Comparative genomics : 15th International Workshop, RECOMB CG 2017, Barcelona, Spain, October 4-6, 2017, Proceedings 978-3-319-67979-2, 3319679791, 978-3-319-67978-5

Table of contents :
Front Matter ....Pages I-VIII
The Similarity Distribution of Paralogous Gene Pairs Created by Recurrent Alternation of Polyploidization and Fractionation (Yue Zhang, David Sankoff)....Pages 1-13
A Tractable Variant of the Single Cut or Join Distance with Duplicated Genes (Pedro Feijão, Aniket Mane, Cedric Chauve)....Pages 14-30
Generation and Comparative Genomics of Synthetic Dengue Viruses (Eli Goz, Yael Tsalenchuck, Rony Oren Benaroya, Shimshi Atar, Tahel Altman, Justin Julander et al.)....Pages 31-52
ASTRAL-III: Increased Scalability and Impacts of Contracting Low Support Branches (Chao Zhang, Erfan Sayyari, Siavash Mirarab)....Pages 53-75
Algorithms for Computing the Family-Free Genomic Similarity Under DCJ (Diego P. Rubert, Gabriel L. Medeiros, Edna A. Hoshino, Marília D. V. Braga, Jens Stoye, Fábio V. Martinez)....Pages 76-100
New Algorithms for the Genomic Duplication Problem (Jarosław Paszek, Paweł Górecki)....Pages 101-115
TreeShrink: Efficient Detection of Outlier Tree Leaves (Uyen Mai, Siavash Mirarab)....Pages 116-140
Rearrangement Scenarios Guided by Chromatin Structure (Sylvain Pulicani, Pijus Simonaitis, Eric Rivals, Krister M. Swenson)....Pages 141-155
A Unified ILP Framework for Genome Median, Halving, and Aliquoting Problems Under DCJ (Pavel Avdeyev, Nikita Alexeev, Yongwu Rong, Max A. Alekseyev)....Pages 156-178
Orientation of Ordered Scaffolds (Sergey Aganezov, Max A. Alekseyev)....Pages 179-196
Finding Teams in Graphs and Its Application to Spatial Gene Cluster Discovery (Tizian Schulz, Jens Stoye, Daniel Doerr)....Pages 197-212
Inferring Local Genealogies on Closely Related Genomes (Ryan A. Leo Elworth, Luay Nakhleh)....Pages 213-231
Enhancing Searches for Optimal Trees Using SIESTA (Pranjal Vachaspati, Tandy Warnow)....Pages 232-255
On the Rank-Distance Median of 3 Permutations (Leonid Chindelevitch, João Meidanis)....Pages 256-276
Statistical Consistency of Coalescent-Based Species Tree Methods Under Models of Missing Data (Michael Nute, Jed Chou)....Pages 277-297
Fast Heuristics for Resolving Weakly Supported Branches Using Duplication, Transfers, and Losses (Han Lai, Maureen Stolzer, Dannie Durand)....Pages 298-320
Back Matter ....Pages 321-321

Citation preview

Joao Meidanis

LNBI 10562

Luay Nakhleh (Eds.)

Comparative Genomics 15th International Workshop, RECOMB CG 2017 Barcelona, Spain, October 4–6, 2017 Proceedings

123

Lecture Notes in Bioinformatics

10562

Subseries of Lecture Notes in Computer Science

LNBI Series Editors Sorin Istrail Brown University, Providence, RI, USA Pavel Pevzner University of California, San Diego, CA, USA Michael Waterman University of Southern California, Los Angeles, CA, USA

LNBI Editorial Board Søren Brunak Technical University of Denmark, Kongens Lyngby, Denmark Mikhail S. Gelfand IITP, Research and Training Center on Bioinformatics, Moscow, Russia Thomas Lengauer Max Planck Institute for Informatics, Saarbrücken, Germany Satoru Miyano University of Tokyo, Tokyo, Japan Eugene Myers Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany Marie-France Sagot Université Lyon 1, Villeurbanne, France David Sankoff University of Ottawa, Ottawa, Canada Ron Shamir Tel Aviv University, Ramat Aviv, Tel Aviv, Israel Terry Speed Walter and Eliza Hall Institute of Medical Research, Melbourne, VIC, Australia Martin Vingron Max Planck Institute for Molecular Genetics, Berlin, Germany W. Eric Wong University of Texas at Dallas, Richardson, TX, USA

More information about this series at http://www.springer.com/series/5381

Joao Meidanis Luay Nakhleh (Eds.) •

Comparative Genomics 15th International Workshop, RECOMB CG 2017 Barcelona, Spain, October 4–6, 2017 Proceedings

123

Editors Joao Meidanis University of Campinas Campinas, São Paulo Brazil

ISSN 0302-9743 Lecture Notes in Bioinformatics ISBN 978-3-319-67978-5 DOI 10.1007/978-3-319-67979-2

Luay Nakhleh Rice University Houston, TX USA

ISSN 1611-3349

(electronic)

ISBN 978-3-319-67979-2

(eBook)

Library of Congress Control Number: 2017953424 LNCS Sublibrary: SL8 – Bioinformatics © Springer International Publishing AG 2017 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, express 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume contains the papers presented at RECOMBCG-17: the 15th RECOMB Comparative Genomics Satellite Workshop, held during October 4–6, 2017, in Barcelona, Spain. There were 32 submissions. Each submission was reviewed by at least three reviewers, and some received up to five reviews. The reviews were conducted by members of the Program Committee as well as by additional reviewers who were sought based on their expertise for specific papers. The committee decided to accept 16 papers. The program also included four invited talks and a poster session. We thank the members of the Program Committee as well as the additional reviewers for their diligent work reviewing the manuscripts and conducting thorough discussions on the manuscripts and their reviews that informed the decision process. We also thank the members of the Steering Committee, Jens Lagergren, Aoife McLysaght, and David Sankoff, for their guidance. Special thanks go to the local organizers, Laura Prat Busquets, Tomàs Marquès-Bonet, and Roderic Guigó Serra. Last but not least, we would like to thank the four keynote speakers who agreed to speak at the conference despite their very busy schedules: Iria Fernández Silva, Toni Gabaldón, Iñaki Ruiz-Trillo, and Wes Warren. The workshop would not have been possible without the generous contribution of our sponsors, whose support is greatly appreciated: – – – – – – – –

FEDER - Fondo Europeo de Desarollo Regional Goberno de España - Ministerio de Economía, Industria y Competitividad Instituto de Salud Carlos III PRB2 - Plataforma de Recursos Biomoleculares y Bioinformáticos ELIXIR Spain INB - Instituto Nacional de Bioinformática Illumina The Company of Biologists

We used the EasyChair system for submissions, reviews, and proceedings formatting. August 2017

Joao Meidanis Luay Nakhleh

Organization

Program Committee Max Alekseyev Lars Arvestad Anne Bergeron Marília Braga Cedric Chauve Leonid Chindelevitch Miklos Csuros Ingo Ebersberger Nadia El-Mabrouk Guillaume Fertin Katharina Jahn Asif Javed Kevin Liu Ketil Malde Joao Meidanis István Miklós Siavash Mirarab Luay Nakhleh Aida Ouangraoua Marie-France Sagot Michael Sammeth Jens Stoye Krister Swenson Eric Tannier Glenn Tesler Tamir Tuller Lusheng Wang Tandy Warnow Louxin Zhang Jie Zheng

George Washington University, USA Stockholm University, Sweden Université du Quebec a Montreal, Canada Bielefeld University, Germany Simon Fraser University, Canada Simon Fraser University, Canada Université de Montreal, Canada Goethe University, Germany University of Montreal, Canada University of Nantes, France ETH Zurich, Switzerland GIS A-STAR, Singapore Michigan State University, USA Institute of Marine Research University of Campinas/Scylla Bioinformatics, Brazil Rnyi Institute, Budapest, Hungary University of Texas at Austin, USA Rice University, USA Université de Sherbrooke, Canada Inria, France Centre d’Anàlisí Genòmica (CNAG) Bielefeld University, Germany CNRS, Université de Montpellier, France Inria, France University of California, San Diego, USA Tel Aviv University, Israel City University of Hong Kong, SAR China The University of Illinois at Urbana-Champaign, USA National University of Singapore, Singapore Nanyang Technological University, Singapore

Additional Reviewers Aganezov, Sergey Alexeev, Nikita Avdeyev, Pavel Bansal, Mukul S. Bulteau, Laurent Coelho, Vitor Doerr, Daniel

Durand, Dannie Fransiskus Xavier, Ivan Labarre, Anthony Moret, Bernard Peterlongo, Pierre Sankoff, David Schulz, Tizian

Sinaimeri, Blerina Viduani Martinez, Fábio Henrique Willing, Eyla Wittler, Roland Zhou, Xinrui

Contents

The Similarity Distribution of Paralogous Gene Pairs Created by Recurrent Alternation of Polyploidization and Fractionation . . . . . . . . . . . Yue Zhang and David Sankoff

1

A Tractable Variant of the Single Cut or Join Distance with Duplicated Genes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pedro Feijão, Aniket Mane, and Cedric Chauve

14

Generation and Comparative Genomics of Synthetic Dengue Viruses . . . . . . Eli Goz, Yael Tsalenchuck, Rony Oren Benaroya, Shimshi Atar, Tahel Altman, Justin Julander, and Tamir Tuller ASTRAL-III: Increased Scalability and Impacts of Contracting Low Support Branches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chao Zhang, Erfan Sayyari, and Siavash Mirarab Algorithms for Computing the Family-Free Genomic Similarity Under DCJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diego P. Rubert, Gabriel L. Medeiros, Edna A. Hoshino, Marília D.V. Braga, Jens Stoye, and Fábio V. Martinez

31

53

76

New Algorithms for the Genomic Duplication Problem . . . . . . . . . . . . . . . . Jarosław Paszek and Paweł Górecki

101

TreeShrink: Efficient Detection of Outlier Tree Leaves . . . . . . . . . . . . . . . . Uyen Mai and Siavash Mirarab

116

Rearrangement Scenarios Guided by Chromatin Structure . . . . . . . . . . . . . . Sylvain Pulicani, Pijus Simonaitis, Eric Rivals, and Krister M. Swenson

141

A Unified ILP Framework for Genome Median, Halving, and Aliquoting Problems Under DCJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pavel Avdeyev, Nikita Alexeev, Yongwu Rong, and Max A. Alekseyev Orientation of Ordered Scaffolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergey Aganezov and Max A. Alekseyev Finding Teams in Graphs and Its Application to Spatial Gene Cluster Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tizian Schulz, Jens Stoye, and Daniel Doerr Inferring Local Genealogies on Closely Related Genomes . . . . . . . . . . . . . . Ryan A. Leo Elworth and Luay Nakhleh

156 179

197 213

VIII

Contents

Enhancing Searches for Optimal Trees Using SIESTA. . . . . . . . . . . . . . . . . Pranjal Vachaspati and Tandy Warnow

232

On the Rank-Distance Median of 3 Permutations . . . . . . . . . . . . . . . . . . . . Leonid Chindelevitch and João Meidanis

256

Statistical Consistency of Coalescent-Based Species Tree Methods Under Models of Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Nute and Jed Chou

277

Fast Heuristics for Resolving Weakly Supported Branches Using Duplication, Transfers, and Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Han Lai, Maureen Stolzer, and Dannie Durand

298

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

The Similarity Distribution of Paralogous Gene Pairs Created by Recurrent Alternation of Polyploidization and Fractionation Yue Zhang and David Sankoff(B) University of Ottawa, Ottawa, Canada {yzhan481,sankoff}@uottawa.ca

Abstract. We study modeling and inference problems around the process of fractionation, or the genome-wide process of losing one gene per duplicate pair following whole genome doubling (WGD), motivated by the evolution of plants over many tens of millions of years, with their repeated cycles of genome doubling and fractionation. We focus on the frequency distribution of similarities between the two genes, over all the duplicate pairs in the genome. Our model is fully general, accounting for repeated duplication, triplication or other k-tupling events (all subsumed under the term WGD), as well as a general fractionation rate in any time period among multiple progeny of a single gene. It also has a biologically and combinatorially well-motivated way of handling the tendency for at least one sibling to survive fractionation. We show how the method reduces to previously proposed models for special cases, and settles unresolved questions about the expected number of gene pairs tracing their ancestry back to each WGD event. Keywords: Whole genome duplication · Gene loss · Birth and death process · Multinomial model · Paralog gene tree · Sequence divergence

1

Introduction

The basic data of comparative genomics and phylogenetics is the degree of identity between homologous genes or proteins, as assessed by any one of a number of quantitative measures. For example, identity among orthologous genes, singly or in combination with other genes, feeds directly into distance-based methods for phylogenetic inference. An important way of visualizing and analyzing the relationship among and within genomes is based on the distribution of similarities (percent identity, KS , 4dTv, etc.) across all pairs of detectible homologs. Similarities between homologs decay over time in a roughly synchronous manner due to sequence-level mutations - largely base replacement, insertion and deletions - so that similarities between orthologs in two genomes tend to be relatively highly concentrated around a value indicative of the time of speciation of the two organisms, while a concentration of similarities between paralogs in a single genome points to the c Springer International Publishing AG 2017  J. Meidanis and L. Nakhleh (Eds.): RECOMB CG 2017, LNBI 10562, pp. 1–13, 2017. DOI: 10.1007/978-3-319-67979-2 1

2

Y. Zhang and D. Sankoff

Fig. 1. Distribution of similarities in poplar, showing two peaks, one reflecting an early core eudicot triplication, the other a more recent duplication in the family Salicaceae. Additional peak near 100% similarity captures heterozygous maternal-paternal pairs, not duplicate genes.

time of a whole genome duplication (WGD) event. The concentration of these homolog similarities are reflected in local peaks, or local modes, in the frequency distributions of all similarities. There may be several peaks if there has been a history of recurrent whole genome doubling, tripling, quadrupling, etc. events (collectively referred to as WGD), as illustrated in Fig. 1. Except for very recent events, however, the evolutionary signal in the distribution may be largely or completely obscured by: • the progressive flattening of the peaks as the genome-wide synchrony of duplicate gene similarity decays; that is, the variance of the duplicate similarities increases as a function of time, and the tails of neighbouring peaks increasingly overlap, • the accumulation of “tandem” duplicate pairs that arise over time, not linked to a genome-wide event like speciation or polyploidization, • the propagation of transposable elements, • bioinformatic limitations in gene identification and gene comparison and, most important, • the process of fractionation whereby either member of a duplicate pair disappears due to DNA excision [1,2] or pseudogenization. All of these processes increase the level of background noise more severely for early events than for more recent ones. The most directly applicable statistical method in wide use treats the distribution of similarities of duplicate pairs as a mixture of distributions, such as

The Similarity Distribution of Paralogous Gene Pairs

3

that implemented in the Emmix software [3]. While this may produce optimum results for the mixture problem, it cannot take into account biological constraints and will often produce biologically uninterpretable results, such as lower variance components of the mixture for early events or inconsistently smaller numbers of pairs for recent events. In this paper, we propose a general model for the repeated cycle of WGD events followed by fractionation. This allows an arbitrary number of events and arbitrary rates of fractionation of the progeny of any gene holding across the entire genome after each event. We calculate expected numbers of duplicate gene pairs at the time of observation originating at each of the WGD events, and from this predict the entire distribution of similarities, with the help of standard models of mutational processes. This distribution becomes the basis for principled inference of the parameters of the WGD and fractionation model, although the details of the statistical procedures are only touched upon here. The main mathematical advance here is the calculation of the expected number of duplicated pairs more efficiently, without explicitly counting pairs in each evolutionary history. The present paper studies only WGD events in full detail rather than the extension to speciation events, although these can be handled by similar models and inference procedures [4,5].

2

The Model

We propose a model for the generation, maintenance and loss of duplicate genes through polyploidization (leading to WGD) and fractionation that is basically a continuous-time birth-and-death process with the entire population synchronized with respect to birth times and progeny number, but with deaths of individuals coordinated among siblings. Thus, the process starts with m1 ≥ 1 genes at time t1 ; at times t1 < · · · < tn−1 for some n ≥ 1, each existing gene is replaced by r1 , . . . , rn−1 progeny, respectively, where each ri ≥ 2. For each gene’s progeny, at least one and at most ri genes survives until time ti+1 , as governed by a probability distribution u(i) (1), . . . , u(i) (ri ). Although the data available for inference consists of similarity measures on pairs of paralogous genes, we must first develop an account of the fate of individual genes and their descendants in the environment of WGD and fractionation. 2.1

The Gene Tree of WGD Paralogs

Let r = (r1 , r2 , . . . , rn−1 ) where each ri ≥ 2. Starting with a single diploid genome, where each of the m1 genes is single copy, suppose we have a series of polyploidizations with ploidies 2r1 , 2r2 , . . . , 2rn−1 at times t1 < t2 < · · · < tn−1 . The results are observed at time tn , namely a measure of residual identity between n−1all pairs of descendants of each gene. Although there may be as many as m1 i=1 ri such descendants in the haploid genome observed at time tn , there

4

Y. Zhang and D. Sankoff

are likely far less, because of the process of fractionation, which potentially deletes single genes and all their descendants from the gene tree. In the model, illustrated in Fig. 2, at each polyploidization time ti , each existing gene, say gene g, that has survived fractionation up to ti , is replaced by ri copies, at least one of which survives until time ti+1 , where ri is called the ploidy.

Fig. 2. Event with ploidy ri = 4, showing population of mi = 5 genes at time ti , (i) each giving rise to 4 progeny, of which 1 ≤ j ≤ 4 survive until time ti+1 . aj is the number of times j progeny survive. In the diagram, thin solid lines represent individual progeny that survive, and thick grey lines represent the total progeny of a gene that do not survive. In this example the only gene, all of whose progeny survive, is g4 . Here (i) (i) (i) (i) a1 = 2, a2 = a3 = a4 = 1, mi+1 = 2 × 1 + 1 × 2 + 1 × 3 + 1 × 4 = 11. (i)

(i)

If there are mi genes at time ti , let a1 , . . . , ari be the number of cases where ri (i) aj = mi . 1, . . . , ri copies survive fractionation until time ti+1 , so that j=1 Note that there is no provision for g to have zero surviving descendants (i.e., (i) a0 ≡ 0); these genes would be considered as leaving no evidence for their existence and are not counted in mi . ri (i) jaj . The initial value is m1 ≥ 1. It follows that mi+1 = j=1 We have N (r) =

n−1  i=1

(i)

where 0 ≤ aj

 mi (i) (i) , a1 , . . . , ari

≤ mi , j = 1, . . . , ri , and, for i ≥ 1, mi =

(1) r i

j=1

(i)

aj

=

ri−1 j=1

(i−1)

jaj , and N (r) represents the number of different gene copy-and-fractionation histories. (i) We use uj to represent the probability that j of the ri potential copies survive to time ti+1 , for j = 1, . . . , ri . To avoid excessive parametrization, we may set ri  (i)j (1 − v (i) )ri −j j v (i) . (2) uj = 1 − (1 − v (i) )ri The parameter v (i) is the survival probability for a gene copy created at time ti until ti+1 . It should be monotone decreasing in ti+1 − ti and, for empirical purposes, we will assume v (i) = α + (1 − α) exp[−β(ti+1 − ti )] for some rate constant β > 0 and “background” level 0 < α < 1.

(3)

The Similarity Distribution of Paralogous Gene Pairs

5

Thus the probability distribution of the evolutionary histories represented by (i) i=1...n−1 r and the variable a = {aj }j=1...r is i P (r; a) =

n−1  i=1

mi (i) (i) a1 , . . . , ari

 ri j=1

(i)

(i)aj

uj

.

The expected number of genes at time tn is then P (r; a) mn E(mn ) =

(4)

(5)

a

Based on Eq. (1), for 1 ≤ j ≤ k we write N (j,k) (r, a) =

k−1  i=j

mi (i) (i) a1 , . . . , ari

 (6)

to represent the number of different partial gene copy-and-fractionation histories starting with mj genes and undergoing polyploidization with ploidies rj , . . . , rk−1 at times tj , . . . , tk−1 . Similarly, based on Eq. (4), we write P

(j,k)

(r; a) =

k−1  i=j

mi (i) (i) a1 , . . . , ari

 ri h=1

(i)

(i)ah

uh

(7)

for the probability measure over all events starting at tj with mj genes, and preceding tk . In this case the expected number of genes at time tk is P (j,k) (r; a) mk . (8) E(j,k) (mk ) = a

Having characterized the origin and survival of individual genes and their descendants in the environment of WGD and fractionation, we may now focus on the main objects of study, the pairs of genes observed at time tn . In previous work [4,5] we simply generated all evolutionary histories, and counted the pairs of various types in each history. This approach was limited computationally as the number of histories quickly becomes prohibitively large for even moderate size ri and n. In the next section, we develop a formula for the number of pairs, which will allow for the automated counting of pairs in larger examples. 2.2

The Synchronous Gene Pairs

 (i) For each of the aj genes with j surviving copies, j ≥ 2, there are 2j surviving pairs of genes. If j = 1 there are no pairs. The total number of pairs created at ti and surviving to ti+1 is thus d(i,i+1) =

ri   j j=2

2

(i)

aj .

(9)

6

Y. Zhang and D. Sankoff

These are called the ti -pairs at time ti+1 . The expected number of such pairs is E(d

(i,i+1)

)=

a

P

(1,i+1)

(r; a)

ri   j j=2

2

(i)

aj .

(10)

Fig. 3. Counting ti -pairs. The three unfractionated progeny of gene g define three ti pairs, as indicated by three ovals. We follow the pair contained in the uppermost oval, as the two members at time ti+1 independently (shaded triangles) evolve into mn and mn genes, respectively, at time tn , defining mn mn ti -pairs at time tn .

As illustrated in Fig. 3, at time th , i + 1 ≤ h ≤ n, any two descendants of the two genes making up a ti -pair with no more recent common ancestor is also called a ti -pair (at time th ). In other words, for any two genes at th , they form a ti -pair if their most recent common ancestor underwent WGD at ti . For a given ti -pair g  and g  at time ti+1 , where i < n − 1 the expected number of pairs of descendants d(i,n) having no more recent common ancestor than g  and g  , will be E(d(i,n) ) = E(d(i,i+1) )E(i+1,n) (mn )E(i+1,n) (mn )

(11)

where mi+1 = mi+1 = 1. This follows from the independence of the fractionation process between ti and ti+1 and both parts of the process starting with g  and g  . Note that the two terms containing mn and mn are identical. They could be collapsed into one quadratic term, but we write them separately just to emphasize the independence of the processes starting from g  and g  .

The Similarity Distribution of Paralogous Gene Pairs

7

The terms E(d(i,i+1) ) and E(i+1,n) (mn ) in Eq. (11) both involve calculating probabilites with Eq. (7). As n and the ri increase, so do the mi , and this becomes computationally very expensive, due to the product of multinomial coefficients in Eq. (6) and the sum of many probabilities with the coefficient in Eqs. (4) and (7). Nevertheless, making use of the recursive nature of these calculations allows for more efficiency than the explicit generation of evolutionary histories and the counting of pairs within each one. 2.3

The Constraints Against Zero Progeny and on Fractionation Among Siblings only

There is strong biological motivation for retaining at least one offspring of most genes after WGD, as in our model. Our mathematical solution for this goes back to early (pre-genomic) work comparing human and mouse gene families [6]. The loss of all progeny of a gene with important functions would generally be lethal (although the cell sometimes has ways of buffering the loss of genes with essential functions). In our model, if zero progeny of a gene g were to survive, no trace of that g would remain in the “population” of genes, which in turn is equivalent to reducing the number of progeny of the parent of g by one. In the probabilities, this phenomenon is handled by truncating the binomial, as in Eq. (2). The idea that a duplication event ends the fractionation among existing genes and replaces it with fractionation only among newly generated “siblings” is somewhat arbitrary, biologically speaking, but is not completely unrealistic. Certainly fractionation will be more intense among siblings than among “cousins”, and though our model exaggerates this to some extent, inference of total fractionation rates should not be substantially affected. Modeling fractionation among successive generations of cousins would increase the complexity of the model and the number of parameters beyond any hope of inferential meaningfulness. These two constraints on our model are the key to its analytical tractability. 2.4

The Distribution

Knowing the expected number of pairs of genes originating at each WGD in the past is the first step in predicting the full distribution F of similarities. The second step is to derive the actual distribution of gene pair similarities, or an appropriate approximation to it, for ti -pairs. Gene pair divergence may be measured in terms of a probability p reflecting similarity – the proportion of nucleotide positions that are occupied by the same base in the two orthologs (or paralogs). Alternative measures are synonymous distance Ks – the proportion of synonymous changes (not affecting translation to an amino acid) over all eligible positions, and fourfold degenerate synonymous distance 4dTv – the transversion rate at fourfold degenerate third codon positions [7], which require somewhat different analyses. Besides p, the other important parameter is G, gene length in terms of the number of nucleotides in the genes’ coding region, setting aside for the moment that this varies greatly from gene to gene. In the simplest case, p is the sum of

8

Y. Zhang and D. Sankoff

G binomial distributions, divided by G, and is related to the time t ∈ [0, ∞) elapsed since the event that gave rise to the pair: 1 3 −λt + e ∈ [0, 1] 4 4 3 (1 + 3e−λt )(1 − e−λt ) , variance : E(p − E[p])2 = 16 G mean : E[p] =

(12)

where λ > 0 is a divergence rate parameter. The binomial distribution may be approximated by a normal distribution or, if p is close to 1.0, by a Poisson. Despite the similarity in form with Eq. (3), the processes of fractionation and divergence are orthogonal, neither influencing the other in our model. Formula (12) holds no matter how many duplications have occurred, and the rates u and v, which are proxies for time, are defined without reference to time, although they have a natural conversion to time rates, if desired, once t and λ are determined. The densities of similarities of ti -pairs can be approximated by a normal distribution N(E[p], E(p − E[p])2 ), and the expected frequency by Fi = E(d(i,n) )N(E[p], E(p − E[p])2 ).

(13)

Finally, we can predict the entire frequency distribution over all ti as: F (s) =

n−1

E(d(i,n) )Fi (s).

(14)

i=1

Although in its simplest interpretation G is gene length in terms of the number of nucleotides in the genes’ coding region, in practice it is a parameter to be estimated.

3

Inference

Duplicate gene frequency distributions may be produced using the genome data base and analysis tools on the CoGe database [8,9]. Genomes for which coding sequences are known may be compared, giving complete lists of duplicate pairs and their similarities. Ks scores are also available. Self-comparison gives the type of data pertinent to the present paper. In practice, p for duplicate gene pairs is generally much greater than 0.25, so we base our analysis on those pairs with similarity greater than, say, 0.5 or 0.6. We also endeavour to remove all tandem gene pairs. In addition, distribution of similarities within heterozygous genomes generally have a peak very close to 100%, as in Fig. 1, but these pairs consist mostly of maternal-paternal variants of a single heterozygous gene, and not pairs created by WGD. The parameters in the model may be estimated using maximum likelihood or other methods. This includes • n − 1: number of events affecting the genome, • the ri : ploidy of each event, • the ti : times of each event,

The Similarity Distribution of Paralogous Gene Pairs

9

• the u(i) or v (i) : fractionation rates, • λ: the divergence rate, and • G: the gene length parameter. Of course, estimation may be computationally difficult if too many parameters are unknown. Moreover the form of Eqs. (3) and (12) suggest that only U = ut and Λ = λt and not all of u, λ and t can be estimated.

4 4.1

Examples Two Successive WGD (1)

The simplest example involves two whole genome doublings. Set u = u1 (2) v = u1 . For t1 -pairs (1)

and

(1)

d(1,2) = 0 × a1 + 1 × a2

E(d(1,2) ) = 0 × (1 − u) + 1 × u =u

(15) 



If m2 = 1, there are no t1 -pairs. Otherwise consider the single pair g , g . Now, m3 = 1 or m3 = 2 with probabilities 1 − v and v, respectively, so that E(m3 ) = 1 + v. Similarly E(m3 ) = 1 + v. So E(d(1,3) ) = E(d(1,2) )E(2,3) (m3 )E(2,3) (m3 ) = u(1 + v)2

(16)

For t2 -pairs, (2)

(2)

d(2,3) = 0 × a1 + 1 × a2

E(d(2,3) ) = 0 × (1 − u)(1 − v) + 2uv 2 + 2uv(1 − v) = v(1 + u)

(17)

For this small example, a breakdown of the set of evolutionary histories, the t1 and t2 -pairs and their probabilities is given in Fig. 4.

Fig. 4. The case of two WGD.

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Three Successive WGD

A mathematically more interesting case adds a third doubling event to the first (1) (2) (3) two. Set u = u1 and v = u1 as before, and w = u1 . Then the expected number of t1 -, t2 - and t3 -pairs works out to be E(d(1,4) ) = 8uv 2 w3 + 6uv 2 w2 + 8uvw3 + 4uv 2 w + 2uv − 6uvw2 + 4uvw + 4uw2 + 2uv 2 E(d(2,4) ) = v + 2uv 2 + 2vw + vw2 − v 2 + 2v 2 w − v 2 w2 − 4uv 2 w3 + 2uv 2 w2 + 4uvw3 E(d(3,4) ) = vw + uw2 + w − 2uvw2 − 3uv 2 w3 + uv 2 w2 + 3uvw3 + 2uv 2 w

(18)

Fig. 5. The case of three WGD. Numbers in parentheses indicate number of evolutionary histories, differing only in the labelling (colouring) of the genes at time t4 in the case of three WGD.

The Similarity Distribution of Paralogous Gene Pairs

11

Fig. 6. Three models differing only in divergence rates. Left: u = 0.1, v = 0.1, w = 0.2. Middle: u = 0.1, v = 0.3, w = 0.1. Right: u = 0.5, v = 0.1, w = 0.2

Figure 5 displays a breakdown of the pairs in each evolutionary history. Beyond this case, n = 4, the number of evolutionary histories for successive WGD becomes too cumbersome to display. The contrived examples in Fig. 6 demonstrate the importance of fractionation rates and divergence time ti to the structure of the distribution of duplicate pair similarities with three successive WGD.

Fig. 7. A triplication followed by a duplication.

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Three successive WGD are exemplified by the evolution of the Poaceae: rice, sorghum, millet, etc. while some of the species in this family have undergone further duplication (corn) or triplication (wheat). One of the earlier WGD may also be a triplication [10]. 4.3

Triplication Followed by Duplication

An important case with application to the “core eudicot” flowering plants involves one whole genome triplication more than 100 million years ago. The descendants have been extraordinarily successful in diversifying and colonizing in virtually all terrestrial regions, and many of them have independently undergone a further duplication, like the Salicaceae lineage including the poplar genome depicted in Fig. 1, and other WGD events (Fig. 7). This event, along with other hexaploidizations (e.g., tomato [11]), octoploidizations and decaploidizations (e.g., strawberry [12]), motivate the generality of our model with respect to the ri .

5

Conclusions

In this paper, we have elaborated a model for WGD and fractionation that is completely general as far as numbers of events, their ploidy, and fractionation rates are concerned. Its main simplification, the restriction of fractionation pressures to sibling genes, not something for which there is empirical evidence in either direction, is a minor, common sense assumption, biologically, and the constraint on the survival of at least one sibling can be seen as a notational convenience. The key result is the expected number of observed pairs dating from each WGD, and the main application is the prediction of frequency curves for the number of gene pairs as a function of similarity. This leads to a variety of inference problems, about times and ploidies of WGD events, and about fractionation rates. It thus represents an advance on mixture of models methods like Emmix, which may optimize statistical fitting criteria in locating the ti , but which do not offer any principled approach to the fractionation process, nor to any of the other biologically important parameters that we have modeled. As a birth-and-death process, the constraint against zero survivors among the progeny of a gene ensures the WGD-fractionation model tends to be super(i) critical for non-trivial fractionation rates (i.e., excluding cases like a1 ≡ mi ), under the definition presented here. However, we have not studied the inevitable disappearance of ti pairs as time progresses so that their similarities are no longer distinguishable from noise. And we have not incorporated constraints on total gene complement which must eventually operate, nor the many other processes of gene gain and loss which are less dramatic in the short term than doubling and fractionation. Acknowledgements. Research supported in part by grants from the Natural Sciences and Engineering Research Council of Canada. DS holds the Canada Research Chair in Mathematical Genomics.

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References 1. Eckardt, N.: A sense of self: the role of DNA sequence elimination in allopolyploidization. Plant Cell 13, 1699–1704 (2001) 2. Freeling, M., Woodhouse, M.R., Subramaniam, S., Turco, G., Lisch, D., Schnable, J.C.: Fractionation mutagenesis and similar consequences of mechanisms removing dispensable or less-expressed DNA in plants. Curr. Opin. Plant Biol. 15(2), 131–139 (2012) 3. McLachlan, G.J., Peel, D., Basford, K.E., Adams, P.: The EMMIX software for the fitting of mixtures of normal and t-components. J. Stat. Softw. 4(2), 1–14 (1999) 4. Zhang, Y., Zheng, C., Sankoff, D.: Evolutionary model for the statistical divergence of paralogous and orthologous gene pairs generated by whole genome duplication and speciation. IEEE/ACM Trans. Comput. Biol. Bioinform. (2017). doi:10.1109/ TCBB.2017.2712695 5. Sankoff, D., Zheng, C., Zhang, Y., Meidanis, J., Lyons, E., Tang, H.: Models for similarity distributions of syntenic homologs and applications to phylogenomics. IEEE/ACM Trans. Comput. Biol. Bioinform. (2017, in press) 6. Nadeau, J.H., Sankoff, D.: Comparable rates of gene loss and functional divergence after genome duplications early in vertebrate evolution. Genetics 147(3), 1259– 1266 (1997) 7. Kumar, S., Subramanian, S.: Mutation rates in mammalian genomes. Proc. Nat. Acad. Sci. 99(2), 803–808 (2002) 8. Lyons, E., Freeling, M.: How to usefully compare homologous plant genes and chromosomes as DNA sequences. Plant J. 53(4), 661–673 (2008). doi:10.1111/j. 1365-313X.2007.03326.x 9. Lyons, E., Pedersen, B., Kane, J., Alam, M., Ming, R., Tang, H., Wang, X., Bowers, J., Paterson, A., Lisch, D., Freeling, M.: Finding and comparing syntenic regions among Arabidopsis and the outgroups papaya, poplar and grape: CoGe with rosids. Plant Physiol. 148, 1772–1781 (2008) 10. Murat, F., Armero, A., Pont, C., Klopp, C., Salse, J.: Reconstructing the genome of the most recent common ancestor of flowering plants. Nat. Genet. 49(4), 490–496 (2017) 11. Tomato Genome Consortium: The tomato genome sequence provides insights into fleshy fruit evolution. Nature 485, 635–641 (2012) 12. Hirakawa, H., Shirasawa, K., Kosugi, S., Tashiro, K., Nakayama, S., Yamada, M., Kohara, M., Watanabe, A., Kishida, Y., Fujishiro, T., et al.: Dissection of the octoploid strawberry genome by deep sequencing of the genomes of Fragaria species. DNA Res. 21(2), 169–181 (2014)

A Tractable Variant of the Single Cut or Join Distance with Duplicated Genes Pedro Feij˜ ao1 , Aniket Mane2 , and Cedric Chauve2(B) 1

School of Computing Science, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada [email protected] 2 Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada {amane,cedric.chauve}@sfu.ca

Abstract. In this work, we introduce a variant of the Single Cut or Join distance that accounts for duplicated genes, in the context of directed evolution from an ancestral genome to a descendant genome where orthology relations between ancestral genes and their descendant are known. Our model includes two duplication mechanisms: single-gene tandem duplication and creation of single-gene circular chromosomes. We prove that in this model, computing the distance and a parsimonious evolutionary scenario in terms of SCJ and single-gene duplication events can be done in linear time. Simulations show that the inferred number of cuts and joins scales linearly with the true number of such events even at high rates of genome rearrangements and segmental duplications. We also show that the median problem is tractable for this distance.

1

Introduction

The analysis of genome evolution by genome rearrangements has been subject to much research in the field of computational biology. Various rearrangement mechanisms explaining these variations have been proposed and studied, leading to a large corpus of algorithmic results [1]. The pairwise genome rearrangement distance problem aims at finding a most parsimonious or most likely sequence of genome rearrangements, within a given evolutionary model, that transforms one given genome into another given genome, thus giving a possible evolutionary scenario between the two given genomes. This problem has numerous applications toward unraveling important evolutionary mechanisms; recent examples include the nature of evolutionary breakpoints in bacteria [2] or the differentiated evolutionary mode of sex chromosomes and autosomes in Anopheles mosquitoes [3]. A number of rearrangement models have been studied from an algorithmic point of view, among them the reversal model was one of the first for which it was shown that the distance problem is tractable [4]; we refer to [1] for a review of the rich literature in this field up to 2009. For most evolutionary models that do not consider gene duplication, computing a rearrangement distance is tractable; additionally, some models can account for unequal gene content due to c Springer International Publishing AG 2017  J. Meidanis and L. Nakhleh (Eds.): RECOMB CG 2017, LNBI 10562, pp. 14–30, 2017. DOI: 10.1007/978-3-319-67979-2 2

A Tractable Variant of the SCJ Distance with Duplicated Genes

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gene gain or loss, for example [6,7]. However, when gene duplication is allowed as an evolutionary event, most rearrangement distance problems become NPhard. For instance, whereas the distance between two genomes can be computed in linear time for genomes without duplicate genes under the Double-Cut and Join (DCJ) model, it becomes NP-hard to compute when duplicate genes are considered [9], even when the gene content in both genomes is the same [11]. So far limited tractability results for computing distances in the presence of duplicated genes exist only for simpler genome rearrangement models, such as the breakpoint (BP) distance and the Single Cut or Join (SCJ) distance [5]. Even in these simpler models, the general problem of computing a distance with duplicated genes is difficult [12–15], although exponential algorithms have been developed for some specific problems such as the Exemplar BP distance for example [16–19], as well as polynomial time algorithms for two extensions of the SCJ model that include large-scale duplications: the double distance [5], where duplicated genes occur through a whole genome duplication, and the SCJ and whole chromosome duplication problem [20]. In the present work, we consider a problem of rearrangement distance with duplicated genes that we prove to be tractable. Our evolutionary model is also an extension of the SCJ model, that includes single-gene duplications. In the problem we consider that one genome, say A, is duplication-free, while the other one, denoted by D, can contain duplicated genes. This setting is inspired by the recent development of algorithms that reconstruct ancestral gene orders along a given species phylogeny using reconciled gene trees that provide, for each gene family within the set of considered genomes, one-to-one or one-to-many orthology relations between each ancestral gene and its descendant gene(s), if any. This general framework was introduced by Sankoff and El-Mabrouk in [21] (see also [23]) and motivated the introduction of the EBD problem [22]; it was later implemented in the DeCo∗ family of algorithms [24] to reconstruct ancestral gene orders in a duplication-aware evolutionary model from data including extant gene orders and reconciled gene trees. In this context, the genome A represents an ancestral genome, reconstructed for example with DeCo∗ , the genome D represents a descendant of A, either extant or reconstructed too, and we are interested in computing a directed distance, from an ancestor to its descendant, where all members of a same gene family present in genome A are considered as distinguishable thanks to the information provided by the reconciled gene tree of this family. In the evolutionary model we consider, rearrangements are either Single Cuts or Single Joins, while duplications can only be single gene duplications, but of two different types, Tandem Duplications (TD) or Floating Duplications (FD) in which a new copy is introduced as a circular chromosome. We show that in this model the distance problem can be simply reduced to deciding, for each gene family with duplicates in D, the length of a tandem array of duplicates to introduce in A and we provide a polynomial time algorithm for this problem. The remaining paper is organized as follows: in Sect. 2, we recall some basic definitions, describe our evolutionary model and the variant of the SCJ distance and its relation with existing problems. We also introduce the median problem

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for this distance. In Sect. 3 we present our theoretical results, a closed equation for the SCJ distance with duplications, a linear time algorithm to find an optimal scenario and an algorithm for the median problem. Finally, we provide preliminary experimental results in Sect. 4.

2

Preliminaries

2.1

Definitions and Problem Statements

Genes and Genomes. A genome consists of a set of chromosomes, each being a linear or circular ordered set of oriented genes1 . In our examples, a circular chromosome is represented using round brackets (e.g. (a, b, c)) while a linear chromosome is represented using square brackets (e.g. [a, b, c]), where a gene b in reverse orientation is denoted by b. Alternatively, a genome can be represented by a set of gene extremity adjacencies. In this representation, gene x is represented using a pair of gene extremities (xt , xh ), xt denotes the tail of the gene x and xh denotes its head, and an adjacency is a pair of gene extremities that are adjacent in a genome. For example (a, b, c) is encoded by the set of adjacencies {ah bh , bt ct , ch at } and [a, b, c] by {ah bh , bt ct }. We assume that a given gene a can have multiple copies in a genome, with its number of occurrences being called its copy number. A genome in which every gene has copy number 1 is a trivial genome [18]. In this context, a non-trivial genome sometimes cannot be represented unambiguously by a set of adjacencies unless we distinguish the copies of each gene, for example by denoting the copies of a gene a with copy number k by a1 , . . . , ak . For example, the genome (a1 , b, c1 , a2 ), [a3 , d, c2 ], with two duplicated genes of respective copy numbers 3 and 2, is represented by {a1h bh , bt c1t , c1h a2t , a2h a1t , a3t dt , dh c2t }. We call a gene family the set of all copies of a gene that is present in both considered genomes. A gene family is trivial if it has exactly one copy in both genomes. From now, we identify a genome with its multi-set of adjacencies. Let A and D be the respective adjacency sets for the genomes [a, b, c¯, d] ¯ c¯]. Then the multiset difference between the two sets is denoted and [a, b, c¯, d][b, by A − D (similarly D − A). Thus, if A = {ah bt , bh ch , ct dt } and D = {ah bt , bh ch , ct dh , bh ch } then A − D = {ct dt } whereas D − A = {ct dh , bh ch }. Evolutionary Model. We consider two types of evolutionary events: genome rearrangements and duplications. Genome rearrangements are modeled by Single Cut or Join (SCJ) operations, that either delete an adjacency from a genome (a cut) or join a pair of gene extremities that are not adjacent to any other gene extremity (a join), thus forming an adjacency. For duplication events, we consider two types of duplications, both creating an extra copy of a single gene: Tandem Duplications (TD) and Floating Duplications (FD). A tandem duplication of an existing gene g introduces an extra copy of g, say g  by adding an 1

We use the generic term “gene” here to identify a genomic locus.

A Tractable Variant of the SCJ Distance with Duplicated Genes

A

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Fig. 1. In this example, a tandem duplicate of gene g is introduced. The adjacency gh yt has been replaced by gh yt and an adjacency gh gt has been introduced. In this case the total number of operations to obtain D from A is 5. Note that the number of cut and join operations is dependent on the adjacencies of the gene g in A and D.

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Floating duplication of g xt

xh

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Fig. 2. In this example, a floating duplicate of gene g is introduced. An adjacency gh gt has been added. In this case the total number of operations to obtain D from A is 5.

adjacency gh gt , and, if there was an adjacency gh x by replacing it by the adjacency gh x, as shown in Fig. 1. A floating duplication introduces an extra copy g  of a gene g as a single-gene circular chromosome by adding the adjacency gh gt , as in Fig. 2. The motivation for this type of duplication is that gene insertions and gene deletions have been modeled with artificial circular chromosomes before, greatly simplifying how to deal with such type of operations. For instance, in the

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Double-Cut and Join (DCJ) model, a deletion of a gene can be seen as a DCJ operation that applies two cuts to remove the given gene from a chromosome, followed by two joins to “repair” the broken chromosome and to circularize the deleted gene. A gene insertion is the inverse of this operation. This idea was effectively used in the DCJ indel model by Compeau [10]. We discuss the possibility of using a single-gene linear chromosome instead of a circular one at the end of Sect. 3.2. The Pairwise Distance and Directed Median Problems. We consider the case of directed evolution from a trivial ancestral genome A to a descendant genome D. The evolutionary model excludes gene loss and de-novo gene creation, so we assume that every gene a in A has at least one descendant in D and conversely every gene D has a unique ancestor gene in A. If so, we say that A and D have the same gene families set. The Directed SCJ-TD-FD (d-SCJ-TD-FD) Distance Problem. Let A be a trivial genome and D be a non-trivial genome, such that no gene family is absent from either A or D. Compute the minimum number of SCJ, TD and FD operations needed to transform A into D, denoted by dDSCJ (A, D). Note that if D is a trivial genome, the usual SCJ distance, denoted by dSCJ (A, D) is defined by the symmetric differences of the adjacencies sets of A and D: dSCJ (A, D) = |A − D| + |D − A| where the first term accounts for the number of cuts and the second term for the number of joins. We now turn to the directed median problem, that is the natural extension of the pairwise directed distance problem towards the small parsimony problem. The Directed SCJ-TD-FD (d-SCJ-TD-FD) Median Problem. Let k ≥ 2 and D1 , . . . , Dk (possibly) non-trivial genomes, such that no gene family is absent A on the same set of gene families as from any Di . Compute a trivial genome  k the non-trivial genomes, that minimizes i=1 dDSCJ (A, Di ). 2.2

Relation to the Exemplar Distance Framework

Sankoff [22] introduced the notion of Exemplar Breakpoint (EBP) distance, where an exemplar of a non-trivial genome is obtained by keeping exactly one gene copy from each gene family. In the directed evolution setting, an exemplar can be assumed to be the original gene from A having evolved into a gene now present in D, all other copies having been created by duplications. So the EBP distance problem aims to find an exemplar for each group of duplicates in D such that the trivial genome that results from deleting all non-exemplar copies minimizes the breakpoint distance to A. The notion of exemplar distance can naturally be used in conjunction with the SCJ distance instead of the BP distance, a problem we denote the ESCJ distance. The EBP distance problem has been shown to be NP-hard even in the directed evolution case where every duplicated gene has exactly two copies in D [12], and it is immediate to extend this hardness result to the directed ESCJ distance problem.

A Tractable Variant of the SCJ Distance with Duplicated Genes

19

Intuitively, the directed ESCJ distance and the d-SCJ-FD-TD distance problems seem very similar. For example in the case of duplicated genes having exactly two copies in D, the later aims at deciding which copy in D is exemplar (i.e. evolved from the original copy in A) and then, for the second copy, if it originates from a TD or a FD, thus resulting in a matching between two genomes with two copies of each duplicate, opposed to the ESCJ setting where the matching is between genomes with one copy of each gene. It is interesting to notice that both problems, although similar, have opposed properties in terms of tractability, and that the d-SCJ-TD-FD distance problem is tractable despite considering a larger solution space. Moreover, one can ask if there is a strong correlation between the distance obtained in both settings. It is not difficult to find examples that show that both distances can be quite different: the ESCJ distance between A = [a, b, c, d] and D = [a, c, b, d, c, a, d, b] is 0, whereas the SCJ-TD-FD distance between the same two genomes is 18 (4 duplications, 7 cuts, 7 joins). However, although the difference between both distances can be arbitrarily large, tight bounds can be derived. Lemma 1. Let A be a trivial genome, D be an arbitrary genome on the same set of gene families than A, dDSCJ (A, D) and dESCJ (A, D) denote the d-SCJFD-TD and the ESCJ distances, respectively. Let k be the difference between the number of genes in D and the number of genes in A. The following bounds k ≤ dDSCJ (A, D) − dESCJ (A, D) ≤ 5k are tight. Proof. First, we obtain the genome A by applying dESCJ (A, D) SCJ operations on A in a way that the duplicated genes in D are in the same order as the corresponding matched genes in A , as given by an optimal exemplar matching. In the SCJ-FD-TD model, we need to apply at least k duplications on A to obtain D, so dDSCJ (A, D) ≥ dESCJ (A, D) + k (otherwise dESCJ (A, D) would not be optimal). To show that this bound is tight, we can see that the trivial case of no duplications holds. But, whenever A and D differ by only k tandem duplications, the bound is tight, since in this case dESCJ (A, D) = 0 and dDSCJ (A, D) = k. Now, from A , we can apply k free duplications, followed by k cuts on these duplications. Also, perform at most k cuts, between any two genes on A if both have more than one copy on D. Since A was ordered in relation to its corresponding copies on D, it is possible to join the “fragments” of A that were created with the previous 2k cuts with 2k joins in a way to transform A in D, and therefore we built a d-SCJ-FD-TD scenario from A to D with dESCJ (A, D) + 5k operations. Any pair of circular genomes A = (1, 2, . . . , n) and D = (1, n, 2, 1, . . . , i, i − 1, . . . , n, n − 1) satisfies the tight bound.  

3

Algorithmic Results

In this section, we show that, after a preprocessing step of removing obvious TD and FD in D, the d-SCJ-TD-FD distance can be calculated with the symmetric

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difference between the adjacency (multi)sets of the input genomes, with an extra factor to account for the gene duplications. We first focus on the preprocessing. Next we describe a linear time algorithm to compute a parsimonious scenario and a polynomial time algorithm for the directed median problem. 3.1

The Directed SCJ-TD-FD Distance

An observed duplication in D is defined as an adjacency of the form gh gt , that defines either a single-gene circular chromosome or a tandem array of two (or more) copies of a gene g that occur consecutively and with the same orientation. We denote by t the number of such adjacencies in D and by D the genome obtained from D by removing first all genes but one from each tandem arrays, and then all single-circular chromosomes for genes from non-trivial families but one if all genes of the family are in such circular chromosomes. D can obviously be obtained from D by t duplications and the following lemma is immediate: Lemma 2. dDSCJ (A, D) = dDSCJ (A, D ) + t. As a consequence, we assume from now on that D has been preprocessed as described above and does not contain any tandem array or any extra copy of a non-trivial family that is in a single-gene circular chromosome. We say that D is reduced. Note that single-gene linear chromosomes are not impacted by this preprocessing as, in our setting, if the considered gene is from a non-trivial family, the linear chromosome it forms required at least a cut to be created. Theorem 1. Given a trivial genome A and a reduced non-trivial genome D such that no gene family is absent from either A or D and where D has nd more genes than A, the d-SCJ-TD-FD distance between A and D is given by dDSCJ (A, D) = |A − D| + |D − A| + 2nd . Proof. First, we show that dDSCJ (A, D) ≥ |A − D| + |D − A| + 2nd . To obtain D from A, we need exactly nd gene duplications. Each duplication of a gene g will create the adjacency gh gt , regardless of the type of the duplication or the timing of the duplication event. Therefore, nd adjacencies of the type gh gt will have to be cut, as D is reduced and has no adjacency of this type. In addition, any adjacency in A − D and D − A defines an unavoidable cut or join respectively. Therefore, we can not transform A into D with less than |A − D| + |D − A| + 2nd operations. Now, we show that dDSCJ (A, D) ≤ |A − D| + |D − A| + 2nd , by induction on nd . For the base case nd = 0, the result follows immediately as both genomes are trivial and dDSCJ (A, D) = dSCJ (A, D). We now assume that nd > 0, and pick a gene g with one copy in A and more than one copy in D. Depending on how the adjacencies of g are conserved or not in D, we have a few different subcases to consider. However, in each subcase the general strategy remains the same, as follows. We build a genome A2 from A by applying one duplication (FD or TD) and also relabeling the original copy

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g as g  , creating an adjacency gh gt in the case of an FD or gh gt in the case of a TD. Then we build a genome D2 from D by also relabeling one copy of g to g  , thus creating a new trivial gene family and an instance of the d-SCJ-TD-FD problem with exactly nd − 1 duplicated gene copies. We can apply the induction hypothesis, leading to the inequality dDSCJ (A2 , D2 ) ≤ |A2 − D2 | + |D2 − A2 | + 2(nd − 1). Also, as D and D2 are identical but for the relabeling of g, there is a scenario from A to D, going from A to A2 and then to D, resulting in the upper bound dDSCJ (A, D) ≤ dDSCJ (A, A2 ) + dDSCJ (A2 , D2 ) = 1 + dDSCJ (A2 , D2 ). We will then show that we can build A2 and D2 in a way that they satisfy |A − D| + |D − A| = |A2 − D2 | + |D2 − A2 | − 1, where the −1 term is due to the extra gh gt adjacency on A2 created with the duplication. Together with the above inequalities this will lead to dDSCJ (A, D) ≤ 1 + dDSCJ (A2 , D2 ) ≤ |A − D| + |D − A| + 2nd and the result follows. To show that we can build A2 and D2 that satisfy the above conditions, we will consider three subcases. Case (i): Assume that g is not a telomere (and so there are two adjacencies involving g in A, say xgt and gh y) and there is a copy of g in D whose extremities for also adjacencies xgt and gh y. We say that the context of g is strongly conserved between A and D. Note that x and y do not need to belong to trivial gene families and there might be several copies of x, y, g in D that conserve the context of g in A. In this case, we build A2 by applying an FD to create an extra copy of g and relabel the original copy of g in A as g  ; we also relabel g  an arbitrary copy of g in D that has the same context than g in A, to obtain D2 (see Fig. 3). Comparing the adjacency sets of A and D with A2 and D2 , we can see that from A to A2 two adjacencies where renamed from xgt and gh y to xgt and gh y, and exactly the same change happened from D to D2 . Also, the adjacency gh gt was added in A2 . As a result, A2 = A − {xgt , gh y} + {xgt , gh y, gh gt }. Similarly, D2 = D − {xgt , gh y} + {xgt , gh y}. Therefore, we have that |A − D| + |D − A| = |A2 − D2 | + |D2 − A2 | − 1. Note that this relabeling only works if we introduce a an extra copy of g in A with an FD here; if instead we introduce it with a TD, it would not be possible to get adjacencies xgt and gh y in D2 , as the copy of g involved in both adjacencies would be different. Case (ii): Assume that g is not a telomere in A, its context is not strongly conserved between A and D, but both adjacencies involving g, xgt and gh y, are present in D on different copies of g. We say that the context of g is weakly conserved between A and D. Again x and y need not to be trivial gene families and there might be several occurrences of adjacencies xgt and gh y in D.

22

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g

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Fig. 3. The context of g is strongly conserved between A and D (Case (i)).

In this case, we build A2 by applying a TD on g, relabeling the gene g that has the adjacency xgt as a new gene g  in both A2 and D2 , as shown on Fig. 4. Comparing the adjacency sets of A and A2 , we notice that the adjacency xgt changes to xgt , and gh gt is added. Thus, A2 = A − {xgt } + {xgt , gh gt }. From D to D2 we also have the same change, and possibly one more, depending if gh is a telomere in D (no change) or if gh has an adjacency gh w. In the former case, D2 = D − {xgt } + {xgt }. Otherwise, D2 = D − {xgt , gh w} + {xgt , gh w}. In either case, the possible adjacency gh w does not exist in A or A2 . Consequently, the equality |A − D| + |D − A| = |A2 − D2 | + |D2 − A2 | − 1 holds. Note also that in this case an FD would not be optimal, because it would force the labeling of the adjacency gh y to gh y, and since the adjacency gh y on D cannot have the label gh y, this would force an extra pair of SCJ operations. x A ...

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Case (iii): We assume now that the context of g in A is neither strongly nor weakly conserved, and so at most one adjacency of g in A is also present in D. This case is similar to case (i), if we assume that either xgt or gh y, are present in D, or neither. In the same way, we apply an FD on g, labeling the original copy as g  , as shown in Fig. 5. On D, we pick a gene g that has an adjacency xgt or gh y if any or, if no adjacency involving g is conserved in D, we pick an arbitrary g, and relabel it as g  . Now, any adjacencies that were conserved between A and D will remain conserved between A2 and D2 , and no new conserved adjacencies have been created. Since, as before, A2 has a new gh gt adjacency, the equality |A − D| + |D − A| = |A2 − D2 | + |D2 − A2 | − 1 holds. These three cases cover all possible configurations for g, so the theorem is proved.  

A Tractable Variant of the SCJ Distance with Duplicated Genes x A ...

g

x D ...

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Fig. 5. At most one adjacency of g is conserved (Case (iii)).

3.2

Computing a Parsimonious Scenario

It follows from Lemma 2 and Theorem 1 that computing the d-SCJ-TD-FD distance can be done in linear time in the size of the considered genomes A and D. Moreover, they define a simple algorithm that computes a parsimonious scenario in terms of duplications, cuts and joins from A to D, described in Algorithm 1 below. Algorithm 1. Compute an SCJ-TD-FD parsimonious scenario between a trivial genome A and a genome D Reduce D into a reduced genome D Let A = A and i = 1 while (A , D ) has a non trivial gene family do Let g be an arbitrary gene from a non trivial family in A ; relabel g by g i . if the context of g is strongly conserved then relabel the corresponding copy of g in D by g i add to A a single-gene circular chromosome g. else if the context of g is weakly conserved then create an extra copy of g i with a TD relabel a copy of g involved in adjacency xgt in D by g i . else if one adjacency of g is conserved in D then relabel the corresponding copy of g in D by g i add to A a single-gene circular chromosome g i . else relabel an arbitrary copy of g in D by g i add to A a single-gene circular chromosome g i . end if i=i+1 end while Compute an SCJ scenario from A to D . Recreate in D , the tandem arrays and single-gene circular chromosomes removed when reducing D into D .

Theorem 2. Given a trivial genome A with nA genes and a possibly non-trivial genome D on the same set of gene families and with nD genes, Algorithm 1 computes a parsimonious SCJ-TD-FD scenario that transforms A into D and can be implemented to run in time and space O(nD ).

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The correctness of the algorithm follows immediately from the fact that it implements exactly the rules described to compute the SCJ-TD-FD distance (Lemma 2 and Theorem 1). The linear time and space complexity follows from the fact that these rules are purely local and ask only to check for the conservation of adjacencies in both considered genomes. Every iteration of the while loop in Algorithm 1 takes place only if there is a non-trivial gene family left in D . The maximum number of iterations is the number of duplicates genes, nd = nD − nA which is O(nD ) when nD ≥ nA . In each iteration, we check if the context of the chosen gene g is strongly conserved, weakly conserved or not conserved. This involves trying to match the adjacencies of g in A with those in the adjacency set of D that involve a copy of g. This can be done in constant time, with a linear time preprocessing of the data. Hence, the worst-case time complexity is O(nD ). Remark 1. We have discussed in Sect. 2.1 the rationale to create duplicate genes with a FD creating a circular single-gene chromosome. However if the evolutionary model of the FD event created a linear single-gene chromosome, this would introduce a dissymetry between TD and FD (namely no adjacency is created with an FD), while in our model each created copy induces a cost of two due to the necessary break of the created adjacency required in the process of obtaining the reduced genome D . We conjecture that the use of linear chromosomes would affect the choice of duplication event (FD or TD) only when the context is not conserved, which would result in a more complicated distance formula. 3.3

The Directed Median Problem

Let us remind that under the SCJ-TD-FD evolutionary model, the directed median problem asks, given k non-trivial genomes D1 , . . . , Dk , k ≥ 2, with the k same gene families, to find a trivial common ancestor A, such that i=1 dDSCJ (A, Di ) is minimized. We first assume that the genomes D1 , . . . , Dk are reduced. We define the score s(A) of a genome A as s(A) =

k  i=1

dDSCJ (A, Di ) =

k 

(|A − Di | + |Di − A| + 2ndi )

i=1

where ndi is the number of extra gene copies in Di compared to A, for i = 1, . . . , k. Using the fact that |A − D| + |D − A| = |A| + |D| − 2|A ∩ D| we derive  k   |A ∩ Di | − k|A| s(A) = Nd − 2 i=1

k

where Nd = i=1 (2ndi + |Di |), and does not depend from A. Therefore, minik mizing s(A) is equivalent to maximizing 2 i=1 |A ∩ Di | − k|A|. For a given adjacency a, let δi (a) be 1 if a ∈ Di , and 0 otherwise. The score    k of a genome with a single adjacency a is s({a}) = Nd − 2 i=1 δi (a) − k .

A Tractable Variant of the SCJ Distance with Duplicated Genes

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This motivates the following approach, similar to the breakpoint median algorithm of [8]. Build a graph G where the vertices are defined as the extremities (head and tail) of a unique copy for each gene family in the considered genomes Di (so a gene family a induces two vertices ah and at ), and weighted edges are defined as follows: for any edge e = (x, y) such that x and y form an adjacency k in at least one of the genomes Di , the weight of e is w(e) = 2 i=1 δi (e) − k. Any matching M on G defines a trivial genome AM , having the adjacencies corresponding to the edges in the matching M . Also, if W (M ) denotes the weight of the matching M , that is the sum of the weights of the edges in M we have that  k   s(AM ) = Nd − 2 |AM ∩ Di | − k|AM | i=1



= Nd −



2

e∈M

k 

 δi (e) − k

i=1

= Nd − W (M ) Therefore, solving a maximum weight matching problem on G solves the directed median problem. To handle the case when some Di is not reduced, we can rely on Lemma 2 that implies that the genomes can be reduced first without impacting the optimality of a trivial genome obtained by a maximum weight matching. Combined with the tractability of computing a maximum weight matching [25], this proves our last theorem. Theorem 3. Let k ≥ 2 and D1 , . . . , Dk be k genomes on the same set of n gene families, having respectively n1 , . . . , nk adjacencies. The directed SCJ-TDFD median problem for these genomes can be solved in time and space O(n(n1 + · · · + nk ) log(n1 + · · · + nk )). Remark 2. In the case of the median of two genomes D1 and D2 , note that the only edges with strictly positive weight in the graph are defined by adjacencies that appear in both D1 and D2 , while edges appearing just once have weight 0. So a median genome can be defined as a maximum matching over the unweighted graph defined only by adjacencies that appear in both genomes, and given such a median, it can be augmented by any subset of edges appearing just once that do not re-use a gene extremity already used in the matching.

4

Experimental Results

We ran experiments on simulated instances with the aim to evaluate the ability of the d-SCJ-TD-FD distance to correlate with the true number of syntenic events. We followed a simulation protocol inspired from [16]. The code itself was programmed in Python and is available via github2 . We first describe the simulation protocol, followed by the results we obtained. 2

https://github.com/acme92/SCJTDFD.

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Fig. 6. Experimental results, for four duplications parameters – single-gene segmental duplication (top row), two-genes segmental duplication (second row), five-genes segmental duplications (third row), variable length segmental duplications (bottom row) – and two measured quantities – inferred cuts and joins (left column) and SCJ-TD-FD distance (right column).

A Tractable Variant of the SCJ Distance with Duplicated Genes

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We started from a genome A composed of a single linear chromosome containing 1000 single-copy genes. Then, we transformed A genome into a genome D through a sequence of random segmental duplications and inversions. We fixed the number N of evolutionary events (from 50 to 500 by steps of 50) and the probability P that a given event is a segmental duplication (from 0 to 0.5 by steps of 0.1). A segmental duplication is defined by three parameters: the position of the first gene of the duplicated segment, the length of the duplicated segment, and the breakpoint where the duplicated segment is transposed into; we considered two models of segmental duplications, one with fixed segment length L (with L taking values in {1, 2, 5}) and one where for each segment, L is picked randomly (under the uniform distribution) in {1, 2, 5, 10}. Inversion breakpoints were chosen randomly, again under the uniform distribution. For each array of parameters, we ran 50 replicates. For each instance, we compared two quantities to the true number of cuts and joins in the scenario transforming A into D, which is roughly four times the number of inversions plus three times the number of segmental duplications: first we compared the full SCJ-TD-FD distance, defined as stated in Theorem 1 and the number of cuts and joins (|A − D| + |D − A|). Figure 6 illustrates the results we obtained. We can make several observations from these results. The first one is a general trend that both measured quantities (the number of cuts and joins and the full SCJ-TD-FD distances) scale linearly with the true number of cuts and joins. The second observation is that, as expected, the slope and a y-intercept of the graphs depend from both the frequency of duplications and the length of the duplicated segments. This leaves open the question of using the SCJ-TD-FD distance as an estimator of the number of cuts and joins in an evolutionary model where the probability of duplication compared to rearrangements (that can be estimated for example from reconciled gene trees and adjacency forests [24]) is given and the length of duplicated segments is expected to follow a well defined distribution.

5

Conclusion

In this work, we introduced a simple variant of the SCJ model that accounts for duplications, and showed that, in this model, computing a directed parsimonious genomic distance from a trivial ancestral genome to a non-trivial descendant genome can be done in linear time and that a directed median can be computed in polynomial time. The tractability stems mostly from the combination of assuming that one genome is trivial and of a simplified model of duplication where gene duplication are single-gene events. However we believe it is interesting to push the tractability boundaries of the SCJ models toward augmented models of evolution (here accounting for duplications). Moreover, our work is motivated by the increasing performance of ancestral gene order reconstruction methods, that can now account for complex gene histories using reconciled gene trees and motivate the directed distance approach, and provides an additional positive result along the line of the research program outlined in [21].

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For example, our algorithm will allow to extend the small parsimony algorithm PhySca introduced in [26] to a duplication-aware framework by allowing to score exactly and quickly an ancestral gene order configuration within a species phylogeny (work in progress). There are several avenues to extend the results we presented in this paper. It will likely be easy to modify our algorithm to work in an extended the evolution model to integrate the loss of gene families and de-novo creation of genes. Our main result provides a simple algorithm that computes a parsimonious scenario, however it is likely one among a large number of parsimonious scenarios, and it is open to see if the results of [27] about counting and sampling SCJ parsimonious scenarios can be extended to our model. An important open question toward a more realistic model of evolution concerns the possibility to include larger scale duplications as unit-cost events. The case of a single whole genome duplication and of whole-chromosome duplications have been shown to be tractable [5,20], but to the best of our knowledge there is no known result including segmental duplications in which a contiguous segment of genes is duplicated either in tandem or appearing as a single chromosome. It also remains to be seen if the directed SCJ-TD-FD distance can be used toward the computation of an estimated distance in a more realistic evolutionary distance, similarly to the use of the breakpoint distance to estimate the true DCJ distance [28]; our experimental results suggest this is a promising avenue, although it might be difficult to obtain analytical results in models mixing rearrangements and duplications. Finally, the question of the tractability of the small parsimony problem in our model is also, to the best of our knowledge, still open. It is known to be tractable in the pure SCJ model (i.e. with no duplications) due to the independence of adjacencies; this assumption does not hold anymore here and the small parsimony problem is thus likely more difficult in our model. Our tractability result for the directed median is a first step toward this goal as it already provides a building block for a bottom-up ancestral reconstruction algorithm. Acknowledgments. CC is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. PF is supported by the Genome Canada grant PathoGiST.

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Generation and Comparative Genomics of Synthetic Dengue Viruses Eli Goz1,2, Yael Tsalenchuck2, Rony Oren Benaroya2, Shimshi Atar1, Tahel Altman2, Justin Julander3, and Tamir Tuller1,2,4(&) 1

3 4

Department of Biomedical Engineering, Tel-Aviv University, Ramat Aviv, Israel [email protected] 2 SynVaccine Ltd., Ramat Hachayal, Tel Aviv, Israel Institute for Antiviral Research, Utah State University, Logan, UT, USA Sagol School of Neuroscience, Tel-Aviv University, Ramat Aviv, Israel

Abstract. Synthetic virology is an important multidisciplinary scientific field, with emerging applications in biotechnology and medicine, aiming at developing methods to generate and engineer synthetic viruses. Here we demonstrate a full multidisciplinary pipeline for generation and analysis of synthetic RNA viruses and specifically apply it to Dengue virus type 2 (DENV-2). The major steps of the pipeline include comparative genomics of endogenous and synthetic viral strains. In particular, we show that although the synthetic DENV-2 viruses were found to have lower nucleotide variability, their phenotype, as reflected in the study of the AG129 mouse model morbidity, RNA levels, and neutralization antibodies, is similar or even more pathogenic in comparison to the wildtype master strain. These results may suggest that synthetic DENV-2 may enhance virulence if the correct sequence is selected. The approach reported here can be used for understanding the functionality and the fitness effects of any set of mutations in viral RNA. It can be also used for editing RNA viruses for various target applications.

1 Background The ability to synthesize and engineer RNA viruses has important applications to biotechnology and human health [1–5]. Although some previously published studies involved generation and analysis of various synthetic RNA viruses (e.g., [6–12]), more studies are required to understand their evolution and population dynamics together with developing efficient synthetic biology approaches for their research [13, 14]. Dengue virus (DENV) is an important RNA flavivirus with 4 genomically diverse serotypes. Its genome is a positive polarity single stranded RNA of approximately

E. Goz and Y. Tsalenchuck—Equal contribution. © Springer International Publishing AG 2017 J. Meidanis and L. Nakhleh (Eds.): RECOMB-CG 2017, LNBI 10562, pp. 31–52, 2017. DOI: 10.1007/978-3-319-67979-2_3

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11 kb composed of 3 parts: the unique coding sequence (CDS) that produces 10 mature viral proteins via a single precursor polyprotein, and two flanking untranslated regions (UTRs) that contain important structural and functional elements [15–17]. Dengue is an important human pathogen, widely recognized as a major public health concern [18]; therefore new insights into biological and evolutionary properties of its wildtype and synthetically generated strains may have significant influence on the development of new antiviral vaccines and therapies. Here we demonstrate, for the first time, a comprehensive analysis of DENV-2 (Dengue virus, serotype 2) synthetic variants using a multidisciplinary pipeline which combines comparative genomics, synthetic biology, next generation sequencing (NGS), and experiments with animal models of viral infection. Specifically, basing on DENV-2 New Guinea C wildtype strain (WT), two synthetic variants were generated; comparative genomics as well as their replication and pathogenesis analyses in AG129 mouse model were performed. The suggested approach can be also directly applied to other RNA viruses.

2 Results 2.1

Study Outline

A full comparative genomics pipeline for synthetic biology of RNA viruses demonstrated in this study is described in Fig. 1 (see Sect. 4 for more details): DENV-2 New Guinea C wild type viral strain (Fig. 1.I) was sequenced (Sanger) (Fig. 1.II). The resulting sequence was used as a master sequence to generate synthetic variants. In order to demonstrate the robustness of the reported results two different approaches were examined: (1) Incellulo construction of subgenomic infectious amplicons (Fig. 1. III), and (2) PCR assembly of full-length infectious DNA (Fig. 1.IV). After three passages of the wildtype and synthetic DENV-2 viruses in Vero E6 cells (Fig. 1.V), viral RNA was extracted and next generation sequencing was performed (Fig. 1.VI). The resulting sequences were verified and analysis of genomic variants (variants calling) with respect to single nucleotide variations (SNV) and inserts was performed (Fig. 1.VII). The results were compared to a population sample of 618 available DENV-2 genomes (Fig. 1.VIII-IV) with respect to several genomic features: nucleotide variability, secondary structure, codon and codon pair bias, CpG and GC contents. In addition, the wildtype and synthetic (generated by the assembly method) viruses were compared in vivo in the AG129 mouse model (Fig. 1.X); the comparison included: weight change, survival rates, RNA levels, and neutralizing antibodies (Ab). In the following we describe the results of our analyses. 2.2

Next Generation Sequencing and Comparative Genomics Analysis of the Synthetic and Wildtype Viruses

The NGS and comparative genomics analysis of the populations of synthetic and WT viruses in the cell line is summarized in Figs. 2 and 3 (further details can be found in the Methods section).

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Fig. 1. A full comparative genomics pipeline for synthetic biology of RNA viruses. Details can be found in the main text.

The positions of the discovered variants (SNVs and inserts) and read coverage along the genome for the WT virus and its synthetic constructs (assembly and incellulo) are shown in Fig. 2A. Genomic coordinates and annotations are specified in the “genome” panel. In total, 38 variants were found: 35 in the CDS and 3 in the UTRs; positions of these variants were marked by vertical bars (blue and red) in the “variants” panel. 32 (91%) of the variants in the CDS were due to SNVs, and the rest 3 (9%) were due to inserts; 23 (72%) of SNVs in the CDS were synonymous. In the UTRs all variants were due to inserts. All the inserts were due to an additional A within a stretch

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of A’s. 64% of all SNVs were found in a WT virus (but not in synthetic variants). Positions of 8 variants (7 SNVs and one insert) with variant allele frequencies (VAF) of at least 15%, that also passed various filters (Position Bias [PB] and Strand Bias [SB] filters; see Methods), were marked by red bars (“variants” panel); 6 of them were found in the WT strain only. Additional analysis of the 32 CDS SNVs is summarized in Fig. 2B: Most of the SNVs were C ! T or A ! G. As expected, the majority of the variations were found to be synonymous (72%). SNVs at positions 3708 and 5061 are substitutions (VAF = 1), the others (VAF < 1) may indicate polymorphisms within the population (panels “SNVs” and “VAF”). These polymorphisms may be related either to errors in viral replication [19] or to sequencing errors (for very low VAF), and may

Fig. 2. Results of NGS study. A. Positions of variants (SNVs and inserts) and read coverage along the genome of the wildtype virus (WT) and its synthetic constructs (assembly and incellulo). Genomic coordinates and annotations are specified in the “genome” panel. Positions of variants are marked by vertical bars (blue and red) in the “variants” panel; summary of variants for all samples (first row in “variants” panel), and sample-specific variants (three different rows for incellulo, assembly, and WT) are shown. 8 Positions (1 indel and 7 SNVs) with VAF at least 0.15 that also passed Position Bias and Strand Bias filters were marked by red bars; 6 of them are found in the WT strain only. B. Analysis of the 32 CDS SNVs. Nucleotide variability at SNV positions based on a set of 618 aligned DENV2 genomes is represented by the corresponding sequence logo in the “variability” panel. The coordinates of SNVs are specified in the x-axis; corresponding genes are specified in the bottom row. The variability at each position is represented by a stack of letters. The relative sizes of the letters indicate their frequency in the alignment at specific position. The total height of the letters depicts the information content of the position, in bits. The SNVs at each position are depicted in the “SNVs” panel in the following format: reference codon (reference AA) ! variant codon (variant AA). The altered nucleotides are marked in red. The Variant allele frequencies (VAF) at each position were specified in the “VAF” panel. Non synonymous SNVs (18%) were marked by “N”. Positions that overlap with regions that undergo a conserved selection for strong or weak mRNA folding (based on [16]) were marked by “s” or “w” respectively. 7 CDS SNVs with VAF of at least 0.15 that also passed Position Bias and Strand Bias filters were marked by red asterisks. (Color figure online)

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Fig. 2. (continued)

Fig. 3. CpG dinucleotides are significantly suppressed in regions of 100 codons around SNVs as compared to genomic regions of the same size that do not contain any SNV.

play a role in improving the fitness of the WT virus [19–21]. 7 significant SNVs (VAF  15% that also passed PB and SB filters) were marked by red asterisks. Further, the positions of the CDS SNVs were compared to the population sample consisting of 618 available DENV-2 genomes (see Methods for additional details). In particular relations to regions with significantly strong/weak RNA structures ([16] and Methods) and to position-wise nucleotide variability (panel “variability”) were

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analyzed. Specifically: 14 out of 32 (44%) of SNVs in CDS were found to overlap with regions that undergo a conserved selection for strong or weak mRNA folding, although this overlap is not statistically significant with respect to random (p-value = 0.14, see Methods); this result suggests that only some of the SNVs are related to secondary structures and may influence the viral fitness via the effect on the RNA folding. Moreover, no accord between the SNVs and the nucleotide variability at the corresponding positions in the alignment of the population sample (quantified by normalized entropy, Methods) was found (i.e. SNV’s do not tend to appear in more or less conserved positions). Also, Spearman correlation between variant frequencies derived from the NGS data and the frequencies of these variants observed in the population sample at SNV positions was found to be low/insignificant (rho = 0.16, p-value = 0.38). These results may/demonstrate that the variability at the population level is a dynamic trait and may differ both at different stages of the replication cycle within the same host and across different hosts. Finally, we analyzed the association of SNVs with additional local different genomic features, namely: Codons Usage Bias as measured by ENC (Efficient Number of Codons), Codon Pairs Bias (CPB), CpG content (CpG), GC content (GC) (see Methods). Various studies related these features to different genomic mechanisms that take part in the viral replication cycle and are related to the viral fitness. For example, it was suggested that viral codons may be under selection to improve the translation efficiency, among others, via adaptation to the host tRNA pool (or other translation resources)(e.g., [22–34]). It was also suggested that translation efficiency is affected not only by single codons, but also by distribution of codon pairs (although it is debated whether this feature is under a direct selection or can be a consequence of distribution of dinucleotides [35–38]. In [39–41] it was shown that CpG pairs are under-represented in many RNA and most small human DNA viruses, in correspondence to dinucleotide frequencies of their hosts. This phenomenon can be related, for example, to the contribution of the CpG stacking basepairs to RNA folding [42] and/or to the enhanced innate immune responses to viruses with elevated CpG [43]. The stability of the RNA secondary structures can be also affected by the genomic GC content [44]. By analyzing local genomic regions of 100 codons we found that regions centered around the positions of SNVs were characterized by a significantly lower CpG content (Wilcoxon ranksum test, p-value = 0.004) compared to regions of the same size that didn’t contain any SNV (Fig. 3). As was explained above, the higher mutation rates in the CpG suppressed regions may be associated with a stronger evolutionary pressure possibly related to RNA folding or to the interaction with the immune system. We didn’t find significant differences between the SNV and non-SNV regions with respect to the rest of the analyzed features (supplementary material, Sect. 1). 2.3

AG129 Mouse Model Study

An AG129 mouse model was used to characterize the pathogenicity and immunization capabilities of synthetic DENV-2 generated by the Assembly method. Infection with the WT virus resulted in around 50% mortality (Fig. 4A, Table 1), which was consistent with intraperitoneal infection of AG129 mice in previous studies in our lab.

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Synthetic wild-type virus injection resulted in 75% mortality and had a similar mortality curve to mice infected with the WT virus. The mortality rates caused by both synthetic and WT infection were significantly higher (p-value < 0.001) than in sham-infected controls (Fig. 4A).

Fig. 4. Results of AG129 mouse model study. A. Percentage of survival of AG129 mice infected with wild-type and synthetic DENV-2 (p-value < 0.001 for 12 dpi and furhter, as compared with sham controls). B. Mean weight change (%) of AG129 mice infected with different DENV-2 variants. C. Virus titers from serum collected 3 dpi from infected AG129 mice (**p-value < 0.01, *p-value < 0.05, as compared with sham controls). D. 50% plaque reduction neutralization titers (PRNT50) from serum taken 14 dpi (***p-value < 0.001, as compared with sham controls). Note that there were not sufficient quantities of serum for 5 samples: 2 from group 1(infected with WT variant), 1 from group 2 (infected with synthetic variant) and 2 from control group for quantification of neutralizing Ab.

Some weight loss was observed at various times over the course of infection in individual animals infected with WT or synthetic DENV-2, although the general trend was for little average weight change beyond baseline within all groups (Fig. 4B). For some reason, vehicle inoculated control animals (6–7 weeks old) did not gain weight as expected (Fig. 4B), although reasons for this are unknown. Viral RNA was detected in all animals infected with synthetic DENV-2 in serum collected 3 days post infection (dpi), which was significantly (p-value < 0.01) higher than sham-infected controls (Fig. 4C). Half of the animals inoculated with the WT virus had detectable titers, while the other half had RNA levels below that of assay

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detection (Fig. 4C). Despite the lower rate of detection of viral RNA in these samples, the mean viral RNA level in this group was significantly (p-value < 0.05) elevated as compared with controls. No viral RNA was detected in the serum of sham-infected mice. As anticipated, animals infected with synthetic DENV-2 elicited significantly (p-value < 0.001) elevated neutralizing Ab levels on 14 dpi as compared with sham-infected controls (Fig. 4D). Infection with WT virus resulted in a variable response, usually including detection of viral RNA and mortality. No neutralizing Ab was observed in sham-infected mice.

Table 1. Morbidity and mortality of AG129 mice infected with wild type and synthetic DENV-2 New Guinea C strain. Animals: Male and female AG129 mice Duration of experiment: 28 days Virus route: Intraperitoneal injection Volume: 0.2 ml Toxicity controls Infected, treated PRNT50 titer Alive/total Virus titer PRNT50 titer Treatment Virus titer (ge/µg RNA) (GM ± SD)b (ge/µg RNA)a (GM ± SD)b DENV-2 – – 4/8 1.3 ± 1.4* 1.5 ± 0.7 WT DENV-2 – – 2/8 2.1 ± 0.3** 2.8 ± 0.2*** synth Vehicle 0.0 ± 0.0 1.0 ± 0.0 – – – control a Virus titer expressed as virus titer equivalents (estimated by CCID50/ml stock) per µg total RNA. b mean day to death of mice dying prior to 31 dpi. PRNT50 – 50% plaque reduction neutralization titers. GM – geometric mean; SD – standard deviation (***p-value < 0.001, **p < 0.01, *p < 0.05, as compared with sham-infected [vehicle] controls)

3 Discussion This study demonstrates a multidisciplinary pipeline for comparing and analyzing synthetic and wildtype viruses. Central aspects/steps of this pipeline includes comparative genomics: First, synthetic viruses can be designed based on comparative genomics analysis of WT viral genomes (e.g., [16, 45]). Second, the synthetic variants can be analyzed based on comparative genomics tools; this includes the comparisons of the variants to each other and to WT genomes. Among others, this study demonstrates the interactions between comparative genomics and other research tools in synthetic virology. Specifically, we report here for the first time a set of SNVs that appear in a master wildtype DENV-2 virus in comparison to its synthetic constructs. In general, the wildtype virus was found to have a higher genomic variability (i.e. more nucleotide variations) than its synthetic strains. It is known that the population of viral variants is

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important for its fitness [20, 21]; thus, this analysis demonstrates an approach for connecting viral fitness to the structure of its population. In addition, our study demonstrates that the synthetic variants may cause more morbidity and mortality than the wildtype virus. Also, infection with the synthetic virus resulted in higher serum viral RNA titers (as measured 3 dpi) and more consistent and high levels of neutralizing Ab (as measured 14 dpi) as compared to infection with the wildtype DENV-2. These findings may be related to the already mentioned fact that the synthetic virus includes lower levels of genomic sequence variability. RNA viruses are often defined as quasispecies [46], collections of closely related mutant spectra or mutant clouds. The fact that a virus exists as a ‘cloud’ of closely related genomes constitutes an evolutionary advantage, among others by enabling the virus to efficiently adjust to different hosts or cells. Therefore, higher level of variability in the wildtype virus may improve its replication rate in hosts such as Vero cells (the origin of the investigated wildtype virus) and/or humans/A. aegypti, but on the other hand, decreas its fitness/replication rate in mice. It is also possible that the wildtype virus undergoes a direct selection for reduced virulence (among others this may be related to its variability) which actually increases its transmissibility and fitness [47]. The approach described here can be performed on any RNA virus and can be used as a platform technology for rational design and synthesis of live attenuated vaccines for RNA viruses. Specifically, in the near future we plan to use this approach for studying the effect of synonymous mutations on the fitness of various types of viruses (e.g. different serotypes of DENV and Zika). To this end, we will generate large libraries of viral genomes with different type of synonymous mutations. The RNA and protein levels of these libraries will be measured in cell lines and model organisms and the mutations will be connected to the observed measurements and to other viral phenotypes (e.g. cytopathic effect).

4 Methods 4.1

Virology

Reagents: Minimum Essential Medium Eagle- Earle`s (EMEM medium), Fetal Bovine Serum (FBS), Penicillin-Streptomycin-Nystatin Solution, and Dulbecco’s Phosphate Buffered Saline (DPBS) were obtained from Biological Industries (Bet-Haemek, IL). Opti-MEM medium was obtained from Gibco-ThermoFisher (Grand Island, NY). TransIT-X2 Mirus reagent was also used (MC-MIR-6003; Madison, WI). Cell Lines and Virus: Vero E6 (ATCC CRL-1586) cells were cultured in EMEM supplemented with 10% Fetal Calf Serum, streptomycin 0.1 mg/ml, penicillin 100 u/ml, nystatin 12.5 U/ml, 0.29 mg/ml at 37 °C in a 5% CO2 incubator. DENV-2, New Guinea-C strain was prepared in Vero E6 cells grown in EMEM medium containing 2% inactive FBS, Strep-Pen-Nys solution and incubated at 37 °C for 10 to 12 days until the appearance of cytopathic effects (CPE). The supernatant was then harvested and divided into aliquots and stored at −80 °C. Viral RNA was quantified using RT-PCR and infectious tissue culture assay.

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Cell Transfection and Infection: Vero E6 cells 20,000 cell/well were seeded in 96-well tissue culture plates and incubated overnight at 37 °C. A final amount of 2000 ng of an equimolar mix of all cDNA was amplified by PCR and incubated with 2.4 ml of TransIT-X2 Mirus reagent in 400 ml of Opti-MEM medium. Following the manufacturer’s instructions, the mix was incubated for 15 min at room temperature, then divided into 8 wells (of a 96-well plate) with Vero E6 cells and incubated for 4 h. After incubation the cell supernatant was removed and cells were washed three times with D-PBS, and 100 ml of EMEM medium supplemented with 2% inactive FBS was added. The supernatant was harvested 10 to 12 days post transfection. The supernatant was centrifuged and then passaged two more times as follows: 0.8 to 1 ml supernatant was added to 25 cm2 culture flask of Vero E6 cells (400,000 cells incubated overnight), and incubated for 4 h. After incubation the cell supernatant was removed and washed three times with D-PBS, and 6 ml of EMEM medium supplemented with 2% inactive FBS was added. The cell supernatant was incubated for 10 to 12 days until the appearance of cytopathic effects (CPE). The supernatant was then harvested divided to aliquots and stored in at 80 °C. 4.2

Molecular Biology

Dengue Virus Sequencing and Construction of Subgenomic Amplicons’ Plasmids: Dengue virus type 2, New Guinea C (Culture collection) RNA was extracted from aliquoted culture supernatants using the QIAamp Viral RNA mini kit (Qiagen). cDNA was synthesized using Maxima H Minus First Strand DNA synthesis kit (Thermo Scientific) with random hexamer primers, according to manufacturer’s protocol. cDNA was amplified by PCR in 500–3000 bp overlapping amplicons using PfuUltra II Hotstart PCR Master Mix(Agilent Technologies). The 500–1000 bp amplicons were sequenced using the Sanger method. The 2500–3500 bp amplicons were cloned into pJET1.2/blunt vectors using the CloneJET PCR Cloning Kit (Thermo Scientific). Cytomegalovirus promoter was inserted before the first fragment (0–2.5 kb). The last fragment (7–10.7 kb) was synthesized using the solid-phase DNA synthesis method (BioBasic), followed by hepatitis delta and simian polyadenylation signal. Subgenomic Infectious Amplicons: Dengue subgenomic amplicons were generated as previously described in [6]. Entire viral genome was amplified by PCR in 4 DNA fragments of approximately 3 kb, each with 75–80 bp overlapping arms. The first fragment was flanked at 5′ by the cytomegalovirus promoter. The last fragment was flanked at 3′ by the hepatitis delta ribozyme followed by the simian polyadenylation signal. The PCR products were purified using MinElute PCR purification kit (Qiagen) and 1000 ng of equimolar mix was transfected to Vero E6 cell line (ATCC). Assembly PCR of Full-Length Infectious DNA: Dengue viruses’ genomes were amplified by PCR in 4 DNA fragments of approximately 3 kb, each with 100–500 bp overlapping ends. The first fragment was flanked at 5′ by the cytomegalovirus promoter. The last fragment was flanked at 3′ by the hepatitis delta ribozyme followed by the simian polyadenylation signal. The overlapping fragments were extended using Q5

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High-Fidelity DNA Polymerase (New England Biolabs), using the neighboring fragment as a primer. NGS Sequencing: Dengue virus type 2, New Guinea C (Culture collection) RNA was extracted from aliquoted culture supernatants using QIAamp Viral RNA mini kit (Qiagen). cDNA was synthesized using Maxima H Minus First Strand DNA synthesis kit (Thermo Scientific) with random hexamer primers, according to manufacturer’s instructions. The cDNA was amplified in seven 1700–2500 bp overlapping amplicons using PfuUltra II Hotstart PCR Master Mix (Agilent Technologies). PCR products were separated in a 1% agarose TAE gel. After confirming the bands of interest were present without non-specific products, the PCR product was purified using the MIniElute PCR purification kit (Qiagen). Sequencing libraries were prepared using an in-house (INCPM) DNA-Seq protocol, and sequenced 2  150 on an Illumina MiSeq nano v2. 4.3

Computational Analysis

NGS Analysis and Variants Calling. Three DENV cDNA samples were sequenced and analyzed: one of the samples was the wildtype (WT) strain, with the WT (Sanger) sequence used as a reference genome for NGS analysis. The two (assembly and incellulo) synthetic variants of the virus synthesized using two different methods (full length assembly PCR and subgenomic infectious amplicons). Sequencing libraries were prepared using the INCPM DNA-seq protocol, and sequenced 2  150 on an Illumina MiSeq nano v2 PE150. Sequenced reads were mapped to a reference genome (genome of the wildtype virus) using BWA MEM v0.75 [48]. Among all read-pairs with the same alignment, a single representative read was chosen using Picard MarkDuplicates (http://picard.sourceforge.net). Variants were called using Freebayes v1.0.2 [49] with the setting -f 0.05–ploidy 5 (in order to increase sensitivity to low allele fraction variants). Freebayes is a haplotype based Bayesian genetic variant detector designed to find small polymorphisms and complex events (composite insertion and substitution events) smaller than the length of a short-read sequencing alignment. It uses short-read alignments (BAM files with Phred+33 encoded quality scores) for individuals from a population and a reference genome to determine the most-likely combination of genotypes for the population at each position in the reference. Positions of putatively polymorphic variants were identified and characterized by Variant Allele Frequency (VAF) - the fraction of variant alleles, out of all observed reads. Two types of biases were tested for each variant: Strand Bias (SB) is a type of sequencing bias in which one (positive/negative) strand is favored over the other, which can result in incorrect evaluation of the amount of evidence observed for one allele vs. the other. We used GATK Strand-Odds-Ratio (SOR) test [50] to determine if there is strand bias between forward and reverse strands for the reference or alternate alleles. Positions with SOR values > 4 may be considered as biased.

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Position Bias (PB) is a composite measure composed of (phred scaled) probabilities for observing positional bias within supporting reads for a variant. Positions with PB values > 40 may be considered as biased. Variants with VAF > 15% that passed SB (  4) and PB (  40) filters were marked as especially significant/strong. Comparative Genomics Analysis of SNVs. Multiple alignments of 618 DENV2 coding regions (population sample), and regions with conserved selection for strong or weak mRNA folding were computed as in [16]. Briefly, the translated amino acid sequences corresponding to the coding regions were aligned by Clustal Omega package [51] with default parameters and then mapped backed to the nucleotide sequences basing on the original nucleotide composition of each coding region. Multiple alignment conservation scores can be found in the supplementary material, Sect. 2. Coordinates of different genomic features (ORFs/UTRs) were obtained from DENV2 reference strain NC_001474.2 and aligned to the genomes used in this study. Local minimum free folding energy (MFE) profiles were constructed by applying a 39 nt length sliding window to each genomic sequence (robustness to the sliding window size and to multiple sequence alignment inaccuracies was discussed in our previous paper [16]): at each step the MFE of a local subsequence enclosed by the corresponding window was calculated by the Vienna (v. 2.1.9) package RNAfold function with default parameters [52]. These profiles were compared to folding profiles of corresponding randomized variants preserving different genomic features of the wildtype sequences which are not necessarily directly related to mRNA folding: amino acid order and content, (“vertical”) distribution of synonymous codons at each position in the alignment, frequency of di-nucleotides. As a result, regions with positions that undergo a significantly conserved (across different genomes) selection at the synonymous/silent level for weak/strong mRNA folding (MFE-selected regions) were identified. More details, including the coordinates of the identified regions with significant folding signals, can be found in [16], and in the supplementary material, Sect. 3. Statistical analysis of the overlap between the positions of SNVs and regions selected for strong/weak RNA structure was performed as follows: from the analysis in [16] the total length of MFE - selected regions (for weak and strong folding) occupies * 1/3 of the entire coding sequence (3368/10176), i.e. a randomly chosen point lies in a MFE-selected region with a probability p * 1/3. On the other hand, in our case, 14 out of 32 SNV positions overlap with some MFE-selected region. In order to compare this number with what is expected in random under a Uniform distribution hypothesis we calculate the Hyper-geometric p-value: Let N (Population size) = 10176 (total DENV2 CDS length); Successes in population (K) = 3368 (The total length of MFE - selected regions); Sample Size (n) = 32 (number of CDS SNV positions); Successes in Sample (k) = 14 (number of SNV positions that overlap with MFE-selected regions). Then, the probability to get at least 14 positions overlapping with MFE selected positions can be calculated as:

Generation and Comparative Genomics of Synthetic Dengue Viruses

K p¼1

13 X

!

Nk

!

nk !

k

 0:14

N n

k¼1

43

For a specific genomic position i, the nucleotide variability with respect to the population sample was quantified by a normalized Shannon entropy of a distribution on nucleotides normalized by the maximal entropy value at this position in the corresponding alignment: n P

Vi ¼

pj log2 ðpj Þ

j¼1

log2 n

Where n = 4, i.e. the number of different possible nucleotides. This variability measure takes values between 0 and 1, and describes how dispersed the distribution of the alphabet elements is: higher values correspond to more uniform nucleotide usage; lower values correspond to more biased nucleotide, indicating that some nucleotides are preferred/positions are conserved. Effective Number of Codons (ENC) is a measure that quantifies how far the codon usage of a coding sequence departs from equal usage of synonymous codons [53]. For each amino acid (AA) let us define xi to be the number of its synonymous codons of each type in the sequence, and n to be the number of times this AA appears in the sequence: n¼

d X

xi

i

The frequency of each codon is therefore: pi ¼ xi =n The ENC for a specific AA is: ENCA ¼ 1=FA ; where FA ¼

d X

p2i

i

ENC for the group of AAs with degeneracy d(Ad): ENCAd ¼ 1=FAd ; where FAd ¼

1 X FA jAd j A2A d

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(when an AA is missing, the corresponding effective number of codons is defined as an average over the given AAs of the same degeneracy). Finally ENC for a gene is defined as an average of the group ENCs over all degeneracy AA groups weighted by the number of AAs in each group computed over the entire coding sequence. ENC ¼ 2 þ

9 1 5 3 þ þ þ FA2 FA3 FA4 FA6

ENC can take values from 20, in the case of extreme bias where one codon is exclusively used for each amino acid (AA), to 61 when the use of alternative synonymous codons is equally likely. Therefore smaller ENC values correspond to a higher bias in synonymous codons usage; consequently, a negative correlation with ENC values means is equivalent to a positive correlation with synonymous codons usage. Codon Pairs Bias (CPB). To quantify codon pair bias, we follow [36] and define a codon pair score (CPS) as the log ratio of the observed over the expected number of occurrences of this codon pair in the coding sequence. To achieve independence from amino acid and codon bias, the expected frequency is calculated based on the relative proportion of the number of times an amino acid is encoded by a specific codon: 0

1 FðABÞ

CPS ¼ log@ FðAÞFðBÞ

FðXÞFðYÞ  FðXYÞ

A;

where the codon pair AB encodes for amino acid pair XY and F denotes the number of occurrences. The codon pair bias (CPB) of a virus is them defined as an average codon pair scores over all codon pairs comprising all viral coding sequences: CPB ¼

k1 1 X CPS½i k  1 I¼1

CpG Content. Following [54] we compute a dinucleotide score (DNTS) for a pair of nucleotides XY as an odds ratio of observed over expected frequencies: DNTS ¼

FðXYÞ ; FðXÞFðYÞ

where F denotes the frequency of occurrences. Specifically, the CpG score is equal to the DNTS corresponding to the CG nucleotide. The Dinucleotide Pair Bias (DNTB) of a virus is defined as an average of dinucleotide scores over all dinucleotides comprising all viral sequences:

Generation and Comparative Genomics of Synthetic Dengue Viruses

DNTB ¼

45

k1 1 X DNTS½i k  1 i¼1

GC content is defined as: GC% ¼

FðGÞ þ FðCÞ FðAÞ þ FðGÞ þ FðCÞ þ FðTÞ

Where F() is a number of occurrences of each one of nucleotides A,G,C, and T. 4.4

AG129 Study

Animals. Male and female AG129 6–7 weeks old mice with an average weight of 20 g were used. Because of the immunocompromised nature of this strain, AG129 mice (produced in house) were housed in pre-sterilized bonneted (HEPA-filtered) cages in a ventilated cage rack during the duration of the experiment. Animals were randomly assigned to cages and individually marked with eartags. All instruments, eartags, gloved hands, or other surfaces that came in contact with the mice were cold-sterilized with a 10% betadine solution and/or 75% EtOH. Facilities. Experiments were conducted in the BSL-2 animal suite at the Utah State University Laboratory Animal Research Center (LARC). All personnel receive continual special training on blood-borne pathogen handling by this university’s Environmental Health and Safety Office. Standard operating procedures for BSL-2 were used. Quantification of Viremia. The virus titers in plasma were assayed using the Brilliant II QRT-PCR Master Mix 1-step kit with samples run on the Mic (Magnetic Induction Cycler) real time PCR machine (Bio Molecular Systems, Inc). Breifly, RNA was extracted from serum samples collected 3 dpi using the QIAamp MinElute Virus Spin Kit (QIAGEN, cat# 57704), and was eluted with 25 µl of elution buffer. A volume of 5 µl of the RNA preparation was added to the appropriate mixture of PCR reagents following manufactures protocol. A virus stock of known titer was also extracted in parallel for use in quantification. Samples were subjected to 40 cycles of 15 s at 95 °C and 60 s at 60 °C following an initial single cycle of 30 min at 50 °C and 10 min at 95 °C. Samples of unknown quantity were quantified by extrapolation of C(t) values using a curve generated from serial dilutions of standard samples. Experimental Design. Animals were block-randomized by cage to two groups, with 8 included in each. Groups of 8 mice were challenged intraperitoneally (i.p.) with wild-type and synthetic viral variant. A third group of mice was also included as normal controls, and were sham-infected with vehicle to monitor handling, inoculation and caging techniques for effects on the immunocompromised AG129 mice. Mice were i.p. infected with 0.2 ml of each virus preparation. Mortality was observed daily for 28 days. Mice were weighed on day 0 and every other day through 11 dpi. Serum was

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collected from all animals on 3 dpi for quantification of viremia. Serum was also collected on 14 dpi for measurement of neutralizing antibody titer. Statistical Analysis of Mice Data. Survival data were analyzed using the Wilcoxon log-rank survival analysis, and all other statistical analyses were performed using one-way ANOVA with a Bonferroni group comparison. Acknowledgment. E.G. is supported, in part, by a fellowship from the Edmond J. Safra Center for Bioinformatics at Tel-Aviv University. T.T. is partially supported by the Minerva ARCHES award.

Appendix: Supplementary Material 1. No difference in various genomic features in 100 codon regions around SNV compared to regions that do not contains SNVs

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2. Multiple alignment of 618 DENV-2 genomes analyzed in this study - conservation scores Multiple alignment conservation score was defined by us as an average sum-of-pair score (SP). For the i-th column in the alignment we define Pijk = 1 for every pair Aij and Aik of elements (either nucleotides of amino acids, depending on the type of the aligned sequences) which are equal to each other and Pijk = 0 otherwise. The score Si for the ith column is Si ¼

N X N X 1 Pijk N ðN  1Þ=2 j¼1 k¼j þ 1

and the SP for the alignment is: SP ¼

M 1X Si M i¼1

The following values summarize the SP scores for the multiple alignment of 618 DENV-2 coding sequences analyzed in this study: SP(amino acids) = 0.97, SP (nucleotides) = 0.94 3. List of regions selected for strong/weak folding energy used in this study Coordinates of regions predicted to be selected for strong/weak folding energy can be found in the following tables (see details in reference [16] in main text): Each row in a file corresponds to one region (number of rows = number of regions) and contains 3 comma separated values x, y, z in the following order: Region start coordinate, region end coordinate, maximum folding selection conservation index (FSCI) in the cluster. The coordinates are given with respect to the start of the polyprotein coding sequence in the reference genome NC_001474.2. E.g., coordinates x, y for some region correspond to the nucleotides at x-th and y-th positions in the coding sequence of NC_001474.2

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start 96 186 332 441 530 781 867 1098 1263 1333 1506 1617 1705 1879 2020 2123 2219 2374 2817 2885 2998 3062 3258 3378 3564 3717 3844 4046 4204 4355 4465 4576 4629 4758 5028

weak folding end 153 247 416 490 686 830 917 1191 1311 1381 1563 1673 1840 1968 2086 2171 2291 2432 2865 2933 3061 3107 3328 3438 3613 3766 3904 4106 4292 4403 4563 4625 4682 4806 5077

FSCI 0.47 0.33 0.43 0.23 0.93 0.29 0.31 0.95 0.49 0.33 0.3 0.4 0.66 0.82 0.66 0.54 0.49 0.28 0.33 0.33 0.38 0.22 0.32 0.38 0.56 0.35 0.34 0.74 0.53 0.32 0.65 0.25 0.41 0.28 0.26

119 266 387 695 820 1010 1179 1287 1353 1521 1585 1662 1964 2157 2244 2345 2523 2689 2839 2932 2991 3263 3333 3419 3481 3572 3793 3845 3972 4029 4122 4197 4347 4407 4680 4869

strong folding 165 353 615 745 872 1142 1272 1332 1520 1583 1635 1739 2078 2231 2313 2413 2688 2805 2903 2980 3038 3312 3406 3464 3545 3631 3838 3941 4020 4080 4171 4251 4399 4525 4732 4914

0.21 0.29 0.74 0.44 0.68 0.48 0.31 0.21 0.74 0.54 0.57 0.85 0.48 0.43 0.36 0.37 0.7 0.48 0.38 0.26 0.2 0.51 0.8 0.27 0.41 0.35 0.2 0.21 0.33 0.35 0.29 0.32 0.35 0.52 0.44 0.21

Generation and Comparative Genomics of Synthetic Dengue Viruses

5239 5299 5458 5626 5772 5844 6129 6186 6338 6475 6649 6752 6895 6995 7240 7332 7412 7658 7731 7785 7913 8022 8207 8279 8409 8470 8580 8747 8920 9046 9156 9214 9344 9395 9578 9829 10019 10109

5290 5387 5557 5674 5823 5940 6179 6231 6386 6589 6718 6815 6983 7183 7299 7379 7461 7709 7780 7835 8004 8079 8266 8331 8463 8556 8624 8821 8969 9102 9213 9292 9385 9532 9691 9887 10069 10167

0.21 0.36 0.45 0.23 0.68 0.62 0.3 0.71 0.24 0.8 0.38 0.29 0.45 0.93 0.65 0.26 0.3 0.32 0.3 0.5 0.29 0.5 0.3 0.51 0.67 0.63 0.25 0.49 0.34 0.6 0.53 0.94 0.2 0.78 0.74 0.56 0.3 0.46

5083 5142 5347 5567 5940 6080 6230 6328 6712 6803 6946 7137 7192 7286 7608 7679 7799 7871 8073 8229 8345 8696 8873 9258 9460 9585 9666 9880 9950

5131 5235 5396 5618 6000 6150 6282 6376 6776 6871 6994 7187 7271 7434 7668 7727 7851 7928 8166 8296 8393 8754 8951 9309 9508 9636 9848 9944 10021

49

0.24 0.87 0.22 0.5 0.48 0.78 0.31 0.26 0.48 0.42 0.37 0.34 0.52 0.68 0.63 0.25 0.25 0.45 0.49 0.41 0.27 0.41 0.98 0.47 0.23 0.67 0.95 0.57 0.51

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ASTRAL-III: Increased Scalability and Impacts of Contracting Low Support Branches Chao Zhang, Erfan Sayyari, and Siavash Mirarab(B) Department of Electrical and Computer Engineering, University of California at San Diego, San Diego, USA {chz069,esayyari,smirarab}@ucsd.edu

Abstract. Discordances between species trees and gene trees can complicate phylogenetics reconstruction. ASTRAL is a leading method for inferring species trees given gene trees while accounting for incomplete lineage sorting. It finds the tree that shares the maximum number of quartets with input trees, drawing bipartitions from a predefined set of bipartitions X. In this paper, we introduce ASTRAL-III, which substantially improves on ASTRAL-II in terms of running time by handling polytomies more efficiently, exploiting similarities between gene trees, and trimming unnecessary parts of the search space. The asymptotic running time in the presence of polytomies is reduced from O(n3 k|X|1.726 ) for n species and k genes to O(D|X|1.726 ) where D = O(nk) is the sum of degrees of all unique nodes in input trees. ASTRAL-III enables us to test whether contracting low support branches in gene trees improves the accuracy by reducing noise. In extensive simulations and on real data, we show that removing branches with very low support improves accuracy while overly aggressive filtering is harmful. Keywords: Phylogenomics

1

· Incomplete lineage sorting. ASTRAL

Introduction

Reconstructing species phylogenies from a collection of input trees each inferred from a different part of the genome is becoming the standard practice in phylogenomics (e.g., [1–5]). This two-step approach stands in contrast to concatenation [6], where all of the sequences are combined into a supermatrix and analyzed in one maximum likelihood analysis. The two-step approach promises to effectively account for discordances between gene trees and the species tree [7] (but see recent literature for ongoing debates [8–11]) and is more efficient than statistical co-estimation of gene trees and the species tree [12] or site-based estimation of the species tree [13]. Among several causes of gene tree discordance [14], incomplete lineage sorting (ILS) is believed to be ubiquitous [15] and is extensively studied. ILS is typically modeled by the multi-species coalescent model (MSCM) [16,17], where branches of the species tree represent populations, and lineages are allowed to coalesce inside each branch; lineages that fail to coalesce at the root of each branch are moved to the parent branch. c Springer International Publishing AG 2017  J. Meidanis and L. Nakhleh (Eds.): RECOMB CG 2017, LNBI 10562, pp. 53–75, 2017. DOI: 10.1007/978-3-319-67979-2 4

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Several methods are proposed to infer a species tree from a collection of input trees (even though these trees need not be inferred from functional genes, following the conventions of the field, we will call them “gene trees”). Examples of summary methods include MP-EST [18], NJst [19], DISTIQUE [20], and STAR [21], which only use the topology of the input gene trees, and GLASS [22] and STEAC [21], which also uses the input branch lengths. While most methods need rooted gene trees as input, NJst and DISTIQUE can take unrooted input. These methods are all proved statistically consistent under the MSCM when the input gene trees are error-free, but no summary method is proved consistent when input trees are inferred from sequence data [23]. One of the statistically consistent methods under the MSCM is ASTRAL [24], which takes as input a collection of unrooted gene tree topologies and produces an unrooted species tree. ASTRAL uses dynamic programming to find the tree that shares the maximum number of induced quartet topologies with the collection of input gene trees. Since this problem is NP-Hard [25], ASTRAL solves a constrained version of the problem exactly, where the search space is limited to a predefined set of bipartitions X. In ASTRAL-I, the set X is the collection of all bipartitions in input gene trees. Showing that this space is not always large enough, ASTRAL-II [26] uses several heuristics to further augment the search space. Using the fact that for unrooted quartet trees the species tree always matches the most likely gene tree [27], ASTRAL is proved statistically consistent, even when solving the constrained problem, and its accuracy has been established in simulations [20,24,26,28,29]. ASTRAL-II has running time O(nk|X|2 ) for n species and k binary genes. Finally, ASTRAL has the ability to compute branch lengths in coalescent units [14] and a measure of branch support called local posterior probability [30]. Perhaps most importantly, ASTRAL and ASTRAL-II have been adopted by the community as one of the main methods of performing phylogenomics, and many biological analyses have adopted them. ASTRAL-II has several shortcomings, some of which we address here by introducing ASTRAL-III. While ASTRAL-II can analyze datasets of 1,000 species and 1,000 genes on average in a day, ASTRAL-II has trouble scaling to many tens of thousands of input trees. Datasets with more than ten thousands genomic loci are already available (e.g., [3]) and with the increase in genome sequencing, more will be available in future. Moreover, being able to handle large numbers of input trees enables using multiple trees per locus (e.g., a Bayesian sample) as input to ASTRAL. The limited scalability of ASTRAL with k is because of a Θ(nk) factor in the running time that corresponds to scoring a potential node in the species tree against all nodes of the input gene trees. This computation does not exploit similarities between gene trees, a shortcoming that we fix in ASTRAL-III. Moreover, while ASTRAL-II can handle polytomies in input gene trees, in the presence of polytomies of maximum degree dm , its running time inflates to O(d3m k|X|2 ) = O(n3 k|X|2 ), which quickly becomes prohibitive for input trees with polytomies of large degrees. ASTRAL-III uses a mathematical trick to enable scoring of gene tree polytomies in time similar to binary nodes.

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55

The ability to handle large polytomies in input gene trees is important for two reasons. On the one hand, some of the conditions that are conducive to ILS, namely shallow trees with many short branches, are also likely to produce gene sequence data that are identical between two species. A sensible gene tree (e.g., those produced by FastTree [31]) would leave the relationship between identical sequences unresolved (tools such as RAxML that output a random resolution take care to warn the user about such input data). On the other hand, all summary methods, including ASTRAL, are sensitive to gene tree estimation error [26,32–36]. One way of dealing with gene tree error, previously studied in the context of minimizing deep coalescence [37], is to contract low support branches in gene trees and use these unresolved trees as input to the summary method. While earlier studies found no evidence that this approach helps ASTRAL when the support is judged by SH-like FastTree support [26], no study has tested this approach with bootstrap support values. We will for the first time evaluate the effectiveness of contracting low support branches and show that conservative filtering of very low support branches does, in fact, help the accuracy. We note that the main competitors to ASTRAL, namely NJst [19] and its fast implementation, ASTRID [38], are not able to handle polytomies in input gene trees. ASTRAL-III makes it efficient to use unresolved gene trees as input to the species tree. Empirically, we observe that ASTRAL-III improves the running time compared to ASTRAL-II by a factor of 3X-4X for binary trees with large numbers of genes. Moreover, ASTRAL-III finishes on a dataset of 5,000 species and 500 genes in 18–30 h (24 on average). The ASTRAL-III software is publicly available at https://github.com/smirarab/ASTRAL.

2 2.1

Background and Notation Notations and Definitions

We denote the set of species by L and let n = |L|. Let G be the set of k input gene trees. The set of quartet trees induced by any tree t is denoted by Q(t). We refer to any subset of L as a cluster and refer to clusters with cardinality one as singletons. We define a partition as a set of clusters that are pairwise mutually exclusive (note that we abuse the term here, as the union of all clusters in a partition need not give the complete set). A bipartition (tripartition) is a partition with cardinality two (three); a partition with cardinality at least four corresponds to a polytomy and is referred to as a polytomy in this paper. Let X (the constraint bipartition set) be a set of clusters such that for each A ∈ X, we also have L − A ∈ X. We use Y to represent the set of all tripartitions examined in the ASTRAL dynamic programming: Y = {(A , A − A , L − A)|A ⊂ A, A ∈ X, A ∈ X, A − A ∈ X}. We use N (g) to represent the set of partitions correspondent to internal nodes in the gene tree g. We use E to denote the set of unique partitions and the number of times they appear in G:

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E = {(M,



|N (g) ∩ {M }|)|M ∈ N (g), g ∈ G}

(1)

g∈G

and we define D as the sum of the cardinalities of unique partitions in gene trees:  D= |M |. (2) (M,c)∈E

Finally, we use [d] to represent the set {1, 2 . . . , d}. 2.2

Background on ASTRAL-I and ASTRAL-II

The problem addressed by ASTRAL is: Given: a set G of input gene trees  Find: find the species tree t that maximizes g∈G |Q(g) ∩ Q(t)|. Lanford and Scornavacca recently proved this problem is NP-hard [25]. ASTRAL solves a constrained version of this problem where a set of clusters X restricts bipartitions that the output species tree may include (note ∀A ∈ X : L−A ∈ X). To solve the constrained version, ASTRAL uses a dynamic programming method with the following recursive relation to obtain the optimal tree. V (A) =

max

(A |A−A |L−A)∈Y

V (A ) + V (A − A ) + w(A |A − A |L − A)

where the function w(T ) scores each tripartition T = (A|B|C) against each node in each input gene tree. Let partition M = (M1 |M2 | . . . |Md ) represent an internal node of degree d in a gene tree. The overall contribution of T to the score of any species tree that includes T is:   1 QI(T, M ) (3) w(T ) = 2 g∈G M ∈N (g)

where, defining ai = |A ∩ Mi |, bi = |B ∩ Mi |, and ci = |C ∩ Mi |, we have:    ai + bj + ck − 3 ai bj ck . QI((A|B|C), M ) = 2

(4)

i∈[d] j∈[d]−{i} k∈[d]−{i,j}

As previously proved [24], QI(T, M ) computes twice the number of quartet trees that are going to be shared between any two trees if one includes only T and the other includes only M . ASTRAL-II requires Θ(d3 ) time for computing QI(.), making the overall running time O(n3 k|Y |) with polytomies of unbounded degrees or O(nk|Y |) in the absence of polytomies.

3

ASTRAL-III Algorithmic Improvements

Noting trivially that |Y | < |X|2 , the previously published running time analysis of ASTRAL-II was O(nk|X|2 ) for binary gene trees and O(n3 k|X|2 ) for input trees with polytomies. A recent result by Kane and Tao [39] (motivated by the question raised in analyzing the ASTRAL algorithm) indicates that |Y | ≤ |X|3/log3 (27/4) . This result immediately gives a better upper bound:

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57

Corollary 1. ASTRAL-II runs in O(nk|X|1.726 ) and O(n3 k|X|1.726 ), respectively, with and without polytomies in gene trees. ASTRAL-III further improves this running time using three new features: 1. A new way of handling polytomies is introduced to reduce the running time for scoring a gene tree to O(n), instead of O(n3 ), in the presence of polytomies, which reduces the total running time to O(nk|X|1.726 ) irrespective of the gene tree resolution. 2. A polytree is used to represent gene trees, and this enables an algorithm that reduce the overall running time from O(nk|X|1.726 ) to O(D|X|1.726 ). 3. An A*-like algorithm is used to trim parts of the dynamic programming DAG. In addition to these running time improvements, ASTRAL-III changes parameters of heuristics described in ASTRAL-II [26] to expand the size of set X for gene trees that include polytomies. We next describe each improvement in detail. 3.1

Efficient Handling of Polytomies

Recall that ASTRAL-II uses Eq. 4 to score a tripartition against a polytomy of size d in Θ(d3 ) time. We now show this can be improved. Lemma 1. Let QI(T, M ) be twice the number of quartet tree topologies shared between an unrooted tree that only includes a node corresponding to the tripartition T = (A|B|C) and another tree that includes only a node corresponding to a partition M = (M1 |M2 | . . . |Md ) of degree d; then, QI(T, M ) can be computed in time Θ(d). Proof. In Θ(d) time, we can compute:   ai and Sa,b = ai bi Sa = i∈[d]

(5)

i∈[d]

where ai = |A∩Mi | and bi = |B ∩Mi |; ditto for Sb , Sc , Sa,c and Sb,c . Equation 4, as proved before [26], computes twice the number of quartet tree topologies shared between an unrooted tree with internal node T and another tree with one internal node M . Equation 4 can be rewritten using these intermediate sums as:  ai  QI((A|B|C), M ) = ((Sb − bi )(Sc − ci ) − Sb,c + bi ci ) 2 i∈[d]   bi  + ((Sa − ai )(Sc − ci ) − Sa,c + ai ci ) (6) 2 i∈[d]  ci  + ((Sa − ai )(Sb − bi ) − Sa,b + ai bi ) 2 i∈[d]

(the derivation is given in the Appendix A). Computing Eq. 6 instead of Eq. 4   clearly reduces the running time to Θ(d) instead of Θ(d3 ).

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ASTRAL needs to score each of the |Y | tripartitions considered in the dynamic programming against each internal node of each input gene tree. The sum of degrees of k trees on n leaves is O(nk) (since that sum can never exceed the number of bipartitions in gene trees) and thus: Corollary 2. Scoring a tripartition (i.e., computing w) can be done in O(nk). 3.2

Gene Trees as a Polytree

ASTRAL-II scores each dynamic programming tripartition against each individual node of each gene tree. However, nodes that are repeated in several genes need not be recomputed. Recalling the definitions of E and D (Eqs. 1 and 2), Lemma 2. The score of a tripartition T = (A|B|C) against all gene trees (i.e., the w(T ) score) can be computed in Θ(D). Proof. In ASTRAL-III, we keep track of nodes that appear in multiple trees. This enables us to reduce the total calculation by using multiplicities:  c × QI(T, M ). (7) w(T ) = (M,c)∈E

We achieve this in two steps. In the first step, for each distinct gene tree cluster W , we compute the cardinality of the intersection of W and sets A, B, and C once using a depth first search with memoization. Let children(W ) denote the set of children of W in an arbitrarily chosen tree g ∈ G containing W . Then, we have the following recursive relation:  |Z ∩ A| (8) |W ∩ A| = Z∈children(W )

(ditto for |W ∩ B| and |W ∩ C|). All such intersection values can be computed in a post-order traversal of a polytree. In this polytree, all unique clusters in the gene trees are represented as vertices and parent-child relations are represented as edges; note that when a cluster has different children in two different input trees, we arbitrary choose one set of children and ignore the others. The polytree will include no more than D edges; thus, the time complexity of traversing this polytree and computing Eq. 8 for all nodes is Θ(D). Once all intersections are computed, in the second step, we simply compute the sum in Eq. 7. Each QI(.) computation requires Θ(d) by Lemma 1. Recalling that D = (M,c)∈E |M |, it is clear that computing Eq. 7 requires Θ(D). Therefore, both steps can be performed in Θ(D).   Theorem 1. The ASTRAL-III running time is O(D|X|1.726 ) for both binary and unresolved gene trees. Proof. By results of Kane and Tao [39], the size of the set Y is O(|X|1.726 ), and for each element in Y , by Lemma 2, we require O(D) to compute the weights, regardless of the presence or absence of polytomies. The running time of ASTRAL is dominated by computing the weights [26]. Thus, the overall running time is O(D|Y |) = O(D|X|1.726 ).

ASTRAL-III: Increased Scalability and Impacts

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Trimming of the Dynamic Programming

Our last feature does not improve theoretical running time but can result in some improvements in the experimental running time. Our main insight is that U (A) = w(A|A|L) − w(A|A|A) is an upperbound of V (A) (see the Appendix A for proof). 2 3 Since V (A) ≤ U (A), for any (A |A−A |L−A ) ∈ Y and (A |A−A |L−A ) ∈ Y , we no longer need to recursively compute V (A ) and V (A − A ) when: U (A ) + U (A − A ) + w(A |A − A |L − A) ≤ V (A ) + V (A − A ) + w(A |A − A |L − A)

(9)

Thus, in ASTRAL-III we trim the DAG of the memoized recursive dynamic programming when this calculation indicates that a path has no chance of improving the final score. To heuristically improve the efficiency of this approach, we order all (A |A − A |L − A) ∈ Y according to U (A ) + U (A − A ) + w(A |A − A |L − A).

4

Experimental Setup

Using simulation studies and on real data, we study two research questions: RQ1: Can contracting low support branches improve the accuracy of ASTRAL? RQ2: How does ASTRAL-III running time compare to ASTRAL-II for large polytomies and many gene trees? Note that addressing RQ1 in a scalable fashion is made possible only through the running time improvements of ASTRAL-III. 4.1

Datasets

Avian Biological Dataset: Neoaves have gone through a rapid radiation, and therefore, have extremely high levels of ILS [3]. A dataset of 48 whole-genomes was used to resolve this rapid radiation [3]. MP-EST run on 14,446 gene trees (exons, introns, and UCEs) produced a tree that conflicted with strong evidence from the literature and other analyses on the dataset (e.g., the Passerimorphae/Falcons/Seriemas grade was not recovered). This motivated the development of the statistical binning method to reduce the impacts of gene tree error [32,33]. MP-EST run on binned gene trees produced results that were largely congruent with the concatenation and other analyses. Here, we test if simply contracting low support branches of gene trees produces a tree that is congruent with other analyses and the literature. This analysis is made possible because ASTRAL-III can handle datasets with a large number of polytomies and large k efficiently. Simulated Avian-Like Dataset: We use a simulated dataset that was previously used in the statistical binning paper [32] to emulate the biological avian dataset. Since estimating the true branch lengths in coalescent units are hard, three versions of this dataset are available: 1X is the default version, whereas

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Fig. 1. Properties of the S100 dataset. (a) The density plot of the amount of true gene discordance measured by the FN rate between the true species tree and the true gene trees. (b) The density plot of gene tree estimation error measured by FN rate between true gene trees and estimated gene trees for different set of sequence lengths.

0.5X divides each branch length in half (increasing ILS) and 2X multiplies them by 2 (reducing ILS). The amount of true discordance (ILS), measured by the average RF distances between true species tree and true gene trees, is moderate at 0.35 for 2X, high at 0.47 for 1X, and very high at 0.59 for 0.5X. Moreover, to study the impact of gene tree estimation error, sequence lengths were varied to create four conditions: very high error with 250 bp alignments (0.67 RF distance between true gene trees and estimated gene trees), high error with 500 bp (0.54 RF), medium error with 1000 bp (0.39 RF) and moderate error with 1500 bp (0.30 RF). We use 1000 gene trees, and 20 replicates per condition. Gene trees are estimated using RAxML [40] with 200 replicates of bootstrapping. SimPhy-Homogeneous (S100): We simulated 50 replicates of a 101-taxon dataset using SimPhy [41] under the MSCM, where each replicate has a different species tree. In order to generate the species trees, we used the birth-only process with birth rate 10−7 , fixed haploid effective population size of 400K, and the number of generations sampled from a log-normal distribution with mean 2.5 M. For each replicate, 1000 true gene trees are simulated under the MSCM (the exact simulation commands are given in Appendix B and parameters are shown in Table 2). The amount of ILS, measured by the false-negative (FN) rate between true species trees and true gene trees, mostly ranged between 0.3 and 0.6 with an average of 0.46 (Fig. 1). We use Indelible [42] to simulate the nucleotide sequences along the gene trees using the GTR evolutionary model [43] with 4 different fixed sequence lengths: 1600, 800, 400, and 200 bp (Table 2). We then use FastTree2 [31] to estimate both ML and 100 bootstrapped gene trees under the GTR model for each gene of each replicate (> 2000200 runs in total). Gene tree estimation error, measured by the FN rate between the true gene trees and the estimated gene trees, depended on the sequence length as shown in Fig. 1 (0.55, 0.42, 0.31, and 0.23 on average for 200 bp, 400 bp, 800 bp, and 1600 bp, respectively). We sample 1000, 500, 200, or 50 genes to generate datasets with varying numbers of gene trees.

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Methods and Evaluation

We compare ASTRAL-III (version 5.2.5) to ASTRAL-II (version 4.11.1) in terms of running time. To address RQ1, we draw the bootstrap support values on the ML gene trees using the newick utility package [44]. We then contract branches with bootstrap support up to a threshold (0, 3, 5, 7, 10, 20, 33, 50, and 75%,) using the newick utility and use these contracted gene trees as input to ASTRAL. Together with the original set, this creates 10 different ways to run ASTRAL. To measure the accuracy of estimated species trees, we use False Negative (FN) rate. Note that in all our species tree comparisons, FN rate is equivalent to normalized Robinson-Foulds (RF) [45] metric, since the ASTRAL species trees are fully resolved. All running times are measured on a cluster with servers with Intel(R) Xeon(R) CPU E5-2680 v3 @ 2.50 GHz; each run was assigned to a single process, sharing cache and memory with other jobs.

5

Results

5.1

Impact of Contracting Low Support Branches on Accuracy

We investigate the impact of contracting branches with low support (RQ1) on our two simulated datasets (avian and S100) and on the real avian dataset. S100: On this dataset, contracting very low support branches in most cases improves the accuracy (Table 1 and Fig. 2); however, the excessive removal of Table 1. Species tree error (FN ratio) for all model conditions of the S100 dataset, with true gene trees (true), no filtering (non), and all filtering thresholds (columns). Genes Alignment true non 0

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