rivier 1984). In a tessellation pattern, four neighboring cells determine a vertex at which four edges meet (Fig. 6.1a). An assembly of vertices connecting to four neighboring vertices can form various networks, including a diamond structure and its variants (Wooten et al. 1985; Wooten 2002). However, among these, we consider only a 3D space tessellation pattern, which we consider to consist of faces rather than edges because we treat a cell aggregate as consisting of cell membranes, not of particles

7

Reproduced from excerpts on pages 440–450 in Honda et al. 2004 with permission of Elsevier.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_6

83

84

6 3D Vertex Model

Fig. 6.1 Configuration of cells, with identification of vertices, edges, interfaces, and polyhedra. Spheres represent cell centers. (a) Four cells determine one vertex at which four edges meet. (b) Three cells (α, β, γ) determine one edge connecting the two vertices i and j. (c) Two cells (α, β) determine an interface involving the two vertices i and j. (d) A cell (α) determines a polyhedron involving the two vertices i and j [Reproduced from Fig. 1 in Honda et al. 2004 with permission of Elsevier]

and bonds as in amorphous materials. This type of modeling is called the coarse graining viewpoint in statistical physics (see the later subsection Fundamentals of vertex dynamics). We treated interfaces between neighboring cells and volumes of polyhedral cells in a 3D cell model. To present interfaces and volumes, we use vertices. Spatial relationships between cells and vertices in a 3D space tessellation are as follows: three neighboring cells (α, β and γ in Fig. 6.1b) determine an edge ij; thus, three pairs of cells, (αβ), (βγ), and (γα), determine boundary surfaces in Fig. 6.1b. These three boundary surfaces determine the edge ij as an intersection of the surfaces. Two neighboring cells (α and β in Fig. 6.1c) in 3D space determine a boundary surface between them. A cell (α) corresponds to a polyhedron consisting of several polygons (Fig. 6.1d). The tessellation pattern of a cell aggregate is completely defined by the coordinates of the vertices (xi, yi, zi) and the vertex relationships, which are expressed by lists of the four cells [ai, bi, ci, di]. The lists of the four cells [ai, bi, ci, di] around each

6.2 Equation of Motion in 3D Vertex Dynamics (Honda et al. 2004)7

85

vertex i (i ¼ 1, 2, . . ., nv where nv is the total number of vertices of the cell aggregate) define vertex relationships as follows. We compare two lists, [ai, bi, ci, di] and [aj, bj, cj, dj], for vertices i and j. If the two lists for i and j show three common cells (e.g., cells α, β, and γ), an edge defined by the three cells α, β, and γ (Fig. 6.1b) directly connects vertices i and j. If two cells (e.g., cells α and β) are common, the two vertices i and j belong to a boundary surface between the two cells α and β (Fig. 6.1c). If one cell (e.g., cell α) is common, the two vertices i and j belong to a polyhedron of cell α (Fig. 6.1d). If the lists show no common cells, the two vertices are isolated from each other, as shown in Fig. 6.1a. The cell identifications (e.g., cells α, β, γ, . . .) define vertex relationships but not the x-, y-, and z-coordinates of vertices.

6.2

Equation of Motion in 3D Vertex Dynamics (Honda et al. 2004)7

Here, for calculation of the vertex coordinates in 3D space, we use a 3D version of the equation of motion (Eq. 4.1 in Chap. 4), where ri and — i are a 3D positional vector and the 3D nabla differential operator of vertex i, respectively. A more general 3D extension of the 2D equations of motion for vertices was given by Nagai et al. (1990) and Fuchizaki et al. (1995). The potential U contains terms of surface energy (US) and elastic volume energy (UV). U ¼ US þ UV:

ð6:1Þ

The potential US denotes the total surface energy of the cells: U S ¼ σ Σ f S f þ σ O Σm Sm




f ¼ 1, 2, . . . , n f ;

 m ¼ 1, 2, . . . , nm :

ð6:2Þ

The first and second terms in Eq. (6.2) are the interface energy between neighboring cells and the surface energy between cells and the external culture medium, where nf and nm are the number of polygons facing an adjacent cell and facing the external culture medium, respectively. Sf is the surface area of polygon f facing adjacent cells, and σ is its interface energy per unit area. Sm is the surface area of polygon m facing the external culture medium, and σ O is its surface energy per unit area. We can compute the areas of interfaces Sf and surfaces Sm by triangulation of polygons. In cell aggregates without external culture medium (i.e., with periodic boundary conditions), the right-hand side of Eq. (6.2) lacks the second term. The potential UV denotes the total energy of cell compression and expansion,

86

6 3D Vertex Model

U V ¼ κ Σα ðV α  V std Þ2 ,

ðα ¼ 1, 2, . . . , nÞ,

ð6:3Þ

where κ is a positive constant (an analog of the elastic constant) and n is the total number of cells. Vα is the volume of cell α, and Vstd is the volume of relaxed cells. Equation (6.3) gives a cell volume constraint. See also Fundamentals of vertex dynamics in this chapter. In addition to the equations of motion, our model involves an elementary process of reconnection of neighboring vertices. When the length of an edge connecting two neighboring vertices is short (less than a critical length δ), the neighbors reconnect, as shown in Fig. 6.2 (see explanation in the following section), and the neighbor relationships of the vertices change. The reconnection is an extension of the procedure of reconnection in 2D cellular patterns (Fig. 4.14a). Other reconnections are possible in general networks consisting of vertices with four edges (Wooten et al. 1985; Wooten 2002), but our network is based not on edges but on faces composed of the cell membrane.

6.3

Reconnection of Neighboring Vertices7

[Step 1] We calculate all edge lengths and list edges with lengths shorter than the critical length δ in order of ascending length. [Step 2] We pick edges ij sequentially from the list of short edges. If vertex i or j connects to edges that have already reconnected, we skip it and repeat Step 2 for the next edge in the list. Otherwise, we count how many triangles the edge belongs to. In general, each edge belongs to three polygons (see Fig. 6.1b). If none of the three polygons is a triangle, the edge is type I (Fig. 6.2c). We reconnect from type I to type H, as shown in Fig. 6.2d, and then repeat Step 2. If one of the three polygons is a triangle, the edge is type H (Fig. 6.2b). We reconnect from type H to type I, as shown in Fig. 6.2e, and then repeat Step 2. If two or more of the three are triangles, we skip the edge and repeat Step 2. If we encounter other unexpected patterns, we also skip this step and repeat Step 2. When we have tried all edges in the list, the steps end. The next procedure begins with the Runge–Kutta calculation of the coordinates of the vertices using the equations of motion.

6.4

Structure of Packed Polyhedra in 3D Space7

We first examined the minimal surface problem using the vertex dynamics of Eqs. (4.1) with (6.1). We required that all edges were straight (not curved). Among straight edged polyhedra that are identical in shape and size and fill 3D space without gaps or overlaps, orthic tetrakaidecahedra are known to have the least surface area (Weaire and Phelan 1996). This polyhedron is the Voronoi cell of the

6.4 Structure of Packed Polyhedra in 3D Space7

87

Fig. 6.2 Reconnection of neighboring vertices in a 3D tessellation. (a) Presentation by lines. When a triangle or an edge length is small, vertex topological connections of type H (left) and type I (right) are interchangeable. (b) Type H configuration presented by faces. Type H consists of 10 faces: three upper faces, three lower faces, three side faces, and a small triangular face. (c) Type I configuration presented by faces. Type I consists of 9 faces: three upper faces, three lower faces, and three side faces. (d) Reconnection of vertices from type I (broken line) to type H (solid line). (e) Reconnection of vertices from type H (broken line) to type I (solid line) [Reproduced from Fig. 2 in Honda et al. 2004 with permission of Elsevier]

body-centered cubic lattice (bcc) structure and has six square faces and eight hexagonal faces. We have already studied a flattened type of the polyhedron in Fig. 3.2a in Chap. 3. A Voronoi tessellation with centers arranged in a bcc structure generates an orthic tetrakaidecahedral tessellation, which does not change under the

88

6 3D Vertex Model

Fig. 6.3 Polyhedral patterns produced by computer simulations under vertex dynamics. A periodic boundary condition on a cube is used. The step size in the Runge–Kutta calculation is h ¼ 0.02. Critical edge length, δ ¼ 0.05. κ ¼ 5. For simplicity, we show only four cells. (a) Orthic tetrakaidecahedra. Voronoi tessellation of the body-centered cubic lattice (bcc) arrangement of centers (n ¼ 54) in the cube (length of box edge ¼3.78). (b) Computer simulation. Initial condition: bcc, with a nearest neighbor distance of 1.09; one center is displaced toward the x-axis (abscissa) by 0.8. After the computer simulation, the pattern was converted to the pattern shown in (a). (c) Computer simulation. Initial condition: bcc, with a nearest neighbor distance of 1.09; all centers were displaced randomly along each axis in the range 0.3 to +0.3. After the computer simulation, the pattern was converted to the pattern shown in (a) [Reproduced from Fig. 3 in Honda et al. 2004 with permission of Elsevier]

vertex dynamics (Fig. 6.3a). The boundary area of this tessellation did not decrease, and thus, it was at least a local minimum energy configuration. Figure 6.3b shows the deformed pattern when we displaced one of the Voronoi centers by 0.8, where the nearest neighbor distance in the bcc was 1.09. Under vertex dynamics (t ¼ 30), the deformed polyhedra recovered the tetrakaidecahedral structure shown in Fig. 6.3a through vertex reconnection around the displaced Voronoi center. If we disturbed the positions of all Voronoi centers at random (displacing along each axis by 0.3 to +0.3, where the average volume of cells is 1), the resulting Voronoi tessellation was irregular, as shown in Fig. 6.3c, with no tetrakaidecahedra. Under vertex dynamics (t ¼ 80), the deformed polyhedra recovered the tetrakaidecahedra shown in Fig. 6.3a through vertex reconnection. Thus, vertex dynamics, including vertex reconnection, can reduce the interface energy of cells.

6.5

Formation of a Spherical Cell Aggregate

Next, we examined an isolated cell aggregate. We constructed an initial cell aggregate of 105 polyhedra as described in Sect. 2.3.4 in Honda et al. 2004. The vertex dynamics using Eqs. (4.1) with (6.1) shows that a cell aggregate in free space rounds up, as shown in Fig. 6.4a, left (t ¼ 10). See also Fig. 6.6 (t ¼ 0). The assumption of a reduction in the total surface area via movement of the vertices leads to a spherical cell aggregate.

6.6 Flattening of a Cell Aggregate by Centrifugal Force7

89

Fig. 6.4 Computer simulation of centrifugal flattening of an aggregate of polyhedra. An initial cell aggregate (radius r ¼ 2.9) was produced. Number of cells, n ¼ 105; Critical edge length for reconnection, δ ¼ 0.05. Step size in the Runge–Kutta calculation, h ¼ 0.002. Parameters, σ o ¼ 5 and k ¼ 5. Bar length, 5.0. (a) Left, A cell aggregate at t ¼ 10 without centrifugation (ρ ¼ 0 and wfloor ¼ 0). Right, A cell aggregate at t ¼ 12 with centrifugation. The change in the cell aggregate was simulated for t ¼ 0–12 under centrifugal force (ρ ¼ 1 and wfloor ¼ 0.5). (b) Interior cells of the aggregates at t ¼ 10 and 12 [Reproduced from Fig. 4 in Honda et al. 2004 with permission of Elsevier]

6.6

Flattening of a Cell Aggregate by Centrifugal Force7

To examine the mechanical properties of cell aggregates, we noted an experiment. When a force was applied on a cell aggregate via centrifugation, the cell aggregate was flattened. Initially, the individual component cells were also quickly flattened, but they rearranged themselves to recover their original shape (Phillips et al. 1977; Phillips and Steinberg 1978; Phillips and Davis 1978). A centrifuge is schematically drawn in Fig. 6.5, inset. A cell aggregate (ellipsoid) is placed at the bottom of a centrifugation tube. The z-axis is perpendicular to the rotation axis. The process of deformation of the cell aggregate is schematically presented in Fig. 6.5. We attempted to perform a computer simulation for flattening of cell aggregates (Fig. 6.4). To recapitulate the experiment of aggregate flattening under centrifugal force (or gravity), we introduce two additional terms, Uz and Ufloor, to the potential U:

90

6 3D Vertex Model

Fig. 6.5 Schematic presentation of cell rearrangement under force deforming a tissue. A tissue (circle) is compressed along the z-axis, and compression is continued. Initially, normal hexagonal cells in a tissue are suddenly flattened and gradually recover their original shape through cell rearrangement. Inset, schematic drawing of the centrifuge. The z-axis of the centrifugation tube is perpendicular to the rotation axis [Reproduced from Fig. 3-15a in Honda 2010]

U ¼ U S þ U V þ U z þ U floor ,

ð6:4Þ

U z ¼ ρ Σ α V α zα ,

ð6:5Þ

and U floor ¼ wfloor Σα 1=½1 þ exp ða zα Þ,

ðα ¼ 1, 2, . . . , nÞ:

ð6:6Þ

Here, Uz is the potential energy induced by centrifugation. A body (mass m) in a centrifugation tube experiences a centrifugal force mω2(R – z), where ω is the angular velocity, R is the centrifugation radius at the bottom floor of the tube, and z is the distance of the body from the bottom floor (Fig. 6.5, inset). When R is much larger than z, the centrifugal force is approximately mω2R (¼ mgC where gC is the centrifugal acceleration, ω2R). Then, centrifugation gives the body a potential energy, mgCz. Therefore, cell α with mass mα has a potential energy mα gCzα, where zα is the z-coordinate of the center of mass of the cell (Fig. 6.5, inset). If the mass density of the cells under the centrifugal force is uniform, then the potential energy of cell α can be written as ρVα zα; with ρ ¼ (mass density)  gC, which leads to Eq. (6.5). Ufloor expresses the hindrance of downward motion of cells along the

6.6 Flattening of a Cell Aggregate by Centrifugal Force7

91

Fig. 6.6 Sectional views of a cell aggregate during application and removal of centrifugal force. An initial cell aggregate was produced. Number of cells, n ¼ 105. Critical edge length for reconnection, δ ¼ 0.05. Step size in the Runge–Kutta calculation, h ¼ 0.002. Parameters, σ o ¼ 5 and k ¼ 5. Bar length 5.0. Inset, views of entire aggregates at corresponding times (the small bar has a length of 5.0). Initial structure (t ¼ 0) to which the centrifugal force is applied (ρ ¼ 1, and wfloor ¼ 0.5). At t ¼ 24, the centrifugal force is removed (ρ ¼ 0 and wfloor ¼ 0). t ¼ 26 (t0 ¼ 2 after removal of the centrifugal force). t ¼ 54 (t0 ¼ 30 after the removal of the centrifugal force) [Reproduced from Fig. 5 in Honda et al. 2004 with permission of Elsevier]

z-axis by the floor below the cell aggregate. Cell α moves without restriction when it is above the floor (zα > 0), but it produces a large potential energy when it crosses below the floor (zα < 0). Instead of a step function, for example, {¼ 1, zα < 0; ¼ 0, zα > 0}, we use an analytic function 1/[1+ exp(a zα)]. In Eq. (6.6), a is the slope of the approximated step function, and wfloor is the strength of the potential. Then, exposing the cell aggregate to centrifugal force using Eqs. (4.1) with (6.4) flattened the cell aggregate, as shown in Fig. 6.4a (right) and B (right). The cells did not rearrange at first. The relative positions of the cells in the aggregate at early stages were similar to those of the initial cell aggregate (compare the cells in Fig. 6.6 t ¼ 12 with those in Fig. 6.6 t ¼ 0). Subsequent application of centrifugal force rearranged the cells, as shown in Fig. 6.6 t ¼ 24, although the shape of the whole cell aggregate did not change. After we removed the centrifugal force, the cell aggregate began to recover its original shape (Fig. 6.6 t ¼ 26) and finally became completely spherical (Fig. 6.6 t ¼ 54). It should be noted that the individual cells elongated vertically without rearrangement (Fig. 6.6 t ¼ 26) and then rearranged (Fig. 6.6 t ¼ 54). We traced three neighboring cells in the simulation (Fig. 6.6) and present snapshots in Fig. 6.7. The upper and lowest cells, which were initially separated from each other (Fig. 6.7 t ¼ 0 and 12), came into contact (Fig. 6.7 t ¼ 24), and again, the upper and lowest cells separated, as shown in Fig. 6.7 t ¼ 54.

92

6 3D Vertex Model

Fig. 6.7 Behaviors of three cells in a cell aggregate during changes in centrifugation. The same computer simulation as described in Fig. 6.6 [Reproduced from Fig. 6 in Honda et al. 2004 with permission of Elsevier]

Figure 6.8 quantitatively shows the change in cell and aggregate shapes characterized by flatness (see the legend of Fig. 6.8). Figure 6.8a shows a schedule of application of centrifugation force. The centrifugation stopped at t ¼ 24 (arrow in Fig. 6.8a). The flatness of the cell aggregate increased and returned to the initial value (open squares in Fig. 6.8b). However, while the individual cells (solid circles in Fig. 6.8b) also flattened under centrifugation (like the whole aggregate), they returned to spherical (isodiametric) shapes at approximately t ¼ 12. This recovery was observed experimentally by Forgacs et al. (1998). When centrifugation was removed at t ¼ 24 (arrow in Fig. 6.8a), the cells (solid circles in Fig. 6.8b) suddenly decreased in flatness and became spherical. Intriguingly the sphere-returning of the cells overshot and then the cells elongated (flatness less than 2. Arrowhead at t ¼ 24 in Fig. 6.8b), which was gradually recovering their original round shape. The behavior of the individual cells depended not only on the cell aggregate shape but also on the relaxation time. This result demonstrates the viscoelastic properties of the cell aggregate based on the deformation and rearrangement of component cells, which have been investigated in several works (Phillips et al. 1977; Phillips and Steinberg 1978; Phillips and Davis 1978; Forgacs et al. 1998; Beysens et al. 2000). Figure 6.8b (solid line) shows the average shape change of individual cells. In comparison with the experimental result (Figs. 3 and 4 on pp. 129–130 in Phillips et al. 1977), the actual morphological change of individual cells was more remarkable than the simulation result. In the simulation, the surface of the aggregate facing the culture medium influenced the shape of the cells within the aggregate. Figure 6.6 t ¼ 0 shows that

6.6 Flattening of a Cell Aggregate by Centrifugal Force7

93

Fig. 6.8 Shape changes of the cell aggregate and individual cells in the simulation described in Fig. 6–6. Abscissa, simulation time (t). (a) Application of stress via centrifugation. Ordinate, ρ, which is proportional to the centrifugal force acting on a unit volume of a cell (see text). Centrifugation was applied at t ¼ 0 and removed at t ¼ 24 (arrow). (b) Open squares, flatness of the cell aggregate obtained by Flatness ¼ Σi[(xi – xG)2 + (yi – yG)2]/Σι(z–zG)2. Solid circles, average flatness of the cells. First, the flatness of each cell within the aggregate was obtained by Σi[(xi – xG)2 + (yi – yG)2]/ Σi(z – zG)2, and then, the flatness of all cells was averaged. Dotted lines with arrowheads indicate sites that should be compared with the analytic results (gray line in Fig. 6.10c) [Reproduced from Fig. 7 and an excerpt of its legend in Honda et al. 2004 with permission of Elsevier]

polyhedral cells that contacted the culture medium (surface cells) had irregular cell shapes (elongated toward the surface of the aggregate), while cells contacting only other cells (inner cells) were isodiametric. Our simulated aggregate contained only 105 cells, so the ratio of surface cells to inner cells was approximately 5:2. In contrast, the number of cells in an actual cell aggregate is much larger, and most

94

6 3D Vertex Model

Fig. 6.9 Simulation of cell flattening and recovery with periodic boundary conditions. Critical edge length for reconnection, δ ¼ 0.025. Step size in the Runge–Kutta calculation, h ¼ 0.005. For simplicity, we drew only selected central cells. Parameter values, σ o ¼ 1 and k ¼ 5. (a) Left, t ¼ 50: The initial condition was a Voronoi tessellation: spheres (radius ρ ¼ 0.45, number n ¼ 54) were distributed at random in a cube of edge length 3.78 with periodic boundary conditions. Under vertex dynamics, the cells relax toward roundish and isodiametric irregular polyhedra. Middle, the zcoordinates in the left figure were reduced by 2/3, and the x- and y-coordinates increased so that the boundary volume remained constant (the box size was 4.63  4.63  2.52). Right, t ¼ 200 after the middle figure under the same conditions. The flattened polyhedra rearranged themselves and returned to be roundish. (b) Left, middle, right, Sections of the figures presented in A. The cell marked by an asterisk in B disappeared by moving vertically out of the sectional plane [Reproduced from Fig. 8 and an excerpt of its legend in Honda et al. 2004 with permission of Elsevier]

cells are far from the aggregate surface. We simulated this situation using a periodic boundary condition (simulating an infinite system of cells), as shown in Fig. 6.9a, left. We then compressed the box height by 2/3, keeping the box volume constant (Fig. 6.9a center). The vertex dynamics shaped the aggregate, as shown in Fig. 6.9a, right, at t ¼ 200. Figure 6.9b shows cross-sectional views of Fig. 6.9a. Roundish cells (Fig. 6.9b, left) initially flattened remarkably after compression (Fig. 6.9b, center) and then recovered nearly to their original shapes (Fig. 6.9b, right).

6.7

Viscoelastic Properties of Cell Aggregates7

When cell aggregates undergo mechanical deformation, they relax like elastic materials on short time scales and like viscous liquids on long time scales (Phillips et al. 1977; Phillips and Steinberg 1978; Phillips and Davis 1978; Forgacs et al. 1998; Beysens et al. 2000). The viscoelastic properties of cell aggregates have been extensively studied and discussed (Foty et al. 1995, 1996; Beysens et al. 2000;

6.7 Viscoelastic Properties of Cell Aggregates7

95

Palsson 2001). Our simulations, which assumed minimization of cell surface area, exhibited cell behaviors in an aggregate (Figs. 6.6 and 6.7) and quantitatively showed shape changes of cells and the aggregate (Fig. 6.8). We will discuss the viscoelastic properties of an aggregate with regard to cell rearrangement caused by a motive force provided by cell interface minimization. Based on the theory of viscoelasticity, we performed an analytic calculation of the viscoelastic properties of the cell aggregate, as described in the legend of Fig. 6.10, and our results are shown in Fig. 6.10. Figure 6.10a–c shows stress Fa(t) on the aggregate (A), deformation u(t) (¼ Fa/Ga) of the aggregate (B), and stress Fc(t) on the individual cells (C), where Ga is an elastic spring constant. The change in the shape (flatness) of the aggregate in the simulation (gray line in Fig. 6.8b) is supported experimentally. This result is close to the change in the roundness of an experimental aggregate of embryonic liver cells (Fig. 10 on page 10 in Phillips and Steinberg 1978). Both the flatness in our definition and the roundness in the definition in the experiment (Phillips and Steinberg 1978) show the creep behavior of viscoelastic bodies (Fung 1981). The change in the flatness of the aggregate in the simulation (gray line in Fig. 6.8b) corresponds to the analytical calculation (gray line in Fig. 6.10b). When stress (e.g., centrifugation) was applied to the aggregate for a certain duration (t ¼ 0 – t1 in Fig. 6.10a), the aggregate, as shown in Fig. 6.10b, deformed from a spherical to a flattened shape just after the stress started (t ¼ 0) and recovered from a flattened to a spherical shape just after the stress ended (t ¼ t1). The thin solid line in Fig. 6.10b represents the case of a perfect elastic solid (relaxation time of deformation, τa ¼ 0). The gray line in Fig. 6.10b represents deformation with a relaxation time (τa) ¼ 1, which is similar to the simulation result shown by the gray line in Fig. 6.8b. The elongation of the cells, that is, the overshoot of sphere-returning of the cells in Fig. 6.10c (arrow), is properly coincident with Fig. 6.8b (arrowhead at t ¼ 24). On the other hand, our simulation showed curious changes in the flattening and elongation of individual cells (solid line in Fig. 6.8b). We will discuss the curious changes with the aid of analytic calculations (Fig. 6.10c) because shape changes of individual cells are difficult to monitor experimentally. We consider two relaxations: a short relaxation time (τa) due to the elastic deformation of a cell aggregate and a long relaxation time (τc) due to the rearrangement of individual cells in the aggregate, according to Forgacs et al. (1998). The aggregate was flattened by an external force, as shown by the thin solid or gray lines in Fig. 6.10b. Flattening the aggregate also compresses individual cells that experience stress Fc0 (at t ¼ 0 in Fig. 6.10c). However, the cells rearrange themselves to recover their original isodiametric shapes. During recovery, the stress on each cell decreases with relaxation time τc (t ¼ 0 – t1 in Fig. 6.10c). Based on the deformation of the aggregate with τa ¼ 0 (thin solid line in Fig. 6.10b) and assuming relaxation time due to cell rearrangement τc ¼ 0.4, we calculated the decrease in the stress, and the result is shown by the thin solid line in Fig. 6.10c. The stress decreased until the cells within the flattened aggregate became isodiametric. Although the whole aggregate remained flat (t  t1 in Fig. 6.10b), the cells within the aggregate became almost isodiametric. Removing

96

6 3D Vertex Model

Fig. 6.10 Analysis of a cell aggregate under gravity force using the theory of viscoelasticity. Stress and deformation of a cell aggregate and individual cells in it are calculated. (a) Stress on a cell aggregate. Stress (Fa0) is applied at t ¼ 0, maintained at a constant force and removed at t ¼ t1. (b) Deformation of the aggregate. We obtain the deformation (called creep) u(t) from the Voigt model (e.g., Fung 1981): u(t) ¼ Fa0/Ga[1– exp(t/ τa)] for  t1, where the relaxation time of the aggregate deformation τa ¼ μa/Ga, Ga is an elastic spring constant and μa is the coefficient of viscosity for the dashpot. When the stress is suddenly released at t ¼ t1, the deformation of the aggregate obeys u(t) ¼ u(t1) exp[(t–t1)/ τa] for t > t1, where u(t1) is obtained for u(t) for t  t1. Thin solid line, the perfect elastic case (τa ¼ 0). Gray line, viscoelasticity with relaxation time τa ¼ 1. (c) Stress on individual cells. When the aggregate is suddenly deformed, as shown according to the thin solid line in (b) (τa ¼ 0), the stress Fc(t) follows a Maxwell model (e.g., Fung 1981): Fc(t) ¼ Fa0exp(t/τc)] for t  t1, where the relaxation time of stress for cells τc ¼ μc/Gc. Gc is an elastic spring constant, and μc is the coefficient of viscosity for the dashpot. When the aggregate suddenly recovers its original shape at t ¼ t1 (arrow in b), the stress is obtained as Fc(t) ¼ [Fc(t1) –Fa0] exp[ (t–t1)/τc] for t > t1, where Fc(t1) is obtained from Fc(t) for t  t1. Thin solid line, stress for viscoelasticity with relaxation time (τc ¼ 0.4) for deformation of the aggregate (τa ¼ 0, solid thin line in b). Gray line, stress for viscoelasticity with a long relaxation time (τc ¼ 2) for the same deformation of the aggregate (τa ¼ 0, the solid thin line in b). Deformation of the individual cells is similar to that shown in (c), with Fc replaced by Fc/Gc [Reproduced from Fig. 9 and an excerpt of its legend in Honda et al. 2004 with permission of Elsevier]

6.8 Modification of 3D Vertex Dynamics

97

the stress (t  t1 in Fig. 6.10a), the flattened aggregate recovers its original spherical shape (arrow in Fig. 6.10b). Simultaneously, the isodiametric cells within the aggregate experience a tensile stress (i.e., extension) and elongate (arrow in Fig. 6.10c). The elongated cells again rearrange to recover their original isodiametric shapes, decreasing the extension stress (t > t1 in Fig. 6.10c). Corresponding to the simulation result (solid line in Fig. 6.8b), we calculated the stress of individual cells assuming a slower relaxation time due to cell rearrangement (τc ¼ 2). The analytic result is shown by the gray line in Fig. 6.10c, which is similar to the simulation result. Differences between the simulation result and the analytic result are indicated by the dotted lines (arrowheads at t ¼ 0–10 and t ¼ 24–30 in Fig. 6.8b), which are due to the assumption that the aggregate is perfectly elastic (i.e., τa ¼ 0). Quantitative comparison of the simulation results (Fig. 6.8) with the analytic results (Fig. 6.10) is difficult because the flatness shown in Fig. 6.8b is not proportional to the mechanical deformation shown in Fig. 6.10b. In addition, the solid line in Fig. 6.8b shows the average over all cells in the aggregate, which contains more surface cells than bulk cells, and thus, the culture medium affects the shapes of the cells. Nevertheless, the analytic results in Fig. 6.10 help us to understand the flattening and elongation of individual cells (solid line in Fig. 6.8b). The result of our computer simulation suggests that elastic deformation of cells and minimization of the boundary surface between cells cause bimodal relaxation and short-term solid-like and long-term liquid-like behaviors, as indicated by Forgacs et al. (1998). Tension of the cell boundary surface causes movement of cells in tissues. Our computer simulation using 3D vertex dynamics visualized shape changes and rearrangements of component cells, which correspond to the mechanical properties of cell aggregates. The 3D vertex dynamics link the constants of the dashpot and spring model (viscosity and elastic constants) to the coefficients of viscosity η, surface energy σ, and elasticity κ in the coarse-grained model.

6.8

Modification of 3D Vertex Dynamics

In contrast to general 3D networks, our network is based not on edges but on faces (Honda et al. 2004). We assumed strong contraction of polygonal faces. Our thought is reasonable because real cells are composed of cell membranes lined with cytoskeletons. We need not consider reconnections of vertices other than those in Fig. 6.2. Although real cell faces sometimes deviate from flat planes during morphogenesis, our vertex cell model is useful to grasp macroscopic shapes and changes during cell morphogenesis because cells fluctuate during morphogenesis. On the other hand, Okuda et al. (2013a) modified the 3D vertex model (for improvement) by replacing a flat simple polygon with a polygonal surface consisting of radially arranged triangles composed of each edge and the center point of the polygon. The modified model was used for tissues involving cell proliferation and epithelial tissues with apical contraction (Okuda et al. 2013b, 2013c, 2015a, 2015b). In

98

6 3D Vertex Model

addition, in our vertex dynamics model, edges are assumed to be straight, but general cell patterns sometimes involve edges with curvature. A vertex cell (2D) model consisting of curved edges has been developed (Ishimoto and Morishita 2014; Mohammad et al. 2022).

6.9

Transition of a Cell Aggregate to a Vesicle (Honda et al. 2008)8

In the development of multicellular organisms, some cell aggregates produce a space within the aggregate, and the space enlarges to form a cavity enclosed by the cell sheet, as shown in Fig. 6.11. In particular, the mammalian embryo develops from a cell aggregate (the morula) to a vesicle (the blastocyst) in which the trophectoderm (TE) surrounds the blastocyst cavity and the inner cell mass (ICM) (Fig. 6.12). The blastocyst is enclosed by the zona pellucida (ZP), a capsule comprising a glycoprotein layer surrounding the plasma membrane. The blastocyst displays obvious asymmetry with the ICM at one end and the blastocyst cavity at the other. Takashi Hiiragi and his colleagues investigated blastocyst development in detail (Motosugi et al. 2005). To investigate blastocyst formation, we applied 3D vertex dynamics to a cell aggregate involving a cavity. We assumed that cells in the aggregate obey the equation of motion (Eq. 4.1). Here, the potential U contains terms of surface energy (US), elastic energy (UEV, UEI), and boundary energy due to restriction (Ubound): U ¼ U S þ U EV þ U EI þ U bound ,

ð6:7Þ

The potential US denotes the total surface energy of the cells: U S ¼ σ S ΣnS k Sk þ σ O ΣnO k Ok þ σ I ΣnI k I k :

ð6:8Þ

The first two terms in Eq. (6.8) represent the interface energy between neighboring cells and the surface energy between cells and the outside, where nS and nO are the numbers of polygons facing an adjacent cell and facing the outside, respectively. Sk is the surface area of a facing adjacent cell, and σ S is its interface energy per unit area (Fig. 6.13). Ok is the surface area of polygon k facing the outside, and σ O is its energy per unit area. The third term in Eq. (6.8) represents the energy between cells and the internal cavity (blastocyst cavity), where nI is the number of polygons facing the blastocyst cavity. Ik is the area of polygon k facing the blastocyst cavity, and σ I is its energy per unit area. The potential UEV contains two terms, energy of compression and expansion of the cells and that of the blastocyst cavity:

8

Reproduced from excerpts in Honda et al. 2008 with permission of The Company of Biologists.

6.9 Transition of a Cell Aggregate to a Vesicle (Honda et al. 2008)

99

Fig. 6.11 Schematic presentation of the change in an aggregate of polyhedra. (a) From a solid aggregate of polyhedra to a vesicle containing a cavity. (b) Figures showing the inside of the aggregate and the vesicle [Original]

Fig. 6.12 Early stages of mammalian embryo development. The morula and the blastocyst are indicated. ZP, zona pellucida; ICM, inner cell mass; TE, trophectoderm [Reproduced from Zu 214b in Honda 1991]

U EV ¼ κV Σn α ðV α  V std Þ2 þ κ VI ðV I  V Istd Þ2 ,

ð6:9Þ

where κ V and κVI are the elastic constants of the volumes of the cells and the blastocyst cavity, respectively, and n is the total number of cells. Vα and VI are the volumes of cell α and the blastocyst cavity, respectively. Vstd and VIstd are the volumes of relaxed cells and of the relaxed cavity, respectively. Equation (6.9)

100

6 3D Vertex Model

Fig. 6.13 Parameters used in the equation of motion (Eqs. 6.8 and 6.9). σ O, σ S, and σ I: surface energy densities of the surface area facing outside, the interface between cells and the surface facing the cavity, respectively. κ V and κ VI: elastic constants of the cell volume and the cavity volume, respectively

imposes a constraint on the volumes of the cells and the blastocyst cavity. Since the presence of the basement membrane was suggested at the surface of the TE cells facing the blastocyst cavity (Klaffky et al. 2001), we introduced the potential UEI. The potential UEI denotes the elastic energy of compression and expansion of the surface area of a polygon k facing the blastocyst cavity: U EI ¼ κI ΣnI k ðI k  I std Þ2

ð6:10Þ

where κ I is the elastic constant of the surface of polygon k facing the blastocyst cavity. Istd is the relaxed surface area of the cells facing the blastocyst cavity. Equation (6.10) provides the cell surface constraint of polygon k facing the blastocyst cavity. Restriction of the embryo by the ZP is described as the additional term Ubound: U bound ¼ wbound Σi nv 1=½1 þ exp ðλ f ðri ÞÞ,

ð6:11Þ

in which λ is a positive constant, ri ¼ (xi, yi, zi) and f ðri Þ ¼ ðxi  x0 Þ2 =a2 þ ðyi  y0 Þ2 =b2 þ ðzi  z0 Þ2 =c2  1:

ð6:12Þ

Ubound expresses the hindrance of outward motion of a vertex from the ellipsoid [the diameters of its main axes are a, b, and c, and its center is (x0, y0, z0)]. When a point ri is outside or inside the ellipsoid, f(ri) is positive or negative, respectively. Vertex i moves without restriction when it is inside the ellipsoid ( f(ri) < 0), whereas it produces substantial potential energy when it crosses the ellipse boundary ( f(ri) > 0). Instead of a step function, i.e., {¼1, f(ri) > 0; ¼ 0, f(ri) < 0}, we use a continuous analytic function, the logistic function 1/[1 + exp{λ f(ri)}], Eq. (6.11),

6.9 Transition of a Cell Aggregate to a Vesicle (Honda et al. 2008)

101

Fig. 6.14 Figures for explanation of computer simulations. (a) A columnar cell with the apical face (k) facing a cavity [Original]. (b) Formation of a small cavity (a tetrahedron) at a vertex at which four edges meet. (c) Increase in the cavity volume (internal space). The volume of the cavity (VIstd) is forced to increase linearly until t ¼ 500 [Reproduced from Fig. 1C and D in Honda et al. 2008 with permission of The Company of Biologists]

where λ/4 is the slope of the logistic function at f(ri) ¼ 0, and wbound is the strength of the potential. In this study, an aggregate of 40 cells was considered to simulate the morphology of mid- to late-stage expanded blastocysts (Chisholm et al. 1985; Dietrich and Hiiragi 2007; Gueth-Hallonet et al. 1993; Smith and McLaren 1977). The number of cells is constant in this simulation, and the calculation is carried out to obtain the most stable (i.e., eventual) structure of the cell aggregate that has the minimum potential energy. A cavity is created in the aggregate by replacing one vertex with a small tetrahedron (Fig. 6.14b; the length of the six edges is 0.1, which is slightly larger than the critical length δ ¼ 0.05). The volume of the tetrahedron (VIstd), which subsequently forms the polyhedron, is forced to increase during t ¼ 0–500 (Fig. 6.14c) to half the total volume of the initial cell aggregate (VIstd ¼ 20), reaching the expanded blastocyst stage. To increase VIstd, we used the method of Adiabatic approximation as will be shown in the later section. The simulation includes an additional phase after the increase in the volume (VIstd) stops (t ¼ 500) until the actual cavity reaches the maximal volume and the structure reaches a stable state. The blastocyst cavity is an intercellular space in the mammalian embryo created by the directional fluid secretion of the outer layer cells (Aziz and Alexandre 1991; Calarco and Brown 1969; Wiley and Eglitis 1981). This secretion most likely occurs in essentially all outer blastomeres with apicobasal epithelial polarity, thus initiating multiple small cavitations (Motosugi et al. 2005), and these multiple cavities coalesce to form one large cavity. The current simulation calculates the most stable structure with the minimal potential energy composed of 40 cells and one cavity that should correspond to the eventual blastocyst morphology. The calculation process to find a stable state is exemplified in Fig. 6.15. In this example, the aggregate is surrounded and restricted by an ellipsoidal (1.2:1) capsule (Honda et al. 2008), corresponding to an ellipsoidal ZP in vivo. A vertex (small solid

102

6 3D Vertex Model

Fig. 6.15 An example of the calculation process to reach a stable state. Viewed from cross-sections of the yz-plane. Numbers indicate the time point of the simulation (t). A vertex (black circle in the sample at t ¼ 0) is replaced by a tetrahedron (see Fig. 6.14b) and is enlarged until its volume reaches half the initial total volume (at t ¼ 500; see Fig. 6.14c). The blastocyst axis continuously changes during t ¼ 500–2000 until it is localized and stabilized at one end of the long axis of the ellipsoidal ZP (t ¼ 2000), when it no longer migrates (t ¼ 2000–3500). Two cells are marked (* and +) to illustrate their movement. To see this figure in color, go online (Honda et al. 2008) [Reproduced from Fig. 4A in Honda et al. 2008 with permission of The Company of Biologists]

circle in Fig. 6.15 at t ¼ 0), replaced by a tetrahedron (see Fig. 6.14b), is enlarged until its volume reaches half the initial total volume (at t ¼ 500; see Fig. 6.14c). The embryonic–abembryonic axis (Em-Ab axis) keeps changing with respect to the outer capsule until the embryonic pole, the position of the ICM, is localized at one end of the long axis of the ellipsoidal capsule (t ¼ 2000). The structure is then “stabilized,” remaining unchanged thereafter for a substantial amount of time (t ¼ 2000–3500), equivalent to the time required to stabilize the structure (t ¼ 0–2000). Tracing of cells (Fig. 6.15, marked with * and +) further demonstrates that cells and the cavity gradually change their position relative to each other during the process. The cell with * migrates to a cluster of cells (t ¼ 500). The cell with + is situated at the center of the cluster, and the cluster (including the cell labeled with +) migrates and stabilizes at one end of the long axis (t ¼ 1000–3500). The enlarging cavity eventually becomes surrounded by an outer single-cell layer, with its position stabilized at one end of the long axis, while a cluster of cells forms on the opposite side (Fig. 6.16). Thus, the simulation recapitulates the blastocyst morphology in vivo well, with the outside cells corresponding to the TE and the inner population to the ICM.

6.9 Transition of a Cell Aggregate to a Vesicle (Honda et al. 2008)

103

Fig. 6.16 A computer simulation result. (a) The simulated blastocyst at t ¼ 2000 in cross-sectional views of the xz- and yz-planes. (b) A stereoscopic view of the blastocyst at t ¼ 2000, in which the ZP and some of the TE cells are removed to provide an internal view [Reproduced from Fig. 4B and C in Honda et al. 2008 with permission of The Company of Biologists]

An experimental observation that the Em-Ab axis tended to be aligned with the longest ZP axis in ellipsoidal capsules of the ZP (Fig. 3 in Honda et al. 2008) confirmed our simulation result in Fig. 6.15 (t ¼ 2000 and 3500), where the ICM and the blastocyst cavity were aligned with the longest ellipsoidal axis. Intriguingly, despite the assumed equivalence in mechanical properties of the 40 cells in the computer simulation (Fig. 6.15), enlargement of the cavity enhances the difference in volume of the inner versus outer cells (Fig. 6.17). This prediction based on computer simulations was again confirmed in another study (Aiken et al. 2004), demonstrating in vivo that the average cell volume of the outer cells is significantly larger than that of the inner cells. The present simulation provides insight into the mechanism by which the inside cells form a distinct cluster (ICM) instead of a continuous cell layer beneath the TE. This would indeed be the case only if the entire structure of the embryo were precisely radially symmetrical and cavity formation was initiated at the very center of such a structure. However, in biological systems, such absolute precision is essentially impossible, and once there is the slightest deviation from the central location, the cavity never returns to the center but stabilizes at the periphery, forming a cluster of inside cells, the ICM. During simulation, equivalent cells form two distinct populations composed of smaller inner cells and larger outer cells. These results reveal a unique feature of early mammalian development: asymmetry may emerge autonomously in an equivalent population with no need for a priori intrinsic differences. The computer simulation in the present study confirms that the mechanical constraint imposed on the embryo is sufficient to orient the blastocyst axis. Furthermore, it illustrates the

104

6 3D Vertex Model

Fig. 6.17 Volume of inner cells and outer cells of the blastocyst after simulations. The data are based on a total of five simulations under various conditions (differing in the direction of the long axis of the ellipsoidal ZP and in the initial position of the cavity). Average volumes (s.d.) of the inner cells (black bars) and the outer cells (gray bars) were 0.702  0.0088 (n ¼ 56) and 0.744  0.0136 (n ¼ 144), respectively [Reproduced from Fig. 5 in Honda et al. 2008 with permission of The Company of Biologists]

minimal cellular mechanical properties required for creating blastocyst morphology, thereby allowing us to gain deep insight into blastocyst morphogenesis from a mechanical point of view. Generally, the formation of an envelope by a collective of cells in which the cells wrap around a luminal space is a remarkable event in the development of animals. The topic will be discussed in further detail in a later section (Chap. 7. The world of epithelial sheets). We succeeded in recapitulating one of the events, the formation of blastocysts, using vertex dynamics. Formation of a cyst enclosing a fluid-filled lumen was simulated using the phase-field method (Akiyama et al. 2018). In the phase-field method, a phase-field variable is introduced to describe individual cells. The interfacial regions (cell boundaries) involve continuous but highly localized variations of the phase field (Nonomura 2012).

6.10

Fundamentals of Vertex Dynamics

We have used and will use vertex dynamics in this book for 2D polygonal patterns (Chap. 4), 3D polyhedral patterns (Chap. 6), and surface sheets in 3D space (Chap. 9). A summarization of fundamental concepts and treatment of vertex dynamics may be appropriate here.

6.10

Fundamentals of Vertex Dynamics

105

Coarse-grained model In the vertex model, we assumed collective cells to be an assemblage of polygons (2D) or polyhedrons (3D), and we expressed it through mathematical words, vertices, edges, and faces. Since the mathematical elements are only approximations of real cells, each element has a finite size. In the model, the finite size defines a minimum length, δ, below which one cannot discuss any event within the model, but one may do so in a microscopic model. This view of nature is called coarse graining in statistical physics, which is explained below. Nature shows a layered structure (hierarchy) in space and time. On the length scale, we can view nature by separating it into three layers, i.e., microscopic stage (atomic scale), mesoscopic stage (mesoscale), and macroscopic stage (macroscale) (Kawasaki 2000). The theme of this book “morphogenesis based on cells” is a work connecting the mesoscopic stage (cell aggregates) to the macroscopic stage (forms of living things). Details of cells and cell–cell adhesions belong to the microscopic stage (molecular level), which is a task of molecular biology. Our vertex model is called a coarse-grained model because it can be considered a model obtained through coarse graining of the microscopic structure of a real system (Nagai et al. 1990). Coarse graining means that one describes the system by averaging each microscopic region statistical-mechanically. The resulting coarsegrained model contains some model parameters that reflect the properties of microscopic elements. The vertex model contains several model parameters, for example, η (friction constant), σ L (interfacial energy per unit length) and κS (elastic constant of cell volume) in Eq. (4.1). If the values of these parameters are determined by experiments, then we can obtain information about microscopic structures in principle. We need the following checks on the vertex model: the results of the vertex model reproduce the morphology observed in experiments and the parameter values are consistent with molecular biology findings. Dissipative motion We then assumed that the vertices obey the equations of motion in classical mechanics for mass-less particles, given by η dri/dt ¼ — i U (Eq. 4.1), as already described in Chap. 4. U(r1(t), r2(t), . . ., rnv(t)) denotes the total potential energy. The vertices move so that the total potential energy decreases, which is derived from Eq. (4.2). The mechanical energy is converted into dissipated heat. The coefficient η in the left-hand side of Eq. (4.1) is a positive constant (an analog of the coefficient of viscosity). The left-hand side represents a viscous drag force proportional to the vertex velocity dri/dt. Vertices do not have mass (inertia), so the motion of the vertices and cells is completely damped. In other words, these vertices are driven by thermodynamic forces to minimize the total potential energy of the system. This is a slow process of dissipative motions going to equilibrium. Edge energy density We have used in Eq. (4.4) and will use the edge energy density σ L in vertex dynamics, which is a constant of proportionality of edge length. In collective polygonal cells, there are edges that are boundaries (junctions) between neighboring cells. We assumed that an edge has potential energy, which functions to shorten the edge length in vertex dynamics. The idea is based on observation of the contractile properties of the circumferential actomyosin bundles with which the cell membrane is lined (Eguchi 1977; Owaribe et al. 1981). Although we assumed that

106

6 3D Vertex Model

the edge energy density is constant at first, we will assume that it is variable depending on its orientation (Eqs. 8.10 and 9.8). Indeed, the edge energy density is not always constant due to remodeling of actomyosin bundles, and recently, one of the underlying mechanisms was experimentally elucidated: the actomyosin bundle is remodeled by mechanical force (Fernandez-Gonzalez et al. 2009; Stephenson et al. 2019). On the other hand, the edge energy density depends not only on the contractile force of actomyosin but also on intercellular adhesion (Chap. 5). When cell–cell adhesion at a cell boundary is strong, the boundary is elongated in the theory of differential cell adhesion in the cell sorting system (see Fig. 5.16). When cell adhesion is weak, the cell boundary becomes short, that is, the boundary contracts (Honda et al. 1986; Graner and Sawada 1993; Katsunuma et al. 2016). Maitre et al. (2012) reported a relationship between the contraction of cell junctions and cell adhesion in which cell adhesion provides a mechanical scaffold for cell cortex tension. The contributions of contraction of cell edges and cell–cell adhesion to the edge energy density should be systematically investigated. Equation of motion expressed by dimensionless quantities Until now, when we performed computer simulations using the vertex model, we scaled all quantities by new units to obtain dimensionless quantities and then solved the equations of motion for vertices expressed by them. We explain a method of expressing the equation of motion in dimensionless quantities, taking the formation of epithelial tissues in Chap. 4 (Eqs. 4.1–4.6) as an example. To reconfirm correct dimensions of all quantities, we now start by modeling epithelial tissues. We approximate an epithelial tissue as a monolayer cell sheet with a constant thickness h, in which each cell is prism shaped with a height h and a base area Sα, and assume that this system contains a sufficiently large number of cells in a uniform medium. Since the cell height h is constant, the system can be described by an assembly of 2D polygons as given below. The total potential energy of the system is the sum of the cell–cell interfacial energy UI and the volume elastic energy of cells UD, i.e., U ¼ UI þ UD:

ð6:13Þ

These potential energies are functions of vertex positions, r1(t), r2(t), . . ., rnv(t), where ri(t) denotes a 2D positional vector of vertex i at time t and nV represents the number of vertices. Since the area of a rectangle [edge  height] is |ri – rj|  h, the cell–cell interfacial energy UI is written as U I ¼ σ I Σ




 h jri  r j j

ð6:14Þ

where σ I is the energy density of the cell–cell interface. Here, denotes an edge between neighboring vertices i and j, and thus, the sum in Eq. (6.14) means summing all edges. Since the volume of cell α is Sα  h, the volume elastic energy UD can be written as

6.10

Fundamentals of Vertex Dynamics

U D ¼ κ D Σn α ðh Sα  h S0 Þ2

107

ð6:15Þ

where κ D denotes the volume elastic constant, and S0 represents the base area of the cell in the relaxed state. The cell–cell interfacial energy UI (Eq. 6.14) is identical to the total edge energy UL (Eq. 4.4), i.e., the coefficient was defined as σ L σ I h, and the edge length was defined as L |ri – rj|. Similarly, the volume elastic energy UD (Eq. 6.15) is identical to the total elastic energy of the polygon area UES (Eq. 4.5), i.e., the coefficient was defined as κS κ Dh2. For computer simulations, we transform all quantities into dimensionless quantities, leading to dimensionless equations of motion for vertices. Equation (4.1) contains two variables, length and time. Therefore, we choose two new units, a length unit R0 and a time unit τ0, which are both characteristic of our system. First, the new length unit is a linear expression of cell size, because our theme is to observe a cell assembly on the basis of a cell. Thus, the new length unit is R0 √S0. Leaving the new time unit τ0 until later, we rewrite Eq. (4.1) with the total potential U given by Eq. (6.13). Writing dimensionless quantities as ones with the symbol (00 ), i.e., ri ¼ ri00 Ro, t ¼ t00 τ0, and — i ¼ — i00 /Ro, we substitute them into Eq. (4.1) to obtain η R0 =τ0 dri 00 =dt 00 ¼ ð1=R0 Þ∇i 00

h i 2 σ I h R0 Σ jr00i r00j jþκD R0 4 h2 Σα ðSα 00  1Þ , ð6:16Þ

where we have used So00 ¼ 1 on the right-hand side. Dividing both sides by the coefficient of the left-hand side η R0/τ0, the coefficient of the first term on the righthand side becomes c1 τ0 σ I h/(η R0). If we set c1 ¼ 1, we then obtain τ0 η R0/(σ I h), which leads to c2 ¼ κD h R03/σ I for the coefficient of the second term on the righthand side. Here, we have required that c2 6¼ 1 but c1 ¼ 1 because we consider that the interfacial energy plays the principal role in morphogenesis. Finally, the dimensionless equation of motion for vertices, omitting primes (00 ), is as follows: h i dri =dt ¼ ∇i Σ jri  r j jþκD Σα ðSα  1Þ2 ,

ð6:17Þ

where κD* c2 ¼ κD h R03/σ I. Now, we have obtained a generalized equation of motion for vertices, Eq. (6.17), in which the explicit parameters η and σI are removed without loss of generality and all quantities are dimensionless. Adiabatic approximation In the preceding section (Chap. 6.9 Transition of a cell aggregate to a vesicle), we used an approximation, adiabatic approximation, when we performed computer simulations. Here we explain it. The adiabatic approximation is used for obtaining molecular states quantum-mechanically in molecular physics. A molecule is a system which consists of electrons and plural nuclei. To describe states of this system we separate both motions: i.e., first, one fixes nuclei and then solves electronic motion; next, one solves nuclear motion in a mean field produced by electronic motion already known. Such a separation is possible since

108

6 3D Vertex Model

electronic motions are very fast compared with nuclear motions. Electrons see nuclei just like them at rest. This comes from that nuclear masses are much larger than electron’s. This approximation method is not only limited to solving the molecular state but also serves to create our picture of the molecular state. The second term of the potential UEV (Eq. 6.9) denotes the elastic energy of the cavity volume, where the cavity volume is a 3D space that is enclosed by the spherical cell sheet. The term is κVI (VI – VIstd)2, where VI is the cavity volume at time t and VIstd is the relaxed state of the cavity volume. VIstd is a constant at every calculation step but is assumed to be forced to increase in proportion to the calculation step. Calculations at every step were divided into two substeps. The adiabatic approximation between the two substeps was confirmed to be applicable as follows: (1) under fixation of VIstd, the averaged velocity of vertices is more than δ/h ¼ 10, where δ (0.05) is the minimum length of the system and h (0.005) is the time interval of the calculation step. (2) The cavity volume VIstd(t) is forced to increase as VIstd(t) ¼ 0.04 t, where VIstd(0) ¼ 0 and VIstd(500) ¼ 20. The velocity of the change in the cavity is 0.04. Since the linear size of the cavity can be defined by LIstd (t) ¼ VIstd1/3, the velocity of the change in the linear size of the cavity is h i  0:04 dLIstd =dt ¼ dLIstd =dV Istd  dV Istd =dt b ð1=3ÞV Istd ðt Þ2=3 t¼t min
2 ¼ 0:04= 3δ ¼ 5:33, where VIstd(tmin) ¼ δ3 is used for the minimum volume because the system is a coarse-grained model and we have the minimum length δ. Then, the velocity of the change in the linear size of the cavity is smaller than the averaged velocity of vertices (> δ/h ¼ 10). The adiabatic approximation can be applied, where a calculation step is divided into two substeps: vertex dynamics and global dynamics. Repetition of the two dynamics leads to lower U states step by step and produces a system with the minimum U. The method has been and will be used frequently in this book, e.g., zigzag pattern formation in the Drosophila wing in Chap. 5, cavity formation within blastocysts in Chap. 6 (here), tube elongations in Chap. 8, and remodeling of heart tubes in Chap. 9.

6.11

Summary

2D vertex dynamics was expanded to 3D vertex dynamics, where x-, y-, and z-coordinates of vertices and relationships between neighboring vertices are necessary. The neighboring vertex relationships were expressed using lists of four cells [ai, bi, ci, di]. The method of expression of vertex relationships was explained. When we encounter minute short edges or triangles during calculation, we have to use a 3D elemental process of reconnection of vertices, which was explained (Fig. 6.2). First, we made a spherical cell aggregate with surface tension. Next, the spheres were

References

109

flattened by external force, and then the viscoelastic properties of the cell aggregates were examined. We also examined a cell aggregate involving a cavity forming a spherical shell (vesicle). The morphogenesis of shell formation was adapted to the formation of blastocysts in mammalian development. Finally, we added a few fundamental comments on vertex dynamics, the coarse-grained model, dissipative motion in the dynamics, edge energy density, a dimensionless equation of motion and adiabatic approximation in computer simulations.

References Aiken, C.E., Swoboda, P.P., Skepper, J.N., Johnson, M.H.: The direct measurement of embryogenic volume and nucleocytoplasmic ratio during mouse pre-implantation development. Reproduction. 128, 527–535 (2004) Akiyama, M., Nonomura, M., Tero, A., Kobayashi, R.: Numerical study on spindle positioning using phase field method. Phys. Biol. 16, 016005 (2018) Aziz, M., Alexandre, H.: The origin of the nascent blastocoele in preimplantation mouse embryos: ultrastructural cytochemistry and effect of chloroquine. Rouxs Arch. Dev. Biol. 200, 77–85 (1991) Beysens, D.A., Forgacs, G., Glazier, J.A.: Cell sorting is analogous to phase ordering in fluids. Proc. Nat. Acad. Sci. 97, 9467–9471 (2000) Calarco, P.G., Brown, E.H.: An ultrastructural and cytological study of preimplantation development of the mouse. J. Exp. Zool. 171, 253–283 (1969) Chisholm, J.C., Johnson, M.H., Warren, P.D., Fleming, T.P., Pickering, S.J.: Developmental variability within and between mouse expanding blastocysts and their ICMs. J. Embryol. Exp. Morphol. 86, 311–336 (1985) Dietrich, J.E., Hiiragi, T.: Stochastic patterning in the mouse preimplantation embryo. Development. 134, 4219–4231 (2007) Eguchi, G.: Cell shape change and establishment of tissue structure. (in Japanese). Saienu. 7(5), 66–77 (1977) Fernandez-Gonzalez, R., et al.: Myosin II dynamics are regulated by tension in intercalating cells. Dev. Cell. 17(5), 736–743 (2009) Forgacs, G., Foty, R.A., Shafrir, Y., Steinberg, M.S.: Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys. J. 74, 2227–2234 (1998) Foty, R.A., Forgacs, G., Pfleger, C.M., Steinberg, M.S.: Liquid properties of embryonic tissues: measurement of interfacial tensions. Phys. Rev. Lett. 72, 2298–2301 (1995) Foty, R.A., Pfleger, C.M., Forgacs, G., Steinberg, S.M.: Surface tensions of embryonic tissues predict their mutual envelopment behavior. Development. 122, 1611–1620 (1996) Fuchizaki, K., Kusaba, T., Kawasaki, K.: Computer modelling of three-dimensional cellular pattern growth. Philos. Mag. B. 71, 333–357 (1995) Fung, Y.C.: Biomechanics. Springer, New York (1981) Graner, F.A., Sawada, Y.: Can surface adhesion drive cell rearrangement? Part II: a geometrical model. J. Theor. Biol. 164, 477–506 (1993) Gueth-Hallonet, C., Antony, C., Aghion, J., Santa-Maria, A., Lajoie-Mazenc, I., Wright, M., Maro, B.: gamma-Tubulin is present in acentriolar MTOCs during early mouse development. J. Cell Sci. 105, 157–166 (1993) Honda, H.: Shiito karano Karada-tsukuri (in Japanese). Chuokoron-sha, Tokyo (1991) Honda, H.: Katachi no Seibutugaku (in Japanese). NHK Shuppan, Tokyo (2010) Honda, H., et al.: Transformation of a polygonal cellular pattern during sexual maturation of the avian oviduct epithelium: computer simulation. J. Embryol. Exp. Morphol. 98, 1–19 (1986)

110

6 3D Vertex Model

Honda, H., Tanemura, M., Nagai, T.: A three-dimensional vertex dynamics cell model of spacefilling polyhedra simulating cell behavior in a cell aggregate. J. Theor. Biol. 226, 439–453 (2004) Honda, H., Motosugi, N., Nagai, T., Tanemura, M., Hiiragi, T.: Computer simulation of emerging asymmetry in the mouse blastocyst. Development. 135, 1407–1414 (2008) Ishimoto, Y., Morishita, Y.: Bubbly vertex dynamics: a dynamical and geometrical model for epithelial tissues with curved cell shapes. Phys. Rev. E. 90, 052711 (2014) Katsunuma, S., Honda, H., Shinoda, T., Ishimoto, Y., Miyata, T., Kiyonari, H., Abe, T., Nibu, K., Takai, Y., Togashi, H.: Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium. J. Cell Biol. 212, 561–575 (2016) Kawasaki, K.: Non-equilibrium and phase transition—mesoscopic statistical physics (in Japanese). Asakura-shoten, Tokyo (2000) Klaffky, E., Williams, R., Yao, C.C., Ziober, B., Kramer, R., Sutherland, A.: Trophoblast-specific expression and function of the integrin alpha 7 subunit in the peri-implantation mouse embryo. Dev. Biol. 239, 161–175 (2001) Maitre, J.L., et al.: Adhesion functions in cell sorting by mechanically coupling the cortices of adhering cells. Science. 338(6104), 253–256 (2012) Mohammad, R.Z., Murakawa, H., Svadlenka, K., Togashi, H.: A numerical algorithm for modeling cellular rearrangements in tissue morphogenesis. Commun. Biol. 5, 239 (2022) Motosugi, N., Tobias, B., Zbigniew, P., Davor, S., Takashi, H.: Polarity of the mouse embryo is established at blastocyst and is not prepatterned. Genes Dev. 19, 1081–1092 (2005) Nagai, T., Ohta, S., Kawasaki, K., Okuzono, T.: Computer simulation of cellular pattern growth in two and three dimensions. Phase Trans. 28, 177–211 (1990) Nonomura, M.: Study on multicellular systems using a phase field model. PLoS One. 7, e33501 (2012) Okuda, S., Inoue, Y., Eiraku, M., Sasai, Y., Adachi, T.: Reversible network reconnection model for simulating large deformation in dynamic tissue morphogenesis. Biomech. Model. Mechanobiol. 12, 627–644 (2013a) Okuda, S., Inoue, Y., Eiraku, M., Sasai, Y., Adachi, T.: Modeling cell proliferation for simulating three-dimensional tissue morphogenesis based on a reversible network reconnection framework. Biomech. Model. Mechanobiol. 12, 987–996 (2013b) Okuda, S., Inoue, Y., Eiraku, M., Sasai, Y., Adachi, T.: Apical contractility in growing epithelium supports robust maintenance of smooth curvatures against cell-division-induced mechanical disturbance. J. Biomech. 46, 1705–1713 (2013c) Okuda, S., Inoue, Y., Adachi, T.: Three-dimensional vertex model for simulating multicellular morphogenesis. Biophys. Physicobiol. 12, 13–20 (2015a) Okuda, S., Inoue, Y., Eiraku, M., Adachi, T., Sasai, Y.: Vertex dynamics simulations of viscositydependent deformation during tissue morphogenesis. Biomech. Model. Mechanobiol. 14, 413–425 (2015b). https://doi.org/10.1007/s10237-10014-10613-10235. On line Owaribe, K., Kodama, R., Eguchi, G.: Demonstration of contractility of circumferential actin bundles and its morphogenetic significance in pigmented epithelium in vitro and in vivo. J. Cell Biol. 90, 507–514 (1981) Palsson, E.: A three-dimensional model of cell movement in multicellular systems. Futur. Gener. Comput. Syst. 17, 835–852 (2001) Phillips, H.M., Davis, G.S.: Liquid-tissue mechanics in amphibian gastrulation: germ-layer assembly in Rana pipiens. Am. Zool. 18, 81–93 (1978) Phillips, H.M., Steinberg, M.S.: Embryonic tissues as elasticoviscous liquids. I. Rapid and slow shape changes in centrifuged cell aggregates. J. Cell. Sci. 30, 1–20 (1978) Phillips, H.M., Steinberg, M.S., Lipton, B.H.: Embryonic tissues as elasticoviscous liquids. II. Direct evidence for cell slippage in centrifuged aggregates. Dev. Biol. 59, 124–134 (1977) Smith, R., McLaren, A.: Factors affecting the time of formation of the mouse blastocoele. J. Embryol. Exp. Morphol. 41, 79–92 (1977)

References

111

Stephenson, R.E., et al.: Rho flares repair local junction leaks. Dev. Cell. 48(4), 445–459.e445 (2019) Tanemura, M., Ogawa, T., Ogita, N.: A new algorithm for three-dimensional Voronoi tessellation. J. Comput. Phys. 51, 191–207 (1983) Weaire, D., Phelan, R.: A counter-example to Kelvin’s conjecture on minimal surfaces. In: Weaire, D. (ed.) The Kelvin Problem, pp. 47–51. Taylor and Francis, London (1996) Weaire, D., Rivier, N.: Soap, cells and statistics-random patterns in two dimensions. Contemp. Phys. 25, 59–99 (1984) Wiley, L.M., Eglitis, M.A.: Cell surface and cytoskeletal elements: cavitation in the mouse preimplantation embryo. Dev. Biol. 86, 493–501 (1981) Wooten, F.: Structure, odd lines and topological entropy of disorder of amorphous silicon. Acta. Crystallogr. A. 58, 346–351 (2002) Wooten, F., Winer, K., Weaire, D.: Computer generation of structural models of amorphous Si and Ge. Phys. Rev. Lett. 54, 1392–1395 (1985)

Chapter 7

The World of Epithelial Sheets9

Outline In the computer simulation of blastocyst formation in the previous chapter, we encountered epithelial cells (cells of the trophectoderm). Generally, epithelial cells form epithelial sheets and have an essential role in animal morphogenesis. Here, the epithelial tissues are globally reviewed from a general perspective. The behavior of the vacuolar apical compartment (VAC) within epithelial cells is important. Based on VAC behavior, epithelial cells are classified into a few types. Furthermore, we understand the mechanism by which a closed envelope is automatically formed from epithelial cells.

7.1

Morphogenesis of Multicellular Animals Is Deformation of a Closed Epithelial Envelope

The epithelium, which is a sheet consisting of epithelial cells, forms various organs in animal bodies, especially early embryonic bodies, guts, neural tubes, lung branches, and vascular networks. An epithelial sheet functions as a partition that divides a space into two parts. Usually, these two spaces are occupied by fluids (liquid or air) and solid materials (Fig. 7.1a). Therefore, each epithelial sheet has two sides, the apical and basal surfaces. The two spaces are characterized by the basal side and apical side substances: the basal side substances are related with extracellular matrix (ECM) involving collagens, proteoglycans, laminin, and fibronectin; the apical side substances involve podocalyxin and chitin (Sigurbjörnsdóttir et al. 2014). The partition function requires that the epithelial sheet does not have a peripheral edge; otherwise, the materials in the two spaces could escape and mingle along the edge. Therefore, it naturally follows that an epithelial sheet has no choice but to form a closed envelope (Fig. 7.1a, right). This is a remarkable property of epithelial sheets. 9

Reproduced from excerpts in Honda 2017 with permission of John Wiley and Sons.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_7

113

114

7 The World of Epithelial Sheets

Fig. 7.1 Properties of an epithelial sheet and elemental processes of its deformation. (a) Epithelial sheets have two surfaces, apical and basal. Usually, the sheet is a closed envelope where the apical surface is inside or outside. Elemental processes include elongation of envelopes (ELG; b and c), invagination (IVG; d), protrusion (PRT; e), reconnection to alter topology via fusion (apical surfaces face each other, aREC; basal surfaces face each other, bREC; f), and inversion (IVS; g). Elongation can be divided into two cases (b) and (c): elongation of the envelope with the outer surface being apical (b) or basal (c). To see this figure in color, go online (Honda 2017). [Reproduced from Fig. 1 and an excerpt of its legend in Honda 2017 with permission of John Wiley and Sons]

Intriguingly, a closed envelope seems to form automatically without other aid. The purpose of this section is to elucidate the autonomous mechanism by which closed envelopes form. During processes of animal morphogenesis, epithelial sheets deform and change shape. We will extract a few processes from such morphogeneses, as shown in Fig. 7.1: specifically, elongation (ELG), invagination (IVG), and protrusion (PRT) of epithelial envelopes, as shown in Fig. 7.1b–e. In addition, sometimes two epithelial sheets reconnect, altering topology via fusion (Fig. 7.1f, right and left). We know two cases of such reconnections: fusion between two apical surfaces that face each other (aREC) and fusion between two basal surfaces that face each other (bREC). We will refer to these events as “elemental processes” of epithelial sheets. We do not know of a fusion between apical and basal surfaces.

7.1 Morphogenesis of Multicellular Animals Is Deformation of a. . .

115

It might be appropriate to add another item to the list of elemental processes, namely, inversion (IVS in Fig. 7.1g). We know two types of inversions. First, the sponge embryo performs an inversion that takes place within a mother body (e.g., Calcareous sponge; Dan 1987). Initially, the outside of epithelial envelope of the sponge embryo is basal. A slit occurs on the envelope, and the sheet goes out through the slit whereby the sheet performs a concave–convex transformation; thus, the envelope is turned inside out. The physical mechanism underlying this inversion remains unclear. A cellular contractile force localized along the periphery of the epithelial sheet has been observed in Madin–Darby canine kidney (MDCK) epithelial sheets (Imai et al. 2015). Such a contractile force may play a role in the late stage of inversion of the sheet of the sponge embryo. The inversion of the embryonic epithelial envelope in the mother body of the sponge may reflect an adaptation to the sponge life cycle, as the inverted embryo is protected against fusion with its mother’s epithelial sheet (Dan 1987). We know of a similar case in algae. Volvox performs a similar inversion during its reproduction (Nishii and Ogihara 1999). Although it is not an animal, it should be mentioned because it may be important for the study of the inversion mechanism to notice such behavior in algae. In the other type of inversion, which is observed in thyroid follicles (Miyagawa et al. 1982) and in envelopes consisting of MDCK cells (Wang et al. 1990), the closed envelopes are suspended in culture medium and invert the apical–basal polarity of their epithelial cells. Such inversion takes place within cells by reconstruction, redistribution, and rearrangement of intracellular components such as apical junctional apparatus and Golgi apparatus. The culture conditions that determine outside–inside polarity have been investigated in detail (Yonemura 2014), and the genetic and molecular mechanisms of the inversion have also been investigated (O’Brien et al. 2001; Yu et al. 2005, 2008). Let us now consider combinations of these elemental processes. First, an envelope is considered in a case in which the apical surface faces outward. The results of the application of elemental processes will be mentioned first, and then, the correspondence between each result of the applications and actual epithelia will be discussed. Application of [ELG] to a spherical envelope produces an elongated envelope, and application of a combination of elemental processes, [ELG + bREC] (i.e., sequential application of [ELG] followed by [bREC]), produces two separate spheres (Fig. 7.2a). Aside from artificial reproduction, we cannot find examples of two envelopes generated by [ELG + bREC]. One process in which an envelope could form two isolated envelopes is division of a multicellular body, which does not occur in nature. We sometimes obtain clones via proliferation of a single cell but not by dissection of a multicellular body. Application of an elemental process, protrusion [ELG] to a spherical envelope produces a sphere with a protrusion (Fig. 7.2b). An actual example is a limb bud or a wing bud (Fig. 7.2b, right). The combination of [IVG + aREC] produces two envelopes nested within one another; that is, the smaller envelope fits into the larger envelope (Fig. 7.2c), which would correspond to the formation of a neural tube and the lens (sectional views in Fig. 7.2c, right). The combination [IVG + bREC] forms an invaginated envelope (Fig. 7.2d). The bottom of the concave structure faces the basal surface of the same

116

7 The World of Epithelial Sheets

Fig. 7.2 Combinations of the elemental processes in epithelial envelopes in which the apical surface faces outward. Products are shown at the right. (a) ELG + bREC. (b) PRT: Protrusion of an envelope, forming a limb bud or wing bud. (c) IVG + aREC: Formation of a neural tube and lens (Honda 1991). (d) IVG + bREC: Formation of the gut, a torus (doughnut shape) (Honda 1991). Facing of two basal surfaces indicated by a rectangle. (e) aREC with two envelopes: two apical surfaces of the palatal shelves fuse and reconnect during facial formation. (f) bREC with two nested envelopes: gut formation in an experimental reconstruction of the starfish embryo (Honda 1991). For abbreviations of elemental processes, see the legend of Fig. 7.1. To see this figure in color, go online (Honda 2017). [Reproduced from Fig. 2 and an excerpt of its legend in Honda 2017 with permission of John Wiley and Sons]

envelope and fuses the two basal surfaces (rectangle in Fig. 7.2d). Reconnection between the concave sheet and outer envelope creates a penetrating tube (a torus), forming a gut. We will now consider the elemental processes that occur between two epithelial envelopes. Between two envelopes of independent individual bodies, [aREC] is prohibited because it would form a chimera body, which cannot maintain its lineage over the course of evolution. However, [aREC] of two epithelia within an individual body can take place. For example, in craniofacial development, the left and right palatal shelves fuse with each other (Fig. 7.2e, right). Another example is the adhesion of mesothelia in the abdominal cavity, where fusion takes place between the peritoneum and mesothelium of an organ. [bREC] between two

7.1 Morphogenesis of Multicellular Animals Is Deformation of a. . .

117

envelopes nested within one another takes place in experimental reconstruction of starfish embryos (Fig. 7.2f), in which the dissociated cells associate into two envelopes that nest within one another (Dan-Sohkawa et al. 1986). In this model, the inner envelope ultimately fuses with the outer envelope to form a gut. This is a process of [IVG + aREC] in the inverse direction (Fig. 7.2c). Epithelial structures that are encompassed by another larger envelope, as shown in Fig. 7.2c (middle) and Fig. 7.2f (left), will be discussed. The basal surface of these epithelial structures faces outward (Fig. 7.3). The large envelopes covering the epithelial structures are not shown in Fig. 7.3 (e.g., the mesothelium or upper layer of the chorion is not shown). Morphogenesis of the gut and the trachea are discussed here, which is not completely encompassed by a larger envelope, but the main morphogenesis takes place inside a larger envelope (villi, folds, and crypts of the intestine and branching of the lung, salivary glands, and mammary glands). Through combination [PRT + aREC], which leads to the same topology as [ELG + aREC], two protruding folds in the outflow duct of the embryonic heart fuse and reconnect with each other. Then, two arteries form, which correspond to the formation of the aorta and the pulmonary artery (Fig. 7.3a). Here, conotruncal swelling ridges fuse with each other to make the conotruncal septum that forms the aorta and the pulmonary artery (e.g., Schoenwolf et al. 2009). The elemental process [PRT] applied to the intestinal tube corresponds to the formation of villi or folds (Fig. 7.3b, left). An obvious case corresponding to [PRT + bREC] (Fig. 7.3b, right) is not known. A new envelope is nested within the previous envelope. This is secondary nesting because the previous envelope has been made by nesting ([IVG + aREC], Fig. 7.2c). This situation is similar to that of a mammalian embryo that is enclosed by blood fluid in the placenta but should be examined in detail. A case corresponding to [PRT + aREC] is not known (Fig. 7.3c); it seems to result in a torus within the connective tissue with an apical–basal polarity opposite that of the early embryo (Fig. 7.2d). Application of [IVG] to an envelope corresponds to the formation of crypts in the intestine (Sato et al. 2011). Iterative invagination causes branching of a cavity, which gives rise to the structure of the lung (Fig. 7.3d, right), salivary glands, and mammary glands. When considering invagination as sprouting, a new sprout from an existing capillary forms a branching structure. Furthermore, additional [bREC] causes fusion to create an anastomosing network. Thus, [IVG + bREC] corresponds to the formation of angiogenesis in capillary networks (Fig. 7.3e, right). An example of an elemental process between two epithelial envelopes is [bREC] of the amnion cavity and the yolk sac in the mammalian embryo (Fig. 7.3f, right), followed by formation of an isolated embryo between them. The results are summarized in Fig. 7.4. This is the world of epithelial sheets. Initially, simple epithelial envelopes form, and several elemental processes operate sequentially on the envelope to produce an epithelial sheet of various shapes that is adapted for the life of living organisms. We would like to again stress the barrier function of the epithelial sheet. Certain diseases are related to the epithelial sheet. Ulcers and erosion are disruptions of the barrier function of the epithelial sheet. An ulcer is an open sore on an epithelial sheet caused by a break in the epithelial sheet

118

7 The World of Epithelial Sheets

Fig. 7.3 Combinations and iteration of the elemental processes in epithelial envelopes in which the basal surface faces outward. These structures are almost covered by epithelial sheets (mesothelium or chorion). Products are shown at the right. (a) PRT + aREC: Formation of the aorta and pulmonary artery. (b) PRT: Protrusion produces villi or folds in the intestine. PRT + bREC: Formation of a closed envelope with an apical outer surface, which does not correspond to an actual biological structure. (c) PRT + aREC: An apical–basal inverted torus (doughnut shape), which does not correspond to an actual biological structure. (d) IVG: Formation of crypts. Iterated IVG: Branching of a cavity and formation of the lung (Honda 1991). (e) IVG + bREC: Angiogenesis to form capillary networks (Honda 2010). (f) bREC with two envelopes: Formation of a mammalian embryo between the amniotic cavity and the yolk sac. The outer layer of the chorion is not shown (Honda 2010). For abbreviations of elemental processes, see the legend of Fig. 7.1. To see this figure in color, go online (Honda 2017). [Reproduced from Fig. 3 and an excerpt of its legend in Honda 2017 with permission of John Wiley and Sons]

that fails to heal, and erosion is the gradual destruction of the epithelial sheet. Dental caries, which result from gradual destruction of tooth enamel that is derived from epithelial cells, also belong in the erosion category. Among cell junctions between epithelial cells, tight junctions bear a responsibility for the barrier function between the apical and basal sides of the epithelium (Hudspeth 1975), which is also true for

7.2 Classification of Epithelial Cells

119

Fig. 7.4 The world of epithelial sheets consists of members produced by application of combinations and iteration of the elemental processes described in Fig. 7.1 involving one or two closed envelopes. There are two types of envelopes: the outer surface of one is apical (apical outside), and that of the other is basal (basal outside). Figures are taken from Figs. 2 and 3. For abbreviations of the elemental processes, see the legend of Fig. 7.1. [Reproduced from Fig. 4 in Honda 2017 with permission of John Wiley and Sons]

mammalian epidermis, where several flat epithelial cells are vertically stacked in several layers. Although the epidermis seems to be a complicated structure, the completely shielding structure of the tight junctions has been observed (Yokouchi et al. 2016). An epithelial envelope is variously deformed and remodeled through combinations of the elemental processes. The principal structures of most organs are regarded as remodeled or deformed epithelial envelopes. However, some of these epithelial envelopes seem to be theoretical and do not correspond to any real organs, for example, an inverted torus in connective tissues (Fig. 7.3c) and an envelope nested in higher order (nested after nesting) (Fig. 7.3b, right).

7.2

Classification of Epithelial Cells

Epithelial sheets consist of epithelial cells, which are prism-like in shape and packed in a two-dimensionally extended monolayer sheet. Each typical epithelial cell has a top and bottom surface. One of these surfaces is apical, the contact-free surface. The apical surface of a cell is surrounded by apical junctional complexes (Fig. 7.5a), which are composed of tight junctions (Zihni et al. 2016) and adherens junctions (AJs; Harris and Tepass 2010). We can say that each epithelial cell has one apical domain on the cell surface. Other types of epithelial cells exist (Honda 2010, 2012). Tubes, such as the trachea and vascular (or lymph) vessels, are epithelia. As shown in Fig. 7.5b–d, most epithelial cells in tubes have one apical domain. However, cells adjacent to the

120

7 The World of Epithelial Sheets

Fig. 7.5 Epithelial cells in tissues with one apical domain (Type 1). Typical prism-like cells (a) and cells that generate a tube (b–d) belong to Type 1 (g). A cell (e) close to the tip of a tube is Type 1 (tunnel or seamless, h). A tip cell (f) of a tube is Type 1 (cup, i) (Honda 2010, 2012). [Reproduced from Figs. 13–15 in Honda 2012]

tip of the tube are of different types, that is, tunnel type (Fig. 7.5e) and cap type (Fig. 7.5f). When the tip of a capillary touches another capillary and the two capillaries link together (Wolff and Bar 1972), the tip cell (Fig. 7.5f) of the capillary becomes a tunnel cell (Fig. 7.5e). The tunnel cell has a penetrated aperture but is distinct from a seamed cell, which is made by seaming two edges of a single cell (Fig. 7.5d). The formation of tunnel cells (seamless cells) has been investigated in detail in Drosophila trachea and zebrafish intersegmental vessels (TanakaMatakatsu et al. 1996; Kamei et al. 2006; Kakihara et al. 2008). On the other hand, a tip cell has one apical domain that is concave (Fig. 7.5f). Next, we will examine the relationship between the apical surface of epithelial cells and the apical surface of the epithelium. We will carefully evaluate which surface of cells is apical during morphogenesis. Figure 7.6a shows the process of capillary angiogenesis. Cell proliferation adds an endothelial cell to a capillary, and fusion of cell membranes takes place twice (thick arrows in Fig. 7.6a). Then, a vacuole in the added cell becomes linked to the lumen of the blood capillary (Fig. 7.6a, bottom). Because the inner surface of the blood vessel lumen is apical, according to this schematic, it is very likely that the membrane of the vacuole in the added cell is apical (Fig. 7.6a, top). This idea has been supported by experiments showing that an antibody specific for the apical membrane becomes localized at the intracellular vacuole in a polarized epithelium (Le Bivic et al. 1988). Therefore, we can conclude that an epithelial cell might contain its apical membrane on a vacuole. Furthermore, the apical membrane of epithelial cells can be converted into the apical surface of the epithelium.

7.2 Classification of Epithelial Cells

121

Fig. 7.6 Identification of the apical domain and classification of epithelial cells. (a) Process of capillary angiogenesis. A cell is added to a capillary through cell proliferation, and fusions of the cell membrane take place twice (thick arrows); ultimately, a capillary branch is formed. The vacuole in the added cell is considered to be enclosed by the apical membrane (Honda 2010) [reproduced from Fig. 3-47c in Honda 2010]. (b) Hepatocytes in the liver form bile canaliculi and have more than one apical domain (Honda 2012) [reproduced from Fig. 16 in Honda 2012]. (c) Classification of epithelial cells: Type 0, epithelial cells with apical domains inside; Type 1, epithelial cells with one apical domain, includes the cup-type and tunnel-type (seamless type); Types 2, 3, . . ., epithelial cells with two, three, or more apical domains (Honda 2012) [reproduced from Fig. 15 in Honda 2012]. [Reproduced from an excerpt of Fig. 6 legend in Honda 2017 with permission of John Wiley and Sons]

Next, we will consider other types of epithelial cells (Honda 2010, 2012). Hepatocytes are classified as epithelial cells, but their structures are not simple (Fig. 7.6b). In the liver, a hepatocyte has multiple apical surfaces of bile canaliculi through which bile is secreted. Thus, each hepatocyte has more than one apical domain. Taking this into account, we compiled a classification table of epithelial cells (Fig. 7.6c). All epithelial cells have one or more apical domains in the cell membrane. Epithelial cells that have apical domains inside are Type 0. Type 1 cells have one apical domain on the outer surface. Type 1 epithelial cells include not only prism-like cells but also cup-type cells and tunnel-type cells. Cup-type cells have a concave apical domain, and tunnel-type cells are referred to as seamless-type cells. Hepatic-type cells have more than one apical domain. We define a type n cell as a cell that has n apical domains on the outer surface.

122

7.3

7 The World of Epithelial Sheets

Crucial Roles of Vacuolar Apical Compartments (VACs) in Epithelization

Above, we classified epithelial cells into several types (Fig. 7.6c). To understand the mechanisms that give epithelial cells these properties, we must introduce an organelle within epithelial cells, the vacuolar apical compartment (VAC). The VAC, an intracellular organelle found in MDCK cells (Fig. 7.7a), exhibits apical biochemical markers and microvilli and excludes basolateral markers (Vega-Salas et al. 1987, 1988). Exocytosis of the VAC occurs on the cell surface near areas of cell–cell contact. Cell–cell contact mediates VAC exocytosis, which plays a role in establishing epithelial cell polarity (apicobasal polarity). Consideration of the role of the VAC enables us to understand the classification of epithelial cells (Honda 2010, 2012). As shown in Fig. 7.7c (left), when an isolated epithelial cell is surrounded by collagen gel, the VACs fuse with each other in the central region of cells to form a large vacuole (Type 0). As shown in Fig. 7.7c (middle), fusion of the vacuole with the outer cell membrane produces a concave apical domain (cup-type cell). Successive fusion of the concave portion with the outer cell membrane leads to penetration of the concave portion through the cell, producing tunnel-type or seamless-type cells (Fig. 7.7c, right). When epithelial cells are within an aggregate and the aggregate is surrounded by collagen gel, VACs near

Fig. 7.7 Behavior of the VAC and its contribution to epithelial morphogenesis. VAC, vacuolar apical compartment. (a) Schematic drawing for explanation of VAC behavior. A VAC adjacent to the region of cell–cell contact fuses with the cell membrane to become the apical membrane. VAC is designated with solid line. (b) Formation of a closed envelope structure from a cell aggregate (top) or a single cell (bottom) [reproduced from Fig. 3–46 in Honda 2010]. (c) Formation of type 1 (tunnel or seamless) cells (Honda 2012). (d) Formation of hepatic type cells (types 2, 3, . . .) within the ECM (Honda 2012). (e) Formation of a prism-like cell, Type 1, on the ECM surface (Honda 2012) [reproduced from Fig. 17 in Honda 2012]. [Reproduced from an excerpt of Fig. 7 legend in Honda 2017 with permission of John Wiley and Sons]

7.3 Crucial Roles of Vacuolar Apical Compartments (VACs) in Epithelization

123

areas of cell–cell contact fuse with the cell membrane; in other words, the apical membrane of VACs is supplied to the cell surface at sites of cell–cell contact (Fig. 7.7d). In this case, the epithelial cells become hepatic-type cells (types 2, 3, . . .). When aggregates of epithelial cells are on the surface of collagen gel (a cell aggregate has two sides, collagen-free sites and collagen-contact sites), as shown in Fig. 7.7e, VACs near areas of cell–cell contact fuse with the cell membrane, and the fused VAC membranes spread on the outer cell surface because the cell membrane is free from collagen gel. The fused VAC membranes ultimately become the large apical domain of a prism-like epithelial cell (Fig. 7.7e, rightmost). Notably, hepatictype cells are produced first, whereas typical prism-like cells are produced later. Indeed, prism-like and hepatic-type cells are interchangeable, and the conversion is controlled by PAR1 (Cohen et al. 2004). A transition from the depolarized state to apicobasal polarity through hepatic polarity has been reported in an experiment using a calcium switch model, in which calcium induces reassembly of apical junctions and F-actin reorganization (Ivanov et al. 2005). Indeed, to maintain stable hepatic cells, a complicated ECM-derived scaffold may be necessary that partitions a space into blood and bile spaces. Bioengineered liver was made using decellularized scaffolds (Higashi et al. 2022). In this section, it has been emphasized that an epithelial sheet has no choice but to make a closed envelope. This seems mysterious because the epithelial sheet behaves as if it has its own intention. However, when VACs are introduced into our model of morphogenesis, we can understand how an epithelial sheet can autonomously form a closed envelope (Honda 2010, 2012). As shown in Fig. 7.7b (top), in aggregates of epithelial cells surrounded by collagen gel, VACs migrate to sites of cell–cell contact, fuse with the cell membrane, and produce an apical space (lumen) in the cell aggregate. Following repeated cell divisions, such an aggregate containing a lumen becomes a closed envelope (cyst or follicle). Such autonomous formation of a closed envelope is considered to take place in the formation of blood islands (Chap. 3) and has actually been observed during capillary formation of endothelial cells and thyroid follicles (Folkman and Haudenschild 1980; Nitsch and Wollman 1980; Toda et al. 1993). As shown in Fig. 7.7b (bottom), a single cell in collagen gel also grew to form a closed envelope with a lumen via several stages: enlarged vacuole, cell divisions, and fusions of the cell membrane, forming a larger vacuole. It should be noted that the closed envelope of a cyst or follicle is surrounded by a basement membrane and contains no cells in its lumen. Cells within the lumen are eliminated by apoptosis (Debnath et al. 2002; Martin-Belmonte et al. 2008; Yamamoto et al. 2015). In contrast, as described in Chap. 6, the cells of the inner cell mass remain alive within the closed envelope of mammalian blastocysts. The blastocyst is surrounded by an apical surface, and inside is a basal space because the inner surface is lined with a basement membrane. Indeed, a previous study demonstrated the presence of a basement membrane at the surface of TE cells facing the blastocyst cavity (Klaffky et al. 2001). A lumen can arise de novo either between cells or within a single cell in a position where there was no space previously. The lumen formation was described in detail with related molecules (Sigurbjörnsdóttir et al. 2014). Anti-adhesive factors, such as

124

7 The World of Epithelial Sheets

the highly sialylated transmembrane protein podocalyxin or polysaccharides such as chitin, force the luminal membranes apart owing to negative charges at apical membrane initiation sites. Then, apical polarity complexes, namely, the PAR3– PAR6–aPKC complex and the Crumbs–PALS1–PATJ complex are established. In addition, along with a lateral complex, the Scribble–DLG–LGL complex are established. (Abbreviations: PAR3, partition defective 3, scaffold protein having PDZ domains; PAR6, partition defective 6, scaffold protein having PDZ domain; aPKC, atypical protein kinase C molecule; PALS1, proteins associated with Lin 7; PATJ, protein associated with tight junction; DLG, discs large; LGL, lethal giant larvae.) These are crucial to the formation of apical junctions such as tight junctions in mammalian epithelial cells (Assémat et al. 2008).

7.4

Evolutionary Merit of Enclosure of the Animal Body Within an Epithelial Cell Sheet

The epithelial sheet covering an animal body functions not only as a boundary to keep materials within the body and a barrier against harmful objects but also has other important functions. During developmental processes, genomes determine individual bodies through morphogenesis. An individual body and its genome correspond to each other. After random mutation of the genome, a body is selected among various mutated bodies on the basis of suitability for living. An individual selected through the body’s characteristic properties must correspond to its genome without ambiguity. Otherwise, genome selection does not work well. That is, the body is strictly prohibited from contaminants from other genomes. The epithelial sheet marks out a responsible area of genomes. The structure of an envelope covering an individual body by the epithelial sheet is suitable from the aspect of evolution.

7.5

Signaling Molecules for Formation of the Apicobasal Polarity of Epithelial Cells

A variety of epithelia have been described, and the formation of epithelial cell types and the formation of a closed envelope of epithelia via the function of a characteristic organelle (VAC) are understood. In addition, how the apical and basal surfaces are related to the behavior of VACs has been discussed. An explanation of these epithelial properties in terms of genes and molecules is outlined as follows. Apicobasal polarity (AB polarity) is crucial for epithelial cells and is profoundly related to formation of the apical and basal surfaces of epithelial sheets (Bryant and Mostov 2008). Here, epithelial cells (MDCK cells) are located on an ECM (Fig. 7.8). Laminin molecules in the ECM contact the cells. Rac1 (RAS-related GTP-binding

7.5 Signaling Molecules for Formation of the Apicobasal Polarity of. . .

125

Fig. 7.8 Signaling pathway to form AB (apicobasal) polarity. Contacts of cells with the ECM, including laminin and the surface of neighboring cells, lead to the formation of AB polarity. See text. Cdc42, Rho family GTP-binding protein; PAR3, PDZ domain-containing scaffold protein; PAR6, PDZ domain-containing scaffold protein; aPKC, atypical protein kinase C. Based on Suzuki and Ohno (2006) and Horikoshi et al. (2009). [Reproduced from Fig. 8A in Honda 2017 with permission of John Wiley and Sons]

protein) in cells controls extracellular laminin assembly, and laminin in turn activates Rac1, forming an autocrine pathway. These molecules interact with integrins via actin filaments and establish the basal side of epithelial cells (O’Brien et al. 2001; Yu et al. 2005). On the other hand, atypical protein kinase C molecule (aPKC) forms a complex with PAR6 (scaffold protein having a PDZ domain). When the aPKC– PAR6 complex is close to the membrane area of cell–cell contact, it interacts with PAR3 (scaffold protein having PDZ domains) (Horikoshi et al. 2009). PAR3 and aPKC–PAR6 activate Cdc42 through PTEN-PtdIns(4,5) P2–Annexin II pathway, resulting in rearrangement of cytoskeletons. (Abbreviations: PTEN, phosphatase and tensin homologue; PtdIns(4,5)P2, phosphatidylinositol 4,5-bisphosphate; Annexin II, a calcium-dependent phospholipid-binding protein.) PAR3 and aPKC-PAR6 cause the area of cell–cell contact to develop into an apical junctional complex. Here, the apical side is established. A pair of apical and basal sides forms AB polarity. VACs are produced in an integrin-dependent manner on the basal side (Davis and Camarillo 1996; Kamei et al. 2006). The VAC forms a complex with aPKC–PAR6 and moves to the apical side (Suzuki and Ohno 2006; Horikoshi et al. 2009). Thus, the ECM attached to cells defines the location of the basal region, and the area of cell–cell contact provides cues that promote development of the apical region. Recently, another polarity domain apical to TJs was defined: the Crumbs complex, composed of Pals1, PatJ, Lin7c, and Crumbs3 (Tan et al. 2020).

126

7.6

7 The World of Epithelial Sheets

Summary

The basic form of the multicellular animal body is a vesicle of the epithelial sheet. Various animal shapes are produced by various deformations of the vesicle. To understand the variety of deformation, we focused on the behaviors of epithelial sheets. An epithelial sheet functions as a partition that divides a space into two parts. This partition function requires that the epithelial sheet does not have a peripheral edge. Thus, it naturally follows that an epithelial sheet has no choice but to form a closed envelope. The epithelial sheet performs a few elemental processes: elongation, invagination, and protrusion of vesicles (Fig. 7.1). When two sheets contact each other, they reconnect (Fig. 7.1f). Morphogenesis in animal development is a combination of these elemental processes. For example, neural tube formation is a combination of invagination and reconnection (Fig. 7.2c), and gut formation is a combination of invagination and another type of reconnection (Fig. 7.2d). Usually, epithelial cells are defined as columnar cells having apical and basal surfaces. However, hepatocytes are classified as epithelial cells. Thus, we extended the concept of epithelial cells. Epithelial cells have one or more apical regions, and therefore, there are a few types of epithelial cells. The cell membrane in the apical region is supplied via vacuolar apical compartments (VACs) inside cells. VACs behave in various ways, remaining inside a cell or fusing with the outer membrane once or twice. Then, cells are classified into a few epithelial cell types. Type 0 is a cell having an apical region inside. Type 1 is a cell having one apical region on its surface. Type 2 is a cell having two apical regions on its surface (e.g., hepatocytes). By introducing VACs into our model of morphogenesis, we understand how an epithelial sheet can autonomously form a closed envelope (Fig. 7.7b), which is a practical way to maintain barrier function even under construction of vesicles. Such vesicle formation has been actually observed in blood vessel capillaries and thyroid follicles. Furthermore, we explained the molecular mechanism of epithelial cell formation. When an epithelial cell contacts the ECM, the contact region is activated via Rac1 and integrins, and the basal region of the cell is established. The region produces VACs. On the other hand, a region in contact with other cells is activated, and there, aPKC–PAR6 complex interacts with PAR3. The complex with PAR3 activates Cdc42, and VACs are fused with the outer membrane. Then, the apical region is established. Thus, the ECM attached to cells defines the location of the basal region, and the area of cell–cell contact provides cues that promote development of the apical region. Apicobasal (AB) polarity is established in epithelial cells. We also described the evolutionary merit of enclosure of the animal body in an epithelial cell sheet. The body is strictly prohibited from containing contaminants originating from other genomes. The epithelial sheet marks out a responsible area of the genome.

References

127

References Assémat, E., Bazellières, E., Pallesi-Pocachard, E., Le Bivic, A., Massey-Harroche, D.: Polarity complex proteins. Biochim. Biophys. Acta. 1778, 614–630 (2008) Bryant, D.M., Mostov, K.E.: From cells to organs: building polarized tissue. Nat. Rev. Mol. Cell Biol. 9, 887–901 (2008) Cohen, D., Brennwald, P.J., Rodriguez-Boulan, E., Musch, A.: Mammalian PAR-1 determines epithelial lumen polarity by organizing the microtubule cytoskeleton. J. Cell Biol. 164, 717–727 (2004) Dan, M.: “Doubutu no Keitou to Kotaihasseiny (Animal Phylogeny and Ontogeny)” (in Japanese). Tokyo Daigaku Shuppankai, Tokyo (1987) Dan-Sohkawa, M., Yamanaka, H., Watanabe, K.: Reconstruction of bipinnaria larvae from dissociated embryonic cells of the starfish, Asterina pectinifera. J. Embryol. Exp. Morphol. 94, 47–60 (1986) Davis, G.E., Camarillo, C.W.: An a2b1 integrin-dependent pinocytic mechanism involving intracellular vacuole formation and coalescence regulates capillary lumen and tube formation in three-dimensional collagen matrix. Exp. Cell Res. 224, 39–51 (1996) Debnath, J., Mills, K.R., Collins, N.L., Reginato, M.J., Muthuswamy, S.K., Brugge, J.S.: The role of apoptosis in creating and maintaining luminal space within normal and oncogene-expressing mammary acini. Cell. 111, 29–40 (2002) Folkman, J., Haudenschild, C.: Angiogenesis in vitro. Nature. 288, 551–556 (1980) Harris, T., Tepass, U.: Adherens junctions: from molecules to morphogenesis. Nat. Rev. Mol. Cell Biol. 11, 502–514 (2010) Higashi, H., Yagi, H., Kuroda, K., et al.: Transplantation of bioengineered liver capable of extended function in a preclinical liver failure model. Ame. J. Transplant. 2022(00), 1–14 (2022). https:// doi.org/10.1111/ajt.16928 Honda, H.: Shiito karano Karada Dukuri (in Japanese). Chuoukouron-sha, Tokyo (1991) Honda, H.: Katachi no Seibutugaku (in Japanese), pp. 1–365. NHK Shuppan, Tokyo (2010) Honda, H.: Essence of shape formation of animals. Forma. 27, S1–S8 (2012) Honda, H.: The world of epithelial sheets. Dev. Growth Differ. 59, 306–316 (2017) Horikoshi, Y., Suzuki, A., Yamanaka, T., Sasaki, K., Mizuno, K., Sawada, H., Yonemura, S., Ohno, S.: Interaction between PAR-3 and the aPKC-PAR-6 complex is indispensable for apical domain development of epithelial cells. J. Cell Sci. 122, 1595–1606 (2009) Hudspeth: Establishment of tight junctions between epithelial cells. Proc. Nat. Acad. Sci. USA. 72, 2711–2713 (1975) Imai, M., Furusawa, K., Mizutani, T., Kawabata, K., Haga, H.: Three-dimensional morphogenesis of MDCK cells induced by cellular contractile forces on a viscous substrate. Sci. Rep. 5, 14208 (2015) Ivanov, A.I., Hunt, D., Utech, M., Nusrat, A., Parkos, C.A.: Differential roles for actin polymerization and a myosin II motor in assembly of the epithelial apical junctional complex. Mol. Biol. Cell. 16, 2636–2650 (2005) Kakihara, K., Shinmyozu, K., Kato, K., Wada, H., Hayashi, S.: Conversion of plasma membrane topology during epithelial tube connection requires Arf-like 3 small GTPase in Drosophila. Mech. Dev. 125, 325–336 (2008) Kamei, M., Saunders, W.B., Bayless, K.J., Dye, L., Davis, G.E., Weinstein, B.M.: Endothelial tubes assemble from intracellular vacuoles in vivo. Nature. 442, 453–456 (2006) Klaffky, E., Williams, R., Yao, C.C., Ziober, B., Kramer, R., Sutherland, A.: Trophoblast-specific expression and function of the integrin alpha 7 subunit in the peri-implantation mouse embryo. Dev. Biol. 239, 161–175 (2001) Le Bivic, A., Hirn, M., Reggio, H.: HT-29 cells are an in vitro model for the generation of cell polarity in epithelia during embryonic differentiation. Proc. Natl. Acad. Sci. USA. 85, 136–140 (1988) Martin-Belmonte, F., Yu, W., Rodriguez-Fraticelli, A.E., Ewald, A.J., Werb, Z., Alonso, M.A., Mostov, K.: Cell-polarity dynamics controls the mechanism of lumen formation in epithelial morphogenesis. Curr. Biol. 18, 507–513 (2008)

128

7 The World of Epithelial Sheets

Miyagawa, J., Fujita, H., Matsuda, H.: Fine structural aspects of inverted follicles in cultured porcine thyroids. Archivum histologicum Japonicum (Nihon soshikigaku kiroku). 45, 385–392 (1982) Nishii, I., Ogihara, S.: Actomyosin contraction of the posterior hemisphere is required for inversion of the Volvox embryo. Development. 126, 2117–2127 (1999) Nitsch, L., Wollman, S.H.: Suspension culture of separated follicles consisting of differentiated thyroid epithelial cells. Proc. Natl. Acad. Sci. USA. 77, 472–476 (1980) O’Brien, L.E., Jou, T.S., Pollack, A.L., Zhang, Q., Hansen, S.H., Yurchenco, P., Mostov, K.E.: Rac1 orientates epithelial apical polarity through effects on basolateral laminin assembly. Nat. Cell Biol. 3, 831–838 (2001) Sato, T.J., van Es, H., Snippert, H.J., Stange, D.E., Vries, R.G., van den Born, M., Barker, N., Shroyer, N.F., van de Wetering, M., Clevers, H.: Paneth cells constitute the niche for Lgr5 stem cells in intestinal crypts. Nature. 469, 415–418 (2011) Schoenwolf, G.C., Bleyl, S.B., Brauer, P.R., Francis-West, P.H.: Larsen’s Human Embryology 4e. Elsevier Inc., Amsterdam (2009) Sigurbjörnsdóttir, S., Mathew, R., Leptin, M.: Molecular mechanisms of de novo lumen formation. Nat. Rev. Mol. Cell Biol. 15, 665–676 (2014) Suzuki, A., Ohno, S.: The PAR-aPKC system: lessons in polarity. J. Cell Sci. 119, 979–987 (2006) Tan, B., Yatim, S.J.M., Peng, S., Gunaratne, J., Hunziker, W., Ludwig, A.: The mammalian crumbs complex defines a distinct polarity domain apical of epithelial tight junctions. Curr. Biol. 30, 2791–2804.e6 (2020) Tanaka-Matakatsu, M., Uemura, T., Oda, H., Takeichi, M., Hayashi, S.: Cadherin-mediated cell adhesion and cell motility in Drosophila trachea regulated by the transcription factor Escargot. Development. 122, 3697–3705 (1996) Toda, S., Yonemitsu, N., Minami, Y., Sugihara, H.: Plural cells organize thyroid follicles through aggregation and linkage in collagen gel culture of porcine follicle cells. Endocrinology. 133, 914–920 (1993) Vega-Salas, D.E., Salas, P.J.I., Rodriguez-Boulan, E.: Modulation of the expression of an apical plasma membrane protein of Madin-Darby canine kidney epithelial cells: cell-cell interactions control the appearance of a novel intracellular storage compartment. J Cell Biol. 104, 1249–1259 (1987) Vega-Salas, D.E., Salas, P.J., Rodriguez-Boulan, E.: Exocytosis of vacuolar apical compartment (VAC): a cell-cell contact controlled mechanism for the establishment of the apical plasma membrane domain in epithelial cells. J. Cell Biol. 107, 1717–1728 (1988) Wang, A.Z., Ojakian, G.K., Nelson, W.J.: Steps in the morphogenesis of a polarized epithelium II. Disassembly and assembly of plasma membrane domains during reversal of epithelial cell polarity in multicellular epithelial (MDCK) cysts. J. Cell Sci. 95, 153–165 (1990) Wolff, J.R., Bar, T.: ‘Seamless’ endothelia in brain capillaries during development of the rat’s cerebral cortex. Brain Res. 41, 17–24 (1972) Yamamoto, H., Awada, C., Matsumoto, S., Kaneiwa, T., Sugimoto, T., Takao, T., Kikuchi, A.: Basolateral secretion of Wnt5a in polarized epithelial cells is required for apical lumen formation. J. Cell Sci. 128, 1051–1063 (2015) Yokouchi, M., et al.: Epidermal cell turnover across tight junctions based on Kelvin’s tetrakaidecahedron cell shape. Elife. 5, e19593 (2016) Yonemura, S.: Differential sensitivity of epithelial cells to extracellular matrix in polarity establishment. PLoS One. 9, e112922 (2014) Yu, W., Datta, A., Leroy, P., O’Brien, L.E., Mak, G., Jou, T.S., Matlin, K.S., Mostov, K.E., Zegers, M.M.: Beta1-integrin orients epithelial polarity via Rac1 and laminin. Mol. Biol. Cell. 16, 433–445 (2005) Yu, W., Shewan, A.M., Brakeman, P., Eastburn, D.J., Datta, A., Bryant, D.M., Fan, Q.W., Weiss, W.A., Zegers, M.M., Mostov, K.E.: Involvement of RhoA, ROCK I and myosin II in inverted orientation of epithelial polarity. EMBO Rep. 9, 923–929 (2008) Zihni, C., Mills, C., Matter, K., Balda, M.S.: Tight junctions: from simple barriers to multifunctional molecular gates. Nat. Rev. Mol. Cell Biol. 17, 564–580 (2016)

Chapter 8

Cells Themselves Produce Force for Active Remodeling

Outline The vertex model demonstrates that cells perform active remodeling of cell collectives to which they belong; that is, the cells themselves produce force for remodeling without other aids. Specifically, convergent extension (CE) of collective cells by contractile force actively elongates cell collectives perpendicular to the orientation of the contraction. Several cases of elongation of tissues have been studied in the present book. Elongation is one of the fundamental processes of morphogenesis. In Chap. 5, we described the zigzag cell pattern on the posterior margin of the Drosophila wing blade, where cells along the wing margin are elongated by external force. A developing wing consists of the wing blade (the distal part) and the wing hinge (the proximal part), and strong contraction of the wing hinge causes elongation of the wing blade (Aigouy et al. 2010; Ray et al. 2015; Diaz de la Loza and Thompson 2017). In Chap. 5, we studied the formation of a fine tracheal tube in Drosophila embryos. There are two types of tracheal tubes, multicellular and single cellular. A multicellular tube consists of more than one cell comprising the tube circumference, and a fine tube consists of a single cell comprising the tube circumference. We investigated the mechanism of transformation from a two-cellular tube to a fine tube using computer simulations. The tracheal tube is passively elongated (Ribeiro et al. 2004; Honda et al. 2009), where a growing body of the embryo pulls the trachea within the body. Another example of tissue elongation is angiogenesis mediated by vascular endothelial cells. A tip cell of the branch pulls trailing cells in the vascular branching system. The endothelial cells in the branch are elongated by the force of the tip cell (e.g., Takubo et al. 2019). In contrast to these passive elongations of tissues, other distinguishing phenomena of active elongation of tissues caused by autonomous activity of cells themselves without external forces exist.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_8

129

130

8.1

8 Cells Themselves Produce Force for Active Remodeling

Cell Rearrangement Involving Cell Intercalation

Living organisms construct themselves without any outside aids during morphogenesis. Tissues in a developing embryo sometimes enlarge remarkably. Some tissues are elongated without additional cell proliferation, during which cells rearrange. Keller et al. (1985) observed gastrulation of Xenopus embryos in detail and found that dorsal convergence of cells results in anteroposterior extension in the ectoderm. Such cell behaviors, simultaneous narrowing in width and longitudinal extension of collective cells, are referred to as convergent extension (CE) (Keller et al. 1985; Tada and Heisenberg 2012). Fristrom and Chihara (1978) observed evagination (turning inside out) of imaginal disks of Drosophila, during which cell rearrangement and CE occur. Active cell movements, in which the cells themselves have force, were distinguished from passive cell movement (Fristrom 1982). In addition, Hardin and Louis (1986) studied continued elongation of the gut during sea urchin gastrulation and suggested that forces generated within the gut itself can cause it to elongate. On the other hand, Irvine and Wieschaus (1994) observed a morphological change in the Drosophila germ band after the onset of gastrulation, and the length of the germ band along the anterior–posterior (A–P) axis increased over two-and-ahalf-fold, while its width along the dorsal–ventral (D–V) axis simultaneously narrowed. Cells intercalate between their dorsal and ventral neighbors during extension, increasing the number of cells along the A–P axis and decreasing the number of cells along the D–V axis. Intriguingly, an enrichment of nonmuscle myosin II at A–P cell borders was discovered in the extending germ band (Zallen and Wieschaus 2004; Bertet et al. 2004). Then the mechanism of CE could be understood as follows: an anisotropic contractile cell boundary formed by myosin II causes CE of tissues. Later, such anisotropic enrichment of myosin II was also found in other developmental processes: neural tube closure in the chick embryo (Nishimura et al. 2012), gastrulation in Xenopus (Shindo and Wallingford 2014), and embryonic heart tube development in chicks (Kidokoro et al. 2018). However, for understanding CE, the existence of an anisotropic contractile force mediated by myosin II is not selfevident. The mechanical process of CE induced by an anisotropic contractile force should be elucidated. We performed computer simulations to examine CE.

8.2

Convergent Extension (CE) Mediated by an Anisotropic Contractile Force in 3D Space10

Let us consider cell rearrangement involving edges with an anisotropic contractile force (Fig. 8.1a). When horizontal edges (solid line) have a strong contractile force, directional intercalation of cells takes place so that the cell assemblage elongates

10

Reproduced from excerpts in Honda et al. 2008 with permission of John Wiley and Sons.

8.2 Convergent Extension (CE) Mediated by an Anisotropic Contractile Force. . .

131

Fig. 8.1 Convergent extension (CE) via anisotropic edge contraction and explanation of the apical face in a vesicle. (a) Cell intercalation through remodeling of cell junctions. Two cells (light gray polygons) are intercalated by two neighboring cells (dark polygons). Cell boundaries whose orientations are close to the horizontal plane have strong contractile forces (indicated by thick solid lines). An aggregate of four cells is elongated vertically, perpendicular to the orientation of the contractile force. (b–d) A spherical shell comprising a monolayer cell sheet is partially represented by a wire flame. One of the cells is enlarged, and the apical surface is presented as a gray polygon (c). Among the edges of the apical polygon, an edge ( j) whose orientation is close to the horizontal plane has a strong contractile force (d, thick solid line)

perpendicularly to the horizontal direction. To investigate the process of cell intercalation leading to an elongated tissue, we used a 3D spherical shell, which is a monolayered cell assemblage containing a cavity (Fig. 8.1b). Cells are approximated by polyhedra, cell behaviors are presented by movement of vertices of the polyhedra, and vertex motion is described by the equation of motion involving potential energy (Chap. 6, 3D vertex model). First, we made an assumption of cell border contraction along a specific direction perpendicular to the elongation axis (parallel to the horizontal plane). This assumption is based on the observation that myosin II is specifically abundant along horizontal cell junctions during Drosophila germ-band extension (Zallen and Wieschaus 2004; Bertet et al. 2004).

132

8 Cells Themselves Produce Force for Active Remodeling

We modified the equation of motion for a vesicle (Eq. 4.1 with Eq. 6.7) into one that involved anisotropic contractile edges. The potential energy described in Eq. (6.7) was replaced by the following equations for potential energy,

U edge

U ¼ U S þ U EV þ U EI þ U edge ,
2
¼ κ edge Σβ Σ j w j E βj  Estd , j ¼ 1, 2, . . . , nβ ,

ð8:1Þ ð8:2Þ

where US, UEV, and UEI are given by Eqs. (6.8–6.10), respectively. The new term Uedge is an elasticity term with respect to the total junction length of each polygon on the apical surface of the spherical shell (Fig. 8.1c, gray polygon). Eβj is the length of edge j of polygon β facing the cavity, and Eβ ¼ Σ j wjEβj is the total length of the edges of polygon β. Estd is the total edge length of the relaxed polygon, and the parameter κ edge is the weight applied to the term. The weight wj in Eβ is a parameter describing the polarized mechanical contraction of cell edge j. When wj is large, edge Eβj becomes short. That is, when wj is large, edge j has contractive properties. When the direction of edge j of polygon β is parallel (within θC, where θC is a small angle) to the horizontal plane that is perpendicular to the elongation axis, we give wj a large value (Fig. 8.1d). Otherwise, wj ¼ 1. Then, the equation of motion shortens edges that are perpendicular to the elongation axis (e.g., arrowhead in Fig. 8.2 at t ¼ 0.05). Assuming that a cell edge parallel to the horizontal plane contracts strongly (Fig. 8.1d), we carried out computer simulations and found that the spherical cell aggregate elongates, as shown in Fig. 8.2 (t ¼ 0 ~ 50). A pair of left and right images at each time shows surface patterns of the cavity and the cell aggregate, respectively. The cell aggregate elongates, and the elongation ratio (length along the elongation axis/initial length) becomes large with time. Figure 8.3 shows cell behaviors during the elongation of the aggregate. The horizontal junction (arrowhead in Fig. 8.3a) became short at the initial stage, was then remodeled among neighboring junctions, and finally was rearranged into a vertical junction (arrowhead in Fig. 8.3a, right). Figure 8.3b shows that cells in an array of four gray cells on the spherical shell separated from each other successively through cell intercalation, where cell intercalations provide CE of cell collectives. The computer simulation showed that contraction of a few specific edges of cells (ca. one edge per cell), although only minor elements of a complex cell structure, could induce elongation of the entire cell aggregate. In addition, a hexagonal prism (Fig. 8.3c) was completely remodeled to a different hexagonal prism as if it rotated around the prism axis. A columnar cell has six lateral faces that are almost parallel to the prism axis. While the two hexagons of the initial prism had two horizontal edges (Fig. 8.3c, left, arrowhead), the two hexagons of the remodeled prism had two vertical edges (Fig. 8.3c, right, arrowhead). The simulation shown in Fig. 8.2 was based on the assumption that the cell surfaces facing the cavity were apical. In the inverse case wherein cell surfaces facing the external space were apical, we confirmed that a similar elongation of the cell aggregate occurred (data not shown).

8.3 CE Mediated by an Anisotropic Contractile Force on a Cylindrical Surface10

133

Fig. 8.2 Elongation of a 3D spherical shell via anisotropic edge contraction. Contraction of apical horizontal edges whose orientations are close to the horizontal plane using Eq. (8.2). The apical surface facing the cavity (left, thick line on a dark gray surface) and the whole view (right, light gray surface) of the cell aggregate are shown at various simulation times (t ¼ 0, 0.05, 0.5, 20, 30, and 50). At t ¼ 0 and 50, crosssectional views are inserted between left and right. κ edge ¼ 20, wj ¼ 3 when |θj|  θC (θC ¼ 10 ); otherwise ¼1, where θj is the angle of edge j to the horizontal plane perpendicular to the elongation axis (see Fig. 8.1d). [Reproduced from Fig. 2A and an excerpt of its legend in Honda et al. 2008 with permission of John Wiley and Sons]

8.3

CE Mediated by an Anisotropic Contractile Force on a Cylindrical Surface10

We demonstrated that contraction of specific cell boundaries is sufficient to achieve tissue elongation in the 3D spherical shell structure. In morphogenesis in the 3D cell model, the anisotropic contractile edges on the apical surface of the sphere seemed to

134

8 Cells Themselves Produce Force for Active Remodeling

Fig. 8.3 Details of the results of the computer simulation in Fig. 8.2. (a) Cell junction remodeling via anisotropic edge contraction. Contraction of an apical edge (left, arrowhead) perpendicular to the elongation axis of the cell aggregate, reconnection of neighboring vertices (middle), and elongation of the new edge (right, arrowhead) parallel to the elongation axis. Two cells connected by the contracting edge become closer to each other (left), elongated horizontally (middle), and thickened vertically (right). Simulation times are t ¼ 0, 1, and 10 (from left) of the simulation in Fig. 8.2. (b) Cell intercalation during elongation of a cell aggregate. Four vertically arranged cells (gray) are separated by intercalation of neighboring cells (white), and the region of the four cells is elongated as a whole. The simulation times are t ¼ 0, 5, 20, 30, and 100 (from left) of the simulation in Fig. 8.2. (c) A hexagonal prism in which two hexagons with horizontal edges (left, arrowhead) are remodeled to form hexagons with vertical edges (right, arrowhead). Simulation times are t ¼ 0, 10, and 75 (from left) of the simulation in Fig. 8.2. [Reproduced from Fig. 3D, C, E and excerpts of the legends in Honda et al. 2008 with permission of John Wiley and Sons]

be essential for CE. For convenience of calculation, we constructed a new cell model in which a sheet consisting of polyhedral cells is replaced by a sheet consisting of polygons without thickness. A sheet without thickness is undulating in 3D space. Furthermore, we found that cells around the top and bottom of the 3D cell aggregate interfered with axial elongation of the spherical shell, leading to limitations of the 3D cell model simulation. To more quantitatively investigate elongation of tissues without perturbations from the top and bottom extremities, we constructed a cylinder from a 2D cell model by rolling a rectangular cyclic boundary to wrap around a lumen and joining the right and left ends, which was already described in Fig. 5.5 (the method of rectangle–cylinder conversion). Strictly, this model describes a cell monolayer torus with the major radius much longer than the minor radius. We found some cell patterns in which cell boundaries are contractile and produce force against the frame of cell patterns. Cells not only rearrange themselves within

8.3 CE Mediated by an Anisotropic Contractile Force on a Cylindrical Surface10

135

Fig. 8.4 Computer-simulated elongation of cylindrical cell aggregates via anisotropic edge contraction. The initial state is a random Voronoi pattern of 96 cells. A computer simulation was performed via anisotropic edge contraction (using Eq. 8.2). Cylinders in the bottom figure are unfolded into rectangles of size bx  by for presentation. Snapshots are shown at simulation times of 0, 5, 10, 70, 90, and 120. Simulation times do not include the time of global dynamics. The elongation ratio of the final cylinder (rightmost) is 2.355, and the corresponding total of time intervals τB in the global dynamics is 0.859. Some cells are gray or light gray to demonstrate cell intercalations. δ ¼ 0.15. The weight of edge k, σ k is expressed by a formula involving the logistic function 1 + 2/(1 + exp[a(|θ|–θC)]), where θ is the angle of edge k with the horizontal plane perpendicular to the elongation axis: a ¼ 0.666, θC ¼ 10 , and the maximum σ k is 3.0. To see this figure in color, go online (Honda et al. 2008). [Reproduced from Fig. 4A and an excerpt of its legend in Honda et al. 2008 with permission of John Wiley and Sons]

polygonal patterns but also remodel the outer frame of polygonal pattern in which they are enclosed. Using polygonal patterns on the cylindrical surface, we carried out computer simulations of elongation of the cylinder (height, by, and circumference, bx). Since we assumed the rectangle by  bx is deformable, the vertices have an effect on the shape of the rectangle. The vertices move fast in this system, while the cylinder shape changes sufficiently slowly. Therefore, we divided the calculation step into two substeps to obtain the final cylinder shape: the dynamics of the vertex maintained a constant cylinder shape (vertex dynamics), and the dynamics of the cylinder maintained a topology of vertex arrangement (global dynamics), as described in the following section. We examined remodeling of the cylinder in two ways. The first is under the condition of anisotropic contractile force along cell boundaries. The other method, under the condition of anisotropic cell stiffness, will be described later. The initial pattern consisted of 96 cells arranged at random in 2D space (Fig. 8.4 t ¼ 0). We

136

8 Cells Themselves Produce Force for Active Remodeling

used potential energy, in which horizontal cell edges are assumed to contract strongly. The vertices of the polygons were rearranged, and the cylinder shape changed so that the potential energy became small. The cylindrical aggregates of cells elongated gradually, as shown in Fig. 8.4 (bottom). Cells in a longitudinal array (light and dark gray) in the initial pattern were separated individually through cell intercalation, and some cells derived from the two different arrays became close. The final elongation ratio of the cylinder (elongation ratio, namely, length along the elongation axis/initial length) depended on the initial conditions. We performed 11 computer simulation trials with initial patterns of different random numbers. The results were 1.977  0.2483 (mean and standard deviation; with a minimum and maximum of 1.731 and 2.482, respectively). Furthermore, we examined the remodeling of cylinders with an initial pattern consisting of 96 regular hexagons arranged regularly in 2D space (Fig. 8.5). The orientation of the edges of the hexagons was restricted to three directions: 0 , 60 , and 120 from the horizontal axis (x-axis). Two edges of each hexagon were horizontal. As the arrangement of the hexagons progressed, the height of the system increased until it reached a definite elongation ratio. In the final cell pattern (Fig. 8.5 t ¼ 4), all hexagons had been rotated by 90 and had no horizontal cell edges. The final elongation ratio was 1.731, which is close to the result determined by analytic calculation, √3. Actual cell patterns usually have some disorder. They contain not only hexagons but also pentagons and heptagons, among other shapes. Moreover, the cell edges close to the horizontal axis disappear one after another. Elongation of the cylinder continued for a while after the elongation ratio reached √3 (¼1.73). Our simulations showed a 1.977-fold increase on average and a 2.5-fold increase at the most. These results correspond to the following experimental results. In a study of Drosophila embryos, polarized cell movements were observed to cause germ-band extension, leading to an increase in length of more than twofold, in which cell proliferation was not involved (Zallen and Wieschaus 2004).

8.4

Global Dynamics of a Cylindrical Surface10

The system was assumed to proceed under the two dynamics, vertex dynamics and the global dynamics, to use the adiabatic approximation (Chap. 6). The vertex dynamics followed Eq. (4.1). The global dynamics were dynamics of the shape of the whole system. Here, we will explain global dynamics. Not only the vertices of polygons but also the shape of the rectangular boundary [bx, by] changes so that the potential U becomes small. That is, the vertices move fast while the whole system changes its shape slowly. We use the equation of motion (Eq. 4.1) with the 2D positional vectors ri(xi, yi) and the 2D vertex reconnection (Nagai and Honda 2001; Honda et al. 1982; Honda 1983). First, we calculate vertex coordinates using Eq. (4.1), with an initial state {ri (0)} under a constant boundary condition (bx and by are fixed), and obtain a developed state {ri(0)(tv)} at time tv where the potential

8.4 Global Dynamics of a Cylindrical Surface10

137

Fig. 8.5 Computer simulation where the initial state is a regular Voronoi pattern. The initial Voronoi pattern consists of 96 cells. The computer simulation was performed via anisotropic edge contraction (using Eq. 8.2). Snapshots are shown at simulation times 0, 0.5, 0.5+ (after τB ¼ 0.389 in the global dynamics), 1, 2, and 4. The elongation ratio of the final cylinder (right) is 1.731, and the corresponding total of time intervals τB in the global dynamics is 0.549. Some cells are light gray or gray to demonstrate cell intercalations. The parameters used are the same as those in the simulation discussed in Fig. 8.4. [Reproduced from Fig. 4B in Honda et al. 2008 with permission of John Wiley and Sons]

U has become small. Then, we perform the global dynamics defined using the following transformation: xi ð1Þ ðt v Þ ¼ ð1=λÞxi ð0Þ ðt v Þ,

yi ð1Þ ðt v Þ ¼ λyi ð0Þ ðt v Þ:

ð8:3Þ

When λ > 1, the x-direction is reduced by 1/λ and the y-direction is expanded by λ. Then, the system size becomes [bx/λ, λby]. The transformation Eq. (8.3) keeps

138

8 Cells Themselves Produce Force for Active Remodeling

each cell area and the system area constant. We repeat this discrete transformation for n steps and obtain xi ðnÞ ðt v Þ ¼ ð1=λÞn xi ð0Þ ðt v Þ,

yi ðnÞ ðt v Þ ¼ λn yi ð0Þ ðt v Þ:

ð8:4Þ

Using λ ¼ 1+ ε with a positive infinitesimal ε, we move from the discrete transformation to a continuous transformation as follows: λn ¼ exp ðn lnλÞ  eðnεÞ ¼ eτ ,

ð8:5Þ

where τ ¼ nε denotes dimensionless time in the global dynamics. Equation (8.4) is then expressed as xi ðt v , τÞ ¼ eτ xi ðt v , 0Þ,

yi ðt v , τÞ ¼ eτ yi ðt v , 0Þ,

ð8:6Þ

where xi(tv, τ) ¼ xi(n) (tv) and yi(tv, τ) ¼ yi(n) (tv). Equation (8.6) gives equations of motion for the global dynamics. The local minimum of the potential U is obtained by ∂=∂τ U ðfri ðt v , τÞgÞ ¼ 0:

ð8:7Þ

We vary the shape of the rectangular boundary [bx, by] for a time interval τB according to Eq. (8.6) so that the potential U decreases. Repetition of the two dynamics leads to lower U states step-by-step and produces a cylinder with the minimum U.

8.5

CE Mediated by Anisotropic Cell Stiffness on a Cylindrical Surface10

Next, instead of contractile force, we consider a stiff rod within cells that pushes against the cell side. When the rod has anisotropic properties, the cell shape may become anisotropic. We assumed a new potential energy term,
2 U rod ¼ κrod Σβ W β  W std ,

ð8:8Þ

where Urod is an elasticity term related to the horizontal width of polygons restricted by the rod whose orientation is horizontal inside polygons. Wβ is the horizontal width of polygon β, Wstd is the length of the horizontal rod in polygons, and the parameter κ rod is the weight applied to the term. We examined remodeling of the cylinder under the anisotropic rods, with the initial pattern consisting of 96 cells arranged at random in 2D space (Figs. 8.6 and 8.7). We used potential energy, Urod, in which the horizontal cell width is restricted

8.5 CE Mediated by Anisotropic Cell Stiffness on a Cylindrical Surface10

139

Fig. 8.6 Elongation of cylindrical cell aggregates by anisotropic rods in cells. The initial state is a random Voronoi pattern of 96 cells, as shown in Fig. 8.4. A computer simulation was performed under the assumption of anisotropic rods (Eq. 8.8). Snapshots of rectangles are shown at simulation times 0, 0.5, 0.5+ (after τB ¼ 0.105 in the global dynamics), 3, 6, 10, 12, and 25. The elongation ratio of the final cylinder (rightmost) is 7.880, and the corresponding total of time intervals τB in the global dynamics is 2.064. Some cells are light gray or gray to demonstrate cell intercalations. κ W ¼ 1, Wstd ¼ 1.5, and δ ¼ 0.04. To see this figure in color, go online (Honda et al. 2008). [Reproduced from Fig. 5B in Honda et al. 2008 with permission of John Wiley and Sons]

by the horizontal rods (Eq. 8.8). The vertices of the polygons were rearranged, and the cylinder shape varied so that the potential energy became small. As a result, the cylindrical aggregates of cells elongated gradually, as shown in Fig. 8.7. According to the cell dynamics and the variation in cylinder shape, the cylinder elongated, as shown in Figs. 8.6 and 8.7. Cells in longitudinal arrays were intercalated with neighboring cells. The cylinder elongated smoothly to become long and narrow, decreasing the number of cells comprising the circumference of the cylinder during its elongation (see the insertion in Fig. 8.7). The elongation ratio was 7.880.

140

8 Cells Themselves Produce Force for Active Remodeling

Fig. 8.7 The same computer simulation as shown in Fig. 8.6 is presented by cylinders. Inset: The partially enlarged snapshot at simulation time 25 (rightmost) showing the narrow cylinder. Two cells comprise the cylinder circumference. The small rectangle on the cylinder (rightmost) indicates the position of the inset. To see this figure in color, go online (Honda et al. 2008). [Reproduced from Fig. 5B in Honda et al. 2008 with permission of John Wiley and Sons]

In conclusion, it has been confirmed that cells with anisotropic contractile cell boundaries lead to CE of a multicellular system, which causes elongation along the axis of the system. Cells with anisotropic stiff rods also lead to CE and elongation of the system. These confirmations were performed through various computer simulations of spheres and cylinders using 2D and 3D vertex dynamics (Honda et al. 2008). The mathematical cell model succeeded in integrating anisotropic force into a multicellular system to form characteristic shapes.

8.6 Planar Cell Polarity Signaling Links CE Orientations of Tissues

8.6

141

Planar Cell Polarity Signaling Links CE Orientations of Tissues11

We have elucidated that anisotropic contractile force produces CE, which leads to axis elongation of animal tissues. However, what controls the anisotropy? Here, we will describe a well-established case of anisotropic morphogenesis induced by planar cell polarity (PCP) signaling. Neural tube development successively proceeds with the formation of the neural plate, bending of the plate, and closure of the bending plate to form a tube. The process includes cellular intercalation, CE of the neural plate, planar cell polarity (PCP) signaling, and actomyosin-dependent contraction of neuroepithelial layers. Most epithelial cells, in addition to their ubiquitous apical–basal polarity, are polarized within the plane of the epithelium, which is called PCP. Tamako Nishimura and Masatoshi Takeichi have investigated and integrated these mechanisms into a coherent story to explain neural tube morphogenesis (Nishimura et al. 2012). The neural plate bends most drastically along its midline. To observe the behavior of cells along the midline, the neural tubes in stage 8 of chick embryos were opened. Figure 8.8a shows a pattern of the bending neural plate immunostained for myosin light chain localized in the adherens junctions, which are located near the apical edge of neuroepithelial cell–cell contacts. Figure 8.8b is a drawing of cell shapes based on the photograph shown in Fig. 8.8a. In contrast to typical epithelial cells, which show a honeycomb-like arrangement, neural plate cells display peculiar shapes at their apical plane and have both shorter and longer borders (Fig. 8.8b). The shorter borders of each cell tend to link together linearly across multiple cells. Phosphorylated myosin light chain (pMLC) is an activated form of myosin II. Staining of pMLC was highly enriched at the shorter borders (Fig. 8.8a) and extended across multiple cells, resulting in a cable-like configuration. Individual cells displayed pMLC-rich and pMLC-poor borders in various patterns. Some of these cells showed a bipolar distribution of pMLC; in these cells, the pMLC-rich borders were generally shorter than the pMLC-poor borders, resulting in a rectangular shape of the cells. Cells with this morphology were arranged in a ladderlike pattern in which the pMLC-poor zones corresponded to the “steps” of the ladder (see the boxed region in Fig. 8.8a for examples). In some cases, two ladders were aligned next to each other, sharing a single pMLC cable. Statistical analysis of the distribution of the pMLC cables showed that they tended to orient in a mediolateral direction (bar graphs in Fig. 8.8c). These observations suggest that the polarized shrinking and extension of adherens junctions (AJs) were responsible for the intercalation-like rearrangement of cells.

11 Reproduced from excerpts on pages 1085–1087 and 1089 in Nishimura et al. 2012 with permission of Elsevier.

142

8 Cells Themselves Produce Force for Active Remodeling

Fig. 8.8 Cells along the midline of a neural tube of chick embryos at stage 8 (see also Figs. 7.2c and 8.10b). (a, b) The neural plate in a flat-mounted stage 8 embryo was stained for phosphorylated myosin light chain (pMLC) and zonula occludens protein-1 (ZO-1) and photographed from the apical side. The merged image of stained pMLC and ZO-1 is shown and traced. The anterior side of the neural plate is oriented toward the top, and the median hinge point is located along the vertical center of the photographs. A portion showing their typical distributions is boxed. Scale bar, 10 μm. (c) The top histogram shows the angular distribution of stained pMLC, quantified as follows: pMLC cables that linearly extended more than 10 μm across multiple cells in a 50 μm square region (20–30 cables per specimen) were selected, and the angles of the cables were measured relative to the A–P axis. The histogram shows the angular distribution determined from four specimens. The lower histogram shows the percentages of the mediolateral (M–L, those angled from 60 to 120 to the A– P axis) and anteroposterior (A–P, those angled from 0 to 30 and from 150 to 180 to the A–P axis) lines of pMLC. Data are the average of four specimens. Error bars, SD. ***p < 0.001 against the M–L-oriented pMLC. To see this figure in color, go online (Nishimura et al. 2012). [Reproduced from Fig. 1B and an excerpt of its legend in Nishimura et al. 2012 with permission of Elsevier]

8.7

Mathematical Modeling of Supracellular Actomyosin Cable Formation11

We wondered how AJ-associated actomyosin organized into linear supracellular cables. We suspected that the mediolateral contraction of AJs might be responsible for this patterning. To verify this idea, we made use of mathematical modeling (Eq. 4.1). In this model, a polygonal pattern, where polygons are packed in a 2D space, imitates the epithelial sheets. We used vertex dynamics and global dynamics alternately, as mentioned in the previous section. In global dynamics, the rectangular boundary was remodeled. We introduced anisotropic contraction of the edges into the vertex dynamics. The potential is used, U ¼ U L þ U ES , where

ð8:9Þ

8.8 Observation of PCP Signaling Proteins in the Neural Plate11

U L ¼ Σk σ k Lk :

143

ð8:10Þ

Lk is the length of edge k, and σ k is the edge energy density. σ k is for anisotropic contraction. When σ k is large under a certain condition, Lk becomes short and vice versa; that is, a small σ k leads to a large Lk. σ k expresses the degree of the strength of the contractile force. We used σ k ¼ 10 when edge k was inclined to 15 from the mediolateral axis (0 ); otherwise, σ k ¼ 1. The anisotropic contraction of edge length causes rearrangement of edges and cell intercalation as shown in Fig. 8.9 (inset). Figure 8.9 (simulation step 0) shows a polygonal pattern with uniform contraction of its edges. We applied stronger contractile forces only to the edges inclined to 15 from the mediolateral axis and collected snapshots of the changing patterns at various steps (0 to 180). By step 120, a certain group of polygons organized into mediolateral arrays, where several edges linked with each other to form a straight chain, with a direction biased toward the mediolateral axis. At step 180, the polygons composing these arrays shortened along the mediolateral axis, whereas they expanded along the anterior–posterior axis, assuming a ladderlike arrangement. In the ladderlike pattern, trijunctions of the edge showed a “T” pattern instead of a “Y” pattern. By analyzing detailed changes in a particular group of polygons, we noted that the polygon pattern changes involve their intercalation as well as rosette-like rearrangement (indicated by ellipse area in Fig. 8.9). The resultant overall pattern of the polygons was similar to the cellular pattern observed at the apical plane of the neural plate (Fig. 8.8b). These results support the idea that contraction of individual AJs along mediolateral directions autonomously reorganizes the AJs into supracellular lines.

8.8

Observation of PCP Signaling Proteins in the Neural Plate11

The pattern of pMLC staining seen in the neural plate (Fig. 8.8a) suggested that PCP signaling may be involved. To examine which mechanism determines the mediolateral activation of actomyosin, Tamako Nishimura and Masatoshi Takeichi examined a series of PCP signaling proteins (Nishimura et al. 2012). Their observations are summarized as follows (Fig. 8.10). Celsr1 is a vertebrate homolog of Drosophila flamingo (a core PCP participant). Immunohistochemical analysis of Celsr1 showed that it is localized at the apical side of the neural tube. This protein is known to be required for neural tube closure and other PCP-related events (Curtin et al. 2003; Copp et al. 2003; Devenport and Fuchs 2008; Ravni et al. 2009). Celsr1 colocalizes with condensed F-actin or pMLC (Nishimura et al. 2012). Celsr1 was typically located at the shorter edges of cells. This polarized distribution of Celsr1 was not seen at stage 5 but began at stage 6 and was correlated with the onset of the polarized arrangement of pMLC.

144

8 Cells Themselves Produce Force for Active Remodeling

Fig. 8.9 Mathematical modeling of cell junctional rearrangement. Polygonal cellular patterns are shown under uniform (step 0) and anisotropic (steps 20 to 180) contraction of the edges. Several distributed polygons were organized into ladder patterns. Ellipse areas indicate rosette-like rearrangements. Inset, detailed changes in four polygons, marked with circles, are shown. Two polygons labeled with asterisks initially contact one another with a horizontal edge, but this edge shortens at steps 6 and 16. This edge is then replaced with a new contact formed between the other pair of polygons, and this edge expands vertically at steps 40 and 120. To see this figure in color, go online (Nishimura et al. 2012). [Reproduced from Fig. 2B and an excerpt of its legend in Nishimura et al. 2012 with permission of Elsevier]

As Celsr1 is expected to cooperate with other core participants in PCP, one of them, Dishevelled, was chosen to analyze its potential involvement in polarized actomyosin activation. Dishevelled deficiencies are known to cause neural tube

8.8 Observation of PCP Signaling Proteins in the Neural Plate11

145

Fig. 8.10 Signaling pathway from anisotropic localization of Celsr1 (the core PCP member) to anisotropic contraction of actomyosin in neural tube formation. (a) Apical part of the neural cell. Initially, the actomyosin contraction pathway does not exhibit distinct polarities. The PCP pathway regulates the polarity of the actomyosin contraction pathway. Fz, Frizzled; Dvl, Dishevelled. Arrows indicate the activation or modulation of a molecule or promotion of molecular interactions. Dotted arrows, hypothetical pathways. RhoA activated by PDZ-RhoGEF might lead to not only ROCK activation but also DAAM1 activation, forming a positive feedback loop. To see this figure in color, go online (Nishimura et al. 2012). (b) 3D view of four neural cells. [Reproduced from Fig. 7 in Nishimura et al. 2012 with permission of Elsevier]

defects (Etheridge et al. 2008; Hamblet et al. 2002; Wang et al. 2006). We found that Dishevelled-2, one of multiple isoforms, is localized at the apical portion of the neural tube in chicken embryos, exhibiting a mediolateral linear orientation (Nishimura et al. 2012). These results suggest that Celsr1 may cooperate with Dishevelled. There are likely mediators that link Celsr1/Dishevelled-dependent PCP signaling and ROCK-dependent myosin II activation. Dishevelled is known to regulate Rho (Habas et al. 2001; Strutt et al. 1997; Winter et al. 2001). To identify such mediators, we screened Rho guanine nucleotide exchange factors (RhoGEFs), which can activate ROCKs via RhoA activation. Among several candidates, we found that PDZ-RhoGEF transcripts were highly expressed in the neural tube and that PDZ-RhoGEF proteins were localized at the apical side of neural plates (Nishimura et al. 2012). On the apical surface of chicken embryonic neural plates, PDZ-RhoGEF was concentrated along pMLC cables (Nishimura et al. 2012). These results suggest that PDZ-RhoGEF is involved in polarized actomyosin contraction in the neural plate. DAAM1 is a formin that plays a role in F-actin polymerization (Lu et al. 2007). DAAM1 is distributed in the apical portion of the neural plate and localizes at cell– cell borders without exhibiting distinct polarities. Usually, DAAM1 is autoinhibited,

146

8 Cells Themselves Produce Force for Active Remodeling

but when it interacts with Dishevelled, the inhibition is blocked (Liu et al. 2008). DAAM1 located along the mediolateral axis is actuated by Dishevelled and can form a complex with PDZ-RhoGEF. DAAM1 can enhance the GEF activity of PDZ-RhoGEF, which activates ROCKs via RhoA activation. Then myosin light chains are phosphorylated to produce pMLC cables. The positioning of PCP signaling proteins in a cell sheet defined the orientation of contractile cables. Then CE took place in the cell sheet, and the sheet actively underwent large-scale remodeling to form a neural tube.

8.9

Invagination of Epithelial Sheets

When an epithelial sheet encloses an inner space, a vesicle forms. The sheet consists of columnar cells, which are packed without gaps or overlaps and have two surfaces (apical and basal). The apical surface of the sheet shows a polygonal pattern, and beneath the apical surface, each cell contains a bundle of contractile fibers (actomyosin filaments), which are anchored to the plasma membrane at the lateral periphery of the columnar cell. Contraction of the bundles shortens the apical circumference of the cell and reduces the apical surface area. Such deformation of columnar cells has been referred to as “drawing the purse string” (Baker and Schroeder 1967). Conversion of flat epithelial sheets into 3D organs is one of the fundamental processes that provides dynamic morphogenesis during the development of multicellular animals. Cells themselves produce force for active remodeling. We sought to construct a mathematical cell model to describe the conversion of sheets, and that involves some identified biological molecules. A mathematical cell model of invagination may act as a bridge between the functions of biological molecules and morphogenesis, as the model of neural tube formation did in the previous section. Odell et al. (1981) studied the mechanical aspects of invagination of epithelial sheets (e.g., the initial gastrulation movements of echinoderms, neural tube formation in urodele amphibians, and ventral furrow formation in Drosophila). Based on observation of apical circumferential microfilament (actomyosin) bundles in epithelial cells, they constructed viscoelastic elements for cell models and analyzed the cell shape using nonlinear dynamics. If the stress of the apical circumference is increased beyond a certain threshold, an active contraction is initiated and suddenly reduces the apical circumference of the cell to a shorter length. Contraction in one cell can stretch the apical circumferences of neighboring cells and, if the threshold is exceeded, cause the neighboring cells to suddenly contract. In other words, the cell sheet performed invagination. Mayuko Nishimura, Yoshiko Inoue, and Shigeo Hayashi examined the initiation of tracheal formation in Drosophila embryos (Nishimura et al. 2007; Kondo and Hayashi 2013). On the embryo surface, a placode forms, which is a platelike thickening of the epithelial layer from which the trachea develops. Before invagination of the tracheal placode on the embryo surface, cells form short rows that encircle the future invagination site (Fig. 8.11a). Myosin accumulates along the

8.9 Invagination of Epithelial Sheets

147

Fig. 8.11 Explanation of computer simulation of epithelial invagination. (a) The ring zone grows from a center in the tracheal placode (designated by a white dot). Among edges on the ring zone, edges whose orientation is perpendicular to the radial direction contract strongly (indicated by a thick line). Then edges rearrange to form arcs of edges with strong contractile force. Inset, tangential edges perpendicular to the radial direction are strongly contractile. (b) A central cell (gray) migrates toward the inside of the embryo. (c) Direction of cell migration. The center of gravity of the columnar cell (Go) migrates to G. A, Center of the apical surface of the columnar cell

boundary of the cell rows, and the cells intercalate with each other. Then, the central cells in the placode form a shallow pit. Extracellular signal-regulated kinase (ERK) mitogen-activated protein (MAP) kinase is activated in an outward circular wave from epidermal growth factor receptor (EGFR) signaling. The pit is depressed, and invagination takes place (Nishimura et al. 2007). Based on the observation of the tracheal placode, we made a cell model of vertex dynamics (Honda et al. 2010) that involved the potential U (Eqs. 4.3 and 4.4). UL in the potential U was replaced by
2 U L ¼ λ Σβ Σk σ k Lβk  Lo :

ð8:11Þ

UL was the elastic energy of the polygonal perimeter of the apical face of epithelial cells, which corresponds to the closed strings of stretched rubber in the toy model (Fig. 4.1). We expect 3D remodeling of “drawing the purse string” in 3D vertex model. A similar term was used for other purpose in 2D vertex model in Farhadifar et al. (2007). We assumed a central position of invagination (white dot in Fig. 8.11a, left), and the placode was constructed from a ring zone growing from the center (Fig. 8.11a, middle). Initially, all edges of polygons on the apical surface were evenly contracted (σ k ¼ 1). However, we assumed (1) that the edges of polygons in the ring zone were especially strongly contracted. When the orientation of the edges

148

8 Cells Themselves Produce Force for Active Remodeling

Fig. 8.12 Snapshots of the 3D view of a computer simulation of the invagination process; t ¼ 0, 200, and 800. [Original]

was close to tangential, σ k ¼ 1.3 (inset in Fig. 8.11a). We also assumed (2) that the initial circumferential edge length of the polygons was Lo ¼ 2 in Eq. (8.11) but that the polygons outside the ring became larger, and their edge length remained large (Lo ¼ 3). According to assumptions (1) and (2), the cells in the placode formed chains of strongly contracted edges that encircled the future invagination site (Fig. 8.11a, right). However, the pit at the central cell was shallow (gray polygon in Fig. 8.11b, left). We assumed (3) migration of the central cell; that is, the gravity center of the cell migrated toward the inside of the embryo (Fig. 8.11c). Thus, a new term UM was added to the potential U (Eq. 8.12). U M ¼ wM Σγ ðGGo  dc Þ2 ,

ð8:12Þ

where GGo is the migration distance of the gravity center. Go and G are the gravity centers before and after migration, respectively. G is on an extended line of the line AGo from Go in Fig. 8.11c, where A is the center of the apical polygon of the cell. dc in Eq. (8.12) is a constant distance expressing the degree of migration. Then the pit was expected to be pressed down (Fig. 8.11b, right). The results of the computer simulation are presented in Fig. 8.12. The invagination process is shown in a vertical sectional view and in a surface view (Fig. 8.13). The central cell (designated by *) was encircled by strong contractile edges and invaginated (wM ¼ 30). The figure in the bottom row also shows the case of wM ¼ 0: no migration of the central cell occurred, and invagination did not take place. The strong contractile edges were still active at t ¼ 800 and showed a “drawing purse string” in the central cell (arrowhead in Fig. 8.13). The polygonal pattern on the apical surface shows that cells in the central area are strongly condensed so that the peripheral cells are elongated toward the center. However, it seems to be difficult for the epithelial sheet consisting of tall columnar cells to perform invagination, that is, 3D large-scale remodeling. Initially, we made one assumption: contraction of the apical circumferential junction of the columnar epithelial cells. In our computer simulation, such contraction was not sufficient for invagination. In addition, we assumed migration of the central cell toward the inside of the embryo, as if the central region of the embryo is a target of cell migration. The mechanism of cell migration is not clear at present, although the FGF pathway is known to be related to tracheal cell migration and the formation of branching patterns (Sutherland et al. 1996). Previous studies have reported that invaginations involve other factors. Actual observation of the

8.9 Invagination of Epithelial Sheets

149

Fig. 8.13 Results of computer simulations of invagination with and without migration of the central cell toward the inside of the embryo. Top row, sectional and horizontal views of the results of computer simulation with migration of the central cell (wM ¼ 30). *, central cell; solid line, strongly contracting edges. Bottom row, sectional and horizontal views of the results of computer simulation without migration of the central cell (wM ¼ 0). Arrowhead, cell with strong contraction of the apical edges. [Original]

Drosophila trachea revealed that mitosis plays an active role in epithelial invagination (Kondo and Hayashi 2013). One report stated that mesodermal forces are crucial for invagination during ventral furrow formation in the Drosophila embryo (Conte et al. 2012). On the other hand, the mechanical processes of neural tube formation were investigated using the 3D vertex dynamics model (Inoue et al. 2016): the neural plate in Xenopus is stratified, and the nonneural cells in the deep layer pull the overlying superficial cells, eventually bringing the two layers of cells to the midline. Then the neural plate is invaginated and forms the neural tube. In other paper using 3D vertex dynamics, the buckling of the epithelial cell sheet caused by cell proliferation was indicated to cause invagination (Inoue et al. 2017). In the formation of the ventral furrow in the early Drosophila embryo, Harmand et al. (2021) succeeded in making a minimal physical model for the shape of epithelial cells in 3D, introducing curvature of substances on which cells have an apical line tension and differential surface energies. Together, the invagination of epithelial sheets appears

150

8 Cells Themselves Produce Force for Active Remodeling

to be a complex event in which species-specific factors function through various mechanisms.

8.10

Summary

Collective cells in some developing tissues converge along an axis and extend perpendicularly to the axis, which is called convergent extension (CE). Cells among the cell collective intercalate between neighboring cells and are simultaneously intercalated by other neighboring cells. Some cells seem to behave in this manner without outside controls. The phenomenon was successfully recapitulated using vertex dynamics under a simple assumption that specific oriented cell boundaries contract strongly (anisotropic contraction). In the computer simulations, each simulation step was divided into two substeps, for calculation of vertex dynamics and for calculation of global dynamics, where the adiabatic approximation was used. To address the next problem of what determines the direction of the anisotropic angle, observation of the formation of the chick neural tube revealed that Celsr1/ Dishevelled-dependent PCP signaling determined the anisotropic orientation. Notably, one of the pathways mediating the genes to shape determination was established. Signal transduction molecules, which are delicate biochemical substances, can link large-scale morphogenesis involving strong mechanical forces. Finally, we investigated the invagination of Drosophila trachea cells.

References Aigouy, B., Farhadifar, R., Staple, D.B., Sagner, A., Roper, J.C., Julicher, F., Eaton, S.: Cell flow reorients the axis of planar polarity in the wing epithelium of Drosophila. Cell. 142, 773–786 (2010) Baker, P.C., Schroeder, T.E.: Cytoplasmic filaments and morphogenetic movement in the amphibian neural tube. Dev. Biol. 15, 432–450 (1967) Bertet, C., Sulak, L., Lecuit, T.: Myosin-dependent junction remodelling controls planar cell intercalation and axis elongation. Nature. 429, 667–671 (2004) Conte, V., Ulrich, F., Baum, B., Munoz, J., Veldhuis, J., Brodland, W., Miodownik, M.: A biomechanical analysis of ventral furrow formation in the Drosophila melanogaster embryo. PLoS One. 7, e34473 (2012) Copp, A.J., Greene, N.D.E., Murdoch, J.N.: The genetic basis of mammalian neurulation. Nat. Rev. Genet. 4, 784–793 (2003) Curtin, J.A., Quint, E., Tsipouri, V., Arkell, R.M., Cattanach, B., Copp, A.J., Henderson, D.J., Spurr, N., Stanier, P., Fisher, E.M., Nolan, P.M., Steel, K.P., Brown, S.D.M., Gray, I.C., Murdoch, J.N.: Mutation of Celsr1 disrupts planar polarity of inner ear hair cells and causes severe neural tube defects in the mouse. Curr. Biol. 13, 1129–1133 (2003) Devenport, D., Fuchs, E.: Planar polarization in embryonic epidermis orchestrates global asymmetric morphogenesis of hair follicles. Nat. Cell Biol. 10, 1257–1268 (2008) Diaz de la Loza, M.C., Thompson, B.J.: Forces shaping the Drosophila wing. Mech. Dev. 144, 23–32 (2017)

References

151

Etheridge, S.L., Ray, S., Li, S., Hamblet, N.S., Lijam, N., Tsang, M., Greer, J., Kardos, N., Wang, J., Sussman, D.J., Chen, P., Wynshaw-Boris, A.: Murine Dishevelled 3 functions in redundant pathways with Dishevelled 1 and 2 in normal cardiac outflow tract, cochlea, and neural tube development. PLoS Genet. 4, e1000259 (2008) Farhadifar, R., Roper, J.C., Aigouy, B., Eaton, S., Julicher, F.: The influence of cell mechanics, cellcell interactions, and proliferation on epithelial packing. Curr. Biol. 17, 2095–2104 (2007) Fristrom, D.K.: Septate junctions in imaginal disks of Drosophila: a model for the redistribution of septa during cell rearrangement. J. Cell Biol. 94, 77–87 (1982) Fristrom, D., Chihara, C.: The mechanism of evagination of imaginal discs of Drosophila melanogaster. Dev. Biol. 66, 564–570 (1978) Habas, R., Kato, Y., He, X.: Wnt/frizzled activation of rho regulates vertebrate gastrulation and requires a novel Formin homology protein Daam1. Cell. 107, 843–854 (2001) Hamblet, N.S., Lijam, N., Ruiz-Lozano, P., Wang, J., Yang, Y., Luo, Z., Mei, L., Chien, K.R., Sussman, D.J., Wynshaw-Boris, A.: Dishevelled 2 is essential for cardiac outflow tract development, somite segmentation and neural tube closure. Development. 129, 5827–5838 (2002) Hardin, J.D., Louis, Y.C.: The mechanisms and mechanics of archenteron elongation during sea urchin gastrulation. Dev. Biol. 115, 490–501 (1986) Harmand, N., Huang, A., Hénon, S.: 3D shape of epithelial cells on curved substrates. Phys. Rev. X. 11, 031028 (2021). https://doi.org/10.1103/PhysRevX.11.031028 Honda, H.: Geometrical models for cells in tissues. Int. Rev. Cytol. 81, 191–248 (1983) Honda, H., Ogita, Y., Higuchi, S., Kani, K.: Cell movements in a living mammalian tissue: longterm observation of individual cells in wounded corneal endothelia of cats. J. Morphol. 174, 25–39 (1982) Honda, H., Nagai, T., Tanemura, M.: Two different mechanisms of planar cell intercalation leading to tissue elongation. Dev. Dyn. 237, 1826–1836 (2008) Honda, H., Nagai, T., Wada, H., Kato, K., Hayashi, S.: Formation of an epithelial tube of single cell-size circumference. Seibutsu Butsuri 2009. In: The 47th Annual Meeting of the Biophysical Society of Japan S62 (2009) Honda, H., Nishimura, M., Kondo, T., Hayashi, S.: Mechanical cell properties causing epithelial invagination. Seibutsu Butsuri 2010. In: The 48th Annual Meeting of the Biophysical Society of Japan S108 (2010) Inoue, Y., Suzuki, M., Watanabe, T., Yasue, N., Tateo, I., Adachi, T., Ueno, N.: Mechanical roles of apical constriction, cell elongation, and cell migration during neural tube formation in Xenopus. Biomech. Model Mechanobiol. 15, 1733–1746 (2016) Inoue, Y., Watanabe, T., Okuda, S., Adachi, T.: Mechanical role of the spatial patterns of contractile cells in invagination of growing epithelial tissue. Dev. Growth Differ. 59, 444–454 (2017) Irvine, K.D., Wieschaus, E.: Cell intercalation during Drosophila germband extension and its regulation by pair-rule segmentation genes. Development. 120, 827–841 (1994) Keller, R.E., Danilchik, M., GimlichI, R., Shih, J.: The function and mechanism of convergent extension during gastrulation of Xenopus laevis. Embryol. Exp. Morph. 89(Suppl), 185–209 (1985) Kidokoro, H., Yonei-Tamura, S., Tamura, K., Schoenwolf, G.C., Saijoh, Y.: The heart tube forms and elongates through dynamic cell rearrangement coordinated with foregut extension. Development. 145, dev152488 (2018) Kondo, T., Hayashi, S.: Mitotic cell rounding accelerates epithelial invagination. Nature. 494, 125–129 (2013) Liu, W., Sato, A., Khadka, D., Bharti, R., Diaz, H., Runnels, L.W., Habas, R.: Mechanism of activation of the Formin protein Daam1. Proc. Natl. Acad. Sci. 105, 210–215 (2008) Lu, J., Meng, W., Poy, F., Maiti, S., Goode, B.L., Eck, M.J.: Structure of the FH2 domain of Daam1: implications for formin regulation of actin assembly. J. Mol. Biol. 369, 1258–1269 (2007) Nagai, T., Honda, H.: A dynamic cell model for the formation of epithelial tissues. Philos. Mag. Part B. 81, 699–719 (2001)

152

8 Cells Themselves Produce Force for Active Remodeling

Nishimura, M., Inoue, Y., Hayashi, S.: A wave of EGFR signaling determines cell alignment and intercalation in the Drosophila tracheal placode. Development. 134, 4273–4282 (2007) Nishimura, T., Honda, H., Takeichi, M.: Planar cell polarity links axes of spatial dynamics in neural-tube closure. Cell. 149, 1084–1097 (2012) Odell, G.M., Oster, G., Alberch, P., Burnside, B.: The mechanical basis of morphogenesis I. Epithelial folding and invagination. Dev. Biol. 85, 446–462 (1981) Ravni, A., Qu, Y., Goffinet, A.M., Tissir, F.: Planar cell polarity cadherin Celsr1 regulates skin hair patterning in the mouse. J. Investig. Dermatol. 129, 2507–2509 (2009) Ray, R.P., Matamoro-Vidal, A., Ribeiro, P.S., Tapon, N., Houle, D., Salazar-Ciudad, I., Thompson, B.J.: Patterned anchorage to the apical extracellular matrix defines tissue shape in the developing appendages of drosophila. Dev. Cell. 34, 310–322 (2015) Ribeiro, C., Neumann, M., Affolter, M.: Genetic control of cell intercalation during tracheal morphogenesis in drosophila. Curr. Biol. 14, 2197–2207 (2004) Shindo, A., Wallingford, J.B.: PCP and septins compartmentalize cortical actomyosin to direct collective cell movement. Science. 343, 649–652 (2014) Strutt, D.I., Weber, U., Mlodzik, M.: The role of RhoA in tissue polarity and Frizzled signalling. Nature. 387, 292–295 (1997) Sutherland, D., Samakovlis, C., Krasnow, M.A.: Branchless encodes a drosophila FGF homolog that controls tracheal cell migration and the pattern of branching. Cell. 87, 1091–1101 (1996) Tada, M., Heisenberg, C.P.: Convergent extension: using collective cell migration and cell intercalation to shape embryos. Development. 139, 3897–3904 (2012) Takubo, N., Yura, F., Naemura, K., Yoshida, R., Tokunaga, T., Tokihiro, T., Kurihara, H.: Cohesive and anisotropic vascular endothelial cell motility driving angiogenic morphogenesis. Scientific Rep. 9, 9340 (2019) Wang, J., Hamblet, N.S., Mark, S., Dickinson, M.E., Brinkman, B.C., Segil, N., Fraser, S.E., Chen, P., Wallingford, J.B., Wynshaw-Boris, A.: Dishevelled genes mediate a conserved mammalian PCP pathway to regulate convergent extension during neurulation. Development. 133, 1767–1778 (2006) Winter, C.G., Wang, B., Ballew, A., Royou, A., Karess, R., Axelrod, J.D., Luo, L.: Drosophila Rho-associated kinase (Drok) links Frizzled mediated planar cell polarity signaling to the actin cytoskeleton. Cell. 105, 81–91 (2001) Zallen, J.A., Wieschaus, E.: Patterned gene expression directs bipolar planar polarity in Drosophila. Dev. Cell. 6, 343–355 (2004)

Chapter 9

Expansion of Shape–Dimension

Outline There are phenomena in which objects are suddenly remodeled into complicated structures under restricted conditions. A flat plate twists and a straight rod bends and loops, which are expansions of shape–dimension, 2D to 3D and 1D to 3D, respectively. These shape–dimension expansions can be recapitulated using vertex dynamics. In this chapter, twisting of wisteria seedpods and left-handed helical looping of embryonic heart tubes are described. Let us bear an elongating rod in mind, a thin straight rod that is fixed at the top and bottom terminals. When the rod grows, it suddenly bends crosswise, which is referred to as buckling. The initial rod is a one-dimensional object, but it becomes a two- or three-dimensional object by buckling. A sheet consisting of cells also expands its dimension under restricted conditions, similar to the elongating rod. These processes are self-construction of structures. Rods and sheets twist themselves, and the orientation of twisting is determined without outside aid. When such a phenomenon occurs in the development of biological bodies, it is called a symmetry break of the body. This is an epoch-making event in a developmental process, and mathematical cell models may contribute much to understanding such an event.

9.1

Twisting of a Strip of Sheet: A Narrow Rectangular Sheet (Honda et al. 2019)

When soybean or wisteria seeds become ripe, their seedpods suddenly burst, scattering beans. An initial seedpod is flat, but it twists; that is, a shape transition takes place from 2D to 3D. We examined deformation of the seedpods of wisteria in detail. When pieces of seedpods after burst are placed in a dish in a humid atmosphere, the moist seedpods become flat. However, they begin to twist during desiccation, as shown in Fig. 9.1. We measured the periphery length (L ) and the area (S) enclosed by the periphery of a seedpod and obtained a mathematical relationship of S ~ L3.8 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_9

153

154

9

Expansion of Shape–Dimension

Fig. 9.1 Twisting of a piece of wisteria seedpod during desiccation. D, L1 and L2, and S: the total length, peripheral lengths and area of a piece of seedpod, respectively. L ¼ L1 + L2 [Original]

Fig. 9.2 Relationship between the periphery length and the area of a seedpod. L, periphery length; S, area enclosed by periphery of a piece of wisteria seedpod. The relationship was S ~ L3.8 during desiccation. Broken line, control (S ~ L2). Abscissa, logarithmic expression of peripheral length, L. Ordinate, logarithmic expression of seedpod area, S [Original]

(Fig. 9.2). The manner of change deviated from the proportional change (S ~ L2). We quantitatively compared the twisted seedpod with the flat seedpod. The area enclosed by the periphery decreased by 84.1%, whereas the periphery length decreased by only 95.2%. The result is reminiscent of the surface of a hyperbolic paraboloid in 3D differential geometry. As shown in Fig. 9.3, the hyperbolic paraboloid within a restricted area (
1.2 < x < 1.2 and
0.2 < y < 0.2) shows a twisting surface, which is bounded by long and short straight boundaries, whose area is considerably small in comparison to the initial flat rectangle (decrease ratio, 0.760). Under the condition of a fixed length of the seedpod periphery, the seedpod that is reducing its area seems to have no choice but to twist. We will perform computer simulations to investigate twisting of a flat rectangular strip under the condition of a reduction in the area enclosed by the rectangle while maintaining its periphery length. A rectangle consists of polygons packed in it without gaps or overlaps. Topologically, it is a part of the closed surface of the

9.1 Twisting of a Strip of Sheet: A Narrow Rectangular Sheet (Honda et al. 2019)

155

Fig. 9.3 A hyperbolic paraboloid. A hyperbolic paraboloid f(x,y) ¼ x y, where
1.2 < x < 1.2 and
0.2 < y < 0.2. The surface area is 1.1616, where the surface area of the rectangle of the long and short edges (2.446  0.6248) is 1.529 (1.1616/1.5293 ¼ 0.760). The surface area of the twisted surface was obtained according to Gray (1993) [Original]

polyhedron in 3D space (Fig. 9.4 inset). A vertex dynamics was used for 2D sheets in 3D space (Trichas et al. 2012; Osterfield et al. 2013; Du et al. 2014). We used the equation of motion (Eq. 4.1). The potential is U ¼ wS U S þ we U e þ wb U b þ wd U d þ wc U c ,

ð9:1Þ

where U S ¼ Σκ ðsκ
so Þ2 ,


2 U b ¼ Σ j b j
Bo =n j ,

Ue ¼ Σ j j e j j ,

 Ud ¼ Σ j 2 θ j= b j þ b


 2 jþ1

,

and

Bo ¼ Σ j b j ,


2 U c ¼ Σ j cos θ j :

ð9:2Þ

US is the summation of the elastic term of each polygon in the sheet, where sκ and so are the polygon area and its relaxed state, respectively. Ue is the summation of edge lengths ej. Ub represents borders of the rectangle frame. bj is the length of edge j in the border of the rectangle, where the border consists of nj edges. Bo and B1 are the length of its border. Ud is the elastic term of the deflection of edges along the frame, where θj is the angle between edge j and edge j + 1. Uc is the elastic term of deflection from the right angle at corner j ( j ¼ 1 ~ 4).

156

9

Expansion of Shape–Dimension

Fig. 9.4 Explanation of the initial condition. Inset: the rectangle consisting of polygons is one face of a polyhedron consisting of five faces in 3D space. (a, b) Figures for explanation of Eqs. (9.1) and (9.2). US represents the elastic term for each polygon. so is the relaxed state of the polygon area. Ue represents the edge length. Ub represents the borders of the rectangle frame. Bo and B1 are the lengths of the border. Ud represents the deflection of the edge along the frame, where the angle is the deflection between neighboring edges. Uc represents deflection from a right angle at the corners

A flat rectangular strip (1  4) was used as an initial condition in computer simulations (Fig. 9.4). The total length of the four borders of the rectangle remains constant. The rectangle consists of 102 polygons, which were packed in it without gaps or overlaps. We investigated the shape change of the rectangle using computer simulations. When we assumed that the total surface area of the rectangle was forced to decrease while the frame of the rectangle was kept constant in length, the flat rectangle had to twist, as already expected. The process of twisting appears to be spontaneous and self-activating. The rectangle seems to twist with bias in handedness, as shown in Fig. 9.5d. We examined what factor determines the outcome of handedness, left- or righthanded. We found that an artificial manipulation that made the rectangle slightly concave or convex was effective in defining the handedness, as shown in Fig. 9.5. A convex rectangle (Fig. 9.5b) has a ridge that is parallel to the long axis of the rectangle, similar to a gable roof (the method of artificial perturbation to rectangles is explained in Fig. 9.6). When we used zH ¼ +0.001 or
0.001, the rectangles became convex or concave, respectively. Computer simulations showed that when the initial rectangle was convex (Fig. 9.5b, zH ¼ +0. 001), the rectangle twisted in a left-handed direction (Fig. 9.5d). When the initial rectangle was concave (Fig. 9.5c, zH ¼
0. 001), the rectangle twisted in a right-handed direction (Fig. 9.5e). The concave rectangle had a groove. Changes in unevenness induced by lateral shift and rotational shift of the groove also altered the handedness. When a groove of the concave rectangle (Fig. 9.7f) was shifted rightward (Fig. 9.7h, yS ¼
0.01), the rectangle became right-handed (Fig. 9.7d). Conversely, when a groove of the

9.1 Twisting of a Strip of Sheet: A Narrow Rectangular Sheet (Honda et al. 2019)

157

Fig. 9.5 Computer simulation of a narrow rectangle with a decrease in the rectangle area. (a~c) Initial conditions: flat, convex and concave rectangles, respectively. (d, e) Results of computer simulations showing left-handed and right-handed helixes, respectively. Left to right, 0, 30,000, 50,000, and 100,000 steps. (f, g) Perpendicular views of the long axis of the initial rectangle at 100,000 steps. The surface area (summation of the total polygon area) decreased from 4.037 to 3.562 (decrease ratio ¼ 0.806) in E [Original] Fig. 9.6 Method of perturbation of the structure of the initial rectangle. Left to right: A flat rectangle is subjected to slight convex/ concave (zH) deformation, and the ridge or groove of the rectangle is slightly shifted (yS) leftward/ rightward and slightly rotated (yR) CCW/CW

concave rectangle (Fig. 9.7f) was shifted leftward (Fig. 9.7g, yS ¼ +0.01), the rectangle became left-handed (Fig. 9.7a). Furthermore, when the groove (Fig. 9.7f) was rotated counterclockwise/clockwise (CCW/CW) (Fig. 9.7b, c, yR ¼ +0.01/ yR ¼
0.01), the rectangle became right-handed/left-handed, respectively (Fig. 9.7a/d). In addition, when the shifted groove shown in Fig. 9.7h was rotated

158

9

Expansion of Shape–Dimension

Fig. 9.7 Computer simulations of a flat rectangle with subtle unevenness. (a, d) Results of computer simulations. (b, c) CCW/CW rotation of groove. (e, f) convex and concave perturbation of the rectangle. Each rectangle has a ridge and a groove. (g, h) Lateral shifts of the groove. (i, j) CCW/CW rotation of the groove with a rightward lateral shift [Reproduced from Fig. 2 in Honda et al. 2019]

CCW/CW (Fig. 9.7 i/j, yR ¼ +0.02/
0.02), the rectangle became right-handed/lefthanded, respectively (Fig. 9.7 a/d). Overall, the handedness of plane twisting was controlled by asymmetric shifts and rotations of unevenness in the rectangle. It should be noted that the asymmetric shifts and rotations used in Fig. 9.7 were extremely subtle biases (0.001 ~ 0.02, where the width of the narrow rectangle was 1). Based on these results, the contributions of deformation of the initial rectangle to the outcome of left-handed or right-handed rectangles are summarized. Figure 9.8 shows a case of the deformation of the concave rectangle. The contribution of the left/right shift of the groove of the rectangle (abscissa) and CW/CCW rotation of the groove of the rectangle (ordinate) is plotted two-dimensionally. Perturbation of the rightward shift and CCW rotation of the concave rectangle leads to a righthanded twist, while perturbation of the leftward shift and CW rotation leads to a lefthanded twist. Wisteria seedpods being dried are well known to burst suddenly and scatter beans (e.g., Terada et al. 1933; Ohmiya 1998, 2009; Honda 2012). A seedpod is separated

9.1 Twisting of a Strip of Sheet: A Narrow Rectangular Sheet (Honda et al. 2019)

159

Fig. 9.8 Phase diagram of determination of handedness of the concave rectangle. Determination of left- or right-handedness of the twist of the rectangle depends on the leftward/rightward shift of the groove of the rectangle (abscissa) and CCW/CW rotation of the groove (ordinate). Perturbation of the rightward shift and CCW rotation of the concave rectangle leads to a right-handed twist, while perturbation of the leftward shift and CW rotation leads to a left-handed twist [Original]

into left and right halves. Here, we define the “left” and “right” half seedpods as follows: a seedpod hangs down from a Wisteria twig; then, its distal end is downward, as shown in Fig. 9.9a. A pair of half seedpods was placed so that we could view the outer surface of the proximal part of the seedpod, as shown in Fig. 9.9a. When the suture of a half seedpod is on the right border, we refer to the half seedpod as the “right half seedpod.” When the suture of a half seedpod is on the left border (the midrib line is on the right border), we refer to the half seedpod as the “left half seedpod.” We can say that the right half of the seedpod formed a right-handed twist and the left half of the seedpod formed a left-handed twist. The results of the computer simulations described in Fig. 9.8 are presented in Fig. 9.9b and correspond to the orientation determined by observation (Fig. 9.9a).

160

9

Expansion of Shape–Dimension

Fig. 9.9 Distinction between left and right halves. (a) Distinction between left and right halves of wisteria seedpods. (b) The initial rectangles and results of computer simulations performed based on the respective initial conditions. The orientations of these two results correspond to that of the whole seedpod [Reproduced from Fig. 3 in Honda et al. 2019]

9.2

Twisting of the Heart Tube12

We have described twisting of a strip of sheet as buckling. Another example of buckling is in animals. It is known to take place in the development of the embryonic heart. The helical looping of the embryonic heart tube is a remarkable event that occurs during mammalian and avian development. This cardiac looping was first identified in mammals and birds a century ago (Schulte 1916; Murray 1919; Patten 1922), and since that time, many scientists have investigated the underlying structure and morphogenesis (Stalsberg 1970; Manner 2000; Harvey 2002; Moorman et al. 2003; Bartman and Hove 2005; Ramsdell 2005). Patten (1922) proposed that looping results from a buckling mechanism in a tube elongating between fixed poles. Later, heart looping was observed to form through a combination of ventral bending and rightward rotation (Manner 2000). To understand the mechanism of heart looping from a simple straight tube, regional differences in the tube were expected. Regional differences in cell proliferation have been found to contribute to helical looping in mice (de Boer et al. 2012). In addition, automatic looping or bending of heart rudiments in the absence of external factors was reported (Manning and McLachlan 1990; Flynn et al. 1991).

12 Reproduced from excerpts on pages 746, 748 left, 748 right and 749 in Honda et al. 2020 with permission of Elsevier.

9.2 Twisting of the Heart Tube

161

Fig. 9.10 Scheme showing formation of a helical structure. (a) A straight chain is shown. (b) A chain with ventral bending (dorsal–ventral asymmetry) is shown. (c) A chain with anterior rightward displacement is shown (left–right and anterior–posterior asymmetries), which is displaced rightward (arrow), but the displacement is confined to the anterior region. (d) The shape is a left-handed helix structure around the axis (vertical white line) [Reproduced from Fig. 1A in Honda et al. 2020 with permission of Elsevier]

These experiments suggested to us that studies on helical loop formation through computer simulations would be beneficial to elucidate cellular behavior in regard to cell proliferation.

9.2.1

Helical Looping in Computer Simulations

We intended to generate a helical loop in computer simulations. A helical loop is a chiral structure. In general, a chiral structure is defined in 3D geometry as a structure that cannot be superimposed on its mirror image and is produced by the combination of three axial asymmetries. In this study, we attempted to understand helical looping of the heart using three axial asymmetries. We propose a simple scheme for the formation of the helical structure of the heart tube, as shown in Fig. 9.10. A straight tube that is shown as a chain of balls (Fig. 9.10a) undergoes ventral bending (dorsal– ventral asymmetry), as shown in Fig. 9.10b. Next, anterior rightward rotational displacement occurs, as shown in Fig. 9.10c, where the bent chain is displaced rightward (left–right asymmetry), but the displacement is confined to the anterior portion of the chain (anterior–posterior asymmetry). As a result, a helical structure

162

9

Expansion of Shape–Dimension

Fig. 9.11 Computer simulations for the dorsal–ventral asymmetry of the model tube are shown. The initial model tube at t ¼ 0 (a) is shown. The inset shows cell division along its polarity. Polarity and cell division lines are shown by arrows and white lines, respectively. The model tube with random cell division at t ¼ 30 (b) and with longitudinal cell division at t ¼ 30 (c) is shown. The polygons painted over with gray or black: faint gray, cells of the ventral half of the initial model tube; gray, cells that are divided during simulation; solid black, the ventral-most cells at the initial stage. The bottom shows the view from the top. A, anterior direction of the initial model tube; R, right direction of the initial model tube; V, ventral direction of the initial model tube. To see this figure in color, go online (Fig. 1D–F in Honda et al., 2020) [Reproduced from Fig. 1D–F and excerpts of the legend in Honda et al. 2020 with permission of Elsevier]

was obtained (Fig. 9.10d). We will now describe our computational investigations according to this scheme. As an initial approximation of the primitive heart, we established a simple straight model tube (Fig. 9.11a) covered with 452 polygonal cells without gaps or overlaps. Cells were assumed to be polygons without thickness. We applied cell divisions and displacement of the model tube and sought conditions to generate a helical structure. For computer simulations of morphogenesis, we used cell-based 3D vertex dynamics for sheets, which predict the mathematically stable shape of a sheet in 3D space under given conditions. We assumed that cells in the aggregate obey the equation of motion (Eq. 4.1). Here, the potential U contains terms of boundary energy between cells (UL), elastic energies (UES and UEV), and restriction energy for a smooth surface (UF) (Honda et al. 2020):

9.2 Twisting of the Heart Tube

163

U ¼ U L þ U ES þ U EV þ U F :

ð9:3Þ

The potential UL denotes the total edge potential energy of the cells: U L ¼ σ L Σ L ,

ð9:4Þ

where i and j are neighboring vertices forming an edge , and L is the length of edge . σ L is the edge energy density. The potential UES denotes the total elastic energy of the polygon area: U ES ¼ κS Σn α ðSα
So Þ2 ,

ð9:5Þ

where Sα and So are the polygon area at time t and the polygon area at the relaxed state, respectively. κS is the elastic constant of the polygon area. n is the cell number, which increases during the process when cell divisions take place. UES works so that the area of each polygon becomes the area in the relaxed state. The potential UEV denotes the elastic energy of the tube volume: U EV ¼ κV ½V ðt Þ
V o 2 ,

ð9:6Þ

where κV is the elastic constant of the tube volume. V(t) and Vo are the tube volume at time t and in the relaxed state, respectively. The tube volume Vo was initially a constant at every calculation step but was later assumed to be forced to increase in proportion to the cell number. The change in Vo is more moderate than the motion of the vertices. Calculations at every step were divided into two substeps. The adiabatic approximation between the two substeps was used as explained in Adiabatic approximation in Chap. 6.10. During calculation according to the equation of motion (Eq. 4.1), we used the elementary step of vertex reconnection in 3D space (Fig. 9.12a). The polygons of the anterior and posterior ends were fixed during computer simulations. The stages we considered were before the looping tube begins to raise its inflow region during mouse heart development.
2 U F ¼ κ F Σ j r j
rG :

ð9:7Þ

The potential UF denotes the elastic deviation energy of vertices from the planes, where vertex j is connected to three vertices by edges and the three vertices form a triangle (Fig. 9.12c). rG is the center of the triangle, rG ¼ (rj1+ rj2+ rj3)/3, where j1, j2, j3 are neighboring vertices of j. Summations of j are performed over all vertices except for those that belong to the top or bottom polygon. κF is the elastic constant of deviation of a vertex from a flat plane. UF in the equation of motion acts to ensure that the vertices are arranged as flat as possible. The term is necessary in the 3D vertex dynamics of sheets because the polygons in the model do not have thickness. Without the term, neighboring polygons abnormally fold with each other.

164

9

Expansion of Shape–Dimension

Fig. 9.12 Explanation of methods using computer simulations. (a) Reconnection of vertices is shown. During reconnection of vertices, neighboring polygons (identified by *) were separated through intercalation of other neighboring polygons [Reproduced from Fig. 2C and an excerpt of its legend in Honda et al. 2020 with permission of Elsevier]. (b) Method to update cell polarity at every calculation step. Initially, cell polarity is defined so that the cell polarity is parallel to the main axis of the initial straight tube. Cell polarities rotate slightly according to twisting and deformation of the tube. The amount of the rotation of polarity is assumed to be determined by the movements of surrounding cells. Centers of cells surrounding cell A (inset) are projected on the plane of cell A (top figure). Lines from the center of cell A to the centers of projected surrounding cells rotate around cell A during the simulation step (small arrows in bottom figure). The average rotation is calculated, and the polarity of cell A is rotated by the averaged rotation, as shown in the bottom figure (long dark arrow) [Reproduced from Fig. 2D and an excerpt of its legend in Honda et al. 2020 with permission of Elsevier]. (c) The deviation of a vertex from its triangular plane, which is expressed by the length between j and G, is shown. A vertex j has three edges. The three edge terminals form a triangular plane. G is the center of the triangle [Reproduced from Fig. 2B and an excerpt of its legend in Honda et al. 2020 with permission of Elsevier]. (d) Displacement of the model tube. Rotation around the model tube axis [Reproduced from Fig. 2D and an excerpt of its legend in Honda et al. 2020 with permission of Elsevier]

We examined the contributions of the three axial asymmetries in the process of helical structure formation. First, we addressed the formation of the dorsal–ventral asymmetry of the straight model tube. The straight heart tube of mouse embryos was reported to form specific sites that are at the outer curvature of the looped heart (Christoffels et al. 2000). Mouse embryos at E8–E8.26 display rapid proliferation in the ventral heart tube, forming the outer curvature (de Boer et al. 2012). These findings suggest a relationship between ventral outer curvature formation and cell proliferation. However, the details of cell behaviors are not well understood at single-cell resolution. Thus, we performed EdU pulse labeling with mouse embryos

9.2 Twisting of the Heart Tube

165

in the early stage of heart looping to identify cells in the S phase as proliferating cells, where EdU is a DNA synthesis monitoring probe. After EdU exposure, the embryos were sliced, and images of the middle section of the primitive ventricle in each heart were acquired. We found more EdU-positive cells in the ventral region than in the dorsal region. As a result, the ratio of EdU-positive cells among the counted cells was approximately five times greater in the ventral region than in the dorsal region (Fig. 9.13a), suggesting that cell proliferation occurred predominantly on the ventral side. Thus, we performed a computer simulation with an assumption that the cells on the ventral side were in the proliferating state. In the computer simulation, the cells were set to divide according to the direction of polarity (Fig. 9.11a inset). When the direction of cell polarity was assumed to be random, the ventral region of the model tube became fat (Fig. 9.11b). Furthermore, to explore the appropriate simulation conditions to bend the model tube, we assumed that the direction of cell polarity was parallel to the longitudinal line of the embryonic heart (the cell division plane was perpendicular to its polarity). As a result, the simulated tube was bent, as shown in Fig. 9.11c. The ventral side of the model tube became the outer curvature (convex surface) of the bent heart. The computer simulation suggested that the cells divided longitudinally. To observe the direction of cell division in mouse embryos, we further performed time-lapse recordings of looping mouse hearts and obtained the orientation angles of cell divisions (Honda et al. 2020). This analysis showed that the maximal population of measured cells divided longitudinally (Fig. 9.13b). This finding is consistent with the results of the computer simulation (Fig. 9.11c). We successfully demonstrated tube bending via differential cell proliferation and the direction of cell division, but the model tube merely bent and was not a helical loop. Next, we further considered other factors, specifically, the second and third axial asymmetries (left–right and anterior–posterior asymmetries, respectively). Rightward displacement in chick and mouse heart tubes has been reported (Manner 2000; Harvey 2002; Voronov et al. 2004; Taber 2006; Le Garrec et al. 2017; Kidokoro et al. 2008, 2018), which corresponded to the second axial asymmetry. Importantly, quantitative data revealed that the mouse heart tube was rotated toward the right side (Le Garrec et al. 2017). Furthermore, in developing mouse hearts, the rightward displacement of the heart tube is confined to the anterior region (Le Garrec et al. 2017). We focused on these asymmetries in cell behaviors in mouse heart tubes. We performed computer simulations in which we applied rightward rotational displacement as described in Fig. 9.12d. We assumed longitudinal orientation of cell division during t ¼ 0–30, where t is the simulation time, and applied rightward rotational displacements (t ¼ 16–30). At t ¼ 16, the model tube actually bent, as shown in Fig. 9.14. We applied rightward rotational displacement exclusively to the anterior region of the model tube (t ¼ 16–30). The model tube twisted and became helical (Fig. 9.14 t ¼ 16–30). To confirm that the model tube was a helical structure, we introduced a chain (a series of dots) that is the central line of the model tube. If the projected pattern of this chain onto the horizontal plane is a roundish closed curve, it indicates that the structure is close to helical. Snapshots of the process of formation

166

9

Expansion of Shape–Dimension

Fig. 9.13 Quantitative data of cell division in the mouse embryonic heart. (a), Bar chart showing the mean ratio and standard deviation of EdU/DAPI spot numbers in the ventral and dorsal half of tubes in each horizontal section of four embryos at approximately E8–E8.5. We analyzed four embryos. EdU is a DNA synthesis monitoring probe, DAPI is a DNA binding molecule, and EdU/DAPI staining shows the relative degree of cell proliferation. The EdU/DAPI spot ratio in the dorsal and ventral sides (D and V) of four embryos was estimated. The mean ratios in the dorsal and ventral halves were 0.0703 (16.77%) and 0.3488 (83.23%), respectively. Numerals in parentheses indicate percentages relative to the total amount. The ratios in the right and left (R and L) of the tube were also estimated. The mean ratios in the right and left sides were 0.230 (55.93%) and 0.1813 (44.07%), respectively. The * symbol indicates a significant difference. The null hypothesis that there is no significant difference in cell proliferation between the ventral and dorsal halves was rejected ( p < 0.01; Welch t test: degrees of freedom 1/4 5.3399; t value 1/4 3.91) [Reproduced from Fig. 1C and an excerpt of its legend in Honda et al. 2020 with permission of Elsevier]. (b) Bar chart showing the distribution of the angles (0 is the vertical direction. Clockwise orientation is positive). The maximum population resulted from longitudinal cell division (
10 ~ +10 ). The mean and s.d. (circular statistics) are 10.95 and 36.29 , respectively. n ¼ 75 (four embryos). Differences in gray strength indicate different embryos. To test the uniformity of the distribution of angles, we performed a Rayleigh test. The null hypothesis “Data obey the uniform distribution” was rejected (Statistics 0.818; p-value 0) [Reproduced from Fig. S1C and an excerpt of its legend in Honda et al. 2020 with permission of Elsevier]

of the helical structure are shown in Fig. 9.14. The application of anterior displacement resulted in helical looping. For comparison of the results of the computer simulation with real mouse heart tubes, we performed experiments with mouse embryonic hearts and cell lineage analyses (see Honda et al. 2020). We observed colony patterns of descendant cells, which were consistent with pattern analysis of linear colonies in the model tube.

9.2 Twisting of the Heart Tube

167

Fig. 9.14 Snapshots of the process of formation of the helical structure of the model tube. Cells in the ventral region were divided (gray polygons). Rightward rotational displacement was applied in the anterior portion after t ¼ 16. Polygons: faint gray, cells in the ventral half of the initial model tube; gray, cells that divided during the simulation; black, the ventral-most cells at the initial stage; dark gray in band, cells to which the rightward rotational displacement was applied. The gray band is at the 17% position from the anterior pole, and its width is 3. Bottom row, viewed from the top. Chains show the central line of the model tube. A, V, R: anterior, ventral, and right directions of the initial model tube, respectively. To see this figure in color, go online (Honda et al., 2020) [Reproduced from Fig.S2 in Honda et al. 2020 with permission of Elsevier]

9.2.2

Mechanism of Determination of Output of the Handedness of Helical Heart Tubes

9.2.2.1

Anisotropic Convergent Extension (CE)

As described above, we showed a case in which rightward displacement of the heart tube led to left-handed helical looping. Then, what causes the rightward displacement? Recently, we adjusted the vertex dynamics of the heart tube and obtained computer simulation results showing that an anisotropic contraction of cell boundaries (cell–cell junctions), i.e., intrinsic chiral properties of cardiomyocytes, leads to left-handed helical looping via CE of collective cells. First, we will explain the CE of collective polygons induced by anisotropic contraction of edges. A sheet consisting of polygons was constructed, and we looked at four neighboring polygons in this sheet (Fig. 9.15). The contractile force of specific edges was assumed, that is, edges whose directions were close to the horizontal direction had a strong contractile force (anisotropic contractile force), and the change in the polygonal pattern was examined using a mathematical model system (Eq. 4.1 in Chap. 4). As shown in Fig. 9.15 (left), several edges of the four

168

9

Expansion of Shape–Dimension

Fig. 9.15 Convergent extension (CE) of collective polygons. Contraction of the horizontal edges (anisotropic contraction, thick line) of polygons causes the rearrangement of four polygons, leading to CE. Two horizontally separated polygons (leftmost, dark gray) converge and become adjacent to each other, and a pattern of four polygons extends vertically (right). Inset: Horizontal contraction of an edge causes vertical force of extension [Reproduced from Fig. 2A in Honda 2021]

polygons expressed a strong horizontal contractile force (thick solid line), and the edges were rearranged among each other, exchanging connections at vertices. This is a topological pattern transformation. Two neighboring stacked polygons (light gray) were intercalated by two dark gray polygons, as shown in Fig. 9.15 (right). The pattern of the four polygons changed from a horizontally elongated to vertically elongated shape. Therefore, the anisotropic contractile force of the edges enables a pattern of polygons to be expanded perpendicularly to the direction of contractile force (Fig. 9.15 inset). This is a collective motion of CE at the cellular level. CE caused by the horizontal contractile force of edges on the cylindrical surface was already demonstrated by computer simulation using the mathematical model described in Chap. 8 (Honda et al. 2008; Nishimura et al. 2012).

9.2.2.2

Peculiar Remodeling of an Artificial Tube via CE of Constituent Cells13

We applied the abovementioned anisotropic contractile force to an artificial model of a straight tube consisting of many polygons (Fig. 9.16, leftmost). Several simulations were performed with various anisotropic angles. The method to define contractile boundaries (junctions) of cells by a given anisotropic angle of cells is described in the legend of Fig. 9.18, where cell polarities have previously been calculated as described in the legend of Fig. 9.12b. First, we applied an anisotropic

13

Reproduced from excerpts in Honda 2021.

9.2 Twisting of the Heart Tube

169

Fig. 9.16 Remodeling of a straight tube (leftmost; t ¼ 0) via CE of constituent polygons. When the angle of contraction force of CE is 230 , polygons on the tube surface rotate in a right-handed screw, forming a barber pole-like pattern (t ¼ 150). When the angle of contraction force is 270 , the tube surface does not rotate, but the tube bends like a hairpin (t ¼ 150). When the angle of contraction force is 280 (almost vertical extension), the tube itself loops in a right-handed helix (t ¼ 150). The figure is revised from Fig. 2 in Honda (2021); the numerals of angles that were initially based on an improper baseline were replaced (215 , 255 and 2 65 in Honda (2021) were replaced with 230 , 270 and 2 80 , respectively). Top and middle, side view and horizontal projection, respectively. Bottom, gray line designates an array of gray polygons. The chain line designates the central line of the tube. A translucent tube image is superimposed [Reproduced with revision from Fig. 2C, D, E in Honda 2021]

angle of
30 (Fig. 9.16), where edges whose directions were close to
30 expressed a strong contractile force. The direction of
30 is the orientation of a short clock hand pointing to 11:00 o’clock. The polygons migrated perpendicularly to the anisotropic angle and formed a right-handed screw pattern on the surface of the tube (Fig. 9.16;
30 ). Indeed, the array of gray polygons that had been vertical

170

9

Expansion of Shape–Dimension

in Fig. 9.16 (leftmost) rotated to form a right-handed screw, similar to a barber’s pole, as shown in Fig. 9.16 (
30 ). When we changed the anisotropic angle from
30 to
70 , the array of gray polygons did not rotate on the tube surface. Instead, the tube itself mechanically deformed to a bent 3D shape, as shown in Fig. 9.16 (
70 ). The array of gray polygons followed the bending of the tube. The bottom figure shows that the central line (chain) of the tube forms a hairpin-like shape, which indicates that the tube is bent. The array of gray polygons runs parallel to the hairpin-like shape of the central line of the tube. Moreover, when we changed the anisotropic angle to
80 , the tube not only bent but its entire shape was deformed into a helical loop (gray arrow in Fig. 9.16,
80 ). The array of gray polygons on the tube surface followed the helical looping of the central line. It should be noted that the orientation of rotation was opposite. The helical tube formed a left-handed helix with an anisotropic angle of
80 , whereas the spiral rotation of the tube surface with an anisotropic angle of
30 was right-handed (Fig. 9.16). We understood these results when we considered the horizontal and vertical components of the extension force separately as follows. As the absolute anisotropic angle increased (from 30 to 80 ), the horizontal component of the extension force decreased. The force was so small that the rotation stopped. Instead, the vertical component of the force increased so that the tube itself deformed and bent. Thus, an anisotropic angle of
70 is a critical angle for deformation of the heart model tube. When the absolute anisotropic angle is less than 70 , the array of gray polygons on the tube rotates around the tube axis. When it is more than 70 , the array is fixed to the surface of the tube and moves in parallel with the twisting tube. In conclusion, the anisotropic contractile force of the edges of the polygons on the tube surface causes either (1) spiral rotation of the polygons on the tube surface (a barber pole-like pattern) or (2) 3D remodeling of the entire tube shape. When the cell arrays are difficult to rotate because of a weak component force in the horizontal direction, the tube changes its 3D shape to become a helical loop. The present results show that we have a powerful mathematical cell model tool by which we can construct 3D tissues that are remodeled on a large scale. Anisotropic forces are already known to cause rotation of the tissue surface. A tube of the Drosophila hindgut twists around the tube axis (Inaki et al. 2018). In addition, the Drosophila male genitalia rotates and is surrounded by the A8 segment, where the A8 segment forms a tube ring at the posterior terminal of the embryo and twists around the body axis (Kuranaga et al. 2011; Sato et al. 2015; Uechi and Kuranaga 2019; Okuda and Sato 2022).

9.2.3

Computer Simulations of the Initial Heart Tube13

The abovementioned computer simulation showed that the anisotropic contractile force of the edges of polygons causes an artificial tube to loop helically under certain conditions. This result encouraged us to elucidate a physical mechanism underlying the helical looping of the real heart tube. We considered a model tube for the initial embryonic heart (Fig. 9.17 inset). The model tube consists of the ventral and dorsal

9.2 Twisting of the Heart Tube

171

sides, where cells on the dorsal side do not have anisotropic properties. In the present computer simulation, we did not consider cell proliferation because cardiac loop formation of the chick embryonic heart is not related to cell proliferation (Soufan et al. 2006). We performed computer simulations with various anisotropic angles of cells on the ventral side (Fig. 9.17). We used Eqs. (4.1) with (9.3), where we used U L ¼ σ L Σ wij Lij

ð9:8Þ

instead of UL ¼ σ L Σ Lij (Eq. 9.4). The method to define contractile boundaries (junctions) of cells by a given anisotropic angle of cells is described in the legend of Fig. 9.18, where cell polarities were previously calculated as described in the legend of Fig. 9.12b. The strong contractile force of edges whose angles were close to the anisotropic angles of
70 ,
75 , and
80 produced left-handed helical looping (Fig. 9.17a–c). These patterns were more or less the same. The results show the degree of robustness of the anisotropic angle. When a horizontal edge expressed a strong contractile force (anisotropic angle ¼
90 ), the model tube simply bent, showing a hairpin pattern (Fig. 9.17d). When we used an inverse angle (anisotropic angle ¼ +75 ), we obtained an expected inverse (right-handed) helix loop (Fig. 9.17e). A course of formation of the helical loop is shown in Fig. 9.19. Before notable changes to the shape of the model tube, the polygonal pattern of cells on the tube surface changed from a hexagonal pattern resembling the pattern of a bee’s nest (t ¼ 0) to a horizontal array of polygons (t ¼ 20), as shown in the insets in Fig. 9.19. The vertical array of the ventral-most cells (dark gray cells) was intercalated by neighboring cells. The model tube began to loop at t ¼ 50 and became a distinct lefthanded helical loop (t ¼ 110 and 150). Horizontally projected figures (bottom, t ¼ 110 and 150) obviously show the left-handed helical loop. Patterns at t ¼ 110 and 150 seem to correspond to the embryonic chick heart tube just before HH12 (Fig. 1G in Manner 2000).

9.2.4

Position-Specific Deformation of Cell Colonies in the Process of Helix Loop Formation13

To examine the relationship between cell deformation in local regions and entire helical looping during left-handed helical loop formation, we divided and analyzed the ventral side of the model tube into four regions: anterior left (aL), anterior right (aR), posterior left (pL), and posterior right (pR). We did not analyze the dorsal cells because we assumed that the dorsal cells have no anisotropic properties in the computer simulations. During the process from t ¼ 0 to 80, Fig. 9.20 shows that the aR region extended longitudinally and that the aL region extended with curving, which was compatible with the observations of the chick embryonic heart by

172

9

Expansion of Shape–Dimension

Fig. 9.17 Shape changes in the heart model tube depending on anisotropic angles of strong contractile edges. Inset: an initial heart model tube consisting of the ventral and dorsal sides. Light gray cells on the ventral side exhibit shape changes due to the anisotropic contractile force of edges; t ¼ 0. Gray polygons: a vertically arranged cell array at the ventral-most region on the heart model tube surface. (a2e) Shapes of the model tube with anisotropic edge angles ¼ 270 , 275 , 280 , 290 , and + 75 ; t ¼ 150. Edges of angles close to the anisotropic angle have strong contractile forces. From top to bottom row: circular presentation of anisotropic edge angle (clockwise is positive); side and top views of the model tube; and presentation of a chain pattern that shows the central line of the model tube, where translucent tube images are superimposed. A, V, and R represent the anterior, ventral, and right directions of the initial model tube, respectively. To see this figure in color, go online (Honda 2021) [Reproduced from Fig. 3 in Honda 2021]

Kawahira et al. (2020). In the posterior region, the results were the opposite. The pR region extended with curving, and the pL region extended longitudinally. Furthermore, we examined the behaviors of each cell and its neighbors in detail. We measured the cell area (S), and polygons of cells were approximated by momental ellipses (ellipses of inertia). We obtained cell shape anisotropy (Acell ¼ (a
b)/a, where a and b are the longest and shortest axes of approximated ellipses, respectively) and cell orientation (Ocell; orientation of the longest axis a). To examine the deformation in the neighborhood of each cell, we observed a colony of surrounding cells around each cell. We analyzed colony shape as described in Fig. 9.21 and its legend. We also measured colony shape anisotropy (Acolony ¼ (a0 – b0 )/a0 , where a0 and b0 are the longest and shortest axes of approximated ellipses, respectively) and colony orientation (Ocolony; orientation of the longest axis of colony a0 ). The results of the measurement are shown in Fig. 9.22, in which the lateral surface of the heart model tube was unfolded in a plane. Each measured value of cells was plotted on the plane by a line segment, whose orientation was Ocell and whose length was relative Acell (Fig. 9.22 left). Cell shapes at t ¼ 80 were plotted as

9.2 Twisting of the Heart Tube

173

Fig. 9.18 Determination of the strong contractile edges. The strong contractile edges depending on their orientation have to be determined. However, we do not consider that an edge has the ability to measure its own orientation in 3D space and judge whether it has strong contractile properties. Rather, a cell determines which edges in the cell should have a strong contractile force, referring to its polarity (the cell polarities should always be updated by the method described in the legend of Fig. 9.12b). Therefore, we determined particular edges whose orientations were close to an anisotropic angle as follows. Method to determine two specific edges in a polygon that are closer to the anisotropic angle than other edges in the polygon. An anisotropic angle line (line A) is drawn that forms the anisotropic angle (designated by arc) with respect to the polarity direction (arrow), where line A passes through the center of the polygon. Next, a line (P1P2) is drawn that is perpendicular to line A and includes the center of the polygon. Edges of a polygon that crosses line P1P2 are designated as specific edges (thick line). The orientations of these two edges are closer to the anisotropic angle than those of the other edges [Reproduced from Fig. 1A in Honda 2021]

Fig. 9.19 Snapshots of the process of helical looping of the heart model tube. Anisotropic edge angle ¼ 275 ; w ¼ 2.7. Top and middle rows, side and top views of the model tube. Bottom row, presentation of the chain pattern on which the translucent tube image is superimposed. Insets of t ¼ 0 and 20, partially enlarged view. Dark gray polygons are the ventral-most cells in a vertically arranged cell array on the tube surface at t ¼ 0. The chain indicates the central line of the tube. A, V, and R represent the anterior, ventral, and right directions of the initial model tube, respectively. To see this figure in color, go online (Honda 2021) [Reproduced from Fig. S3 in Honda 2021]

174

9

Expansion of Shape–Dimension

Fig. 9.20 Regional changes in the cell patterns during helical loop formation of the heart model tube. Cell patterns of the aR, aL, pR, and pL regions are shown. Right and left in each figure, heart model tubes at t ¼ 0 and 80, respectively. Cells in each region are light gray. Flames of rectangles are drawn for recognition of pattern changes. Each view direction in the four figures is different. The central cell in each region is dark gray, the normal of which is a view line that is perpendicular to the page. A, R, and V: anterior, right, and ventral, respectively. aL, aR, pL, and pR: anterior left, anterior right, posterior left, and posterior right regions, respectively. To see this figure in color, go online (Honda 2021) [Reproduced from Fig. 5A in Honda 2021]

polygons. Each measured value of the colony was also plotted in the other plane by a line segment, whose orientation was Ocolony and whose length was relative Acolony (Fig. 9.22 right). In the comparison between Fig. 9.22 left and right, colony orientation and anisotropy (Ocolony and Acolony) differed significantly from cell orientation and anisotropy (Ocell and Acell). Ocolony and Acolony are more sensitive indicators of shape change than Ocell and Acell because the cells in a colony migrate individually and dynamically through repetition of cell intercalation. Statistical analyses of these measurements were performed in Honda (2021). This result was in agreement with the observations reported by Kawahira et al. (2020).

9.2 Twisting of the Heart Tube

175

Fig. 9.21 Method to analyze changes in colony shape during morphogenesis. Left and right patterns are at t ¼ 0 and 75, respectively. A colony of closely neighboring cells (gray cells and faint gray cells at t ¼ 0) is deformed to become a concave colony at t ¼ 75. Colony i consists of cell i (gray cell) and ni cells that surround cell i at t ¼ 0. ni is the number of surrounding cells at t ¼ 0. The surrounding cells are designated by a faint gray and referred to as cell k (k ¼ 1, 2, 3, . . ., ni) throughout the analysis. Their central points at t ¼ 75 are designated by circles. Note that cells designated by + and # do not belong to cell k, although they are neighbors of cell i at t ¼ 75 because the cells designated by + and # were not the neighbors of cell i at t ¼ 0. The central points of cell k (k ¼ 1, 2, 3, . . ., ni) at t ¼ 75 are normalized (black dots) as follows: polygon i, whose vertices are the central points of colony member k, was normalized using the ratio of distances rk /r0k, where r0k and rk are distances of cell k from cell i at t ¼ 0 and t ¼ 75, respectively. The positions of the central points of the normalized colony polygon were (xk0 , yk0 ), where xk0 ¼ xi + (xk
xi) rk /r0k and yk0 ¼ yi + (yk
yi) rk /r0k (black dots in Fig. 9.21, right). This normalization enabled us to obtain net changes in colony shape anisotropies at t ¼ 75, regardless of diverse colony shapes at t ¼ 0. A polygon whose vertices are the normalized central points (solid black segment line) was analyzed using the approximation method of a momental ellipse as described in the text. The orientations of thin and thick segment lines represent cell orientation (Ocell) and colony orientation (Ocolony), respectively. The lengths of segment lines represent cell shape anisotropy (Acell) and colony shape anisotropy (Acolony). To see this figure in color, go online (Honda 2021) [Reproduced from Fig. 1C in Honda 2021]

9.2.5

Mechano-physical Mechanism That Determines the Handedness of the Helical Loop13

Among the abovementioned changes in cells and colonies, which term made the largest contribution to determination of the handedness of helical looping? To answer this question, we superimposed line segments expressing cell orientations (Ocell) on the left side on the left column and line segments expressing cell orientations on the right side on the right column (Fig. 9.23a). We also superimposed line segments representing colony orientations (Ocolony) in the left and right regions in a similar manner (Fig. 9.23b). The superimposed line segments of colony orientations should be noteworthy. The distribution of line segments of colony orientations close to the horizontal direction was outstandingly different between the left and right

176

9

Expansion of Shape–Dimension

Fig. 9.22 Analyses of orientation and shape anisotropy of cells (left) and colonies (right). Left, Cell orientation (Ocell) and cell shape anisotropy (Acell) of a heart tube with a 275 anisotropic angle. The direction and length of the line segments indicate the orientation of Ocell and the relative strength of Acell, respectively. Black and gray lines: angles of the line segment from the vertical direction are CW (positive angle) and CCW (negative angle), respectively. The gray horizontal line indicates the horizontal direction for reference. Polygons represent polygonal cells at t ¼ 80 on a relative scale. These are plotted on a plane, which is an unfolded sheet of the lateral surface of the initial heart model tube (t ¼ 0). Vertical gray cell array in the center: cells that were at the ventralmost position in the heart tube. Gray zone on left and right sides: dorsal region in the heart model tube. A, P, L, and R: Anterior, posterior, left, and right sides, respectively. Right, Colony orientation (Ocolony) and colony shape anisotropy (Acolony) of a heart tube with a 275 anisotropic angle. The direction and length of the line segments indicate the direction of Ocolony and the relative strength of Acolony, respectively. For other notes, see the Left legend. To see this figure in color, go online (Honda 2021) [Reproduced from Fig. 5D, E in Honda 2021]

sides. In the anterior half of the left column, many roughly horizontal line segments were present, whereas few were present in the anterior half of the right column. The LR asymmetric distribution was opposite in the posterior half. Longitudinal expansion close to the vertical direction was also remarkable in the anterior half of the right column and the posterior half of the left column. These results show that the colony orientation (Ocolony) can be considered the primary factor in determination of the

9.2 Twisting of the Heart Tube

177

handedness of the helix. Such regional differences produced the chiral structure of the helical loop. We also performed a similar analysis of the right-handed helical loop with a +75 anisotropic angle, and the result is shown in Fig. 9.23c. Inversed asymmetric results between the two anisotropic angles (
75 and +75 ) were confirmed; that is, the left/right bias of the initial heart model tube was negligible. On the other hand, the difference in the cell orientation (Ocell) between the left and right sides was not as clear as shown in Fig. 9.23a. Cell shapes seemed to not be deeply correlated with the handedness of helical looping. Notably, despite the assumption that the distribution of edges with anisotropic contractile force was entirely uniform on the ventral side, the response of colonies in

Fig. 9.23 Superimposed presentation of line segments indicating cell orientation and colony orientation. (a) The anisotropic angle is 275 . Line segments indicating cell orientation in the left and right regions of the heart model tube are arranged on each column. (b) Line segments indicating colony orientation with a 275 anisotropic angle are arranged similarly [Reproduced from Fig. 5F, G in Honda 2021]. (c) Line segments indicating colony orientation with a + 75 anisotropic angle are arranged similarly. Lines whose directions are close to that of the horizontal line (i.e., absolute angle from the vertical direction, |Ocell| > 60 , |Ocolony| > 60 ) are drawn with thick lines. Lines whose directions are close to that of the vertical line (i.e., absolute angle from the vertical direction, |Ocell| < 15 , |Ocolony| < 15 ) are drawn with thin lines [Reproduced from Fig. S6F, G in Honda 2021]

178

9

Expansion of Shape–Dimension

the heart model tube was different in each region. In other words, in the anterior regions of the ventral heart tube, longitudinal extension close to the vertical direction was remarkable in the right region in comparison with that in the left region. This result of the computer simulation corresponds to actual observations of the chick embryonic heart tube. The observations revealed that the orientation of cell edges enriched in phosphorylated myosin II (p-myoII) is uniformly biased anterorightward on both the right and left sides of the heart tube (Fig. S11 in Ray et al. 2018). In addition, analysis of detailed measurement of the chick heart tube showed that orientation of the extension of cell rearrangement is vertical on the right side of the heart tube but horizontal on the left side (Kawahira et al. 2020). Anisotropic contraction of cell edges that are uniform in the right and left regions determines the outcome of the handedness of helical heart looping via longitudinal extension of collective cells in the right or left region of the ventral heart tube. That is, anterior rightward anisotropic contraction of cell edges (
75 in Fig. 9.17b) in the ventral heart tube via the longitudinal extension of collective cells in the anterior right region of the ventral heart tube (Fig. 9.23b) led to left-handed helical looping (
75 in Fig. 9.17b). Conversely, anterior leftward anisotropic contraction of cell edges (+75 in Fig. 9.17e) in the ventral heart tube via the longitudinal extension of collective cells in the anterior left region (Fig. 9.23c) led to right-handed helical looping (+75 in Fig. 9.17e). Although the cells are uniform, the cells show different region-specific appearances as if they have developmentally differentiated. This conclusion could not have been obtained without computer simulation based on a mathematical model.

9.2.5.1

Distinctive Feature of the Cell-Based Vertex Dynamics Model13

We succeeded in connecting the chirality of myocardial cells and the handedness of the helical heart tube via anisotropic cell behavior. To investigate the physical mechanism producing handedness, mathematical models are indispensable. Previously, a few mathematical models have been used in investigation of the looping of the heart tube. Shi et al. (2014b) constructed a finite-element analysis model and recapitulated bending and torsion of the heart tube. Computer simulations using another finite-element analysis model were performed and demonstrated a recapitulation of large-scale dynamic heart looping (Le Garrec et al. 2017). We would like to note the advantage of cell-based vertex dynamics over the finite-element analysis model. In the finite-element analysis, the heart tube was assumed to be a sheet of continuous material rather than an assembly of discrete cells. In the cell vertex dynamics used in our simulation, it was instead possible to assume cell polarity, anisotropic edge properties, chiral properties of individual cells, and orientation of cell division in individual cells.

9.2 Twisting of the Heart Tube

9.2.5.2

179

Intrinsic and Extrinsic Factors Causing Left-Handed Helical Looping13

The chirality of anisotropic cell edge properties that was based on the observation by Ray et al. (2018) is the intrinsic factor for chiral helical looping. They confirmed a clockwise (CW) rotational chirality of cells in the developing myocardium. They examined the following sequentially: the chirality of the rotational behavior of myocardial cells, Golgi rightward polarization within cells (rightward means the orientation of a clock hand pointing to approximately 9 o’clock), the rightwardbiased alignment of the cell boundary and LR asymmetry in cell shape, and enrichments of N-cadherin and p-myoII in the rightward biased cell boundaries. Strong p-myoII intensity was predominantly aligned toward the anterior–rightward direction, which produces anisotropic force along cell boundaries. In the present paper, we showed that the anisotropic force leads to left-handed helical looping of the heart tube. Overall, we demonstrated the possibility that the intrinsic factor alone, without external factors, determines the handedness of heart looping. However, Shi et al. (2014a) have extensive experience investigating real heart tubes and have noticed that the left and right omphalomesenteric veins (OVs) are connected caudally to the heart tube. Normally, the left OV is larger and exerts more pushing force than the right OV, causing the heart tube to form with left-handed helical looping. Recently, based on detailed observations of developing mouse heart tubes, computer simulations were performed (Le Garrec et al. 2017). Recapitulation of large-scale dynamic heart looping was demonstrated. In this study, rightward rotation of the arterial pole and asymmetric cell ingression at the venous pole were observed. On the basis of observation of the formation of the initial heart tube, Kidokoro et al. (2018) proposed that left heart cells may more actively rearrange than right heart cells, driving asymmetric heart elongation and looping. In the previous section, we introduced rightward displacement of the anterior portion of the heart tube and succeeded in forming left-handed helical looping (Honda et al. 2020). The abovementioned observations are all extrinsic factors. We have two questions about the intrinsic and extrinsic factors in heart tube looping. (1) What is the major cause of left-handed helical looping, intrinsic or external factors? (2) The outcomes of the handedness of the helical looping induced by these two factors seem to be the same. Why are the effects of the two factors consistent? For the first question, we considered that intrinsic and external factors synergistically determine the handedness of heart looping; that is, an initially subtle asymmetry is amplified via a positive feedback interaction between the intrinsic and external responses. In fact, in this section, we showed that the individual properties of cell chirality caused left-handed helical looping. On the other hand, the external rightward displacement of the heart model tube has been previously suggested to cause asymmetrical arrangements of individual cells (Honda et al. 2020). The individual cells and the global deformation of the heart tube may synergistically interact with one another. The internal and external factors may not be redundant. For the second question, we think of the Nodal signaling pathway, which is known

180

9

Expansion of Shape–Dimension

to be a global molecular signaling pathway establishing embryonic laterality of the left–right bias. Nodal signaling may be related to the abovementioned external factors that lead to left-handed helical looping (the larger left OV, the rightward rotation of the arterial pole, and the LR asymmetric cell ingression in the venous pole). On the other hand, according to Ray et al. (2018), myocardial cells constructing the heart tube initially have CW chirality, and Nodal signaling reverses the chirality of myocardial cells from CW to CCW. The cells on the right side of the heart tube originating from the Nodal-negative lateral plate mesoderm (LPM) exhibit dominant CW chirality, whereas the cells on the left side of the heart tube, receiving contributions from the Nodal-positive LPM, exhibit a more randomized cellular bias. Ray et al. (2018) reported that such a heart tube forms a left-handed helix. When we consider the effect of Nodal signaling, we understand that intrinsic and extrinsic factors consistently function in the heart tube. Desgrange et al. (2018) also mentioned that intrinsic and extrinsic mechanisms are not mutually exclusive and may occur synergistically to drive morphogenesis.

9.2.5.3

Consideration of CE of Collective Cells Across Different Animal Species13

Behaviors of myocardial cells in the chick heart have been observed in detail (Kidokoro et al. 2018). The cells intercalate with each other, and p-myoII is enriched in cell edges aligned along the convergence axis and perpendicular to the direction of tissue extension, indicating that CE occurs. The myocardial cells in the chick heart tube show asymmetry with rightward biased edges on which N-cadherin and p-myoII are enriched (Ray et al. 2018). These data suggest that CE occurs in the heart tube. Under the assumption of the anisotropic contractile force of edges, we performed computer simulations using a mathematical model. The computer simulation suggested that CE occurs in the heart model tube and that the heart tube was remodeled into a chiral structure of a left-handed helix. Generally, CE is known to be exerted in axial developmental processes across different animal species, i.e. chicks, mice, Xenopus, and zebrafish (Cast et al. 2012). Thus, we will discuss the formation of the chiral heart structure across these animal species. We have investigated the looping of the mouse embryonic heart (Honda et al. 2020). Contrary to the heart of the chick embryo, the looping of the mouse embryonic heart is deeply related to the proliferation of cardiomyocytes (Honda et al. 2020). The mouse heart tube bends in a hairpin fashion through the localized proliferation of cardiomyocytes on the ventral side. Through successive anterior– rightward displacement of the tube, we succeeded in remodeling the bent tube to induce looping, leading to formation of a left-handed helix. The mechanism of the anterior–rightward displacement of the heart tube was unclear in a previous paper. On the other hand, our chick heart model tube results in the present paper suggest the possibility that anisotropic edge contractile force contributes, through rightward displacement of the heart tube, to loop formation of the mouse embryonic heart. We thus performed an additional computer simulation. We assumed that the

9.2 Twisting of the Heart Tube

181

anisotropic edge contractile force works in the mouse heart tube in addition to cell proliferation, and we performed computer simulations, as shown in Fig. 9.24. Figure 9.24a shows the result of the model tube with only cell proliferation; the model tube was simply bent. When we added an anisotropic contractile force of edges (anisotropic angle,
75 ), the model tube became a left-handed helix, as shown in Fig. 9.24b. The ventral view in Fig. 9.24b shows the anterior–rightward displacement of the tube. Conversely, when the anisotropic angle was +75 , we obtained a right-handed helical loop, as shown in Fig. 9.24c. It is plausible that the anisotropic contractile force of edges (
75 ), via CE of the myocardium, caused anterior–rightward displacement of the mouse heart tube. The Xenopus heart has a left-handed heart tube shape, similar to chick and mouse hearts (Mohun et al. 2000). CE in the Xenopus heart tube was experimentally investigated, and CE-defective mutants showed heart abnormalities (Cast et al. 2012). Since the heart looping in amniotes (birds, mice, and frogs) appears to be similar, there may be a common role for CE in the formation of heart looping. In contrast to the heart of amniotes, fish do not show a well-defined helix structure of the heart (Baker et al. 2008; Bakkers et al. 2009; Bakkers 2011; Grant et al. 2017; Ocana et al. 2017), and the mature fish heart has a flat S shape (Desgrange et al. 2018). In the early stage of zebrafish heart tube development, the heart tube forms via fusion of the bilateral cardiac cell populations of the lateral plate mesoderms (LPMs), which are assembled into a disc that rotates clockwise (Desgrange et al. 2017). The disc is remodeled to form a cone-shaped intermediate in which ventricular precursors forming the venous end are at the tip and atrial precursors forming the arterial end are at the base. The cone then telescopes out into a tube (Stainier et al. 1993). As the tube undergoes elongation, its venous end is displaced toward the left, accompanied by rotation around the axis of the venous portion (Desgrange et al. 2017). The ventricular chamber starts to bend rightwards, a process referred to as cardiac looping. The outer curvature of the cardiac chambers expands under constriction at the atrioventricular region. The axis of the looped heart in zebrafish then takes the shape of a flat S. Formation of the zebrafish heart involves rotation of the disk and cone, elongation of the cone and bending of the cone axis; thus, the CE process is expected to be exerted in zebrafish heart formation. Indeed, loss-offunction analysis was performed using gene knockdown, which demonstrated significant impairment of CE (Cast et al. 2012). In contrast to the normal rightward looping of the heart observed in controls, the heart often failed to loop or instead showed a mirror reversal in the knockdown zebrafish. The CE process seems to contribute to the formation of the chiral structure of the zebrafish heart. Further investigation of CE is expected to be performed at the cellular level.

182

9

Expansion of Shape–Dimension

Fig. 9.24 Application of anisotropic contractile force of edges to the model tube of the mouse embryonic heart. The computer simulation of the mouse heart model tube in the previous section involved the assumption of rightward displacement of the anterior portion of the heart model tube (Honda et al. 2020). Here, instead of rightward displacement of the tube, we assumed anisotropic contractile forces of the edges of the cells. Mouse heart model tubes in the figure involved the same cell divisions as in the previous model. Ventral cells divided longitudinally between t ¼ 80 and 100 (daughter cells just after division were arranged longitudinally). Division planes were parallel to the horizontal plane, as shown by the gray line. An anisotropic edge contractile force was applied with anisotropic angles of 290 (a), 275 (b) and + 75 (c), as shown by the black line. The weight of the strength of a strong edge contractile force, w ¼ 2.5; t ¼ 100. The bottom shows patterns of central chains of the heart model tube (the horizontal view from the top). Polygons: faint gray ¼ cells of the ventral half in the initial model tube; gray ¼ cells that were divided during simulation; black ¼ the ventral-most cells at the initial stage. A, R, and V: Anterior, right, and ventral directions of the initial model tube, respectively. To see this figure in color, go online (Honda 2021) [Reproduced from Fig. S8 in Honda 2021]

9.3 Summary

9.2.6

183

Conclusion Regarding the Mechanism Underlying Left-Handed Helical Looping of the Heart Tube

The results of the present computer simulations led to the conclusion that the anisotropic contractile force of cell edges causes cell rearrangements, which consequently produces CE of cell colonies. Cells slipped out of neighboring cells in the convergent region, and cells intercalated between neighboring cells in the extension region. Such deformation took place differently in the orientation of the CE between the left and right regions and between the anterior and posterior regions of the heart tube, although the orientation of the anisotropic contraction was uniform between the left and right sides of the heart tube (Fig. 9.25). When the uniform orientation of the anisotropic contraction is
75 , the helical loop is left-handed (Fig. 9.25a). We also confirmed in Honda (2021) that when the uniform orientation of the anisotropic contraction is +75 , the helical loop is right-handed (Fig. 9.25b), and positionspecific longitudinal and horizontal elongation mediated by CE is shown in Fig. 9.23c. Thus, the direction of the edge contractile force is considered to determine whether a left-handed or right-handed helix loop is formed. In addition, we would like to stress the suggestion of our computer simulations that anisotropic properties of cardiomyocytes are necessary for handedness of helical heart looping, but cell proliferation does not contribute directly to the handedness of helical looping.

9.3

Summary

The dimension of a shape expands through buckling, e.g., a straight rod bends into a curved rod in 3D space, resulting in shape expansion from a one-dimensional object to a two- or three-dimensional object. When such a phenomenon happens in a biological system, it is a symmetry break—an epoch in the developmental process. We observed twisting of wisteria seedpods. A piece of a burst seedpod pair was flat in a humid atmosphere and became twisted during desiccation. The area of the seedpod decreases faster than its peripheral frame. The relative decrease in the seedpod area may cause twisting. When there is a rectangular frame consisting of four bars and the joints of its corners are flexible, we can mathematically make a surface of a hyperbolic paraboloid that is enclosed by the four bars of the deformed rectangular frame (Fig. 9.3). The area of the surface is considerably smaller than that of the original rectangle. Keeping the properties of the hyperbolic paraboloid surface in mind, we performed a computer simulation of remodeling of a flat narrow rectangle. The flat narrow rectangle was under the condition that the area enclosed by the four boundaries was forced to decrease, while the frame lengths of the four boundaries were maintained. As a result, the rectangle twisted, and the shape dimension of the rectangle expanded from two to three.

184

9

Expansion of Shape–Dimension

Fig. 9.25 A heart tube is remodeled into helical loops via anisotropic contraction of cell boundaries (a and b, 275 and + 75 anisotropic angles, respectively), where convergent extension (CE) takes place in the heart model tube. (a) Formation of a left-handed helical tube with a 275 anisotropic angle. Although orientations of anisotropic contraction are even on the ventral right and left sides (275 ), the orientations of CE in the anterior right and anterior left regions are longitudinal and horizontal, respectively. On the other hand, the orientations of CE in the posterior right and posterior left regions are inverse, e.g., horizontal and longitudinal, respectively. Difference of CE orientation between the left and right sides causes helical looping. (b) Formation of a right-handed helical tube with a +75 anisotropic angle. Orientations of anisotropic contraction are even in the ventral right and left sides (+75 ), and orientations of CE in anterior, posterior, right and left regions are also position-specific. R, L, right and left sides of the ventral region of the tube, respectively [Original]

We showed another example of the emergence of twisting. In the development of embryos in mammals, the initial heart tube is straight, but it twists and forms a chiral structure, a left-handed helix loop. Generally, the chiral structure is made by combinations of the axis asymmetries. The straight heart tube bends ventrally via differential cell proliferation between the ventral and dorsal sides of the tube (ventral–dorsal symmetry break). The anterior portion of the heart tube can be distinguished from the posterior portion because the anterior portion develops earlier than the posterior portion (anterior–posterior symmetry break). Then, we assumed that the anterior portion shifts rightward (left–right symmetry break). Under these conditions, we performed computer simulations using vertex dynamics and confirmed the formation of a left-handed helical loop. Finally, we challenged ourselves

References

185

to solve the last problem of what causes the rightward shift of the heart tube. Cardiomyocytes were found to have chiral properties, which produced anisotropic contraction of cell boundaries. This anisotropic contraction provided convergent extension (CE) of the heart tube, leading to the rightward shift of the heart tube and then left-handed helical looping; however, cell proliferation did not contribute directly to handedness. The chirality of cardiomyocytes is an intrinsic factor for determination of the handedness of heart looping. Extrinsic factors for the determination of handedness were also discussed.

References Baker, K., Holtzman, N.G., Burdine, R.D.: Direct and indirect roles for Nodal signaling in two axis conversions during asymmetric morphogenesis of the zebrafish heart. Proc. Natl. Acad. Sci. USA. 105, 13924–13929 (2008) Bakkers, J.: Zebrafish as a model to study cardiac development and human cardiac disease. Cardiovasc. Res. 91, 279–288 (2011) Bakkers, J., Verhoeven, M.C., Abdelilah-Seyfried, S.: Shaping the zebrafish heart: from left-right axis specification to epithelial tissue morphogenesis. Dev. Biol. 330, 213–220 (2009) Bartman, T., Hove, J.: Mechanics and function in heart morphogenesis. Dev. Dyn. 233, 373–381 (2005) Cast, A.E., Gao, C., Amack, J.D., Ware, S.M.: An essential and highly conserved role for Zic3 in left–right patterning, gastrulation and convergent extension morphogenesis. Dev. Biol. 364, 22–31 (2012) Christoffels, V.M., Habets, P.E., Franco, D., Campione, M., de Jong, F., Lamers, W.H., Bao, Z.Z., Palmer, S., Biben, C., Harvey, R.P., Moorman, A.F.: Chamber formation and morphogenesis in the developing mammalian heart. Dev. Biol. 223, 266–278 (2000) de Boer, B.A., van den Berg, G., de Boer, P.A., Moorman, A.F., Ruijter, J.M.: Growth of the developing mouse heart: an interactive qualitative and quantitative 3D atlas. Dev. Biol. 368, 203–213 (2012) Desgrange, M.G., Patterson, V.L., Grimes, D.T., Burdine, R.D.: Modeling syndromic congenital heart defects in zebrafish. Curr. Topics Dev. Biol. 124, 1–40 (2017) Desgrange, A., Le Garrec, J.F., Meilhac, S.M.: Left-right asymmetry in heart development and disease: forming the right loop. Development. 145, 1–19 (2018) Du, X., Osterfield, M., Shvartsman, S.Y.: Computational analysis of three-dimensional epithelial morphogenesis using vertex models. Phys. Biol. 11, 066007 (2014) Flynn, M.E., Pikalow, A.S., Kimmelman, R.S., Searls, R.L.: The mechanism of cervical flexure formation in the chick. Anat. Embryol. (Berl). 184, 411–420 (1991) Grant, M.G., Patterson, V.L., Grimes, D.T., Burdine, R.D.: Modeling syndromic congenital heart defects in zebrafish. Curr. Topics Dev. Biol. 124, 1–40 (2017) Gray, A.: Modern differential geometry of curves and surfaces. CRC Press, Boca Raton (1993) Harvey, R.P.: Patterning the vertebrate heart. Nat. Rev. Genet. 3, 544–556 (2002) Honda, H.: Essence of shape formation of animals. Forma. 27, S1–S8 (2012) Honda, H.: Left-handed cardiac looping by cell chirality is mediated by position-specific convergent extensions. Biophys. J. (2021). Accepted on October 8, 2021 Honda, H., Nagai, T., Tanemura, M.: Two different mechanisms of planar cell intercalation leading to tissue elongation. Dev. Dyn. 237, 1826–1836 (2008) Honda, H., Hatta, H., Yoshida, A., Kodama, R.: Orientation of twist of biological planes. In: Symmetry: Art and Science, 2019 – 11th Congress and Exhibition of SIS 106–109 (2019)

186

9

Expansion of Shape–Dimension

Honda, H., Abe, T., Fujimori, T.: The Chiral looping of the embryonic heart is formed by the combination of three axial asymmetries. Biophys. J. 118, 742–752 (2020) Inaki, M., et al.: Chiral cell sliding drives left-right asymmetric organ twisting. eLife. 7, e32506 (2018) Kawahira, N.D., Ohtsuka, N., Kida, K.I., Hironaka, Y.: Morishita, quantitative analysis of 3D tissue deformation reveals key cellular mechanism associated with initial heart looping. Cell Rep. 30, 3889–3903 (2020) Kidokoro, H., Okabe, M., Tamura, K.: Time-lapse analysis reveals local asymmetrical changes in C-looping heart tube. Dev. Dyn. 237, 3545–3556 (2008) Kidokoro, H., Yonei-Tamura, S., Tamura, K., Schoenwolf, G.C., Saijoh, Y.: The heart tube forms and elongates through dynamic cell rearrangement coordinated with foregut extension. Development. 145, dev152488 (2018) Kuranaga, E., et al.: Apoptosis controls the speed of looping morphogenesis in Drosophila male terminalia. Development. 138, 1493–1499 (2011) Le Garrec, J.F., Dominguez, J.N., Desgrange, A., Ivanovitch, K.D., Raphael, E., Bangham, J.A., Torres, M., Coen, E., Mohun, T.J., Meilhac, S.M.: A predictive model of asymmetric morphogenesis from 3D reconstructions of mouse heart looping dynamics. eLife. 6, e28951 (2017) Manner, J.: Cardiac looping in the chick embryo: a morphological review with special reference to terminological and biomechanical aspects of the looping process. Anat. Rec. 259, 248–262 (2000) Manning, A., McLachlan, J.C.: Looping of chick embryo hearts in vitro. J. Anat. 168, 257–263 (1990) Mohun, T.J., Leong, L.M., Weninger, W.J., Sparrow, D.B.: The morphology of heart development in Xenopus laevis. Dev. Biol. 218, 74–88 (2000) Moorman, A., Webb, S., Brown, H.A., Lamers, W., Anderson, R.H.: Development of heart: (1) Formation of the cardiac chambers and arterial trunks. Heart. 89, 806–814 (2003) Murray Jr., H.A.: The development of the cardiac loop in the rabbit, with especial reference to the bulboventricular groove and origin of the interventricular septum. Am. J. Anat. 26, 28–39 (1919) Nishimura, T., Honda, H., Takeichi, M.: Planar cell polarity links axes of spatial dynamics in neural-tube closure. Cell. 149, 1084–1097 (2012) Ocana, O.H., et al.: A right-handed signalling pathway drives heart looping in vertebrates. Nature. 549, 86–90 (2017) Ohmiya, T.: Pericarp anatomy of Wisteria floribunda. Bull. Bot. Garden Toyama. 3, 25–33 (1998) Ohmiya, T.: Mamegara no Shokubutugaku. Toyama to Shizen. 32(127), 2–5 (2009) (In Japanese) Okuda, S., Sato, K.: Polarized interfacial tension induces collective migration of cells, as a cluster, in a 3D tissue. Biophys. J. 121, 1856–1867 (2022) Osterfield, M., Du, X., et al.: Three-dimensional epithelial morphogenesis in the developing Drosophila egg. Dev. Cell. 24, 400–410 (2013) Patten, B.M.: The formation of the cardiac loop in the chick. Am. J. Anat. 30, 373–397 (1922) Ramsdell, A.F.: Left-right asymmetry and congenital cardiac defects: getting to the heart of the matter in vertebrate left-right axis determination. Dev. Biol. 288, 1–20 (2005) Ray, P., Chin, A.S., Worley, K.E., Fan, J., Kaur, G., Wu, M., Wan, L.Q.: Intrinsic cellular chirality regulates left-right symmetry breaking during cardiac looping. Proc. Natl. Acad. Sci. USA. 115, E11568–E11577 (2018) Sato, K., Hiraiwa, T., Shibata, T.: Cell chirality induces collective cell migration in epithelial sheets. Phys. Rev. Lett. 115, 188102 (2015) Schulte, H.: The fusion of the cardiac anlages and the formation of the cardiac loop in the cat (Felis domestica). Am. J. Anat. 20, 45–72 (1916) Shi, Y., Yao, J., Young, J.M., Fee, J.A., Perucchio, R., Taber, L.A.: Bending and twisting the embryonic heart: a computational model for c-looping based on realistic geometry. Front. Physiol. 5, 297 (2014a)

References

187

Shi, Y., Yao, J., Xu, G., Taber, L.A.: Bending of the looping heart: differential growth revisited. J. Biomech. Eng. 136(081002), 1–15 (2014b) Soufan, A.T., et al.: Regionalized sequence of myocardial cell growth and proliferation characterizes early chamber formation. Circ. Res. 99, 545–552 (2006) Stainier, D.Y.R., Lee, R.K., Fishman, M.C.: Cardiovascular development in the zebrafish I. Myocardial fate map and heart tube formation. Development. 119, 31–40 (1993) Stalsberg, H.: Mechanism of dextral looping of the embryonic heart. Am. J. Cardiol. 25, 265–271 (1970) Taber, L.A.: Biophysical mechanisms of cardiac looping. Int. J. Dev. Biol. 50, 323–332 (2006) Terada, T., Hirata, M., Utigasaki, T.: On the mechanism of Spontaneous expulsion of Wisteria seeds. Scientific Pap. Instit. Phys. Chem. Res. 21, 234–245 (1933) Trichas, G., Smith, A.M., et al.: Multi-cellular rosettes in the mouse visceral endoderm facilitate the ordered migration of anterior visceral endoderm cells. PLoS Biol. 10, e1001256 (2012) Uechi, H., Kuranaga, E.: The tricellular junction protein sidekick regulates Vertex dynamics to promote bicellular junction extension. Dev. Cell. 50, 327–338.e325 (2019) Voronov, D.A., Alford, P.W., Xu, G., Taber, L.A.: The role of mechanical forces in dextral rotation during cardiac looping in the chick embryo. Dev. Biol. 272, 339–350 (2004)

Chapter 10

Mathematical Cell Models and Morphogenesis

Outline At the end of this book, we would like to emphasize the indispensable role of mathematical cell models for a comprehensive understanding of the pathway from genes to shape formation. Furthermore, we noticed the self-construction of cells. Especially for the long-term developmental process, we take into account successive self-organization processes, i.e., the self-construction operations are successively iterated.

10.1

Mathematical Cell Models Are a Bridge to Link Shape Formation with Genes

We intended to elucidate a pathway from genes to shape formation in biological bodies. Information on genes defines proteins that have enzymatic activity. Proteins include (1) enzymes involved in synthesizing and degrading the materials of bodies, (2) signaling molecules participating in signal transduction, and (3) cytoskeletonrelated molecules producing physical forces and supporting configuration for morphogenesis. With the combined progress in biochemistry, molecular biology and cell biology, materials, signaling networks and contractile forces have been elucidated in detail. However, knowledge of these processes does not automatically lead to an understanding of shape formation in biological bodies. Shape formation is a largescale remodeling of collective substances in which mechanical forces have a critical function. To integrate materials, signaling networks, and forces into a coherent understanding of shape formation, mathematical models are indispensable, e.g., we studied neural tube formation (Fig. 8.8). For example, anisotropic localization of Celsr1 (a core participant in PCP) and the contractile protein myosin II were integrated into mediolateral convergence of tissue in the vertex dynamics model, leading to formation of a neural tube (Nishimura et al. 2012). Another PCP pathway, Fat/Dachsous/Four-Joined, is known to remodel the Drosophila epithelium via © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 H. Honda, T. Nagai, Mathematical Models of Cell-Based Morphogenesis, Theoretical Biology, https://doi.org/10.1007/978-981-19-2916-8_10

189

190

10

Mathematical Cell Models and Morphogenesis

oriented cell rearrangement and cell division (Segalen and Ballaiche 2009; Bosveld et al. 2016).

10.2

Self-Construction of Shapes Recapitulated via Mathematical Models

We established the cell center model and the vertex dynamics model for morphogenesis, allowing us to recapitulate self-constructing systems. That is, under a given condition, components in a system automatically construct a stationary or stable shape, just as water in high places flows downward and a water drop becomes spherical due to surface tension, which can be described using mathematical models. Once cell properties (e.g., the contractile force of cell edges, differential adhesion forces between cells, reconnection of neighboring vertices in collective cells, and cell disappearance among collective cells) were introduced into cell models involving vertex dynamics, we succeeded in understanding phenomena in development processes as automatic self-construction or self-organization operations, for example, the viscoelasticity of cell aggregates (Chap. 6), autonomous neat arrangement of epidermal cells (Chap. 3), and autonomous construction of epithelial vesicles, which are the initial structures in almost all animal developmental processes (Chap. 6). Mathematical models also aid in understanding the autonomous formation of capillary networks (Chap. 3), blood branching systems (Chap. 3), and the AB polarity of epithelial cells (Chaps. 6 and 7). After we introduced PCP signaling system, the mathematical model elucidated the mechanism of CE of collective cells (Chap. 8). We can understand the active remodeling of cells, where cells themselves produce anisotropic force (Chap. 8). Until an understanding of anisotropic force production by cells was achieved, active cell rearrangements involving cell intercalation were mysterious, and cells seemed to behave as if they had their own intention. Finally, we noticed expansion of the shape dimension through buckling-like twisting events (Chap. 9). Twisting is a buckling-like event that provides conversion to higher dimensional shapes. A mathematical model recapitulated twisting of wisteria seedpods and development of embryonic heart tubes in animals.

10.3

Successive Self-Construction

We have been investigating many self-construction processes that can be described by cell models, including the formation of a sphere of cell aggregates mediated by the surface tension of individual cells, the formation of mammalian blastocysts comprising a large cavity, and the formation of an epithelial cell sheet accompanying an apical polygon consisting of contracting edges. After formation of the epithelial cell sheet, cell division, apoptosis and cell intercalation take place, and the sheet is

10.4

Summary

191

Fig. 10.1 Successive selfconstruction of cells leading to morphogenesis [Reproduced from Fig. 4 in Honda and Nagai 2015 with permission of Oxford University Press]

remodeled (e.g., elongates or invaginates). Each phenomenon is simple and minimal in comparison with global morphogenesis. We would like to consider these phenomena as elementary processes in morphogenesis. The large-scale and remarkable morphogenesis observed in developmental processes comprises a chain of these elementary processes, which take place successively for complete morphogenesis (Fig. 10.1). Initially, genes define the properties of cells, and the cells themselves construct a structure according to the given cell properties. In the next stage, other genes work to modify the cell properties, after which the modified cells undergo further self-construction processes. Various self-construction events take place, and they can combine with each other, as shown in “The world of epithelial sheets” (Chap. 7). The process is repeated many times, resulting in large-scale and complicated morphogenesis.

10.4

Summary

At the end of this chapter, we stress the indispensability of mathematical cell models, which bridge genes and shape formation. Mathematical models describe various self-construction processes involving cell assemblies. Some of the self-construction events are combined with other self-construction events. Furthermore, successive self-construction operations lead to the dynamic and large-scale morphogenesis that we observe during developmental processes.

192

10

Mathematical Cell Models and Morphogenesis

Reference Bosveld, F., et al.: Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis. Nature. 530, 495–498 (2016) Honda, H., Nagai, T.: Cell models lead to understanding of multi-cellular morphogenesis consisting of successive self-construction of cells. J Biochem. 157, 129–136 (2015) Nishimura, T., Honda, H., Takeichi, M.: Planar cell polarity links axes of spatial dynamics in neural-tube closure. Cell. 149, 1084–1097 (2012) Segalen, M., Bellaiche, Y.: Cell division orientation and planar cell polarity pathways. Semin. Cell Dev. Biol. 20, 972–977 (2009)