Biological Systems: Nonlinear Dynamics Approach [1st ed.] 978-3-030-16584-0;978-3-030-16585-7

Table of contents :
Front Matter ....Pages i-ix
Time-Delay Feedback Control of an Oscillatory Medium (Michael Stich, Carsten Beta)....Pages 1-17
Electrophysiological Effects of Small Conductance Ca\(^{2+}\)-Activated K\(^+\) Channels in Atrial Myocytes (Angelina Peñaranda, Inma R. Cantalapiedra, Enrique Alvarez-Lacalle, Blas Echebarria)....Pages 19-37
Spontaneous Mirror Symmetry Breaking from Recycling in Enantioselective Polymerization (David Hochberg, Celia Blanco, Michael Stich)....Pages 39-57
Self-organized Cultured Neuronal Networks: Longitudinal Analysis and Modeling of the Underlying Network Structure (Daniel de Santos-Sierra, Inmaculada Leyva, Juan Antonio Almendral, Stefano Boccaletti, Irene Sendiña-Nadal)....Pages 59-85
Onset of Mechanochemical Pattern Formation in Poroviscoelastic Models of Active Cytoplasm (Sergio Alonso)....Pages 87-106

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20 Jorge Carballido-Landeira Bruno Escribano Editors

Biological Systems: Nonlinear Dynamics Approach

Se MA

SEMA SIMAI Springer Series Volume 20

Editor-in-Chief Luca Formaggia, MOX–Department of Mathematics, Politecnico di Milano, Milano, Italy Pablo Pedregal, ETSI Industriales, University of Castilla–La Mancha, Ciudad Real, Spain Series Editors Mats G. Larson, Department of Mathematics, Umeå University, Umeå, Sweden Tere Martínez-Seara Alonso, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain Carlos Parés, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Lorenzo Pareschi, Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Ferrara, Italy Andrea Tosin, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Elena Vázquez-Cendón, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, A Coruña, Spain Jorge P. Zubelli, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil Paolo Zunino, Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series (SEMA SIMAI Springer Series) aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance. Mathematical and numerical modeling is playing a crucial role in the solution of the complex and interrelated problems faced nowadays not only by researchers operating in the field of basic sciences, but also in more directly applied and industrial sectors. This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level. Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series. THE SERIES IS INDEXED IN SCOPUS

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Jorge Carballido-Landeira Bruno Escribano •

Editors

Biological Systems: Nonlinear Dynamics Approach

123

Editors Jorge Carballido-Landeira Department of Physics University of Oviedo Oviedo, Spain

Bruno Escribano Department of Modelling and Simulations in Life and Materials Sciences Basque Center for Applied Mathematics Bilbao, Spain

ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISBN 978-3-030-16584-0 ISBN 978-3-030-16585-7 (eBook) https://doi.org/10.1007/978-3-030-16585-7 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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 Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book brings together recent achievements in the field of biological systems from a nonlinear dynamics perspective. During the compilation of this book, we have received contributions of recent results obtained by the group of international scientists who attended the 2nd BCAM Workshop on Nonlinear Dynamics in Biological Systems, held at the Basque Center for Applied Mathematics, Bilbao in September 2016. These contributions embrace diverse disciplines and use multidisciplinary approaches-including theoretical concepts, simulations and experiments-that emphasize the nonlinear nature of biological systems in order to be able to reproduce their complex behavior. The results included in this book represent recent progress and not necessarily what was presented at the conference. The topics included in the book relate to medical applications as well as more fundamental questions in biochemistry. One such question is the ability to control chemically driven reaction–diffusion systems that lead to periodic patterns, which are often observed in biological problems. Here, the authors present a study of this question as a mathematical control problem, taking into account time-delay feedback for both standing waves and travelling waves. Closer to medical applications, we include a study about the prevention and treatment of heart diseases such as tachycardia, ventricular fibrillation and arteriosclerosis. In particular, this problem is discussed from the perspective of a simple dynamical model that describes electrical wave propagation through the heart tissue. This approach uses a reaction–diffusion model to simulate the contraction of the electrical pulse as it propagates through cardiac tissue, a phenomenon that can lead to tachycardia and fibrillation. Another important topic with medical applications is neural network growth. The work presented here involved the performance of a longitudinal graph theory-based study of in vitro neuronal networks, comparing results from both simulations and experiments. This is an example of self-organization through an optimization process that leads to a random small-world network with positive degree–degree correlations. The phenomenon can be qualitatively described by a spatial network growth model based on random growth and neuronal migration.

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Preface

This book also includes recent results in fundamental biochemical questions such as chiral symmetry breaking in polymers, including a brief description of the mechanical forces at play and a feasible dynamical model for enantioselective polymerization in closed systems. The presented model accounts for spontaneous mirror symmetry breaking as a consequence of competing recycling processes, also in the absence of chiral or mutual inhibition. Lastly, we present the application of mechanochemical pattern formation in the cytoplasm. This problem is studied using viscoelastic models for solid and fluid cases and in the particular case of cardiac cells. The editors and the authors wish to express their gratitude to the Basque Government (Eusko Jaurlaritza) for their financial support in the organization of the 2nd BCAM Workshop on Nonlinear Dynamics in Biological Systems. We would like to thank the Basque Center of Applied Mathematics (BCAM) for their valuable assistance in logistics, administrative duties and creation of a good atmosphere for knowledge exchange. We are also thankful to the Berlin Center for Studies of Complex Chemical Systems (BCSCCS) for providing further economic support. Oviedo, Spain Bilbao, Spain September 2018

Jorge Carballido-Landeira Bruno Escribano

Contents

Time-Delay Feedback Control of an Oscillatory Medium . . . . . . . . . . . . Michael Stich and Carsten Beta

1

Electrophysiological Effects of Small Conductance Ca2 +-Activated K+ Channels in Atrial Myocytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Angelina Peñaranda, Inma R. Cantalapiedra, Enrique Alvarez-Lacalle and Blas Echebarria Spontaneous Mirror Symmetry Breaking from Recycling in Enantioselective Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 David Hochberg, Celia Blanco and Michael Stich Self-organized Cultured Neuronal Networks: Longitudinal Analysis and Modeling of the Underlying Network Structure . . . . . . . . . . . . . . . . 59 Daniel de Santos-Sierra, Inmaculada Leyva, Juan Antonio Almendral, Stefano Boccaletti and Irene Sendiña-Nadal Onset of Mechanochemical Pattern Formation in Poroviscoelastic Models of Active Cytoplasm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Sergio Alonso

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About the Editors

Jorge Carballido-Landeira received his Ph.D. degree in Physics from the University of Santiago de Compostela (Spain) in 2011, where he worked on nonlinear physical dynamics emerging in active confined chemical systems. He is currently an Assistant Professor at the Department of Applied Physics at the University of Oviedo, Spain. His research focuses on a multidisciplinary field that includes nonlinear dynamics, self-organization processes, colloidal and polymer science, environmental constraints and fluid mechanics in order to gain an understanding of complex biological systems. Bruno Escribano has been a researcher at the Basque Center for Applied Mathematics since 2014. He holds a Ph.D. in Physics and Mathematics from the University of Granada, Spain (2010). His highly multidisciplinary research focuses on such diverse topics as astrophysics, quasi-crystals, biomineralization and pattern formation in geophysics. His current research interests are related to the modeling and simulation of biological systems using nonlinear dynamical methods.

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Time-Delay Feedback Control of an Oscillatory Medium Michael Stich and Carsten Beta

Abstract The supercritical Hopf bifurcation is one of the simplest ways in which a stationary state of a nonlinear system can undergo a transition to stable self-sustained oscillations. At the bifurcation point, a small-amplitude limit cycle is born, which already at onset displays a finite frequency. If we consider a reaction-diffusion system that undergoes a supercritical Hopf bifurcation, its dynamics is described by the complex Ginzburg-Landau equation (CGLE). Here, we study such a system in the parameter regime where the CGLE shows spatio-temporal chaos. We review a type of time-delay feedback methods which is suitable to suppress chaos and replace it by other spatio-temporal solutions such as uniform oscillations, plane waves, standing waves, and the stationary state.

1 Introduction 1.1 Oscillatory Reaction-Diffusion Systems and Spatio-Temporal Chaos The spontaneous emergence of patterns is a fascinating phenomenon observed in many physical, chemical, and biological systems far from thermal equilibrium. Such patterns may show complex temporal or spatio-temporal dynamics, including chaotic behavior. Since these patterns are created by the internal dynamics of the system, this process is called spatio-temporal self-organization and is referred to as pattern formaM. Stich (B) Non-linearity and Complexity Research Group, System Analytics Research Institute, School of Engineering and Applied Science, Aston University, Aston Triangle B4 7ET, Birmingham, UK e-mail: [email protected] C. Beta Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Carballido-Landeira and B. Escribano (eds.), Biological Systems: Nonlinear Dynamics Approach, SEMA SIMAI Springer Series 20, https://doi.org/10.1007/978-3-030-16585-7_1

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tion [1–4]. Typical examples are the patterns found in chemical reaction-diffusion systems [2, 3], hydrodynamic and liquid crystal systems [3, 5], electrochemical systems [6], semiconductors and gas-discharge systems [7, 8], optical systems [9], granular matter [10], the heart [11], the central nervous system [12], and many other biological and ecological systems [13–18]. The complexity and diversity of selforganizing systems is mathematically reflected by nonlinear equations, which arise in a natural way when systems with many interacting elements, biological systems, or chemical reactions are considered. One common framework to describe these phenomena is provided by the research field called nonlinear dynamics [19]. An important class of pattern-forming systems are reaction-diffusion systems, where the coupling of nonlinear reaction kinetics with a diffusive transport process leads to complex dynamical behavior. The most prominent example of a chemical pattern-forming reaction-diffusion system is the Belousov-Zhabotinsky reaction. It consists of the oxidation of malonic acid by bromate ions in an acidic medium, catalyzed by metal ions. Boris Belousov discovered self-sustained oscillations of this reaction under stirring conditions [20] and Anatol Zhabotinsky and Art Winfree later reported target, spiral, and scroll waves in the unstirred system [21–23]. Later, major advances have been achieved with a modification of the Belousov-Zhabotinsky reaction using microemulsions, which shows inward traveling spiral and target waves, Turing structures, standing waves, oscillatory clusters, and other patterns [24, 25]. Another example of a well-studied chemical reaction giving rise to complex temporal and spatio-temporal behavior is the oxidation of carbon monoxide on platinum single crystal surfaces under low pressure conditions [26, 27]. A special feature of this reaction is the occurrence of spatio-temporal chaos and its control [28], which we will comment on later. The concept and mathematical structure of reaction-diffusion models are quite general, and therefore such models may also be applied to pattern-forming physical and biological systems. In the context of physical systems, reaction-diffusion models are for instance used to describe the dynamics of localized patterns found in gasdischarge systems [29]. In the field of living systems which show oscillations or wave phenomena, reaction-diffusion models are for example successfully applied to explain excitation waves (and their breakdown) in the heart [11, 30] and aggregation patterns in slime mold colonies [31]. Depending on the local kinetics in the absence of diffusive coupling, different types of reaction-diffusion systems can be distinguished, such as multistable, excitable, or oscillatory systems [32]. Here, we will focus on oscillatory media that are characterized by temporally periodic dynamics and may give rise to wave trains, spirals, or concentric wave patterns. The simplest bifurcation in which stable limit cycle oscillations emerge from a stationary fixed point is the Hopf bifurcation. Close to the bifurcation point, it is possible to derive a general model, which holds for all systems undergoing such a bifurcation, regardless of their specific nature. This universal equation is the complex Ginzburg-Landau equation [33]. The complex Ginzburg-Landau equation has the interesting property that it displays not only the aforementioned regular wave patterns, but also different states of spatio-temporal chaos. In particular, a state of phase turbulence has been described,

Time-Delay Feedback Control of an Oscillatory Medium

3

where oscillation phases behave chaotic while amplitudes are still regular, and a state of defect (or amplitude) chaos, where phases and amplitudes are displaying turbulent dynamics. Along with other properties of this equation, these dynamical states have been characterized in detail [34]. The complex Ginzburg-Landau equation (CGLE) [33, 35] reads ∂t A = (1 − iω)A − (1 + iα)|A|2 A + (1 + iβ)∇ 2 A,

(1)

where A is the complex oscillation amplitude, ω the linear frequency parameter, α the nonlinear frequency parameter, β the linear dispersion coefficient, and ∇ 2 the Laplacian operator. While the equation contains a term with the parameter ω, it could be scaled out with a transformation A → A exp(−ωt) to leave the model with two parameters only. However, we prefer to keep this additional parameter in the model because in this way we keep the relation to a real reaction-diffusion system as ω scales with the distance μ to the Hopf bifurcation point: ω ∝ μ−1 .

1.2 Time-Delay Feedback: Global and Local Control Schemes Among the most challenging questions in this field is the control of chaotic behavior. Chaos control has evolved into a rapidly expanding domain of research in its own right [36]. Inspired by the work of Ott, Grebogi, and Yorke (OGY method) [37], chaos control was first realized for low-dimensional systems. While the OGY method becomes too laborious when implemented in high-dimensional systems, empirical control schemes were designed that can be readily applied to spatially extended systems. The most widely known variant of these schemes was proposed by Pyragas in his seminal paper in 1992 [38] that has been modified and extended in various ways (see e.g. [39]). By generating a control signal from the difference between the actual system state and a time-delayed one, powerful methods can be designed to influence the dynamics of a nonlinear system in a subtle, self-generated way. Different approaches based on delay differential equations, nonlinear dynamics, and control theory meet in this interdisciplinary research field [36, 40]. Using such methods, control of spacetime chaos has been studied for a number of different systems such as optical devices [41], lasers [42], and chemical systems [28]. A large body of work on the control of spatio-temporal chaos has been performed with the catalytic CO oxidation on Pt(110), a surface catalytic reaction that has served as a model system to investigate pattern formation in reaction-diffusion systems. Space-time chaos in the CO oxidation is characterized by the statistics of topological phase defects [43, 44], and it can be suppressed by global time-delayed feedback [28, 45, 46] as well as periodic forcing [47, 48]. Also exotic variants of mode-dependent feedback have been considered [49]. Moreover, the laser-induced generation of localized wave sources has been used to suppress spatio-temporal chaos in the CO oxidation system [50–52].

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In general, we may distinguish global and local feedback schemes. In the first case, the same feedback force is uniformly applied to the system at each point in space. In the case of a local scheme, the feedback acts in a spatially resolved fashion and may take different values at different points in space. Local and global feedback schemes can be seen as limiting cases of a more general situation, where the control signal contains both local and global contributions. Such intermediate cases are found in many experimental systems, in particular in the biomedical context. A prominent example is spreading depolarization in the cerebral cortex during migraine and stroke. In an effort to model this system, both local and nonlocal feedback has been considered [53, 54]. A second widely studied example is the control of chaotic states in excitable media that underlie cardiac arrhythmias [55]. Global control (conventional defibrillation) as well as spatially resolved techniques are studied in this field; for an overview see [56]. Also in other systems, combinations of global and local feedback terms may appear. A prototypical system, in which delayed feedback methods have been studied in great detail, is charge transport in semiconductor devices [7]. Here, global feedback typically arises from the global nature of the voltage drop across the device, whereas the space-dependent interface charge density allows for the implementation of either global or local feedback terms [57–60]. Also in the context of the CGLE, both global and local feedback schemes of different kinds have been studied with the aim of controlling native chaotic states. The CGLE under the effect of global time-delay feedback has been studied by Battogtokh et al. [61, 62]. A global feedback scheme of Pyragas type (time-delay autosynchronization – TDAS) was considered by Beta and Mikhailov [63]. The effect of local feedback on the CGLE was first investigated by Socolar and coworkers [64, 65]. Later, Silber and coworkers have extended the study of local feedback in the CGLE to a generalized form of the Pyragas scheme that includes spatial shifts in addition to time delay to stabilize traveling wave solutions [66, 67]. Here, we will discuss the CGLE for a one-dimensional medium with a combination of local and global time-delayed feedback. It has been introduced in Ref. [68] and reads ∂t A = (1 − iω)A − (1 + iα)|A|2 A + (1 + iβ)∂x x A + F,   ¯ − τ ) − A(t)) ¯ F = μeiξ m l (A(x, t − τ ) − A(x, t)) + m g ( A(t , where ¯ = 1 A(t) L



(2) (3)

L

A(x, t) dx

(4)

0

denotes the spatial average of A over a one-dimensional medium of length L. The parameter μ describes the feedback strength and ξ characterizes a phase shift between the feedback and the current dynamics of the system. The parameters m g and m l denote the global and local contributions to the feedback, respectively. If m l = 0, the case of global time-delayed feedback is retrieved, which has been studied in Ref. [63]. We adopt the notation used in Ref. [68], which slightly differs from the one in Ref. [63]. Obviously, if m g = 0, then we have a case of purely local feedback.

Time-Delay Feedback Control of an Oscillatory Medium

5

In the simulations shown below, we typically choose m g + m l = 1 constant, thus reducing the number of parameters by one. If μ = 0, the model reduces to the standard complex Ginzburg-Landau equation. In the following we will review our research on this model [68–72].

2 Uniform Solutions We anticipate that the system introduced in Eq. (2) shows a range of regular and irregular solutions, and we start our discussion with the simplest ones, namely uniform oscillations and the uniform stationary state.

2.1 Existence of Uniform Oscillations The uniform periodic solution (“uniform oscillations”) is given by A0 (t) = ρ0 e−iΩt .

(5)

It is a solution of Eq. (2) with the amplitude and frequency given by ρ0 =



1 + μ(m g + m l )χ1 ,

Ω = ω + α + μ(m g + m l )(αχ1 − χ2 ) .

(6) (7)

Here, χ1,2 denote effective modulation terms that can be positive or negative. They arise from the feedback and hence depend on ξ and τ , χ1 = cos(ξ + Ωτ ) − cos ξ,

(8)

χ2 = sin(ξ + Ωτ ) − sin ξ.

(9)

In general, no explicit analytic solution for Eqs. (6) and (7) can be given because χ1,2 also depend on Ω. Nevertheless, the solution can be computed using root-finding algorithms, as done in [69]. For μ = 0, the solution reduces to ρ0 = 1 and Ω = ω + α which is the wellknown uniform oscillatory solution of the standard CGLE. It is well-known for the latter equation that if 1 + αβ < 0 (the Benjamin-Feir-Newell criterion), the homogeneous periodic solution A0 = e−i(ω+α)t is unstable and a regime of spatio-temporal chaos is observed. Below, we will consider the case where this criterion is fulfilled. Solving Eqs. (6) and (7) as a function of τ for all other parameters fixed, one observes that depending on the value of the feedback strength μ, there may be multiple solutions for an interval of τ . This is shown in Fig. 1 where the frequency (a) and amplitude (b) of oscillations are shown for a set of parameter for which the

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M. Stich and C. Beta 1.2

1.6

(a)

(b)

1.4

1.1

1.2 1

T

1

0.9

0.8 0.6

0.8

0.4 0.7 0.6 0.6

0.2 0.7

0.8

0.9

1

1.1

1.2

0

0

τ

0.2

0.4

0.6

0.8

1

1.2

τ

Fig. 1 a The period of oscillations T = 2π/Ω as determined by Eq. (7) as a function of τ for μ = 0.3 (dashed), μ = 0.66 (solid), μ = 1.0 (dotted-dashed). b Periods T = 2π/Ω (solid) and amplitudes ρ0 (dashed, from Eq. (6)) as a function of τ are shown for μ = 1.5. For more information see text. The other parameters are α = −1.4, ω = 2π − α (hence the unperturbed period is equal to 1), ξ = π/2. Figure from Ref. [71]

period of oscillations without feedback is equal to unity. If the feedback strength is small and we vary τ , the oscillation frequency is not constant, but always relatively close to the value of the frequency of oscillations for μ = 0. However, if the value of μ is larger than a critical threshold, there is an interval of τ for which the Eq. (7) has three solutions. If an even larger value of μ = 0 is chosen, the interval of τ in which multiplicity of solutions occurs becomes larger. In Ref. [71] we showed that as τ becomes larger, also the multiplicity areas become larger.

2.2 Stability of Uniform Oscillations Above we have stated that there are parameter areas with multiplicity of oscillatory solutions. If we are interested in the stability of such solutions, we have to take this multiplicity into account. Nevertheless, the treatment of the stability analysis is quite general, so there is no need to introduce the solution until a given numerical experiment. To perform a linear stability analysis of uniform oscillations with respect to spatiotemporal perturbations, we express the complex oscillation amplitude A as the superposition of a homogeneous mode H with spatially inhomogeneous perturbations, A(x, t) = H (t) + A+ (t)eiκ x + A− (t)e−iκ x .

(10)

Inserting Eq. (10) into Eq. (2), and assuming that the amplitudes A± are small, we obtain

Time-Delay Feedback Control of an Oscillatory Medium

7

∂t H = (1 − iω)H − (1 + iα)|H |2 H + μ(m l + m g )eiξ (H (t − τ ) − H (t)), (11) ∂t A+ = (1 − iω)A+ − (1 + iα)(2|H |2 A+ + H 2 A∗− ) − (1 + iβ)κ 2 A+ +μm l eiξ (A+ (t − τ ) − A+ ), ∂t A∗− = (1 + iω)A∗− − (1 − iα)(2|H |2 A∗− + H ∗2 A+ ) − (1 − iβ)κ 2 A∗− +μm l e−iξ (A∗− (t − τ ) − A∗− ).

(12) (13)

See Appendix A of Ref. [69] for details of this derivation. The solution of Eq. (11) is given by uniform oscillations H (t) = H  exp(−iΩ  t) identical to Eq. (6), i.e., H  = ρ and Ω  = Ω. The equations for A+ and A∗− include terms proportional to m l . They represent the local contributions to the feedback term and constitute the difference between Eqs. (12), (13) and Eqs. (7)–(9) of Ref. [63]. To investigate linear stability of uniform oscillations with respect to spatio-temporal perturbations, we make the ansatz A+ = A0+ exp(−iΩt) exp(λt),

(14)

A∗−

(15)

=

A∗0 −

exp(iΩt) exp(λt),

where λ = λ1 + iλ2 is a complex eigenvalue. The sign of its real part determines stability. After substituting Eqs. (14), (15) into Eqs. (12) and (13) we arrive at the following eigenvalue equation: F = (A + iB − iλ2 + D1 + iD2 )(A − iB − iλ2 + C1 + iC2 ),

(16)

where we have defined F = (1 + α 2 )ρ 4 ,

(17)

A = 1 − λ1 − 2ρ − κ , B = Ω − ω − 2αρ 2 − βκ 2 , C1 = μm l e−λ1 τ cos(ξ + Ωτ + λ2 τ ) − μm l cos ξ,

(18) (19) (20)

C2 = −μm l e−λ1 τ sin(ξ + Ωτ + λ2 τ ) + μm l sin ξ, D1 = μm l e−λ1 τ cos(ξ + Ωτ − λ2 τ ) − μm l cos ξ,

(21) (22)

D2 = μm l e−λ1 τ sin(ξ + Ωτ − λ2 τ ) − μm l sin ξ.

(23)

2

2

For details of this derivation the reader is referred to Appendix B of Ref. [69]. This system of equations can then be evaluated to check for the stability of the oscillatory solution. As usual, the criterion for the first instability is that the real part of the largest eigenvalue crosses zero while all others are still negative. When this happens, the imaginary part reveals whether the instability itself is of oscillatory nature. At the same time, the wavenumber of the most unstable perturbation tells us whether the instability is associated with a spatial periodicity.

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In extensive investigations performed in Ref. [69], a variety of stability diagrams were shown and discussed and examples of different instabilities were presented. As the main result it was found that there is a wide range of possible delay times τ such that as the feedback strength μ is increased, the chaotic state is indeed replaced by a uniform oscillatory state. For nonzero μ, the feedback-induced solution is different from the uniform oscillation without feedback and hence the magnitude of the feedback term is nonzero as the feedback-induced solution is stabilized. This is characteristic for a so-called invasive control [68]. It is worth mentioning that the stability analysis captures not the transition from the fully chaotic state to unifrom oscillations. Rather, it describes the first instability encountered from the uniform oscillatory state upon a decrease of the feedback strength. While instabilities of different kinds (imaginary parts nonzero, wavenumbers zero, etc.) are possible, arguably the most common instability is the one with vanishing imaginary part and nonzero wavenumber. This corresponds to the onset of a pattern with spatial periodicity but with no additional temporal component besides the underlying uniform oscillations. Such a pattern describes standing waves, as shown in Fig. 2 and investigated in more detail in Sect. 3.1. Figure 2 also shows for comparison space-time plots for uniform oscillations and developed defect (amplitude) chaos.

(a)

(b)

(c)

x

t

Fig. 2 Main spatio-temporal solutions: a uniform oscillations, b standing waves, d spatio-temporal chaos. Shown are space-time diagrams in gray scale for |A| (top panels) and ReA (bottom panels) for a time interval of t = 25 in the asymptotic regime and system size L = 256. The delay time is τ = 0.5 and the values of μ are μ = 0.50 (a), μ = 0.15 (b), μ = 0 (c). Black (white) denotes low (high) values of the respective quantity (rescaled for each simulation). For |A|, these values are (|A|min , |A|max ) = (0.94, 1.13) (b), (|A|min , |A|max ) = (0.15, 1.2) (c). For (a), the amplitude is constant |A| = 1.085. The other parameters are as in Fig. 1, and β = 2. These patterns were first reported in Ref. [68]

Time-Delay Feedback Control of an Oscillatory Medium

9

2.3 Existence and Stability of the Stationary State The CGLE has been derived to describe the onset of oscillations in a Hopf bifurcation from a stationary state (fixed point). In the standard version of the CGLE that we utilize, the stationary state is simply represented by a vanishing amplitude |A| = ρ ≡ 0 (the phase is no longer defined). This trivial solution exists always, so we only have to investigate its stability. Following the approach used in Ref. [69], we use the ansatz (10) and perform a linear stability analysis with respect to spatio-temporal perturbations. After separating uniform and periodic modes, we obtain ∂t H = (1 − iω)H + μ(m l + m g )eiξ (H (t − τ ) − H (t)), ∂t A+ = (1 − iω)A+ − (1 + iβ)κ 2 A+ + μm l eiξ (A+ (t − τ ) − A+ ),

(24) (25)

∂t A∗− = (1 + iω)A∗− − (1 − iβ)κ 2 A∗− + μm l e−iξ (A∗− (t − τ ) − A∗− ),

(26)

where as usual higher order terms in H and A± have been neglected. These equations are decoupled and can be studied independently. In order to explore the linear stability of the state H = 0 with respect to uniform perturbations, we set H = H 0 exp(λt),

(27)

with H 0 an initial amplitude and λ a complex eigenvalue. Inserting Eq. (27) into Eq. (24), yields the following characteristic equation λ = 1 − iω + μ(m l + m g )eiξ (e−λτ − 1).

(28)

Separating real and imaginary parts, Eq. (28) can be written in the form   λ1 = 1 + μ(m l + m g ) e−λ1 τ cos(ξ − λ2 τ ) − cos ξ ,   λ2 = −ω + μ(m l + m g ) e−λ1 τ sin(ξ − λ2 τ ) − sin ξ .

(29) (30)

These equations can be solved numerically, searching for λ1 = 0 and determining the parameter area where the stationary state is stable. Some analytic operations can be made using the Lambert W function [73]. In the literature on control of oscillations, the stabilization of this stationary state is also called amplitude death, since oscillations are no longer observed [74]. In particular, Eq. (28) is equivalent to the amplitude death condition for a single Hopf oscillator.

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3 Spatio-Temporal Solutions In this section, we consider patterns that show a space-dependence. In particular, we study standing wave patterns observed in a large parameter region between uniform oscillations and turbulence. Then, we give a short overview over other patterns observed in this system.

3.1 Standing Waves One of the ways uniform oscillations can become unstable is when spatially periodic perturbations can increase and give rise to standing waves with wavelength 2π/kc . We follow Ref. [70] to describe the derivation of the solution of standing waves. To be more precise, at onset we have λ1 (kc ) = 0 , ∂k λ1 (kc ) = 0 , and λ2 (kc ) = 0 , with kc = 0, where λ is the leading eigenvalue of the stability analysis of uniform oscillations, and λ1 and λ2 its real and imaginary part. The ansatz to capture such patterns analytically is therefore (31) A(x, t) = H (t) + Bk (t)(eikx + e−ikx ), with H being the uniform mode and Bk the complex amplitude of the mode with wavenumber k. This ansatz can also be interpreted as a rightgoing wave (with wavenumber k) and a leftgoing one (with wavenumber −k) sharing the same amplitude and thus describing a standing wave pattern. Before inserting ansatz (31) into Eq. (2), let us express several terms appearing in Eq. (2) for the case that A(x, t) is given by Eq. (31), of particular relevance being |A|2 A = |H |2 H + 4|Bk |2 H + 2H ∗ Bk2 + (eikx + e−ikx )(H 2 Bk∗ + 2|H |2 Bk + 3|Bk |2 Bk ) + (e2ikx + e−2ikx )(2H |Bk |2 + H ∗ Bk2 ) + (e3ikx + e−3ikx )|Bk |2 Bk ,

(32)

and Fl = μm l eiξ [(H (t − τ ) − H (t)) + (eikx + e−ikx )(Bk (t − τ ) − Bk (t))], Fg = μm g eiξ (H (t − τ ) − H (t)).

(33) (34)

Equation (33) assumes that the space-dependent part of A(x, t) does not contribute ¯ to the average A(t). This is fulfilled if L → ∞ or L = 2π n/k (with n = 1, 2, ...). In

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the following, we will neglect the higher-order harmonic contributions in Eq. (32), which close to onset is well-justified. Inserting finally ansatz (31) into Eq. (2), and using Eqs. (32)–(34), we obtain d Bk dH + (eikx + e−ikx ) = (1 − iω)(H + (eikx + e−ikx )Bk ) dt dt −(1 + iα)(|H |2 H + 4|Bk |2 H + 2H ∗ Bk2 +(eikx + e−ikx )(H 2 Bk∗ + 2|H |2 Bk + 3|Bk |2 Bk )) −(1 + iβ)k 2 (eikx + e−ikx )Bk +μ(m l + m g )eiξ (H (t − τ ) − H (t)) +μm l eiξ (Bk (t − τ ) − Bk (t))(eikx + e−ikx ).

(35)

Now, we separate the equation into space-dependent and space-independent parts and arrive at dH = (1 − iω)H − (1 + iα)(|H |2 H + 4|Bk |2 H + 2Bk2 H ∗ ) dt +μ(m l + m g )eiξ (H (t − τ ) − H (t)), d Bk = (1 − iω)Bk − (1 + iα)(H 2 Bk∗ + 2|H |2 Bk + 3|Bk |2 Bk ) dt −(1 + iβ)k 2 Bk + μm l eiξ (Bk (t − τ ) − Bk (t)).

(36)

(37)

In the case of Bk = 0, the first equation reduces to the equation describing the mode of the homogeneous periodic solution. However, the coupling terms in the first equation of Eq. (36) tell us that if Bk = 0, the uniform mode H is different from the uniform oscillations (5) and we cannot use Eq. (6). To solve Eq. (36), we assume the solution to be of the following form [75]: H = H0 e−iΩ0 t , Bk = Bk0 e

−i(Ω0 t+γ )

(38) .

(39)

Thus, both modes oscillate at the same frequency Ω0 , while there is a phase shift γ between the modes. The real amplitudes are given by H0 and Bk0 , respectively. After inserting (38), (39) into (36), we obtain 2 2 −2i(Ω0 t+γ ) 2iΩ0 t + 2Bk0 e e )H (−iΩ0 )H = (1 − iω)H − (1 + iα)(H02 + 4Bk0 iξ iΩ0 τ +μ(m l + m g )e (e − 1)H, (40) 2 (−iΩ0 )Bk = (1 − iω)Bk − (1 + iα)(H02 e−2iΩ0 t e2i(Ω0 t+γ ) + 2H02 + 3Bk0 )Bk 2 iξ iΩ0 τ −(1 + iβ)k Bk + μm l e (e − 1)Bk . (41)

Assuming H nonzero in the first equation, and Bk in the second, this simplifies to

12

M. Stich and C. Beta 2 2 −2iγ 0 = 1 + iΩ0 − iω − (1 + iα)(H02 + 4Bk0 + 2Bk0 e )

+μ(m l + m g )eiξ (eiΩ0 τ − 1), 2 0 = 1 + iΩ0 − iω − (1 + iα)(H02 e2iγ + 2H02 + 3Bk0 ) 2 iξ iΩ0 τ −(1 + iβ)k + μm l e (e − 1).

(42) (43)

These equations can be separated into real and imaginary parts. We write them as 2 2 (2 + cos 2γ )) − 2α Bk0 sin 2γ + μ(m l + m g )χ1s , (44) 0 = 1 − (H02 + 2Bk0 2 2 2 s 0 = Ω0 − ω − α(H0 + 2Bk0 (2 + cos 2γ )) + 2Bk0 sin 2γ + μ(m l + m g )χ2 , (45) 2 0 = 1 − k 2 − (H02 (2 + cos 2γ ) + 3Bk0 ) + α H02 sin 2γ + μm l χ1s , 2 ) − H02 sin 2γ + μm l χ2s , 0 = Ω0 − ω − βk 2 − α(H02 (2 + cos 2γ ) + 3Bk0

(46) (47)

s are given by where χ1,2

χ1s = cos(ξ + Ω0 τ ) − cos ξ, χ2s = sin(ξ + Ω0 τ ) − sin ξ,

(48) (49)

For a given k, Eqs. (44)–(47) can be solved numerically through root-finding algorithms, giving solutions for H0 , Bk0 , Ω0 , and γ . There, the wavenumber k is provided by the eigenvalue problem studied in [69], where we use either kc (at threshold) or kmax (away from threshold). Combining Eqs. (31), (38), (39), the family of standing wave solutions can be written as A SW = e−iΩ0 t (H0 + 2Bk0 cos(kx)e−iγ ).

(50)

This solution is plotted in Fig. 3 for an example set of parameters together with the result of a simulation of the CGLE for the same parameters. The agreement is striking, given that the mode separation involves several approximations (neglecting higher-order harmonics, assuming that the space-dependent part of A(x, t) does not ¯ contribute to the average A(t), and that the wavenumber is determined by the linear stability analysis of uniform oscillations).

3.2 Other Patterns The CGLE admits a range of different spatio-temporal solutions, so it should not come as a surprise that also the system in the chaotic regime subjected to a timedelay feedback is able to show the stabilization of multiple solutions with simultaneous space and time dependence. Simulations revealed, among others: (a) traveling waves that arise as standing waves undergo an instability to translational motion or

Time-Delay Feedback Control of an Oscillatory Medium 1

0.8

oscillation amplitudes

Fig. 3 Amplitude profiles for μ = 0.7 (other parameters as in Fig. 2). Shown are a simulation (solid black curve) and the theoretical result using the theoretically obtained wave number (dashed blue curve). We display only a part of the medium (total size, L = 128). This solution was first presented in Ref. [70]

13

0.6

0.4

0.2

0 50

full simulation analytical solution 60

70

80

90

100

space

breathing; (b) plane waves, i.e. waves with a constant amplitude; (c) traveling phase flips on a background of standing waves; and (d) bound states of pacemakers [68]. A theoretical description of these patterns is still lacking.

4 Summary and Discussion Control of chaotic states has been investigated intensively in the last decades within the realm of nonlinear dynamics, control theory, and dynamical systems theory, among others fields. In this chapter, we have reviewed a particular case of spatiotemporal chaos and its control, namely the amplitude chaotic state in the complex Ginzburg-Landau equation (CGLE) and its control via a time-delay feedback term. The latter has global and local contributions and the terms are constructed in the spirit of the Pyragas idea of a feedback that vanishes in the case of control. As the CGLE describes a system with underlying small-amplitude, sinusoidal oscillations, the main solution that can be induced by the feedback are uniform oscillations. We have described the corresponding solution analytically. As the controlled solution is different than the solution without feedback, the control is invasive, i.e., the control force does not vanish as the solution is stabilized. The solution without feedback is unstable and not found in the simulations. Between the uniform oscillations and the turbulent state, standing waves are found. The corresponding solution has been described analytically and the onset of the standing waves has been confirmed to be an instability of the uniform oscillations with respect to spatial perturbations. Again, the feedback is invasive, and the standing wave solution is not a solution of the CGLE without feedback. Other, more complicated patterns like traveling waves, phase flips and bound states have been found numerically but an analytic description has not been performed yet. One pattern that in the case of control has vanishing feedback is the steady state. This noninvasive state is also called amplitude death, as the oscillations disappear

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completely. It can be interpreted as time-delay feedback pushing the system below the supercritical Hopf bifurcation. Together with the homogeneous oscillations it is one of the two uniform patterns studied. For such solutions, the global and local feedback terms become identical. However, as we have seen, this does not imply that global and local terms act equally with respect to stability of these solutions. The description of standing waves as uniform oscillations that become unstable with respect to perturbations of a certain wavelength is an example of this. The validity of the existence and stability criteria of the above-mentioned patterns is limited to one spatial dimension. While the existence criteria for the uniform patterns generalize into two dimensions, the stability analysis will be fundamentally different as the allowed perturbations can depend on both dimensions simultaneously. A study of this situation would be an interesting subject of future work. Another limitation of the work presented in this chapter is that the fundamental CGLE parameters were chosen to correspond to spatio-temporal chaos and are not systematically varied. While we believe that we have now understood central parts of the process of controlling amplitude chaos in the CGLE via time-delay feedback, the same need not to be true for phase chaos or other fundamental solutions of the CGLE. Relevant questions to be asked include how the time-delay feedback – global, local, or mixed – could change the stability area of uniform oscillations, which are stable for the unperturbed CGLE.

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Electrophysiological Effects of Small Conductance Ca2+ -Activated K+ Channels in Atrial Myocytes Angelina Peñaranda, Inma R. Cantalapiedra, Enrique Alvarez-Lacalle and Blas Echebarria

Abstract Atrial fibrillation (AF), a cardiac arrhythmia characterized by an abnormal heart rythm originated in the atria, is one of the most prevalent cardiac diseases. Although it may have diverse causes, genetic screening has shown that a percentage of pacients suffering of AF present a genetic variant related to disregulation of calciumactivated potassium (SK) channels. In this paper we review the main characteristics of these channels and use several mathematical models of human atrial cardiomyocytes to study their influence in the form of the atrial action potential. We show that an overexpression of SK channels results in decreased action potential duration and, under some circumstances, it may give rise to alternans, suggesting a pro-arrhythmic role of this current. This effect becomes more important at higher pacing rates. Nevertheless, we also find it to protect against spontaneous calcium release induced afterdepolarizations, acting in this case as an antiarrhythmic factor.

1 Introduction In the heart, contraction is driven by an electrical wave that propagates through the atria and ventricles. This wave has its origin in the ion fluxes that cross the cardiomyocytes’ cell membrane, producing a transient change in electrical polarity of the membrane, known as an action potential, that propagates from cell to cell. The cardiac action potential is the result of a complex interplay of several transmembrane A. Peñaranda · I. R. Cantalapiedra · E. Alvarez-Lacalle · B. Echebarria (B) Departament de Física, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain e-mail: [email protected] I. R. Cantalapiedra e-mail: [email protected] E. Alvarez-Lacalle e-mail: [email protected] B. Echebarria e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Carballido-Landeira and B. Escribano (eds.), Biological Systems: Nonlinear Dynamics Approach, SEMA SIMAI Springer Series 20, https://doi.org/10.1007/978-3-030-16585-7_2

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currents. Sodium, potassium and calcium ions are involved in determining the appropriate shape of the action potential. Sodium flow into the cell is responsible for the initial depolarization current, while potassium and calcium ions move in and out afterwards, balancing each other in order to sustain the depolarization during the duration of a typical action potential (AP). In the cell, most calcium ions are kept in a store, known as the sarcoplasmic reticulum (SR). When calcium enters the cell (due, for example, to an increase in transmembrane potential), it binds to some calcium-activated receptors at the surface of the SR (the ryanodine receptors, RyR), that open, releasing the content of the SR in a proccess known as calcium induced calcium release (CICR). After relaxation, calcium is reuptaken into the SR and brought out of the cell by the Na–Ca exchanger. Thus, the duration of the action potential is closely linked to the calcium transient and dysfunctions in one affect the other. In effect, under anomalous conditions, calcium release from the SR can occur spontaneously, resulting in an increase of transmembrane voltage (through the action of the Ca–Na exchanger), that may give rise to a depolarization, making it a possible source of ectopic heartbeats and cardiac arrhythmias. In recent years, there has been increased attention into another feedback mechanism between calcium transients and transmembrane potential, in this case mediated by small-conductance Ca2+ -activated potassium channels, or SK channels [31, 37, 50]. These channels belong to a large family of Ca2+ -activated potassium channels, that also include big (BK) and intermediate (IK) conductances (see Sect. 2.1 for more details). The three types of channels differ in their pharmacological properties. SK channels, for instance, and contrary to BK or IK channels, are blocked by apamin, a neurotoxin found in bee venom. SK channels were first discovered in skeletal muscle [3], and subsequently they have been found in a wide range of cell types: neuronal, endothelia, epithelia, etc. [40, 49]. In the heart, the first evidence of the presence of SK channels and a Ca2+ activated K + current (I K Ca ) was found in a rat ventricular cell line [45], but these channels have been found in a wide variety of species and types, from rabbit to rat and human [34, 48], and in the pulmonary vein, ventricle, atria and in pacemakers [7, 8]. SK channels have been found to be preferentially expressed in atria compared to the ventricles [47]. In the atria, there is ample evidence that the SK current contributes to action potential repolarization [43, 47]. Inhibition of SK currents has been shown to prolong AP duration (APD) in mouse and human atrial myocytes [39, 47]. Similarly, ablation of SK channels resulted in a significant prolongation of APD [25] while overexpression of SK showed a significant APD shortening [51]. Following their role in the atrial action potential, SK channels have been recognized to play an important role in atrial fibrillation. The relation between expression of SK and AF has been confirmed by genome-wide association studies, revealing that polymorphisms in the calciumdependent potassium-channel gene KCNN3 are an important risk factor for atrial fibrillation (AF) [14, 33, 35]. This result suggests that SK channels may represent a potential therapeutic target for the treatment of atrial arrhythmias.

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SK channels expression may also play a role in atrial remodelling, i.e., changes in size, shape, structure, and function of the atria occurring in patients with AF. The first work to point in this direction was by Ozgen et al. [34] where fast pacing increased the presence of SK channels, as manifested in the APD abbreviation and the sensitivity of the current to apamin. Other works link this remodelling with enhanced trafficking of SK channels to the membrane during rapid atrial tachycardia [36]. However, SK channel expression has been observed to be significantly reduced in right atrial appendages recovered from chronic AF patients [48] and in patients with permanent AF [26]. A possible explanation for these contradictory results proposes that AF may result in the up-regulation of SK channels as the initial response but, with progression of the disease, they might be down-regulated [50]. Despite the evidence relating disfunctions in SK channels with AF, their proor anti-arrythmic role is not clear. For instance, a decrease in the SK current in a SK knock-out mouse model resulted in action potential prolongation and atrial fibrillation [25]. Similar results have been observed in the canine left atrium [21]. In these studies the prolongation of APD was accompanied by increased occurrences of early after depolarization (EAD), increased APD heterogeneity, occurrences of electrical alternans, and wave breaks. This protective role of SK current against AF agrees with the observation that SK current was decreased in human atria tissue in patients with AF [39]. These results, however, seem to be in contradiction with the observation that inhibition of SK channels significantly reduced AF inducibility [12, 13, 35, 38] and SK channel overexpression resulted in predisposition to induced atrial arrhythmias [51]. Most probably, these contradictory results stem from the various possible mechanisms leading to atrial arrythmias and AF. Since SK current helps repolarization it is expected to protect against afterdepolarization related arrhythmias, while its role in decreasing the APD may help initiation and maintainence of reentries. In this regard, little has been done regarding modeling the effect of SK channels in cardiac electrophysiology in order to unveil different pro- or anti-arrhythmic effects of inhibition of SK channels. Recently, Kennedy et al. [22] studied the effect of calcium-sensitive potassium currents on voltage and calcium alternans in a ventricular cell model but no similar study has been performed for atrial cell models. In this paper we will perform a first analysis of its effect on atrial action potential models. We review existing models of I K Ca currents (not necessarily in cardiac cells) and then study the effect of this current in the cardiac atrial action potential. Due to the undeterminancy in the details of some features of the SK current, we will study its effects under different values for its half-activation calcium concentration, its maximal conductance, and the calcium binding cooperativity. This should be the starting point in order to test whether SK channels could avoid or limit the effect of spontaneous calcium release (SCR) on the generation of afterdepolarizations.

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Table 1 Properties of Ca2+ activated potassium channels. Adapted from [19] Channel type Property BK IK SK Ca2+ needed for activation 1–10 µM (at −50 mV) Voltage dependence Yes Single-channel conductance 100–250 pS

50–900 nM No 20–80 pS

50–900 nM No 4–20 pS

2 Small Conductance Ca2+ -Activated Potassium Channels 2.1 Electrophysiology As described in the introduction, the small-conductance Ca2+ -activated potassium channels (SK) are part of a family of potassium channels activated by calcium. In humans they are divided into three subtypes: large conductance or BK channels, which have very high conductance in the range from 100 to 300 pS, intermediate conductance or IK channels, with intermediate conductance ranging from 25 to 100 pS, and small conductance or SK channels with small conductances around 2–25 pS. Their properties are summarized in Table 1. The IUPAC nomenclature of the channels is K Ca1.1 (BK), K Ca3.1 (IK, or SK4) and K Ca2.1 , K Ca2.2 , K Ca2.3 (SK1, SK2, SK3). The BK channel is encoded in the gen KCNMA1 and, besides calcium activation dependence, it presents voltage-activated dependence. The small and intermediate conductance channels (SK1-4) are encoded by the family of genes KCNN1-4. Pharmacologically, SK channels (SK1-3) but not SK4 (IK) are sensitive to blockade by the bee toxin apamin. The SK channels are tetrameric proteins formed of four subunits. Three subtypes of SK α-subunits exists (SK1-3), which may form homomeric and heteromeric SK channel complexes, the latter consisting of more than one SK channel subtype. Each of the subunits has six hydrophobic alpha helical domains that insert into the cell membrane (Fig. 1). A loop between the fifth and sixth transmembrane domains forms the potassium ion selectivity filter. The SK channel gating mechanism is controlled by intracellular calcium levels [1, 46]. Calcium does not bind directly to the channel α-subunits, rather it binds to the protein calmodulin (CaM), through which the channels sense the intracellular Ca2+ . When bound to calcium, CaM binds to the CaMbinding domain on the intracellular subunit of the SK channel (Fig. 1). When each of the four CaM-binding domain subunits is bound to calmodulin, the SK channel changes conformation, causing the mechanical opening of the channel gate [1]. The time constant of SK channel activation is strongly dependent on Ca2+ concentration and lasts approximately 5 ms at [Ca2+ ] = 10 µM [2, 20]. When calcium levels are depleted, the time constant for channel deactivation is independent on calcium and ranges in a slower time scale from 15 to 60 ms [2]. These gating properties endow SK channels with a short-term memory for [Ca2+ ], i.e., they remain active for more than 100 ms after calcium concentration has returned to resting levels.

Electrophysiological Effects of Small Conductance …

23

Fig. 1 SK subunit, showing the six α-helical domains. The calmodulin (CaM) molecule functions as the channels Ca2+ sensor. Figure taken from [9]

SK channels are found close to L-type calcium channels at the cellular membrane [27]. However, besides calcium entering through the L-type channels, recent studies suggest that sarcoplasmic reticulum Ca2+ release is also required for the activation of cardiac SK channels [30, 41].

2.2 Models Calcium activated potassium (KCa) channels have been modeled in the literature with a sigmoidal dependence on intracellular calcium [23, 24] and typically assuming an ohmic dependence on transmembrane voltage (although a Goldman-Hodgkin-Katz current formulation has also been used by some authors [5]). The dependence with calcium is usually considered to be instantaneous, although some authors introduce a delay [15, 22]. The current can be written as: I K Ca = g K Ca f (Ca)(V − VK )

(1)

being VK the Nernst potential for the potassium ions and f (Ca) =

Ca q + Ca q

q K K Ca

(2)

The half activation Ca concentration K K Ca varies in the 0.05−0.9 µM range, with a single-channel conductance of 4–20 pS ([19], p. 144, with data based on [44]). The value K K Ca = 0.1 µM was found for SK3 channels by Carignani et al. [4]. Values used in the literature are summarized in Table 2. Some modifications of the previous formulation can be found in the literature. In the model by Cha et al. [5] the authors consider the constant field approximation or GHK formulation

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Table 2 Ca2+ dependence of the small conductance Ca 2+ -activated potassium channels, given by a sigmoidal dependence (Eq. (2)), with K K Ca the half activation Ca concentration and q the Hill coefficient References Cell type K K Ca q Engel et al. [15] Goforth et al. [17] Cha et al. [5] Fridlyand et al. [16] Chay et al. [6] Mears et al. [29]

neuronal cells pancreatic β-cells pancreatic β-cells pancreatic β-cells

I K Ca = PK Ca

2.5 µM 0.6 µM 0.74 µM 0.1 µM 1 µM 0.55 µM

2 5 2.2 4 3 5

  F V [K ]i − [K ]o exp − FRTV   f (Ca) RT 1 − exp − FRTV

(3)

with PK Ca = 0.2pA mM−1 , instead of an ohmnic dependence on the voltage, as in Eq. (1). The model is used to study pancreatic β-cells. The Ca2+ dependency for activation of I K Ca (SK) was adopted from Hirschberg et al. [20]. In the model used by Engel et al. [15] a time dependent calcium gate is introduced, with the current (4) I K Ca = g K Ca f 2 (Ca)(V − VK ) and the gate f (Ca) satisfying the equation: df = ( f ∞ − f )/τ dt

(5)

with a time constant τ = 3 ms. In this model the maximal conductance g S K ,max varies between 0 and 0.04 S/cm2 . This model was used to study the effect of the small conductance (SK) Ca2+ -dependent K+ channel on spike frequency adaptation in neuronal cells. The description they make of this current is a modified version of that for thalamic reticular neurons [11]. They use a value of the mid activation point K K Ca lower than in the original model in [11], based on experiments that gave values of K K Ca of less that one [24] and between 0.4–0.7 µM [23]. A K K Ca of 2.5 µM corresponds also well to the half-maximum activation of the SK current measured in hair cells (2 µM) [42]. A time dependency is also used in [20], where they consider the current given by Eq. (1), but with the gate f (Ca) satisfying Eq. (5). In this case the time constant is made dependent on calcium concentration and, following the experiments by Hirschberg et al. [20], it is considered to be: τ=

76 3.572Ca + 1

(6)

Electrophysiological Effects of Small Conductance …

25

This model was used to study the influence of calcium-sensitive potassium currents in the appearance and nature of calcium alternans.

3 Effect on the Cardiac Action Potential In order to investigate the effect of the SK current on the atrial action potential, we considered the computational model described by Eq. (1) for I K Ca , and integrated it into a electrophysiology detailed ionic model of an atrial myocyte. We selected the Lugo et al. atrial model [28] as a benchmark to show the main effects of the SK current, although three other atrial models developed by Nygren et al. [32], Courtemanche et al. [10] and Grandi et al. [18] have also been analyzed. We first investigate how changes in SK physiology may affect the action potential in the Lugo model (see Fig. 2). As standard parameters of the I K Ca current, we set the conductivity in the upper range considered in the bibliography, at g˜ = g/Cm = 0.005 nS/pF, and the gating variable to have a time dependence, with a fast time scale τ = 5 ms. Regarding calcium dependence, we take cooperativity to be given by the Hill coefficient q = 2 and set the calcium half-activation constant at K K Ca = 700 nm, in the middle range of the values reported (see Table 2). Clearly, the effect is larger at faster pacing rates, where we observe both a decrease in the action potential duration, and a shift of the resting potential to more negative values. As there is a large disparity in the values of the parameters of the I K Ca current (see Table 2), we have studied the sensitivity of the results under variations of these parameters. In particular, we study the effects of variations in the calcium activation process, as well as changes in the conductance and gate dynamics. We center our analysis here in changes in APD90 , this is, the time elapsed from depolarization until 90% repolarizations is reached. At a pacing rate of 1 Hz, there is a bimodal dependence of APD90 with SK current conductance g˜ (Fig. 3a). For instantaneous ˜ before calcium activation (τ = 0), the APD90 first increases as a function of g, it decreases, as one would expect from its role as a repolarization current. When

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Fig. 3 APD variation for different I K Ca parameters, in the Lugo model, at a stimulation period of T = 1000 ms. a APD versus conductivity of SK channel for time constant activation τ = 0, 5 and 75 ms, with q = 2 and K K Ca = 700 nM. b APD as function of τ for different values of the conductivity g˜ = 0.001, 0.003 and 0.01 nS/pF, with q = 2 and K K Ca = 700 nM. c, d APD versus half-activation Ca constant, K K Ca , for τ = 0, 5 and 75 ms, with q = 2 and conductivity c g˜ = 0.001 nS/pF, d g˜ = 0.01 nS/pF

calcium activation is made time dependent (at sufficiently slow gate dynamics) this initial increase in APD90 is not observed, but there is still an increase in APD90 at a value of g˜  0.02 − 0.03 nS/pF. In fact, from the change of APD90 with τ (Fig. 3b) one can conclude that there are two very different regimes, one at large τ > 50 ms, i.e., for slow gate kinetics, and another one, for small τ < 10 ms, i.e., for fast gate kinetics. These results give us characteristic values for the effects of the SK conductances on the APD. Conductances around g˜ = 0.001 nS/pF are basically small and their effects barely noticeable. Meanwhile, conductances around g˜ = 0.01 nS/pF have strong effects and correspond to values of conductivity which are within the range observed in experimental data. We should thus test the influence of calcium activation for both relatively small and relatively large conductances. For small conductances g˜ = 0.001 nS/pF (Panel C in Fig. 3) APD90 remains almost constant for all values of the half-activation constant, confirming that this is indeed a small value of the I K Ca conductance. Very different is the situation for g˜ = 0.01 nS/pF (Fig. 3d). Again, different behaviour is observed at slow and fast gate dynamics. While a higher value of τ = 75 ms smooths out the effect of changes in calcium

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Fig. 4 APD variation for different I K Ca parameters, in the Lugo model, at a stimulation period of T = 500 ms. a APD versus conductivity of SK channel for time constant activation τ = 0, 5 and 75 ms, with q = 2 and K K Ca = 700 nM. b APD as function of τ for different values of the conductivity g˜ = 0.001, 0.003 and 0.01 nS/pF, with q = 2 and K K Ca = 700 nM. c, d APD versus half-activation Ca constant, K K Ca , for τ = 0, 5 and 75 ms, with q = 2 and conductivity c g˜ = 0.001 nS/pF, d g˜ = 0.01 nS/pF

half-activation, fast dynamics unveils a drastic drop in APD90 as half-activation is reduced below 500 nM. Finally, we have checked the effect of higher cooperativity (with q up to 6), finding that it does not affect appreciably the results. We have repeated the previous analysis for a stimulation rate of 2 Hz, as shown in Fig. 4. The general behavior is the same with one important exception. In this case, APD90 does not show the initial increase as a function of g˜ at fast gating kinetics. Rather, it appears at values of g˜ ∼ 0.02 nS/pF and results into a bifurcation giving rise to alternans at values g˜ ∼ 0.025 nS/pF. This bifurcation does not appear for τ = 75 ms, showing again that there are two very different regimes for fast and slow gate dynamics. Let us notice that alternans appears at rather large values of the conductivity which would correspond to overexpression of SK in real cells. The rate dependence of the changes in APD can be observed in the dynamic restitution curves, i.e., the dependence of APD90 on the diastolic intervals, this is, time elapsed since the last repolarization occurred. As already shown in Fig. 2, the effect of the I K Ca current is larger at fast stimulation rates (Fig. 5a). This could explain the appearance of alternans for large values of g, ˜ as the slope of the restitution curve

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Fig. 5 Dynamical restitution curves with the Lugo model. a For conductivity values g˜ = 0.001, 0.003 and 0.005 nS/pF, with τ = 5 ms, K K Ca = 700 nM, and q = 2. b For τ = 0, 5 and 75 ms, with g˜ = 0.003 nS/pF and the same values of q and K K Ca . c For half activation Ca constants, K K Ca = 300 and 700 nM, for τ = 5 ms, q = 2 and g˜ = 0.003 nS/pF. Black lines correspond to restitution curves of the Lugo model when I K Ca is not considered

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becomes large. Interestingly, at short pacing periods, the slope does not depend on the gate dynamics (Fig. 5b), but is sligthly dependent on the Ca half-activation (Fig. 5c). We can get a better understanding on the effect of the I K Ca current, plotting its form under different values of the parameters (Fig. 6). First we show that for typical conductances the current is around 0.2–0.3 pA/pF and that a slow gating dynamics smooths the profile of the current (see panel 6d). Changes produced by calcium halfactivation are not very strong as long as its value is around 500 nM (see panel 6f). Generaly speaking, we notice that the current, at least for fast activation of its gate, produces appreciable variations in the phases two and three of the AP, increasing the notch and decreasing the dome. The last phase of repolarization only changed at fast stimulation rates. One may wonder how dependent these results are on the specific atrial model considered. In order to answer this question, we have also incorporated the I K Ca current to the models of Nygren, Courtemanche and Grandi. The resulting changes

Electrophysiological Effects of Small Conductance …

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Fig. 6 Action potential and I K Ca current for a stimulation period of T = 500 ms. Each pannel represents de AP and I K Ca current for the corresponding point in the pannels of Fig. 5. a For efective conductivity values g˜ = 0.001, 0.003 and 0.005 nS/pF. Time constant activation τ = 5 ms, half activation Ca concentration, K K Ca = 700 nM, Hill coefficient q = 2. b For τ = 0, 5 and 75 ms. Efective conductivity g˜ = 0.003 nS/pF. The same values than in a for q and K K Ca . c For half activation Ca concentration, K K Ca = 300 and 700 nM, for τ = 5 ms, q = 2 and g˜ = 0.003 nS/pF

in AP duration and other properties of the action potential, such as APD50 , minimum and maximum of the potential and the maximum depolarization speed, are shown in Table 3, for a pacing rate of 1 Hz, and in Table 4, for 2 Hz. The results shown in the tables indicate that the effect in Lugo model is not particularly different from that in Nygren or Courtemanche. The three models present a very small change in APD90 at a pacing of 1 Hz, and a moderate effect (a change of ∼ 10% or less) at 2 Hz. This effect is more acused on the APD50 but, apart from this, the rest of the

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Table 3 Properties of action potential for different atrial models, at T = 1000 ms, corresponding to g˜ = 0.005 nS/pF, τ = 5 ms, q = 2 and K K Ca = 700 nM. In parenthesis we show, either the change with respect to the original models, or the values obtained with the original models Action Potential Property Channel type Lugo Nygren Courtemanche Grandi Action Potential duration (ms) at 90% repolarization at 50% repolarization Maximum membrane potential (mV) Minimum membrane potential (mV) Maximum rate of variation of membrane potential (mV/ms)

225.99 (0.25%) 209.39 (0.46%) 267.74 (6.07%) 248.85 (17.03%) 96.44 (0.90%) 25.86 (21.37%) 139.31 (6.73%) 49.51 (23.13%) 40.39 (40.53) 41.54 (41.65) 37.56 (37.55) 30.09 (29.73) −75.86 (−75.19) 167.77 (165.53)

−74.89 (−74.42) 166.11 (164.63)

−80.84 (−80.66) 220.18 (220.18)

−73.85 (−73.68) 124.54 (115.8)

Table 4 Properties of Action Potential for different atrial models, at T = 500 ms, corresponding to g˜ = 0.005 nS/pF, τ = 5 ms, q = 2 and K K Ca = 700 nm. In parenthesis we show, either the change with respect to the original models, or the values obtained with the original models Action potential property

Channel type Lugo

Nygren

Courtemanche

Grandi

at 90% repolarization

192.41 (9.0%)

138.06 (4.27%)

243.53 (10.61%)

203.66 (18.32%)

at 50% repolarization

66.65 (17.43%)

22.20 (10.23%)

94.95 (19.20%)

45.19 (26.89%)

Maximum membrane potential (mV)

37.95 (38.73)

40.00 (40.46)

37.20 (37.05)

29.57 (26.88)

Minimum membrane potential (mV)

−76.65 (−74.48) −73.95 (−72.99) −79.54 (−79.10) −76.07 (−75.28)

Maximum rate of variation of membrane potential (mV/ms)

163.78 (156.45)

Action potential duration (ms)

166.07 (164.62)

220.18 (220.18)

129.24 (95.28)

characteristics of the AP are barely affected by the inclusion of the I K Ca current. The effect in the Grandi model is quite different from the rest, with a large variation in APD and the other characteristics of the AP, even at a slow pacing rate of 1 Hz. Experiments with inhibition or knock-out of these channels show a large variation in the value of the APD90 , ranging from 12–14% [35] to 30% [47] in human atria up to ∼ 40% in mice [25, 51]. Thus, for the standard parameters considered the Grandi model is the one that gets a better agreement with the observed values. For the other models, the condutance neccessary to obtain this change in APD is larger than the one described in the literature.

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Fig. 7 Comparison of the effect of the I K Ca current in the action potential of the Lugo et al. (a) and Grandi et al. models (b). The black line corresponds to the original models, while the red and blue lines correspond to the modified Lugo and Grandi models, respectively. We set the T = 1000 ms, g˜ = 0.005 nS/pF, τ = 5 ms, q = 2 and K K Ca = 700 nM. In panels c and d we show subsarcolemmal calcium and the I K Ca current, for the two models

To get a better idea of the reason behind these differences, in Fig. 7 we show how the I K Ca current modifies the action potential in the models by Lugo [28] and Grandi [18]. The SK current produces a noticeable, but rather small, effect when introduced in the model developed by Lugo et al. The main effect occurs just after the depolarization process, precisely when the SK current is expected to be larger given the rise in calcium levels, which activate the SK channels. This, however, has very little effect in the action potential duration. A very different situation occurs for the model by Grandi et al., where I K Ca has a very important effect in repolarization. As shown in Fig. 7d, I K Ca currents are roughly equally in magnitude (∼ 0.3 pA/pF) in the Grandi and Lugo models. The difference in the effect of the I K Ca current in both cases is related to their calcium transients (Fig. 7c), that in the Grandi model lasts for longer. Since I K Ca is a calcium activated current, it persists longer than the other, voltage-activated, potassium currents (Fig. 8). This way, it becomes comparable to the other currents in the last stages of repolarization. This is particularly noticeable in the Grandi model, where the calcium transient is much longer than typical times for voltage inactivation of the potassium currents. In the Lugo model, on the other

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Fig. 8 Sum of potassium currents (except I K Ca ) and sodium-calcium exchanger I N aCa for the Lugo (left pannels) and Grandi (right pannels) models

hand, its effect is largest during the notch and phase 2, when the peak in intracellular calcium is largest. Although the focus of the present work is directed towards measures of APD variations, which are important in themselves, one point to consider is the response to spontaneous calcium release (SCR). As has been mentioned, I K Ca currents could reduce the effect of SCR. In order to analyze this posibility we simulated situations in which a spontaneous release occurred due to an opening of ryanodine receptors. We compared similar conditions with and without I K Ca current. Results for a stimulation period of T = 1000 ms are presented in Fig. 9. The picture on the left shows a situation in which the I K Ca current reduces the elevation of the membrane potential that a spontaneous calcium release would produce. The one on the right shows a case in which, without the effect of the I K Ca current, an extra depolarization would occur, which would give rise to an extrasystole beat. This clearly shows how SK channels avoid it.

Electrophysiological Effects of Small Conductance …

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Fig. 9 Effect of I K Ca current on spontaneous calcium release for a stimulation period of T = 1000 s. q = 2, K K Ca = 700 nM and g˜ = 0.005 nS/pF and τ = 5 ms a The fraction of open RyR channels reaches a value of 5% during 10 ms at a time in the midle of the beat. The same case without I K Ca current is represented for comparison. b The fraction of open RyRs reaches a fraction of 6% during 3 ms at the same instant of time

4 Discussion and Conclusions Studies on SK channels lead to controversy regarding its functionality and its possible pro/anti-arrhythmic effect. As a first attempt to broaden the knowledge of these channels, we performed numerical simulations of whole-cell atrial myocyte models where we introduce this current through a general expression for the SK channels. The channel gate is modeled with a sigmoidal function that depends on calciumactivation with different possible cooperativities and activation levels. Results obtained with the Lugo et al. model [28] show a big dependence of the I K Ca current with its conductivity and the type of gate dynamics. It behaves very differently whether gate dynamics is fast (below 10 ms) or slow (above 30 ms). A change in the calcium sensitivity of the channels, by changing the half activation Ca constant, K K Ca , also shows two different regimes. For K K Ca above ∼ 500 nM, lowering the sensitation of the channel (increasing K K Ca ) does not change much the value of the APD. On the contrary, decreasing it below this value produces a fast reduction of the APD. By introducing the I K Ca current in other atrial models, like Nygren, Courtemanche and Grandi [10, 18, 32], we have observed that the effects of the current are qualitatively the same. Quantitatively, however, each model has its own characteristics. For instance, in the models that present a dome (Lugo, Courtemanche) there is an initial increment of APD as a function of SK channel conductance, until the AP loses the dome, followed by a rapid decrease in APD. For the others (Nygren and Grandi) the APD reduction is monotonical with channel conductance. Another interesting finding is the occurrence of alternans when pacing with a period around 500 ms at high channel conductances and fast gating kinetics. This, together with the reduction in APD, could be related to the pro-arrhythmic effect that has been observed in cases of channel overexpression.

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It has been speculated than one of the protective effects of I K Ca against AF could be to avoid afterdepolarizations resulting from spontaneous calcium release, as I K Ca is a repolarization current activated by calcium. In this case, I K Ca current acts as antiarrhythmic factor, reducing the membrane potential elevation and avoiding extra depolarizations. From our preliminary simulations, it is not clear whether this protective effect stems from the current itself that counteracts the effect of the sodiumcalcium exchanger, or from the change in the resting potential, that presumably affects the refractory period. In any case, the presence of the I K Ca current only prevents SCR induced afterdepolarizations in a narrow region of parameters. Clearly, further work has to be done in order to elucidate this important matter. There are further details of the model that could be taken into account in future work. For instance, the I K Ca current is modeled as linear, while it is well known that it presents positive rectification, such that it saturates for transmembrane potentials above ∼ −50 mV. We have neither considered the calcium dependence of the channel gate time constant. In this work we have considered a model for a general SK channel, without any distinction of channels subtype (SK1-3), although it is known that they are expressed differentially in different types of tissues. We have centered on the effect of the I K Ca in atrial cells, where it has been observed to be preferentially expressed. It could be interesting to study also its effects on ventricular cells, where, during heart failure, it has been shown to be overexpressed and to play an important role. Even with these limitations, the present work represents a first step to try to understand the effects of the small conductance calcium-activated potassium channel, that has been shown to play an important role in the transition to atrial fibrillation. Acknowledgements We thank L. Hove-Madsen for fruitful discussions. The authors acknowledge financial support from Fundació La Marató de TV3 and from the Spanish Ministerio de Economía y Competitividad (MINECO) under grant numbers SAF2014-58286-C2-2-R, SAF2017-88019-C32-R and FIS2015-66503-C3-2P. IRC also acknowledges financial support from the Generalitat of Catalonia under Project 2009SGR878.

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26. Ling, T.-Y., Wang, X.-L., Chai, Q., Lau, T.-W., Koestler, C.M., Park, S.J., Daly, R.C., Greason, K.L., Jen, J., Wu, L.-Q., et al.: Regulation of the SK3 channel by microRNA-499—potential role in atrial fibrillation. Heart Rhythm 10(7), 1001–1009 (2013) 27. Lu, L., Zhang, Q., Timofeyev, V., Zhang, Z., Young, J.N., Shin, H.-S., Knowlton, A.A., Chiamvimonvat, N.: Molecular coupling of a Ca2+-activated K+ channel to l-type Ca2+ channels via α-actinin2. Circ. Res. 100(1), 112–120 (2007) 28. Lugo, C.A., Cantalapiedra, I.R., Peñaranda, A., Hove-Madsen, L., Echebarria, B.: Are SR Ca content fluctuations or SR refractoriness the key to atrial cardiac alternans?: insights from a human atrial model. Am. J. Physiol.-Heart Circ. Physiol. 306(11), H1540–H1552 (2014). https://doi.org/10.1152/ajpheart.00515.2013 29. Mears, D., Sheppard Jr., N., Atwater, I., Rojas, E., Bertram, R., Sherman, A.: Evidence that calcium release-activated current mediates the biphasic electrical activity of mouse pancreatic β-cells. J. Membr. Biol. 155(1), 47–59 (1997) 30. Mu, Y.-H., Zhao, W.-C., Duan, P., Chen, Y., Wang, Q., Tu, H.-Y., Zhang, Q., et al.: RyR2 modulates a Ca2+-activated K+ current in mouse cardiac myocytes. PloS one 9(4), e94905 (2014) 31. Nattel, S., Qi, X.Y.: Calcium-dependent potassium channels in the heart: clarity and confusion. Cardiovasc. Res. 101(2), 185–186 (2014). https://doi.org/10.1093/cvr/cvt340 32. Nygren, A., Fiset, C., Firek, L., Clark, J., Lindblad, D., Clark, R., Giles, W.: Mathematical model of an adult human atrial cell. Cardiovasc. Res. 82(1), 63–81 (1998) 33. Olesen, M.S., Jabbari, J., Holst, A.G., Nielsen, J.B., Steinbrüchel, D.A., Jespersen, T., Haunsø, S., Svendsen, J.H.: Screening of KCNN3 in patients with early-onset lone atrial fibrillation. Europace 13(7), 963–967 (2011) 34. Özgen, N., Dun, W., Sosunov, E.A., Anyukhovsky, E.P., Hirose, M., Duffy, H.S., Boyden, P.A., Rosen, M.R.: Early electrical remodeling in rabbit pulmonary vein results from trafficking of intracellular SK2 channels to membrane sites. Cardiovasc. Res. 75(4), 758–769 (2007) 35. Qi, X.-Y., Diness, J.G., Brundel, B.J., Zhou, X.-B., Naud, P., Wu, C.-T., Huang, H., Harada, M., Aflaki, M., Dobrev, D., et al.: Role of small-conductance calcium-activated potassium channels in atrial electrophysiology and fibrillation in the dog. Circulation 129(4), 430–440 (2014) 36. Rafizadeh, S., Zhang, Z., Woltz, R.L., Kim, H.J., Myers, R.E., Lu, L., Tuteja, D., Singapuri, A., Bigdeli, A.A.Z., Harchache, S.B., et al.: Functional interaction with filamin a and intracellular Ca2+ enhance the surface membrane expression of a small-conductance Ca2+-activated K+ (SK2) channel. Proc. Natl. Acad. Sci. 111(27), 9989–9994 (2014) 37. Skibsbye, L.: Antiarrhythmic principle of SK channel inhibition in atrial fibrillation. Channels 57, 672–681 (2011) 38. Skibsbye, L., Diness, J.G., Sørensen, U.S., Hansen, R.S., Grunnet, M.: The duration of pacinginduced atrial fibrillation is reduced in vivo by inhibition of small conductance Ca2+-activated K+ channels. J. Cardiovasc. Pharmacol. 57(6), 672–681 (2011) 39. Skibsbye, L., Poulet, C., Diness, J.G., Bentzen, B.H., Yuan, L., Kappert, U., Matschke, K., Wettwer, E., Ravens, U., Grunnet, M., et al.: Small-conductance calcium-activated potassium (SK) channels contribute to action potential repolarization in human atria. Cardiovasc. Res. 103(1), 156–167 (2014) 40. Stocker, M.: Ca2+-activated K+ channels: molecular determinants and function of the SK family. Nat. Rev. Neurosci. 5(10), 758–770 (2004) 41. Terentyev, D., Rochira, J.A., Terentyeva, R., Roder, K., Koren, G., Li, W.: Sarcoplasmic reticulum Ca2+ release is both necessary and sufficient for SK channel activation in ventricular myocytes. Am. J. Physiol.-Heart Circ. Physiol. 306(5), H738–H746 (2014) 42. Tucker, T.R., Fettiplace, R.: Monitoring calcium in turtle hair cells with a calcium-activated potassium channel. J. Physiol. 494(Pt 3), 613 (1996) 43. Tuteja, D., Xu, D., Timofeyev, V., Lu, L., Sharma, D., Zhang, Z., Xu, Y., Nie, L., Vázquez, A.E., Young, J.N., et al.: Differential expression of small-conductance Ca2+-activated K+ channels SK1, SK2, and SK3 in mouse atrial and ventricular myocytes. Am. J. Physiol.-Heart Circ. Physiol. 289(6), H2714–H2723 (2005)

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44. Vergara, C., Latorre, R., Marrion, N.V., Adelman, J.P.: Calcium-activated potassium channels. Curr. Opin. Neurobiol. 8(3), 321–329 (1998) 45. Wang, W., Watanabe, M., Nakamura, T., Kudo, Y., Ochi, R.: Properties and expression of Ca2+-activated K+ channels in H9c2 cells derived from rat ventricle. Am. J. Physiol.-Heart Circ. Physiol. 276(5), H1559–H1566 (1999) 46. Xia, X.-M., Fakler, B., Rivard, A., Wayman, G., Johnson-Pais, T., Keen, J., Ishii, T., Hirschberg, B., Bond, C., Lutsenko, S., et al.: Mechanism of calcium gating in small-conductance calciumactivated potassium channels. Nature 395(6701), 503–507 (1998) 47. Xu, Y., Tuteja, D., Zhang, Z., Xu, D., Zhang, Y., Rodriguez, J., Nie, L., Tuxson, H.R., Young, J.N., Glatter, K.A., et al.: Molecular identification and functional roles of a Ca2+-activated K+ channel in human and mouse hearts. J. Biol. Chem. 278(49), 49085–49094 (2003) 48. Yu, T., Deng, C., Wu, R., Guo, H., Zheng, S., Yu, X., Shan, Z., Kuang, S., Lin, Q.: Decreased expression of small-conductance Ca2+-activated K+ channels SK1 and SK2 in human chronic atrial fibrillation. Life Sci. 90(5), 219–227 (2012) 49. Zhang, L., McBain, C.J.: Potassium conductances underlying repolarization and afterhyperpolarization in rat CA1 hippocampal interneurones. J. Physiol. 488(Pt 3), 661 (1995) 50. Zhang, X.-D., Lieu, D.K., Chiamvimonvat, N.: Small-conductance Ca2+-activated K+ channels and cardiac arrhythmias. Heart Rhythm 12(8), 1845–1851 (2015) 51. Zhang, X.-D., Timofeyev, V., Li, N., Myers, R.E., Zhang, D., Singapuri, A., Lau, V.C., Bond, C.T., Adelman, J., Lieu, D.K., et al.: Critical roles of a small conductance Ca2+-activated K+ channel (SK3) in the repolarization process of atrial myocytes. Cardiovasc. Res. 101(2), 317–325 (2013). https://doi.org/10.1093/cvr/cvt262

Spontaneous Mirror Symmetry Breaking from Recycling in Enantioselective Polymerization David Hochberg, Celia Blanco and Michael Stich

Abstract A key challenge for origin of life research is understanding how the homochirality of extant biological systems may have emerged during the abiotic phase of chemical evolution. Living systems depend on bio-macromolecules made from chiral building blocks and a crucial question is the relationship of polymerization with the emergence of homochirality. We present a reaction scheme demonstrating how spontaneous mirror symmetry breaking (SMSB) can be achieved in enantioselective polymerization without chiral inhibition and without autocatalysis. The model is based on nucleated cooperative polymerization: nucleation, elongation, dissociation, fusion and fragmentation and monomer racemization. These are microreversible processes subject to constraints dictated by chemical thermodynamics. To maintain this closed system out of equilibrium, we model an external energy source which induces the irreversible breakage of the longest polymers in the system. Simulations reveal that SMSB can be achieved starting from the tiny intrinsic statistical fluctuations about the idealized mirror symmetric composition.

D. Hochberg Department of Molecular Evolution, Centro de Astrobiología (CSIC-INTA), Carretera Ajalvir Km 4, 28850 Torrejón de Ardoz, Madrid, Spain e-mail: [email protected] C. Blanco Department of Chemistry and Biochemistry, University of California, Santa Barbara, CA 93106-9510, USA e-mail: [email protected] M. Stich (B) Non-linearity and Complexity Research Group, System Analytics Research Institute, School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. Carballido-Landeira and B. Escribano (eds.), Biological Systems: Nonlinear Dynamics Approach, SEMA SIMAI Springer Series 20, https://doi.org/10.1007/978-3-030-16585-7_3

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Fig. 1 A chiral molecule is one that is not superposable on its mirror image; the molecule and its mirror image are then called enantiomers. A homochiral macromolecule is one whose constituents (the smaller molecular building blocks) have the same chirality. Biology uses only homochiral molecules, such as amino acids and sugars. Fundamental questions are how this mirror, or chiral, symmetry is broken and how the tiny initial enantiomeric excesses can be amplified in chemical evolution

1 Introduction Theoretical proposals in prebiotic chemistry suggest that homochirality emerged in nature in abiotic times via deterministic or chance mechanisms [1, 7, 12]. The abiotic scenario for the emergence of single homochirality in the biological world implies that single asymmetry could have emerged provided a small chiral fluctuation with respect to the racemic state can be amplified to a state useful for biotic evolution (see Fig. 1). For this reason, experiments and theoretical modeling that demonstrate the feasibility of stochastic mirror symmetry breaking are particularly important [3, 10, 13, 15, 16, 20, 21, 24, 26, 27]. Once generated by chance or an initial chiral fluctuation, chirality can then be transmitted to the rest of the system provided that the symmetry breaking step is coupled to a sequential step of efficient amplification either via self-replication reactions or via other alternative non-linear transformations. Relevant features common to all such systems are that they take into account the small fluctuations about the mirror symmetric (racemic) state and they display nonlinear kinetic effects [22]. The observed bias in biopolymers composed from homochiral L-amino acids and D-sugars towards a single handedness or chirality is a remarkable feature of biology. In this chapter, we review a mathematical model for achieving spontaneous mirror symmetry breaking (SMSB) in chiral polymerization in which the homochiral interactions dominate over the heterochiral ones [5], as is supported experimentally in reports on the polymerization of amino acids and nucleotides [14, 17, 18, 25]. Most interestingly, mechanical forces in chemistry have been demonstrated to provide selection pressures in various experimental settings [6, 23]. We thus consider mechanically induced fragmentation of the longest polymers, which here serves a

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two-fold purpose. On the one hand, SMSB can only take place in far from equilibrium systems, and external sources of energy applied to closed systems (closed to matter exchange with the environment) can serve this purpose. On the other hand, these external sources must act selectively on some, but not all, of the chemical species of the system.1 In accord with this latter requirement, the chain breakage modeled herein acts on only the longest chains in the polymer population. The irreversible chain breakage in concert with the reversible chain fusion establishes a unidirectional recycling mechanism which is the key for making SMSB thermodynamically feasible in this system. The other model processes treated here namely, monomer racemization, nucleation, chain-elongation, chain-fusion, are all micro-reversible reactions subject to the pertinent thermodynamic constraints.

2 Absolute Asymmetric Synthesis We define spontaneous mirror symmetry breaking (SMSB) as synonymous with thermodynamically controlled absolute asymmetric synthesis, that is, as a chemical transformation where the racemic or mirror symmetric state is metastable and a pair of energetically degenerate chiral states are the genuine stable states of the system [22]. Clearly, this can occur only under experimental conditions that can maintain the system, whether open or closed,2 out of equilibrium, otherwise chemical thermodynamics determines the thermodynamic racemic equilibrium as the final stable state. It should be noted however, that in closed systems able to evolve or relax towards thermodynamic equilibrium, kinetically controlled absolute asymmetric synthesis is possible, that is, a temporary amplification of chirality can take place that may be useful in applied synthesis. These temporary chiral excursions can have reasonably long durations, the excursion time scale depending of course on the system parameters [4, 8]. In SMSB the entire system breaks mirror/chiral symmetry leading to a transition away from the racemic mixture which is triggered by intrinsic statistical fluctuations around the ideal racemic composition. This symmetry breaking can be described as a bifurcation where the racemic mixture is a metastable state and the chiral states correspond to free energy wells of minimum entropy production [2]: chiral reaction noise suffices to take the system out from the racemic composition and towards one of the two stationary enantiomeric states, i.e. left-right symmetry (parity or space inversion followed by a rotation) is broken in the entire system. In this respect, a variety of thermodynamic system configurations (known as “architectures”) can be contemplated, for example: (a) systems open to matter transfer of some of the 1 If

external energy, e.g., mechanical grinding, is supplied to all the species, then the system can equilibrate with its surroundings with a corresponding shift in its equilibrium constants and reaction rate constants. The system merely evolves to a new thermodynamic equilibrium. Only in the case that such energy is supplied to some of the species can the system be kept out of equilibrium. 2 As defined in chemistry, closed systems are those which do not exchange matter with their surroundings, open systems can exchange both matter and energy with their surroundings.

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chemical species, for example, via semi-permeable membranes; (b) compartmentalization of specific reactions, for example by anchoring specific catalysts on surfaces; (c) closed systems with permanent non-isotropic temperature distributions or; (d) specific energy uptake by only some of the chemical species. The case we treat here is a prime example of this latter category: indeed, the external mechanical energy supplied to the system selectively breaks only those oligomer species within a specific size range, typically the longest polymers in the system, and it does so irreversibly.

3 The Polymerization Model Let L k , Rk denote the handedness of the enantiomers, where k denotes the number of monomer units in each one.3 The steps defining our polymerization model can be written as the set of following reactions with their associated reaction rate constants (see Table 1 for definitions): L 1  R1

(1)

L 1 + L j  L j+1 , R1 + R j  R j+1 (1 ≤ j ≤ N − 1), (2) Li + L j  Ln, Ri + R j  R n , (3) (i ≤ j, i min f ≤ i ≤ N − jmin f , jmin f ≤ j ≤ N − i min f , i + j = n). The monomer racemization (1) proceeds in both directions with the rate constant kr . The stepwise monomer attachment reactions (2) proceed with forward, reverse rate constants k1 , k−1 , respectively, within the isodesmic nucleation phase (1 ≤ j ≤ n c − 1), and with forward, reverse rate constants k2 , k−2 , respectively, within the isodesmic chain-elongation phase (n c ≤ j ≤ N − 1). The binary chain fusion and the chain fragmentation reactions (3) proceed with rate constants ka , k−a , respectively. Most importantly, the rates of fusion/fragmentation and elongation/ dissociation are constrained by chemical thermodynamics [5]: k2 ka = , k−a k−2

(4)

and must be taken into account in all the simulations. We point out that there is no mutual or chiral inhibition in this scheme, that is, no reaction involving both an L i and an R j on the same side of a chemical transformation, and there is no autocatalysis. The two homochiral populations of oligomers are coupled together only through the monomer racemization reaction (1), see (a) in Fig. 2. Both chiral populations compete for the limited monomer resources through this step. 3 These are compositional enantiomers, i.e, homochiral linear structures. We make no statement regarding their spatial conformation, nevertheless, since L k , Rk are energetically degenerate structures, they will have equal access to the same set of available folding patterns.

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Table 1 Model parameters and derived quantities. Length refers to the number of monomer units within the oligomer. The overall free energy profile for cooperative nucleated supramolecular polymerization, is given by the schematic diagram Fig. 24 of Ref. [11]. We treat cooperative nucleated polymerization as k1 < k−1 , k2 > k−2 . See also Fig. 2. The rate constants k2 , k−2 and ka , k−a obey a thermodynamic constraint, see (4). Table adapted and reprinted with permission from J. Chem. Phys. B 2017, 121, pp. 942–955. Copyright 2017 American Chemical Society Parameters nc N n min i min i min f jmin f γ k1 , k−1 k2 , k−2 ka , k−a kr ckL ,R (t) Derived N − i min N − jmin f N − i min f i min f + jmin f

Critical nucleus (n c ≥ 2) for cooperative nucleated polymerization Maximum length polymers Minimum length polymers subject to mechanical breakage Smallest length polymer fragment due to mechanical breakage Minimum length polymers that can fuse with the long polymers Minimum length longer polymers fusing with the short polymers Rate of irreversible mechanical breakage of the polymers Forward, reverse rates of monomer addition (nucleation) Forward, reverse rates of monomer addition (elongation) Rate of chain-chain fusion, rate of binary fragmentation Racemization rate of the monomers Time dependent concentration of the k-mer, for chirality L , R Maximum length breakage fragment Maximum length shorter polymers fusing with the long polymers Maximum length longer polymers fusing with the short polymers Minimum length of polymers formed by binary fusion

The processes (b), (c), (d) and (e) depicted in Fig. 2 can take place generally over various oligomer size ranges (see Table 1), and for this reason, it is worthwhile to consider in detail each one individually.

3.1 Racemization, Nucleation and Chain Growth We first consider the nucleation (a) and linear stepwise elongation processes (b) (see Fig. 2) represented by (2). Chain growth by stepwise monomer attachment takes place over the entire size range from the dimer up to the maximum length polymer minus one (N − 1); and so covers both the nucleation and elongation phases. The two distinct isodesmic ranges of the free energy profile (see Table 1) leads naturally to dividing the associated differential kinetic rate equations into the four size range groupings listed below, where α = L or R is a chirality label.

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Fig. 2 Schematic illustrating the microscopic processes of nucleated enantioselective polymerization treated in this chapter a: The two chiral populations of oligomers, built up from either blue (L) or red (R) enantiomeric monomer units, are brought into competition for limited resources in closed systems via the solution phase racemization of the monomers. b: Primary isodesmic nucleation in which a nucleus of size n c ≥ 2 is formed by successive monomer attachments, every step being reversible. c: The post-critical nucleus sized oligomers grow by stepwise attachment of monomers of the same chirality, and they can lose a monomer via dissociation. d: Oligomers within a certain range of lengths can fragment into two smaller pieces and two smaller oligomers can also fuse together forming a longer chain. e: Mechanical forces can break irreversibly those chains greater than a certain minimum length into two smaller fragments. The breakage e and fusion d steps lead to a recycling mechanism for the oligomers. The rate constants for all the microscopic processes in a, b, c, d and e are identical for both chiral populations. Thus the scheme is mirror symmetric. See also Table 1 for definitions of all the model parameters

α α c˙kα (t) = k1 c1α (ck−1 − ckα ) + k−1 (ck+1 − ckα )

c˙nαc (t) c˙kα (t) c˙αN (t)

= = =

2 ≤ k ≤ n c − 1,

k1 c1α cnαc −1 − k−1 cnαc − k2 c1α cnαc + k−2 cnαc +1 α α k2 c1α (ck−1 − ckα ) + k−2 (ck+1 − ckα ) nc + α α α k2 c1 c N −1 − k−2 c N (k = N ).

k = nc , 1 ≤ k ≤ N − 1,

(5) (6) (7) (8)

The first size group (5) includes the dimer up to critical nucleus minus one monomer unit, followed by the critical nucleus itself (6), the third size group (7) includes the critical nucleus plus one unit up to the maximum length oligomer minus one, and then the final size N (8) is the maximum length oligomer by itself. The steps in (5) are endergonic while those in (7), (8) are exergonic. The monomer dynamics follow directly from the conservation of total system mass and from the solution-phase racemization Eq. (1) thus:

Spontaneous Mirror Symmetry Breaking from Recycling …

c˙1L ,R (t) = −

  d   L ,R  k ck (t) + kr c1R,L − c1L ,R dt k=2 N

= −2k1 (c1L ,R )2 + 2k−1 c2L ,R − k1

n c −1 k=2

− k2

45

N −1 

c1L ,R ckL ,R + k−2

k=n c

N 

c1L ,R ckL ,R + k−1

nc 

ckL ,R

k=3

  ckL ,R + kr c1R,L − c1L ,R .

(9)

k=n c +1

The second and third lines in (9) follow from substituting (5)–(8) into the total derivative term in the first line.

3.2 Chain Fusion and Fragmentation We next turn to the fusion and fragmentation of the chains, as represented in Eq. (3). Consider first the contributions to the overall set of rate equations due to binary annealing or fusion of short with long polymers. Then, we compute the contributions due to binary fragmentation of the chains, treating fragmentation as the inverse process of fusion. Note: we herein employ the term “breakage” to refer to mechanically induced irreversible breakage, and “fragmentation” to refer to the inverse process of fusion. The general fusion/fragmentation scenario is depicted schematically by (d) in Fig. 2. As before, we consider both n c and N to be fixed, so the three size intervals implied by binary fusion: (i) the range of the smaller and (ii) the larger polymers subject to fusion as well as (iii) the range of the resultant largest chains, are completely specified once the lower limits i min f and jmin f are chosen, see Table 1 for definitions of all these size limits. We remark that the fusion/fragmentation zones can overlap in qualitatively different ways. For example, we can have a partial overlap of the size zone of the large polymers that fuse, with the size zone of the largest fusion-product polymers by lowering i min f and increasing jmin f . Then the zone of the smallest polymers that can fuse is well separated from the latter two. In the other extreme, we can have a maximal overlap of all three size zones by simply letting i min f = jmin f = n c . This latter situation corresponds to fusion and fragmentation acting over the widest size ranges possible. We note that in all allowed cases the three size intervals always have the same width: Δ = N − (i min f + jmin f ).

3.2.1

Binary Fusion of Chains

We divide the polymer sizes n into a “small” n ∈ (i min f , N − jmin f ) and a “large” n ∈ ( jmin f , N − i min f ) group. Then, the rate of loss to the population of the small polymers that fuse with the large ones is given by

46

D. Hochberg et al.

c˙nα (t) = −ka cnα (t)

N −n 

ciα (t),

i min f ≤ n ≤ N − jmin f .

(10)

i= jmin f

Next, the rate of loss to the population of the large polymers that fuse with the small ones is N −n  ciα (t), jmin f ≤ n ≤ N − i min f . (11) c˙nα (t) = −ka cnα i=i min f

These two expressions (10), (11) are related by the limit interchange i min f ↔ jmin f in the corresponding limits and size ranges, and is a symmetry of this model. Lastly, the rate of gain to the population of the largest polymers (the resulting fusion products) is n− jmin f  α α ciα (t)cn−i (t), jmin f + i min f ≤ n ≤ N . (12) c˙n (t) = +ka i=i min f

3.2.2

Binary Fragmentation

The rate of gain to the small polymers due to fragmentation of the largest ones is c˙nα (t)

= k−a

N 

ciα (t),

i min f ≤ n ≤ N − jmin f .

(13)

i= jmin f +n

The rate of gain to the large polymers due to fragmentation of the largest ones is c˙nα (t)

= k−a

N 

ciα (t),

jmin f ≤ n ≤ N − i min f .

(14)

i=i min f +n

Note that these two expressions (13), (14) are related by the interchange symmetry i min f ↔ jmin f in the summation limits and size ranges. The rate of loss to the largest polymers due to their own fragmentation is given by: 

n− jmin f

c˙nα (t)

=

−k−a cnα (t)

1,

jmin f + i min f ≤ n ≤ N ,

i=i min f

= −k−a cnα (t)[n − ( jmin f + i min f ) + 1].

(15)

Finally, introducing fusion fluxes Ψ (see (29) below) allows us to write the all the above fusion and fragmentation processes together in a more compact form as follows:

Spontaneous Mirror Symmetry Breaking from Recycling …

c˙nα (t) = −

N −n 

47

α Ψn,i (t),

i min f ≤ n ≤ N − jmin f ,

(16)

α Ψn,i (t),

jmin f ≤ n ≤ N − i min f ,

(17)

i= jmin f

c˙nα (t)

=−

N −n  i=i min f



n− jmin f

c˙nα (t) = +

α Ψi,n−i (t),

jmin f + i min f ≤ n ≤ N .

(18)

i=i min f

4 Mechanical Forces in Chemistry In an early pioneering experiment on chiral symmetry breaking and deterministic chiral sign selection, hydrodynamic vortex motion was demonstrated to act as an environmental pressure to select one or the other handedness between two otherwise energetically degenerate mirror symmetric supramolecular populations [23]. In this experimental set-up, achiral diprotonated porphyrins, forming homoassociates in aqueous solution, lead to spontaneous chiral symmetry breaking. The unexpected result found by these authors is that the chirality sign of these homoassociates can be selected by vortex motion during the aggregation process. A more recent experimental report incorporating polymer breakage puts forward the hypothesis that mechanical forces can also act as a selection pressure in the competition between replicators [6]. Here, which of the two replicator populations, competing for limited monomer resources within a dynamic combinatorial library, becomes dominant depends on whether the sample is shaken or stirred. Mechanical forces can act as a selection pressure in the competition between replicators. From this perspective afforded by [6, 23], we are thus led to consider the impact that mechanically induced chain-breakage, in conjunction with chain fusion, has on the mirror symmetry breaking and subsequent chiral amplification in polymerization when combined with the basic polymerization processes (1), (2), (3) outlined above.

4.1 Irreversible Breakage To include these effects, chain breakage must be first modeled explicitly in terms of conventional reaction rate equations. The following will be implemented in our scheme (see (e) in Fig. 2), where γ is the rate constant for a chain to break into two smaller fragments: γ

Ln → Li + L j , γ

Rn → Ri + R j (n min ≤ n ≤ N , i min ≤ i, i ≤ j, i + j = n).

(19)

48

D. Hochberg et al.

The reason for introducing specific upper and lower length limits over which mechanically induced breakage operates is essentially physical. Thus for example, forces originating by shaking the reaction domain or by the action of stirring rods will be transmitted via hydrodynamic shear stresses and these forces ought to act preferentially on the longest chains present. Thus we introduce a minimum chain length n min ≤ N below which these physical forces no longer have any appreciable effect. On the other hand, the chain fragments so produced should also be larger than a certain minimum size i min ≥ n c . The general scenario is depicted in (e) in Fig. 2. As remarked earlier, we consider both n c and N to be fixed, so these two size intervals: the range of larger “donor” polymers subject to breakage and the “acceptor” population augmented by the incoming smaller breakage products, are completely specified once n min and i min are chosen. Note that partial overlaps of the donor and acceptor range zones are also possible, depending on the choices of i min , n min . A maximal size zone overlap holds for i min = n c and n min = 2(i min − 1), allowing for breakage to act over the maximum range of polymer sizes, while the smallest possible breakage fragment corresponds to the critical nucleus itself. In our approach, all these size limits can be varied freely and their effects observed and monitored through the simulations. The general expression for the loss rate of the longest polymers lying within a certain size range due to the mechanical breakage into two smaller chains is: c˙nα (t) = −

1 2

= −γ



γi, j cnα ,

n min ≤ n ≤ N ,

i+ j=n; i, j≥i min

 n

2

 ] − (i min − 1 cnα ,

(20)

where [..] denotes the Floor function. We assume a constant breakage rate γi, j = γ over the indicated range of polymer lengths. The sum, if we omit the factor of one-half, is then carried out over all distinct integer partitions of n such that the minimum fragment length is i min ; see Table 1. Note furthermore that the condi] > (i min − 1). This latter tion [ n2 ] − (i min − 1) > 0 will hold for all n provided [ n min 2 inequality establishes a weak bound relating the lower limit n min for the range of large polymers susceptible to breakage and the lower limit i min of the size range of the smaller polymers populated by the breakage fragments thus generated (see (e) in Fig. 2). This inequality is needed merely to ensure that the overall breakage rate is negative: the rate of change of the concentration in Eq. (20) must be negative: c˙nα (t) < 0, whereas γ > 0, cnα ≥ 0: clearly, breakage depletes the population of the longest polymers. The general expression for the rate of gain of the short polymer population (the acceptors) due to the incoming fragments produced by the breakage of the longer polymers (the donors) is:

Spontaneous Mirror Symmetry Breaking from Recycling …

c˙nα (t) = +

N −n 

α γn, j cn+ j,

49

i min ≤ n ≤ N − i min ,

j≥i min

=γ

N 

q(k) ckα ,

q(k) = IF[k ≥ n min , 1, 0],

k≥i min +n N 

=γ

  1 + δk,2n ckα .

(21)

k=max{n min ,n+i min }

In this case, the population of the shorter polymers is augmented, c˙nα (t) > 0, by the incoming breakage fragments. The breakage of the long polymers having exactly twice the length of the shorter polymers k = 2n is handled by the delta function.

5 Numerical Results The complete model described above is left-right symmetric, that is, possesses a discrete Z 2 symmetry (parity followed by a rotation) when all species are interchanged with their mirror image counterparts: L i → Ri , Ri → L i . This exact symmetry can be broken spontaneously by the dynamical solutions of the differential rate equations. The model is thus apt for investigating spontaneous mirror symmetry breaking. Combining all the above mechanisms leads to the differential rate equations summarized in Eq. (32) in the Appendix. These are integrated numerically using a high level of numerical precision, typically thirty significant digits, to ensure the numerical significance of the initial concentrations and tiny initial enantiomeric excesses employed (see below). The results were monitored and verified to assure that total system mass remained constant in time. The concentration units are mol l −1 , and the different reaction rate constants have the appropriate units to yield rate values in units of mol s −1 . The initial percent enantiomeric excess ee0 (%) for the entire system, from the monomer up to the largest polymer, is defined as N ee0 (%) = n=1 N

([L n ]0 − [Rn ]0 )

n=1 ([L n ]0

+ [Rn ]0 )

× 100.

(22)

In order to study the sensitivity of the reaction scheme, Eqs. (1), (2), (3), (19), to tiny initial enantiomeric excesses, an initial concentration of a scalemic (non racemic) monomer composition was employed in the calculations: these initial monomeric concentrations are [L 1 ]0 = (0.1 + 1 × 10−11 )M and [R1 ]0 = 0.1M. The maximum polymer length was fixed to N = 100, and the critical nucleus size to n c = 5. The full set of the remaining initial concentrations are as follows: [L n ]0 = [Rn ]0 = 0.1M, 2 ≤ n ≤ N , i.e., the strict racemic composition for the remainder of the oligomers (and at a level of numerical working precision that ensures the significance of the

50

D. Hochberg et al.

initial enantiomeric excess). Inserting these initial concentration values into Eq. (22) yields an initial percent chiral excess of ee0 = 5 × 10−11 %. It is roughly two orders of magnitude greater than the expected enantiomeric excess due to purely statistical fluctuations in a racemic sample [19]: eestat =

0.68(%) = 2.0 × 10−13 %, √ M

(23)

where M = 12 × 1024 is the total number of oligomers in the system (Avogadro’s number multiplied by the total initial concentration of the system). Values of the reaction rate constants used in all the simulations are: k1 = 104 , k−1 = 4 × 104 , k2 = 1000, k−2 = 900, ka = 10, k−a = 9, k R = 102 , γ = 50. As mentioned above, the forward and reverse rates of monomer attachment (k2 , k−2 ) in the isodesmic elongation phase and those of binary fusion (ka , k−a ) are constrained by chemical thermodynamics, (4). Note that since we keep both n c and N fixed, we have four free size parameters i min f , jmin f , n min and i min which we can vary, see Table 1. These upper and lower bounds for the breakage and fusion size ranges are specified in the corresponding simulation figure captions. In all the numerical simulations, the character of the final asymptotic stationary state was inferred from the constant concentration values obtained at t ≈ 1015 s. Standard chiral measures are used to quantify and present the results. The percent enantiomeric excess values of the individual homochiral oligomers (1 ≤ n ≤ N ) are calculated according to : een (%) =

[L n ] − [Rn ] × 100. [L n ] + [Rn ]

(24)

The importance of the enantiomeric excess is that it is an order parameter for the symmetry breaking transition: |een (%)| ≥ 0 is strictly zero for mirror symmetric states and nonzero otherwise. In the latter case, the Z 2 symmetry is broken. A distinct global measure of the degree of symmetry breaking is provided by the total mass within each chiral population η L , η R : ηL =

N  n=1

n[L n ],

ηR =

N 

n[Rn ],

(25)

n=1

as well as the total chiral mass contained in both chiral populations: ηT otal = η L + η R . It is important to keep in mind that een (t), η L (t), η R (t) are all time-dependent quantities. An illustrative example of an initial racemic configuration (subject to a tiny chiral perturbation) evolving to a chiral final state transition is shown in Fig. 3 for the ranges of fusion and breakage size limits i min f = 20, jmin f = 30, n min = 85 and i min = 25 (see Table 1 and Fig. 2). Panel (a) shows the time-dependent oligomer enantiomeric excesses, Eq. (24), for all the species collectively. The dependence on the size n of

Spontaneous Mirror Symmetry Breaking from Recycling … 100

(a) enantiomertic excess

enantiomeric excess

100 80 60 40 20

0 0.001

1

1000

106

109

1012

51

(b)

80 60 40 20 0 0

1015

20

20

(c) ηL

800

masses n Ln Rn

total masses ηL , ηR

1000

600 400

ηR

200 0 0.001

1

1000

106

time s

40

60

80

100

80

100

chain length n

time s

109

1012

1015

(d)

15 10 5 0

0

20

40

60

chain length n

Fig. 3 Simulation for i min f = 20, jmin f = 30, n min = 85 and i min = 25. a Time-dependent oligomer enantiomeric excess Eq. (24) showing the gradient dependence of een on the size n of the oligomer: the curves correspond to n = 1 (bottommost) to n = 100 (topmost) and in increasing sequential order. b This gradient in n is resolved showing individual oligomer enantiomeric excesses as a function of chain size n and evaluated at t ∼ 1015 s. c Total mass in each chiral population Eq. (25) as a function of time. d Total chiral mass ηT otal as a function of chain length n evaluated at t ∼ 1015 s. Time is measured in seconds, concentrations and masses are in standard SI units, as for the remainder of this chapter. Reprinted with permission from J. Chem. Phys. B 2017, 121, pp. 942–955. Copyright 2017 American Chemical Society

the n-mer: where 1 ≤ n ≤ N indicates that the een depends strongly on the size of the oligomer. The important aspect to be appreciated from panel (a) is the gradient in een versus oligomer size n after mirror symmetry is broken (at  105 s) and that all the een > 0 are positive. Since a total of 100 individual curves are plotted together, there is a “pile-up” compression or bunching together of these curves for chain sizes with n > 6, whereas the first few are well separated. This explicit size dependence in een is resolved in the top right hand graph, panel (b), indicating the each oligomer enantiomeric excess as a function of chain size n, evaluated at asymptotic times t ∼ 1015 s. That is, panel (b) results from taking a temporal “slice” of panel (a), and well after the symmetry breaking event. We see that the gradient in oligomer size is nonlinear. The degree of symmetry breaking depends on size n in a nontrivial fashion. The mirror symmetry breaking is also manifested in the dynamics of the net mass contained within each chiral population (η L , η R ), Eq. (25), (panel (c)). The chiral masses in each population evolve indistinguishably and then split apart dramatically at t ≈ 105 s as shown. The way the total chiral mass ηT otal , summed over both chiral populations is distributed as a function of chain length n, in the complex nonlinear

52

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manner, is shown in panel (d) and evaluated at t ∼ 1015 s. Note that the individual oligomer enantiomeric excesses, panel (a), all exhibit a sigmoidal time dependence. Since this is plotted on a logarithmic scale, the symmetry breaking transition appears rather abruptly. We can resolve this transition thus making the sigmoidal character manifest using a linear time scale (see Fig. 5). The characteristic time scale in which the rapid acceleration phase begins depends on the rate of racemization kr . Increasing (decreasing) kr decreases (increases) this time scale. The time dependence of the mass in each chiral species also display sigmoidal behavior (panel (a), Fig. 5), as does the net mass contained within each chiral population for the majority chirality (in this case, for L). The minority enantiomers have their masses correspondingly diminished, in an “anti-sigmoidal” fashion: there is a rapid deceleration phase which exactly mirrors the acceleration phase of the majority enantiomers, followed by a rapid slowdown. This chiral “mass splitting” feature, whereby the initial mass degeneracy [L n ]0 = [Rn ]0 (which implies η L (0) = η R (0)) is lifted for all the species, and on the same time scale, is a hallmark feature of spontaneous mirror symmetry breaking (SMSB) in chiral polymerization schemes Varying the upper and lower size limits of the fusion and breakage windows can lead to similar results as far as the overall qualitative collective symmetry breaking features are concerned. Thus for example in Fig. 4: i min f = 6, jmin f = 6, n min = 100 and i min = 50. In contrast to the previous Fig. 3, the tight “clustering” of the individual oligomer enantiomeric excess curves een is here pronounced for the entire range of the oligomers sizes greater than the critical nucleus, 6 ≤ n ≤ 100. The detailed resolution of the way these een ’s are distributed according to chain length (b) confirms this aspect unambiguously. Again the overall qualitative aspects of both the collective set of time-dependent oligomer enantiomeric excesses een and the asymptotic chiral masses [Fig. 4a, c] are qualitatively similar to those corresponding to the previous examples. The salient differences are made clear by examining the details of the precise manner in which the oligomer een ’s and chiral masses are distributed over the individual chain lengths [Fig. 4b, d]. From these two examples it is clear that the chain enantiomeric excesses and chain masses are complicated non-linear functions of the upper and lower limits of the fusion and breakage size windows. The results shown in Figs. 3, 4 and 5 are indicative of an important trend. When SMSB and chiral amplification occur, they do so as a collective and coherent phenomena, involving all of the species and on the same time scale, such that the symmetry breaking experienced in each individual chiral species (oligomer) is produced with the same sign (n ∈ [1, N ]): in other words, the SMSB is homochiral. Employing a logarithmic time scale to display the results in the panels (a), (c) of the above figures affords a compact and concise way to appreciate the collective behavior of all the oligomer enantiomeric excesses in a glance, as well as the total chiral masses, over the entire time range of the simulations. The familiar sigmoidal response of the oligomer enantiomeric excesses can be exposed easily by “zooming” in on the transition time scale using a linear time axis. Thus, for the same simulation parameters leading to Fig. 3, we highlight this sigmoidal behavior for the individual oligomer masses: see panel (a) of Fig. 5. Since we must deal with N = 100 species, for clarity we plot a selection of oligomer sizes as shown. It is clear from these curves that the een of the

Spontaneous Mirror Symmetry Breaking from Recycling … 100

(a)

(b) enantiomertic excess

enantiomeric excess

100 80 60 40 20

0 0.001

53

1

1000

106

109

1012

80 60 40 20 0 0

1015

20

15

(c)

ηL

800

masses n Ln Rn

total masses ηL , η R

1000

600 400

ηR

200 0 0.001

1

1000

106

time s

40

60

80

100

80

100

chain length n

time s

109

1012

1015

(d)

10

5

0

0

20

40

60

chain length n

Fig. 4 Simulation for i min f = 6, jmin f = 6, n min = 100 and i min = 50. a Time-dependent oligomer enantiomeric excess Eq. (24) showing the gradient dependence of een on the size n of the oligomer: the curves correspond to n = 1 (bottommost) to n = 100 (topmost) and in increasing sequential order. b This gradient is resolved showing individual oligomer enantiomeric excesses as a function of chain size n and evaluated at t ∼ 1015 s. c Total mass in each chiral population Eq. (25) as a function of time (in seconds). d Total chiral mass ηT otal as a function of chain length n evaluated at t ∼ 1015 s. Reprinted with permission from J. Chem. Phys. B 2017, 121, pp. 942–955. Copyright 2017 American Chemical Society

majority enantiomers follow a sigmoid, whereas the minority enantiomers follow an “anti-sigmoid” (that is, having a rapid deceleration phase). We note moreover the chiral splittings all take place on a similar time scale for all the oligomers, also the saturation phase (the leveling off of the curves) occurring on a similar time scale for all the oligomers. In panel (b) of Fig. 5 we expose the sigmoidal behavior in terms the total chiral masses. The latter should be compared to panel (c) of Fig. 3. An indication of the polymeric relaxation processes taking place before and after the symmetry breaking event is provided by the panels (c), (d) of Fig. 5 in terms of the oligomer masses. The panel (c) shows the individual oligomer masses over the full size range in n. The distribution at t = 104 s, is racemic, at t = 106 s, the distributions of all the L and R oligomers have separated as shown (and maintain these profiles for the remainder of the simulation). Panel (d) shows the total chiral distribution in terms of the masses n([L n ] + [Rn ]) and for the same two time scales. The trends and features discussed above have been observed in numerous simulations and mapped out over a wide range of breakage and fusion length parameters; see Ref. [5] for further examples.

54

D. Hochberg et al. 1000

(a)

total masses ηL , ηR

masses nLn , nRn

20 15

n = 41 n = 61 n = 21 n = 81 5 n=1

10

0 50 000

70 000

90 000

(b)

ηL

800 600 400

ηR

200 0 50 000

110 000

70 000

20

(c)

(t=106 s)

nLn

15

(t=104 s) nLn

10 5 0

masses n Ln Rn

masses nLn , nRn

20

(t=106 s)

nRn 0

20

40

90 000

110 000

time s

time s

60

chain length n

80

100

t = 106s

(d)

15

t = 104s

10 5 0

0

20

40

60

80

100

chain length n

Fig. 5 Details of the behavior of the oligomer masses: same parameters as for simulation of Fig. 3. a Time-dependent oligomer masses n[L n ] (monotonic increase) and n[Rn ] (monotonic decrease) for the oligomers with length n = 1, 21, 41, 61, 81 showing a close-up of the sigmoidal behavior on a linear time scale (in seconds). b Sigmoidal behavior of the total chiral masses η L (monotonic increase) and η R (monotonic decrease) as functions of time in seconds (linear scale). c Individual oligomer masses n[L n ], n[Rn ] for 1 ≤ i ≤ 100, before (t = 104 s) and after (t = 106 s) the symmetry breaking. d Sum of the chiral masses η L + η R , as a function of chain length n evaluated before (t = 104 s) and after (t = 106 s) the symmetry breaking. Reprinted with permission from J. Chem. Phys. B 2017, 121, pp. 942–955. Copyright 2017 American Chemical Society

6 Conclusions We have constructed a simple polymerization model based on nucleation, chain elongation and binary chain fusion, all micro-reversible transformations. In the isodesmic approximation used here, the forward and reverse rate constants of elongation and binary fusion are constrained by chemical thermodynamics. We introduce chain breakage as the only irreversible process in the scheme. We find numerical evidence for spontaneous mirror symmetry breaking (SMSB) for nucleated enantioselective polymerization in closed systems kept out of equilibrium by selective input of external energy (mechanical breakage). This SMSB is a consequence of the recycling of chiral material through competing unidirectional cycles driven by irreversible breakage and reversible fusion. Moreover, this SMSB is achieved without appealing to chiral or mutual inhibition, and so is markedly distinct from the Frank paradigm [9]. There is no explicit catalysis in our scheme. The binary fusion is the only nonlinear mechanism, and is capable of amplifying the initial tiny enantiomeric excesses up to final large excesses. The mechanical breakage, or selective energy input,

Spontaneous Mirror Symmetry Breaking from Recycling …

55

establishes unidirectional cycles in both chiral oligomer families which compete via the racemization of the monomers. SMSB is observed to be a coherent phenomenon (same final sign for all species), as evidenced in the dynamic curves. Chirality constitutes a unifying feature of the living world and is presumed to be a prime driving force for molecular selection and genetic evolution. The fundamental questions concerning how spontaneous mirror symmetry breaking leads to biological homochirality make up a research topic of interdisciplinary activity which benefits from collaboration between physical, chemical and biological experimentalists and theoreticians. In addition to its role in origin of life research, it is worth mentioning potential applications of chirality in complex chemical systems, and in the preparation of novel molecular materials. As demonstrated in this chapter, there are non-equilibrium thermodynamic scenarios where SMSB can occur in enantioselective polymerization. These scenarios are sensitive to external chiral polarizations; thus, weak natural chiral fields can transform stochastic chiral outcomes of SMSB into deterministic outcomes of a fixed chiral sign. A fundamental question is whether the emergence of chirality is a simple consequence of this process, that is, a random breaking of mirror symmetry followed by a deterministic selection of final chiral sign, or is, in fact, the triggering factor for the emergence of complex abiotic materials. This would distinguish biological homochirality as much more than an advantageous mechanism for the exchange of information between biological compartments and systems. According to this hypothesis, chirality would be the necessary element to drive chemical evolution towards the emergence of complex soft materials working cooperatively and in a complementary fashion, i.e. the primordial factor that allows the transition from evolutionary chemistry to systems with the capacity for Darwinian evolution. Theoretical studies such as the one presented here could assist in the preparation of new synthetic molecular materials and complex networks that exploit understanding how chirality originates, mimicking biological responses and life-like transformations [7]. Acknowledgements The research of CB, MS and DH is supported in part by the grant CTQ201347401-C2-2-P (MINECO). CB is an Otis Williams Postdoctoral Fellow in Bioengineering. MS and DH form part of the COST Action CM1304: Emergence and Evolution of Complex Chemical Systems.

Appendix The complete set of rate equations taking into account all the above processes can be expressed in terms of reaction rate fluxes and stoichiometric matrices. Introduce the reaction rate fluxes, where α = L , R is a chirality label:

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D. Hochberg et al.

φ αj =



k1 c1α cαj − k−1 cαj+1 , k2 c1α cαj − k−2 cαj+1 ,

1 ≤ j ≤ n c − 1, n c ≤ j ≤ N − 1,

(26)

φrL ,R = kr (c1R − c1L ), Φkα = γ ckα ,

(27) (28)

α α Ψm,n = ka cmα cnα − k−a cm+n ,

(29)

for linear chain growth (φ), racemization (φr ), mechanical breakage (Φ) and fusion (Ψ ), respectively. Although the individual fusion fluxes, Eq. (29), depend on two indices m, n, for purposes of counting they can be enumerated in a sequential fashion by imposing, and without loss of generality, that m ≤ n. The number of independent fusion fluxes is calculated to be N − jmin f

Nψ =



N −k 

1,

(30)

k=i min f l=max(k, jmin f )

the lower limit of the second summand ensures that the flux Ψm,n = Ψn,m is counted just once. We can group all these fluxes together into a single flux vector:  Nψ  N −n min +1 −1 . , φrα , {Φkα }k=1 , {Ψmα }m=1 f α = {φ αj } Nj=1

(31)

Then the differential rate equations for the i-th species can be written as follows: r dciα (t)  = Si, j f jα , dt j=1

n

1 ≤ i ≤ N,

1 ≤ j ≤ nr ,

(32)

and S is the stoichiometric matrix with elements Si, j . The sum is over the number n r of reactions: (33) n r = 2N − n min + Nψ + 1. Equations (32) represent the model to be solved numerically.

References 1. Blackmond, D.G.: The origin of biological homochirality. In: Deamer, D., Szostak, J.W. (eds.) The Origin of Life. Cold Spring Harbor Laboratory Press, New York (2010) 2. Blanco, C., Hochberg, D.: Chiral polymerization: symmetry breaking and entropy production in closed systems. Phys. Chem. Chem. Phys. 13, 839–849 (2011) 3. Blanco, C., Ribó, J.M., Hochberg, D.: Modeling spontaneous chiral symmetry breaking and deracemization phenomena: discrete versus continuum approaches. Phys. Rev. E 91, 022801 (2015) 4. Blanco, C., Stich, M., Hochberg, D.: Temporary mirror symmetry breaking and chiral excursions in open and closed systems. Chem. Phys. Lett. 505, 140–147 (2011)

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5. Blanco, C., Stich, M., Hochberg, D.: Mechanically induced homochirality in nucleated enantioselective polymerization. J. Phys. Chem. B 121, 942–955 (2017) 6. Carnall, J.M.A., Waudby, C.A., Belenguer, A.M., Stuart, M.C.A., Peyralans, J.J.-P., Otto, S.: Mechanosensitive self-replication driven by self-organization. Science 327, 1502–1506 (2010) 7. Cintas, P. (ed.): Biochirality: Origins, Evolution and Molecular Recognition. Springer, Heidelberg (2013) 8. Crusats, J., Hochberg, D., Moyano, A., Ribó, J.M.: Frank model and spontaneous emergence of chirality in closed systems. Chem. Phys. Chem. 10, 2123–2131 (2009) 9. Frank, F.C.: On spontaneous asymmetric synthesis. Biochim. Biophys. Acta 11, 459–463 (1953) 10. Gherase, D., Conroy, D., Matar, O.K., Blackmond, D.G.: Experimental and theoretical study of the emergence of single chirality in attrition-enhanced deracemization. Cryst. Growth Des. 14, 928–937 (2014) 11. De Greef, T.F.A., Smulders, M.M.J., Wolffs, M., Schenning, A.P.H.J., Sijbesma, R.P., Meijer, E.W.: Supramolecular polymerization. Chem. Rev. 109, 5687–5754 (2009) 12. Guijarro, A., Yus, M.: The Origin of Chirality in the Molecules of Life. RSC Publishing, Cambridge (2009) 13. Hein, J.E., Cao, B.H., Viedma, C., Kellogg, R.M., Blackmond, D.G.: Pasteur’s tweezers revisited: on the mechanism of attrition-enhanced deracemization and resolution of chiral conglomerate solids. J. Am. Chem. Soc. 134, 12629–12636 (2012) 14. Hitz, T., Luisi, P.L.: Chiral amplification of oligopeptides in the polymerization of alpha-amino acid N-carboxyanhydrides in water. Helv. Chim. Acta 86, 1423–1434 (2003) 15. Iggland, M., Mazzotti, M.: A population balance model for chiral resolution via Viedma ripening. Cryst. Growth Des. 11, 4611–4622 (2011) 16. Iggland, M., Mazzotti, M.: Solid state deracemisation through growth, dissolution and solutionphase racemisation. Cryst. Eng. Comm. 15, 2319–2328 (2013) 17. Illos, R.A., Bisogno, F.R., Clodic, G., Bolbach, G., Weissbuch, I., Lahav, M.: Oligopeptides and copeptides of homochiral sequence, via beta-sheets, from mixtures of racemic alpha-amino acids, in a one-pot reaction in water; relevance to biochirogenesis. J. Am. Chem. Soc. 130, 8651–8659 (2008) 18. Lee, D.H., Granja, J.R., Martinez, J.A., Severin, K., Reza Ghadiri, M.: A self-replicating peptide. Nature 382, 525–528 (1996) 19. Mills, W.H.: Some aspects of stereochemistry. J. Chem. Technol. Biotechnol. 51, 750–759 (1932) 20. Noorduin, W.L., Izumi, T., Millemaggi, A., Leeman, M., Meekes, H., van Enckevort, W.J.P., Kellogg, R.M., Kaptein, B., Vlieg, E., Blackmond, D.G.: Emergence of a single solid chiral state from a nearly racemic amino acid derivative. J. Am. Chem. Soc. 130, 1158–1159 (2008) 21. Noorduin, W.L., van Enckevort, W.J., Meekes, H., Kaptein, B., Kellogg, R.M., Tully, J.C., McBride, J.M., Vlieg, E.: The driving mechanisms behind attrition-enhanced deracemization. Angew. Chem. Int. Ed. 49, 8435–8438 (2010) 22. Ribó, J.M., Blanco, C., Crusats, J., El-Hachemi, Z., Hochberg, D., Moyano, A.: Absolute asymmetric synthesis in enantioselective autocatalytic reaction networks: theoretical games, speculations on chemical evolution and perhaps a synthetic option. Chem. Eur. J. 20, 17250– 17271 (2014) 23. Ribó, J.M., Crusats, J., Sagués, F., Claret, J., Rubires, R.: Chiral sign induction by vortices during the formation of mesophases in stirred solutions. Science 292, 2063–2066 (2001) 24. Ricci, F., Stillinger, F.H., Debenedetti, P.G.: A computational investigation of attrition-enhanced chiral symmetry breaking in conglomerate crystals. J. Chem. Phys. 139, 174503 (2013) 25. Sczepanski, J.T., Joyce, G.F.: A cross-chiral RNA polymerase ribozyme. Nature 515, 440–442 (2014) 26. Soai, K., Shibata, T., Morioka, H., Choji, K.: Asymmetric autocatalysis and amplification of enantiomeric excess of a chiral molecule. Nature (London) 378, 767–768 (1995) 27. Viedma, C.: Chiral symmetry breaking during crystallization: complete chiral purity induced by nonlinear autocatalysis and recycling. Phys. Rev. Lett. 94, 065504 (2005)

Self-organized Cultured Neuronal Networks: Longitudinal Analysis and Modeling of the Underlying Network Structure Daniel de Santos-Sierra, Inmaculada Leyva, Juan Antonio Almendral, Stefano Boccaletti and Irene Sendiña-Nadal Abstract This work analyzes the morphological evolution of assemblies of living neurons, as they self-organize from collections of separated cells into elaborated, clustered, networks. In particular, we introduce and implement a graph-based unsupervised segmentation algorithm that automatically retrieves the whole network structure from large scale phase-contrast images taken at high resolution throughout the entire life of a cultured neuronal network. The network structure is represented by an adjacency matrix in which nodes are identified as neurons or clusters of neurons, and links are the reconstructed connections (neurites) between them. The algorithm is also able to extract all other relevant morphological information characterizing neurons and neurites. More importantly and at variance with other segmentation methods that require fluorescence imaging from immunocytochemistry techniques, our measures are non invasive and entitle us to carry out a fully longitudinal analysis during the maturation of a single culture. In turn, a systematic statistical analysis of a group of topological observables grants us the possibility of quantifying and tracking the progression of the main networks characteristics during the self-organization process of the culture. Our results point to the existence of a particular state corresponding to a small-world network configuration, in which several relevant graphs’ micro- and meso-scale properties emerge. Finally, we identify the main physical processes taking place during the cultures morphological transformations, and embed them into a simplified growth model that quantitatively reproduces the overall set of experimental observations.

D. de Santos-Sierra · I. Leyva · J. A. Almendral · I. Sendiña-Nadal (B) Center for Biomedical Technology, Technical University of Madrid, Madrid, Spain e-mail: [email protected] I. Leyva · J. A. Almendral · I. Sendiña-Nadal Center for Biomedical Technology, Technical University of Madrid, Madrid, Spain S. Boccaletti CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy © Springer Nature Switzerland AG 2019 J. Carballido-Landeira and B. Escribano (eds.), Biological Systems: Nonlinear Dynamics Approach, SEMA SIMAI Springer Series 20, https://doi.org/10.1007/978-3-030-16585-7_4

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1 Introduction Along the past decades, cultured neuronal networks (CNNs) have constituted a fundamental tool for scientists as one of the benchmark models for the study of the central nervous system. They, indeed, allow conducting very well controlled laboratory experiments, thus providing a systematic way to approach fundamental questions, as for example unveiling the principles and mechanisms underlying learning and memory, connectivity, and even information processing of their in vivo counterparts [9, 20, 32, 53]. CNNs have also important practical applications as models for the study of diseases [49], for the discovery and evaluation of drugs [34], or when connected through computers to a real or a simulated robot (to create what is called an hybrot [7, 39] or an animat [18, 57], respectively). In those situations, scientists are then endowed with the possibility of studying some basic neuronal processes in realistic contexts, such as learning and plasticity. Possibly, the most relevant advantage of CNNs is the unique option they offer of following the footprints of the self (or induced) organization of the network’s functionality and dynamics (usually by means of a micro-electrode array (MEA) or calcium fluorescence, recording the CNN electrophysiological data, or inducing electrical stimulations in given spatial positions) together with the monitoring and tracking of the structural organization of the neurons’ connectivity along the entire course of the culture’s growth [13, 19, 37]. Although culturing neurons on top of a MEA equipped chamber implies, in general, only mild constraints, following the development of the culture’s structure is a far more delicate issue. Indeed, image-based biology systems essentially require to gather sequential imaging of the culture and its processing to seek the evolution of the main network’s indicators and measures along the CNN’s maturation [40, 41, 48, 58]. The main drawback of image processing tools in segmenting neurons and neurons’ connections is that they need pictures with a high level of contrast. This traditionally leds experimentalists to rely on immunocytochemistry techniques, which however implies cell fixation and therefore death [33]. Our work comes up with a novel approach for a non-invasive image procedure. We developed a graph-based segmentation algorithm which operates on large scale images acquired by phase-contrast microscopy, and therefore by a fully non-invasive technique, that is, without the need of adding chemicals to the culture. The algorithm accurately identifies the relevant network’s units, and reconstructs the wiring of the network connectivity. With the aim of studying the network’s topology and why and how an assembly of isolated (cultured) neurons self-organizes to form a complex neural network, we carry out a study based on complex network theory [20]. Some previous studies highlighted the fact that the structuring of a neuronal cultured network before the attainment of its mature state is not random, being instead governed and characterized by processes eventually leading to configurations which are comparable to many other real complex networks [10]. In particular, networking neurons simultaneously feature a high overall clustering and a relatively short path-length [48]. Such configurations, which in graph theory are termed small-world [54], are ubiquitously found in real-world net-

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working systems. Small-world structures have been shown to enhance the system’s overall efficiency [1, 29], while concurrently warranting a good balance between two apparently antagonistic tendencies for segregation and integration in structuring processes, needed for the network’s parallel and yet synthetic performance [38]. Most of the existing investigations in neuronal cultures restricted their attention to functional networks (statistical dependence between nodes activities) and not to the physical connections supporting the functionality of the network [21]. The reason behind this drawback is that the majority of investigations focused on excessively dense cultures, hindering the observation of their fine scale structural connectivity. Although there are works striving to indirectly infer the underlying anatomical connectivity from the functional network, it has been shown that strong functional correlations may exist with no direct physical connection [27]. Only few studies dealt with the physical wiring circuitry. However, on one hand, only small networks were considered; on the other hand, how the network state evolves during the course of the maturation process has not been yet deeply investigated [48]. Here, instead, we focus on intermediate neurons’ densities, and provide a full tracking of the most relevant topological features emerging during the culture’s evolution. In particular, we show experimentally that in vitro neuronal networks tend to develop from a random network state toward a particular networking state, corresponding to a small-world configuration, in which several relevant graph’s micro- and meso-scale properties emerge. Our approach also unveils the main physical processes underlying the culture’s morphological transformation, and allows us using such information for devising a proper growth model (in silico networks), qualitatively and quantitatively reproducing our experimental evidence, confirming several results of previous works on functional connectivity [19, 52], or morphological structuring at a specific stage of the cultures’ evolution [48]. The performed longitudinal study of the network structure highlights self-organization properties of cultured neural networks, such as (i) a large increase in both local and global network’s efficiency associated to the emergence of the small-world configuration, and (ii) the setting of assortative degree-degree correlation features.

2 Cultured Neuronal Networks as a Model System Since 1907, neuronal cultures, a technique to keep neurons alive outside the animal [25], are conceived as a research tool for the in vitro study of in vivo phenomena like neuronal plasticity, connectivity, information processing, and so forth (for a review on the subject see Refs. [5, 20, 46]). There are two main different techniques to culture neurons: primary cultures and cell lines. Cell lines are populations of almost identical neurons cloned from a single cell while a primary neuronal culture involves the dissection of a specific tissue (cortex, hippocampus, spinal, ganglion, etc) and dissociation of its neurons which are then plated on top of a substrate. Within a few hours, cultured neurons start growing new connections creating a fully developed neuronal network in a few days. Although these preparations do not preserve the topological

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traits of the original source, neuronal primary cultures maintain the cell variability and functionality, turning themselves into valuable and controllable models for the understanding of the fundamental mechanisms behind the self-organization of functional neuronal circuits [9, 20–22, 32, 53]. In addition, neuronal cultures present many benefits in comparison to other in vivo model systems. Cultures are easy to prepare and maintain, and allow for a wide range of experiments highly reproducible, unlike brain networks studies where the different experimental conditions usually depend on appropriate subject’s availability. There are two main approaches when dealing with cultured neuronal networks. The first approach involves studies of the functional connectivity by means of recordings of the neuronal activity (micro-electrode arrays [35] and fluorescence or calcium imaging [6, 24]) while in the second one, studies focus on the extraction of the neuronal morphology from microscopic imaging [33]. Information flows and network functionality depend on the structural connectivity and, therefore, cultured neuronal networks constitute an excellent platform to understand how the anatomical network determines the circuit’s activity and vice versa (see Ref. [21] for a revision work on structure-function studies of neuronal networks). There are many works, combining tools from statistical physics [15, 20] and graph theory [10], which studied the topological properties of neuronal circuits at different stages of development. For instance, using concepts from percolation theory, the emergence of a giant connected component during the network growth has been characterized and connected to the emergence of spontaneous activity in the culture [14, 31, 50]. Other works focused more on aspects related to the number [28] and connections distribution [15, 17, 48], clustering [15], assortativity [40], small-world property [40], and the existence of functional hubs [51]. All these properties are determinant in driving the dynamical behavior of neuronal circuits to process and propagate electrical signals or to synchronize neuronal populations. Disentangling the physical wiring (anatomical) of a neuronal circuit is not an easy task even for a simple neuronal culture whose connectivity is built ex novo. There are few works trying to infer the underlying structural connectivity from functional measurements of activity correlations [26]. However, a recent study [27] shows that functional correlations may exist between groups of neurons without the evidence of a direct physical connection between them. Therefore, in our research, we take as a model system cultured networks of neurons from the frontal ganglion of locusts [5]. In the rest of the section, we describe this experimental setup as a convenient framework for non-invasive optical observations and for following the dynamics of the neuronal growth process and network organization from single cells to elaborate networks.

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2.1 Methods 2.1.1

Experimental Setup

Primary neuronal cultures are prepared from the frontal ganglion (FG) of adult locusts Schistocerca gregaria. In brief, neuronal cells from dissected FG are isolated (and removed from their original neurites) by enzymatic and mechanical dissociation. Cells are then re-suspended in Leibovitz medium (L-15) with L-glutamine (Sigma-Aldrich, L4386), supplemented with 0.01% penicillin-streptomycin (Biological Industries, Israel), and seeded on a Concanavalin A (Sigma-Aldrich, C0412) pre-coated circular area (r = 2.5 mm) of a 35 mm Petri dish at a density about 150 cells per mm2 , and left eventually for 2 h to allow adherence. Neurons are then incubated with 2 ml conditioned medium L-15 enriched with 5% locust haemolymph, and cultured in darkness for 18 days in vitro (DIV) under controlled temperature (29 ◦ C) and humidity (70%). No medium changes are done to no affect the network topology. High-resolution and large scale phase-contrast images are acquired daily with an inverted microscope (Eclipse Ti-S, Nikon Instruments) equipped with a motorized XYZ stage (H117 ProScan, Prior Scientific), and using a charge coupled device camera (DS-Fi1, Nikon Instruments) with a 10× objective (Achromat, ADL, NA 0.25). The automated control of the motorized XYZ stage and camera is performed using the NIS-Elements software (Nikon Instruments Software, Nikon). Mosaic images with a pixel size of 1.34 µm are captured with the large image method implemented in NIS-Elements, which does automatic blended stitching with an overlap of 25%. Therefore, at each day of measurement, the result is a large high-resolution jp2 image file consisting of an ensemble of images each one acquired with 10× magnification. The whole process of the experimental setup from the FG dissection to the culture preparation, image acquisition and analysis, is sketched in Fig. 1.

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Image Segmentation

Acquired culture images give information about how the cells are physically interconnected through dendrites and axons allowing us to reconstruct the whole physical network. The connectome reconstruction from images is still a work in progress. In this section, we apply a multi-layer graph-based segmentation algorithm [39, 42] to phase-contrast images from invertebrate neuronal cultures. Graph-based segmentation algorithms are powerful tools to process an image without loosing the spatial information. The developed algorithm is based on a multilayer graph [11, 43] representation of the image which groups neighbor pixels with similar properties creating communities of pixels. In each step, communities can be merged with other neighbor communities creating a bigger one. The algorithm merges communities until a goal, or stop condition like a given number of steps or clusters, is reached. Figure 2 illustrates this layer aggregation process.

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Fig. 1 Schematic representation of the experimental pipeline for the longitudinal study of a CNN development. Following the arrows, cultures are prepared using N frontal ganglia (Y shaped) dissected from the brain of N animals. Phase-contrast images are daily acquired and processed to recover the network structure and to perform a longitudinal tracking of its main topological properties

In that sense, each step can be considered as a layer of the segmentation procedure while the relationship between communities and pixels creates the graph structure. The algorithm can be broken down into the following actions: • Initialization. Each image pixel is considered as a graph’s node, and the relationship between nodes is defined. Usually a 4- or 8-neighborhood layout is considered. • Aggregation. A similarity function is used to group those nodes matching a certain criterion. Typically, the adopted rule is such that the similarity value between two nodes is below a given threshold. • Communities. The new nodes and their relationship (neighborhood) are defined. • Stop condition. The aggregation and community creation actions are repeated until a stop condition is reached. The aggregation criterion can be changed at each step in order to group those communities whose relationship does not meet the above criterion. The image-processing algorithm for the network structure extraction of Schistocerca gregaria neuronal culture consists of three main tasks [40]. Figure 3 shows the algorithm’s pipeline for the network structure extraction. In the first task, the algorithm takes the red layer from a phase-contrast RGB-image (Fig. 3a) to segment and separates the background pixels from the foreground (Fig. 3b). The second task applies morphological operations to the foreground to differentiate between pixels belonging to neurons and pixels from neurites. After that, a skeletonization operation to the neurites pixels is carried out to extract the neurite bifurcation points (Fig. 3c). Finally, in the third task, the adjacency matrix representing the network structure is created (Fig. 3d).

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Fig. 2 Schematic illustration of the layer aggregation process in the graph-based algorithm for image segmentation (left and right columns are equivalent representations). The multi-layer algorithm starts defining the first layer (the zero-layer L 0 ) as an input image (10 × 10 pixels of blue, yellow, red, green and white color) which is mapped into a graph whose nodes are the pixels of the image and each node’s neighborhood is formed by its 8 nearest neighbors. Solid lines in each layer depict the relationship (connectivity) between nodes which, for L 0 is defined only by the spatial location. For the next layers, the algorithm merges those nearby nodes whose relationship (similar color) is below a given threshold. The merged nodes in layer L 0 (for example the blue pixels which are neighbors in the grid) are mapped into one node (or region in the representation on the left) in the next layer L 1 and two nodes are said to be neighbors if the regions they are coming from are neighbors in L 0 . In the representation on the left, nodes sharing similar features are represented as regions while in the one on the right, each region is represented by the first node in that region. Dashed lines between layers account for those nodes in the lower layer that are grouped into the same node in the upper one. Notice that, for clarity, not all the dashed lines are drawn. Finally, the algorithm stops when a condition criterion is met (e.g. a given number of layers or number of regions). In this example, the algorithm transforms a grid representation into a single fade violet node (or region)

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Fig. 3 Image processing steps. a Enlarged area of a culture 8 DIV old. b Output of the segmentation algorithm. The region of interest, where neurons and processes are located, is highlighted in yellow (boundaries are in black). c Output of the neuron and neurite searching algorithm. Identified clusters of neurons are marked in green, neurites in blue and processes forks in yellow. d Graph representing the network structure where circles (clusters of neurons) and diamonds (forks and processes endings) are the nodes and blue straight lines (neurites) are the links between nodes whenever there is a process connecting them. Scale bar 500 µm

2.1.3

Network Structure Representation and Measures

The obtained CNN mask (Fig. 3b) is used in a twofold way: to extract morphological parameters characterizing both neuronal clusters (number, size, centroid, roundness, etc.) and neurites (length, orientation, etc.), and to extract the actual adjacency matrix encoding the topology of the neuronal network, as shown in Fig. 3d. The adjacency matrix there is constructed as a binary and undirected graph, whose nodes are either branching points or cluster centroids, and two nodes are linked if there is a neurite process connecting them. The algorithm also provides a network structure representation whose links can be unweighted or weighted by the neurite length connecting two nodes (cluster/branching point) (as shown in the example of Fig. 4b–c). Here we focus on the network statistical properties at the level of the neuronal clusters, ignoring the evolution of both neurite connections and branching points (Fig. 4d–e). Therefore, we end up with a subgraph defining the connectivity among the neuronal clusters in such a way that two of them are linked either directly or through a connected path of branching points. Treating all links as identical, i.e. ignoring edge length and edge directionality, such a graph can be described in terms of a symmetric matrix A whose elements ai j are equal to 1 if nodes i and j are linked, and 0 otherwise as shown in Fig. 4f. Such an object allows us to calculate all classical parameters characterizing the topology of a network (degree distribution, shortest path, node clustering, etc.) and the network morphology (cluster areas and

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Fig. 4 Extracted adjacency matrices. Circles and diamonds stand for neuronal clusters and branching points respectively and blue lines represent a physical connection between two nodes. a Small area of a larger neuronal network. b Weighted graph. The link’s thickness stands for the link’s weight. This weight is computed as the neurite length connecting two nodes. c Unweighted graph. d Weighted cluster graph. Links are weighted by the minimum neurite length between nodes (clusters). e Unweighted cluster graph formed by nodes representing clusters and links representing the existence or not of a path between cluster nodes through bifurcation nodes. f Adjacency matrix corresponding to the previous case

cluster centroids, culture area, neurite lengths, etc). We will report the evolution along days in vitro of these measures in the next Sect. 3.

3 Results In this Section we experimentally investigate the self-organization into a network of the cultures during the course of development, and explore the changes of the main morphological and topological features characterizing the anatomical connectivity during the network’s growth. We report the results on 6 cultured networks grown from independent sets, using the protocol described in Sect. 2. The density at which cultures are seeded determines the maturation rate and the spatial organization at the mature state [41, 45, 48]. For the purpose of this analysis, cultures with around 800 neurons after seeding are monitored during 20 days in vitro (DIV).

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3.1 Morphological Characterization From the seeding of the somas to the culture’s death, a neuronal culture grows with the aim of reaching a stage where neurons are interconnected among them. A typically observed growth evolution is shown in Fig. 5 through snapshots between 0 and 12 DIV. Isolated neuron somas, whose size ranges from 10 to 50 µm, are initially randomly scattered in a two dimensional substrate and end up organizing into a clustered network. To characterize and quantify the morphological changes that a culture experiences during the evolution we need to distinguish at this point between the terms cluster and node. We define a cluster as a group consisting of one or more neurons physically connected to one another through gap junctions (electrical synapses) while we refer to a node in the network as a cluster centroid that is connected with other nodes through neurites establishing chemical synapses. While the former is the result of the image segmentation process and may contain errors due to debris in the culture, the latter ensures that such clusters have developed processes and they are connected to other nodes. In the next subsections, we identify two completely different and almost consecutive developmental stages in the growth process: a first random spreading stage where all the connectivity forms, and a second stage of optimization of the circuitry formed in the previous stage. We characterize these two phases by monitoring the evolution of morphometric parameters of single clusters in cultures: the number and average cluster area (Fig. 6) and the mean number of nodes.

Fig. 5 Evolution of a neuronal culture into a clustered network. The left upper and lower frames represent the initial (DIV 0) and mature (DIV 12) configurations of the whole cultured area (size 7.7 × 6.0 mm2 ). Rectangles identify a specific area, whose intermediate evolution stages are reported in the other frames, ordered clockwise (see arrows). These latter snapshots correspond to DIV 0, 3, 5, 7, 10 and 12, respectively. Black (white) bars correspond to 500 (100) µm

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3.1.1

Random Growth

During the first days of culture and following the enzymatic and mechanical dissociation, neuronal bodies try to adapt to the new environment recovering their spherical shape and increasing their sectional areas as shown in Fig. 5 when comparing the enlarged panels from days 0 and 3. This subtle increase at DIV 3 is about 25.8% as shown in Fig. 6 where we monitor the average area of the clusters (Fig. 6a) and the number of clusters (Fig. 6b) as a function of age. We observe that from DIV 0 to 6, both the average cluster area and the number of clusters stay approximately constant despite errors due to culture debris. From DIV 3 to DIV 6 (top row of Fig. 5), cultures display a very intense phase of network development, in which neurons grow neurites in order to reach not only neighboring neurons or clusters but also neurites. During this growth process, neurites also split and reach other processes forming loops with no evidence for self-avoidance, giving rise to complex connective patterns with neurite-neurite and neurite-neuron synapses.

3.1.2

Optimization Phase

After the initial stage of neurite formation, the growth rate decreases and a different mechanism starts shaping the network: tension is generated along the neurites as they stretch between neurons or bifurcation points to form straight segments [4, 40, 41, 45] and the resulting network is characterized by a random distribution of few clusters of aggregated neurons linked by thick nerve-like bundles. When this force (tension) is stronger than the attachment force (adhesion) to the substrate, the neuronal migration is favored by giving rise to the formation of clusters of neurons. This response takes place throughout the culture’s life since the first connection is formed. However, it becomes more important when the neurite outgrowth stops and the culture reaches a stage where most of the neurons are connected, as evidenced

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from the panels at 7, 10 and 12 DIV of the bottom row in Fig. 5. This aggregation process is clearly outlined in Fig. 6 by the exponential increase of the average cluster area (Fig. 6a) and the corresponding decrease in the number of detected clusters (Fig. 6b). Due to tension forces, some parallel or close neurites merged into a single and thicker connection together with the retraction of those branches which did not target any neuron, slightly lowering the average connectivity. We consider that two neurites are parallel when the links ends are the same, i.e. two neurons connected by two neurites.

3.2 Topological Characterization and Percolation During the growth phase, spanning from DIV 0 to DIV 6, the average number of nodes N  with at least one connection slowly increases with age, while the average number of links L rises exponentially, reaching a maximum at DIV 6 (Fig. 7a). After this moment, the convergence of parallel neurites and neuronal clusterization induces a more gentle decrease in the number of links, accompanied by a slight reduction in the number of nodes. As illustrated in Fig. 7a, at any stage of development, the cultured networks are far from being fully connected (only about 2% of all possible connections exist between nodes), and thus operate in a low-cost regime of sparse anatomical connections. In such a sparse connectivity regime, we quantify how our networks constrained in 2D space percolate [30, 44]. To do so, we measure the size S1 of the giant connected component (GCC) and the size S2 of the second largest component (GCC2) as a function of the age [12, 30]. Figure 7b shows that the number of nodes forming such connected components smoothly increases at the same rate during the first days of

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the network development, up to the DIV 6 when the difference in size between them suddenly and consistently starts to grow. From that point on, percolation is reached as the GCC2 starts collapses and merges into the GCC, and the establishment of an almost fully connected network of clusters characterizes the rest of the culture’s life.

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The associated network statistics sheds light on the transition from random to nonrandom properties with a progression of both the clustering coefficient and the average path length (normalized by the GCC size) as a function of the age (see Fig. 8a). The first significant result is the simultaneous increase in the clustering coefficient (C) and decrease in the mean path length (L), a clear fingerprint of the emergence of a small-world network configuration. This configuration becomes prominent at DIV 6 and stays relatively stable throughout the last two weeks in vitro. To properly asses the significance of this finding and isolate the influence of the variable network size, we calculated the values of C and L normalized to the corresponding expected values for equivalent random (and lattice) null model networks (Fig. 8b). In doing so, we follow the approach used in similar circumstances for the obtainment of null models [58]. According to Watts and Strogatz’s model [54], a small-world network simultaneously exhibits short characteristic path length, like random graphs, and high clustering, like regular lattices. We found a clear change in the trend at DIV 6 where L rand /L ≤ 1, and the average path length of the cultured network starts to be close to that of a random graph and much smaller than that of a regular graph (L r eg is calculated as L r eg = S1 /(2k)). At the same time, the clustering coefficient is much higher (between 30–50 times) than that of the corresponding random graphs. These results are in agreement with previous morphological characterizations of in vitro neuronal networks at a single developmental stage [48], where a similar small-world arrangement of connections was evidenced at DIV 6. However, to reinforce the evidence of the emergence of a small-world configuration during the graph development (as well as the fact that here the small-world metrics are not influenced by network disconnectedness), in Fig. 9 we also measure the local (Fig. 9a) and global (Fig. 9b) efficiency, as introduced by Latora and Marchiori in Ref. [29]. These latter quantities, indeed, are seen as alternative markers of the small-world phenomenon, in that small-world networks are those propagating information efficiently both at a global and at a local scale. The efficiency curves of the cultured networks are reported as a function of the age, and compared to the efficiency of the equivalent random graphs. Our results indicate that the connectivity structure of the neuronal networks evolves towards maximizing global efficiency (making it similar to the value of random graphs), while promoting fault tolerance by maximization of local efficiency (which is, instead, larger than the local efficiency of a random graph), and both properties are carried out at a relatively low cost in terms of number of links.

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Emergence of Assortative and Single Scale Networks

Turning now our attention to network statistics at the micro-scale, we investigated how the node degree distribution evolved during maturation process. At all ages, cultures appeared to belong to the class of single-scale networks, displaying a well defined characteristic mean node degree. Figure 10a shows that the cumulative degree distributions P(k) for DIVs 3, 6, 7, and 12 have a fast declination with a non monotonous increase in the average connectivity, with most of the nodes having a similar number of connections and only a few ones with degrees deviating significantly from such a number. The data is fitted to an exponential scaling law y(k) ∼ exp(−k/b) with a level of confidence larger than 95%. It has to be remarked that the distribution of node connections, although always homogeneous, shifts during culture maturation toward much broader distributions, with few highly connected nodes appearing at DIVs 6 and 7. These hubs at the peak days of the culture evolution result from a branching process, allowing each single neuron to reach a larger neighborhood. Thus, at variance with scale-free networks [2, 8], our cultured and clustered networks are identified as a single-scale homogeneous population of nodes. This is in agreement with reports on many other biological systems like the neuronal network of the worm Caenorhabditis elegans [55, 56], and suggests the existence of physical costs for the creation of new connections and/or nodes limited capability [3]. While the number of neighbors (the degree) is a quantity retaining information at the level of a single node, one can go further to inspect degree-degree correlations, i.e. to quantify whether the degrees of two connected nodes are correlated. It is known, indeed, that biological networks generally display dissaortativity patterns [36], that is, connections are more likely to be established between high-degree and

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low-degree nodes. The assortativity coefficient r was calculated as the Pearson corp relation coefficient corresponding to the best fit of log(k) ∼ p log(ki ). If r > 0 (r < 0), the network is said to be assortative (disassortative), while depending upon the obtained value of p, the degree correlation properties are said to be of a linear ( p = 1), sub-linear ( p < 1), or super-linear ( p > 1) nature. Figure 10b shows the age evolution of the Pearson coefficient r and of the exponent p of the degree correlation (knn  = ak p ). At one hand, as r stays positive during the whole development we can generally conclude that our networks are assortative [16] and, on the other hand, as the degree correlation p is all days under 1 this indicates that it keeps a sub-linear ( p ∼ 0.6) degree-degree correlation regime during the small-world stage.

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4 Data Driven Model A series of previous studies [45, 47] singled out tension along neurites and adhesion to the substrate as the two main factors conditioning the neuronal self-organization into a clustered network. Here we go a step ahead, and propose a simple spatial model which not only incorporates migration of neurons but also explicitly considers neurite growth, and the establishment of synaptic connections.

4.1 Isotropic Growth Model This model is schematically illustrated in Fig. 11. We start by considering a set of N0 cells. Cells are small disks of radius a randomly distributed in a 2-dimensional circular substrate of area S. The algorithm then makes the connections and positions of such disks evolve at discrete times, each time step t corresponding to a DIV of the culture. The complex process of neurite growth and the establishment of synaptic connections is modeled by associating to each cell a time growing disk in such a way that two cells are linked at a given time step if their outer rings intersect as shown for DIVs 2 and 3 in Fig. 11. At each time step t, the radius ri ≥ a of cell i increases by a quantity δri , which decays as δri (t) =

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Fig. 11 Schematic representation of how cells get connected in the isotropic model. At DIV 0, 4 cells of radius a are located at random positions. The first iteration of the algorithm, DIV 1, assigns to each cell i a disk of radius ri (blue shade). At the next iteration, DIV 2, the disks’ growth rate decreases, ri , and a link between two cells is established when their disks intersect (DIV 3). This process continues for Ts steps

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The wiring process is iterated up to a given time step Ts , at which the formation of new connections is stopped. As for the process of cell migration and clusterization, cells or clusters whose distance is less than 2a are then merged into the same, new, cluster. Therefore, the number of clusters Nt is time dependent. Furthermore, whenever two cells are connected, an initial tension Ti j = 0.1ui j is created between them, and it is incremented in 0.1 force units at each time step, being ui j the unit vector along the direction connecting the two cells.  The total force acting on a cell or cluster i is given by Fi = j Ti j with j running over the cell indexes connected to i and not belonging to the same cluster. Furthermore, each cell is “anchored” to the substrate by a force Fa = 0.9 force units, and the ith cell can only be detached if there is a net force Fi acting on it with a magnitude larger than Fa . In the case of a cluster of cells, the adhesion force to the substrate is considered to be the sum of the individual adhesion of the cells composing the cluster. Therefore, cells and clusters move in a certain direction in all circumstances in which the net force acting on them overcomes the adhesion force, and an equilibrium point is reached at a new position in which the new net force balances (or is smaller than) the adhesion to the substrate (see Fig. 12). Despite the simplicity of the model, the isotropic model fits qualitatively the trends of the features at morphological and structural levels of the networks described in Sect. 3.1 [41]. However, an isotropic model for the neurite growth is not quite realistic, due to the fact that in this model neurons cannot connect to distant neurons without linking first to the nearest ones, and this results in a quantitative discrepancy with the experimental results in Sect. 3.1. Nevertheless, a close inspection of the model reveals that the only change we need to consider, in order to improve the overall performance of the model, is that the neurite growth velocity V is not isotropic.

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4.2 Anisotropic Growth Model To overcome the above problem, we start from the model previously reported in Sect. 4.1 that considers a set of N0 cells randomly distributed with a uniform cell density ρ and each cell is associated with an interaction disk of radius ri with an initial radius a. But, now, the neurite growth velocity V is no longer isotropic. Each cell disk i has a different radius with respect to cell j, meaning that the radius ri j ≥ a increases by a quantity δri j which decays as ξi j V δri j (t) = t

  ki (t − 1) 1− , Ki

where ξi j is a random number drawn from a uniform distribution between 0 and 1. This leads to the fundamental consequence that two disks initially located very close to each other do not necessarily establish a connection (Fig. 13), but they have instead an associated probability of linking, as observed in the experiment. The balance between tension and adhesion forces is established as in Sect. 4.1. To calibrate our model we ran a large number of simulations for different values of the model parameters N0 , V , and Ts . Remarkably, when comparing the simulated features to those measured from the experiments (the number of nodes, number of edges, average shortest path and clustering), we found that high correlation values exist only in a very narrow window of V and Ts . Precisely, the parameter values that better fit the experimental observations are N0 = 700 neurons, V = 65 ± 5 and Ts = 10 ± 5. Once the model is calibrated, we find that our model accurately reproduces the experimental values for the shortest path normalized by the GCC size and for the clustering coefficient, Fig. 14. In addition, as an example of how the anisotropic model outperforms the isotropic one, if we focus on the distance distribution, depicted in Fig. 15, we observe no abrupt tail distribution as with the isotropic model. On the opposite, as in the anisotropic model long-range connections can be established without making all close range connections, the distance distribution has a smooth tail as in the experimental data.

Fig. 13 Schematic representation of how cells get connected in the anisotropic model. At each step, neurites from each cell growth looking for other neurons or neurites of other cells. This neurite growth is different from each simulated neurite given an anisotropic growth. This process continues until Ts steps

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Fig. 15 Comparison between cluster distances distributions corresponding to both the iso (dashed line) and anisotropic (dotted line) models and those from the experiments (solid line). Histograms are calculated at DIV 5 and experimental data are averaged for the six cultures. Simulation parameters are N0 = 700, Ts = 12, and V = 65 in the case of the anisotropic model, and N0 = 700 neurons, Ts = 12, and V = 85 for the isotropic model, which yield their best calibrations

4.3 Model Analysis: Transition to Clustered Networks The neuron migration and clustering play a key role in the development of the network’s structure. According to Ref. [45], “clustering instability is a phenomenon exhibited by many living and nonliving systems” and it is considered to be related to an optimal self-organization. In turn, those processes in neuronal cultures are favored by the tension forces between clusters and the cell adhesion to the substrate [4, 45]. In this section, we examine the relationship between the initial cell density and the adhesion force and how these two parameters impact the network’s morphology

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and topology. For that purpose, we use the in silico model presented in Sect. 4.2 by fixing model’s parameters V and Ts to 65 and 12, respectively, and by varying both the initial cell density N0 (or the number of seeded cells) and cell adhesion to the substrate Fa (previously fixed to Fa = 0.9). Specifically, parameter values are N0 = {50, 100, 150, . . . , 850} and Fa = {0.1, 0.2, 0.3, . . . , 1.9}. Each parameter combination is simulated three times. Figure 16 depicts some morphological measures revealing two behaviors in the phase diagram N0 − Fa . For instance, we show in Fig. 16a the fraction of clusters at a mature state N19 at DIV 19 with respect to the initial number of cells N0 , and we observe that the average percentage of aggregated neurons 1 − N19 /N0 reaches values of the order of 100% for low adhesion forces and high N0 , while for high adhesion forces and low N0 , cells keep their initial positions. The reader may also notice from Fig. 16b that a poor cell adhesion to the substrate causes a massive migration as shown by the relatively high values of the ensemble average of the cell’s displacement Δd with respect the substrate diameter D, while high adhesion

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causes moderate migration (∼2%) and aggregation (∼10%). Regarding the effect on the network connectivity, Fig. 16c shows the ensemble average of the fraction of clusters to which each cluster is connected to at DIV 19. We observe that connectivity is absent for low N0 (lower left corner of the phase diagram) due to the fact that neurons are so far away that they are not able to reach each other within the Ts time, and in case they get connected, the adhesion force is so small that is not able to cancel the tension collapsing in one cluster. However, for large enough N0 , if the adhesion force is small, the network undergoes an instability in which cells aggregate very fast (Δd/D large) giving rise to an increase (∼2%) of the normalized mean degree k/N19 . In the rest of the phase diagram, each cluster is connected in average to the 1% of the relatively large number of available clusters.

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We also analyze the self-organization process in terms of the topological properties of the emergent networks in the phase diagram N0 − Fa . To that end, in Fig. 17 we compare, at DIV 19, the average shortest paths and clustering coefficients of the in silico networks with the theoretical values corresponding to the randomized and regular network realizations. We report Crand /C in Fig. 17a, L rand /L in Fig. 17b, and L/L r eg in Fig. 17c, and from the distribution of these values, we distinguish four regions, which are depicted in Fig. 17d: • In region (1), morphologically characterized by a high aggregation rate, the clustering is larger than in an equivalent ER network while the shortest path is of the

Fig. 18 Representative network states at DIV 19 of the different regions identified in the phase diagram Fa − N0 of Fig. 17 and depicted here at the center of the figure. Each network is also accompanied by the initial (DIV 0) network state for comparison. The four selected points in the diagram correspond to high (blue and orange boxes) and low (green and purple boxes) N0 with strong (blue and green boxes) and weak (orange and purple boxes) adhesion forces. Each dot represents a cell and links between cells are depicted as blue lines

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same order, compatible with a small-world network with a very small number of clusters. • In region (2) networks are moderately aggregated and parameter space yield network topologies also compatible with a small-world structure. • In regions (3) and (4), connected components are very sparse, with very small clustering coefficients and large shortest paths. For the sake of illustration, Fig. 18 shows some representative networks corresponding to the above identified regions. Each box corresponds to one point in the phase diagram and plots the network structures at the initial (DIV 0) and final (DIV 19) instants of the automata model generations. Notice how a high N0 almost guarantees the formation of a large connected component (blue and orange boxes) while low N0 preparations give rise to very sparse networks with no evidence of cluster aggregation.

5 Conclusions Despite the existence of some functional studies of CNNs [19, 52] and also of a few theoretical-numerical attempts to model the relationships between CNNs structure and function [21, 23], no experimental verifications are available in terms of scrutinizing that relationship, basically due to the lack of tools allowing a simultaneous tracking of the cultures dynamical activity and morphological/topological changes. To that purpose we have developed an algorithm and a set of tools to enable automatic location of neurons and tracing of neurites in non-invasive phase-contrast images, which can be acquired simultaneously with electrophysiological measures (with e.g., micro-electrode arrays), hence potentially allowing for a combined study of network structure and dynamics. The relevance and value of our work is not only to make a first step in the direction of unveiling and uncovering the structure/function relationships during the evolution of a CNN but also to provide a full longitudinal characterization of the successive network states from immature networks to the final wiring map of adult circuits. We provided a large scale experimental investigation of the morphological evolution of in vitro primary cultures of dissociated invertebrate neurons from locust ganglia. At all stages of the cultures development, we were able to identify neurons and neurites location in automated way, and extract the adjacency matrix that fully characterizes the connectivity structure of the networking neurons. A systematic statistical analysis of a group of topological observables has later allowed tracking of the main network characteristics during the self-organization process of the culture, and drawing important conclusions on the nature of the processes involved in the culture structuring. Early stages of development ( 0 for some value of k. In such a case, if I m(λ) = 0 for all k with Re(λ) > 0 the instability produces static patterns typically associated to a Turing bifurcation [32]. On the other hand, if I m(λ) = 0 for any value with Re(λ) > 0 the instability produces waving behaviors and non-static patterns [11].

3.1 Active Poroviscoelastic Solid The linearization of Eqs. (18) produces the next characteristic equation for the growth rates λ:   ξ(∂ f ) − ρ E ρ2 E ρ 1 c 2  + Dk 4   = 0;  λ2 + λDk 2 1 − (21) 2 2 Dβ 1 +  k β 1 + 2 k 2 where we define 2 = η/β, and obtain (∂c f ) = 1/4 for the case chosen in Eq. (7). If the values of λ are always negative the homogeneous steady state is stable. However, an increase of the active tension of the system, produces the cross of the bifurcation point and λ becomes positive for a certain small window of values of k.

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With the Kelvin–Voigt viscoelastic model there is a non-zero complex component of the growth rate λ and therefore a temporal dynamics is expected, see an example in Fig. 4a. The real part is maximum for k p ∼ 0.3 µm−1 which corresponds to a size of L = 2π/k p ∼ 20 µm, approximately the size of a cell. The temporal frequency of the pattern is expected to be ω p ∼ 1 s−1 which gives rise to a period of T = 2π/ω p ∼ 6 s. Similar spatial and temporal scales are expected for reduced elastic modulus, see Fig. 4b. However, in the limit E = 0 the imaginari component of λ becomes null and no frequency can be defined in the linear stability analysis, see Fig. 4c. In such case static patterns are expected. Note that Eq. (21) reduces to a pure viscous dynamics in the limit E → 0:   ρ1 ξ(∂c f ) 2  ;  (22) λ = −Dk 1 − Dβ 1 + 2 k 2 For more details about the viscous limit see [7] for a very similar set of equations. The detailed phase diagram of the Eqs. (18) obtained by the linear stability analysis is shown in Fig. 5. A straight line divides the diagram in a region where the homogeneous state is stable, typically associated to large elastic modulus and small active tension, and a region with an unstable homogeneous state with a non-zero imaginari growth rate, region associated to small elastic modulus and large active tension. For small values of the Elastic modulus E ∼ 0 the frequency of oscillations becomes zero and the patterns stationary (not shown in Fig. 5). In summary, spatio-temporal patterns are expected in the poroviscoelastic solid model of the cell cytoplasm with the characteristic temporal periods T ∼ 1 − 6 s and spatial scales L ∼ 10–20 µm.

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3.2 Active Poroviscoelastic Fluid The linearization of Eqs. (20) produces the next characteristic equation for the growth rates λ:     ρ ξ(∂ f ) 2k2 2 2 1 + 2 k 2 − 1 Dβc 1 +  ξ(∂ f )k E ρ E Dk 1 c  +   λ2 + λ Dk 2 + −  =0 η1 1 + 22 k 2 η1 β 1 + 22 k 2 1 + 22 k 2 (23) where we define 22 = ρ1 η2 /β. In Fig. 6 two characteristic dispersion curves λ(k) are shown. Two different cases are obtained in the linear stability analysis. First, non-zero imaginari growth rates are obtained, typically associated with waves. The main difference with respect the waves obtained in the previous section is the large value of the imaginari component ω ∼ 103 − 104 s−1 , which corresponds to periods around T = 1 − 10 ms, see Fig. 6a. Furthermore, the maximum wave number k p ∼ 20 µm−1 , corresponds to waves with characteristic length L ∼ 0.3 µm much smaller than the waves observed in the previous section. Second, for higher values of E, a region for small wave numbers k with zero imaginari component appears. If the band of unstable modes is restricted to this region static patterns will be expected, see Fig. 6b, where the characteristic length is again of the order of the cell size L ∼ 10 − 20 µm. In contrast with the previous model, Eq. (23) reproduces the viscous active fluid dynamics Eq. (22) in the opposite limit E → ∞. The detailed phase diagram of the Eqs. (20) obtained by the linear stability analysis is shown in Fig. 7. The stable homogeneous solution becomes unstable when the active tension crosses a threshold value. Waves and Turing patterns are separated by a straight line, while the bifurcation to static Turing patterns is associated to larger values of E, wave bifurcation is related with small values of E, see Fig. 7.

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4 Numerical Simulations We perform one-dimensional numerical simulations of the two viscoelastic models of active cytoplasm to evaluate the nonlinear effects of uniaxial stresses. We employ the finite differences method for the spatial and temporal derivatives and an explicit Euler method for the temporal evolution of the concentration and velocity profiles. The total time of integration depends on the particular numerical simulation and the total size is kept fix to 20 µm representing the characteristic size of an individual living cell. At both ends of the cell, we consider non-flux boundary conditions for the concentration and keep the velocities equal to zero, mimicking a non-moving membrane.

4.1 Active Viscoelastic Solid We integrate numerically Eqs. (18) using parameter values inside the region of waves in the phase diagram, see Fig. 5. We employ parameter values close to the bifurcation line to avoid unphysical deformations |∂x u| > 1, condition which appears further inside the region of waves in the phase diagram and indicates the requirement of non-linear elasticity. For small values of the fraction of phase 2, ρ2  1 the whole phase diagram can be explored [28]. An example of mechano-chemical waves is shown in Fig. 8. It is observed an alternation with a period of the order of several seconds (T ∼ 8 s) of high concentration in the opposite limits of the one-dimensional system representing the cell. These biochemical oscillations interplay with mechanical deformations and the velocity of the two phases oscillates, as it is shown in Fig. 8, giving rise to velocities close to 10 µm/s of both phases. More complex waves are obtained for larger values of E and ξ following the transition to waves in the phase diagram of Fig. 5. In the opposite limit, E → 0 stationary patterns can be also obtained.

4.2 Active Viscoelastic Fluid We employ two different set of parameter values for the numerical integration of Eqs. (20). These two conditions correspond to an unstable homogeneous steady state with and without imaginari growth rate, which are related to waves and Turing instabilities, respectively. The most typical instability with the Maxwell model is the Turing instability, see Fig. 7. An example of such static pattern of concentration in a one-dimensional system is shown in Fig. 9. The concentration of the regulator c accumulates in one side of the system while there is two continuous flows of the two phases in opposite directions

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Fig. 8 Spatio-temporal plots resulting from a one-dimensional numerical simulation of an active viscoelastic solid. The evolution of the concentration c and velocities of both phases 1 and 2 are shown for each point of the domain. Parameter values of the model are E = 2 kPa and ξ = 5 kPa. Values of the numerical parameters: Δt = 10−4 s and Δx = 0.2 µm

Fig. 9 Spatio-temporal plots resulting from a one-dimensional numerical simulation of an active viscoelastic fluid. The evolution of the concentration c and velocities of both phases 1 and 2 are shown for each point of the domain. Parameter values of the model are E = 1 kPa and ξ = 1.4 kPa. Values of the numerical parameters: Δt = 2 · 10−5 s and Δx = 0.1 µm

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Fig. 10 Spatio-temporal plots resulting from a one-dimensional numerical simulation of an active viscoelastic fluid. The evolution of the concentration c and velocities of both phases 1 and 2 are shown for each point of the domain. Parameter values of the model are E = 0.1 kPa and ξ = 1.4 kPa. Values of the numerical parameters: Δt = 10−5 s and Δx = 0.05 µm

with velocities close to 5 µm/s. Such flows keep the incompressibility condition but induces a unrealistic continuous displacement of material for long simulations. For small values of the elastic modulus the homogeneous steady state becomes unstable due to a wave bifurcation when the value of the active tension crosses the bifurcation line, giving rise to dynamical pattens. An example of such transition is shown in Fig. 10. We observe fast oscillations with periods of the order of T ∼ 10−2 s, which are three orders of magnitude smaller than the waves shown in Fig. 8. The characteristic wave length is around half of micron L ∼ 0.6 µm and, as in the case of the temporal oscillations, it is also much smaller than in the waves shown in Fig. 8, where the wave length coincides approximately with the size of a cell L ∼ 20 µm.

5 Conclusions The viscoelastic properties of the cytoplasm depend on the particular type of cell and experimental realization. Simple viscoelastic models are used in the study of active cytoplasm of living cells. The particular viscoelastic properties may be tunned by the activity of the living cell and the environment. However, recently has been demonstrated that some cells are poroviscoelastic [25]. We have shown in this chapter

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Fig. 11 One-dimensional profiles of the active tension, the concentration of the regulator and the velocity of both phases obtained from the numerical simulations of the two types of poroviscoelastic media with uniaxial stresses: Poroviscoelastic solid with E = 5 kPa and ξ = 11 kPa at time t = 40 s (a), poroviscoelastic fluid corresponding to Fig. 9 at time t = 100 s (b), and poroviscoelastic fluid corresponding to Fig. 10 at time t = 10 s (c). Arrows show the direction of the motion for the corresponding phase

two simple models of poroviscoelasticity of the cell. They can give rise to very different active behavior under the similar biochemical conditions. It is important to note that there is not correct model of the viscoelastic properties of the cell. The most convenient model depends on the particular situation and on different factors like the cell type and even the experimental technique. The particular choice of the viscoelastic model may give rise to very different responses of the cell, for example, see the comparison in Fig. 11 among behaviors with different models and parameter values. For comparison, the profile of the active tension, concentration and velocities are shown keeping the same values on the axis in the figure for the three cases under consideration. For example, waves in Kelvin–Voigt and Maxwell viscoelastic models appear with very different spatial scales, see the particular profiles in Fig. 11, and different temporal scales, compare Figs. 8 and 10. The two different scales appear already in the linear stability analysis of both models. Compare, for example, the values in the axis on Figs. 4 and 6. The maximal wave number kmax for the case of the Kelvin–Voigt model, with parameter values used in Fig. 11, is around kmax ∼ 0.4 µm−1 corresponding to a wale length of

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L max ∼ 15 µm, which is close to the typical size of a cell L si ze ∼ 10 − 20 µm. For the case of the Maxwell model, see Fig. 6, the maximal wave number is around kmax ∼ 15 µm−1 which corresponds to a wale length of L max ∼ 0.4 µm corresponding to values comparable with the size of fine structures of the cell. The use of springs and dashpots for the modeling of the viscoelastic properties is only a phenomenological approach, they do not correspond to any particular biomechanics of an organelle of the living cell. Computational tensegrity models are also phenomenological models based in the description of prestresses of the interconnected filament structure, which produce better descriptions for dynamical rheology of living cells [15]. There are other models beginning from the microscopical structure of the cytoskeleton where the parameters have a better comparison with the physical quantities [19]. The poroviscoelastic model for an active cytoplasm presented in this chapter is an oversimplification. Several extensions are possible and some of them already has been implemented to model certain cells. Next, we list some of the possible extensions: • The biochemistry employed in the two models is the same: a simple relation between active stress and the concentration of the regulator [C], see Eq. (6). More complex biochemical models can include non-linear reactions in the equation of [C], see Eq. (5), or include more reaction-advection-diffusion equations for additional biochemical concentrations. Such extension has been already applied for the modeling of patterns in protoplasmic droplets of Physarum polycephalum [2, 29]. • In the numerical simulations with the Maxwell model of the viscoelasticity there are continuous flows of both phases in opposite directions. Despite such flows, the local fraction of the phases are kept constant ρ1 = ρ2 = 0.5. A more detailed description may include the flow of the two phases and dynamical equations for the fractions ρ˙1 and ρ˙2 . A version of these equations was already employed in the formation of membrane protrusions in living cells [4]. • We have performed numerical simulations of the active poroviscoelastic models only close to the onset of the bifurcation to avoid unphysical deformations, which can break the condition |∂x u 1 | < 1. To perform numerical simulations deep inside the region of the instability we can employ more complex models of linear viscoelasticity (coupling more dashpots and springs into the description) or include non-linear viscoelasticity [3]. In summary, there is a large variety of types of cells and environments. Evolution may have driven cells to develop different viscoelastic properties to adapt to particular features of the environment. It implies that there is not a clear procedure for the choice of the most suitable viscoelastic model for the description of a particular living cell. The most convenient description has to be obtained from the particular results of experiments in cell rheology.

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Acknowledgements I acknowledge fruitful discussions with Markus Bär, Markus Radszuweit, Harald Engel and Marcus J.B. Hauser. I thank financial support by MINECO of Spain under the Ramon y Cajal program with the grant number RYC-2012-11265 and FIS2014-55365-P.

Appendix For linear viscoelastic models the strain ε is the deformation by unit of length of the material and the stress σ is the tension of the material under deformation. For a solid, both quantities are linearly related σs = Eεs , by the elastic modulus E. Such dependence is equivalent to the Hooke law, therefore, the spring is a good phenomenological model for a solid. For a fluid the stress is only proportional to the velocity and, therefore, to the rate of change of the strain σ f = η∂t ε f , where the constant of proportionality is the viscosity or viscous damp η. The strain ε corresponds to an infinitesimal change of the deformation field u and therefore ε = ∂x u in the continuous models used in the chapter.

Stress-Strain Relation for the Kelvin–Voigt Model For the Kelvin–Voigt model, a spring and a dampshot are coupled in parallel, see Fig. 2a. Strains of both branches are the same ε = εs = ε f and the total stress σ is the addition of both stresses: σ = σs + σ f = Eε + η∂t ε;

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Stress-Strain Relation for the Maxwell Model For the Maxwell model, a spring and a dampshot are coupled in series, see Fig. 2b. While the stress is the same along the branch σ = σs = σ f , the individual displacements of the spring and the dashpot may be different. The total strain is the addition of both strains ε = εs + ε f , and its derivative gives rise to the relation ∂t ε = ∂t εs + ∂t ε f . The stress-strain relation can be then obtained:

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which, in the case of continuous rate of change of strain ∂t ε = ε˙ o produces a saturation of the stress with a characteristic time τ = η/E:   σ (t) = η˙εo 1 − e−t/τ ;

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which produces to the dynamics shown in Fig. 2b.

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