Sampling Ancestral Recombination Graphs and Reconstruction of Phylogenetic Trees for ...

154 Pages·2016·2.13 MB·English

UNIVERSITA` DEGLI STUDI DI MILANO - BICOCCA
DIPARTIMENTO DI INFORMATICA, SISTEMISTICA E COMUNICAZIONE
DOTTORATO DI RICERCA IN INFORMATICA - CICLO XXVIII
Ph.D. Thesis
Sampling Ancestral Recombination
Graphs and Reconstruction of
Phylogenetic Trees for Explaining
Evolution
Author: Anna Paola Carrieri
Supervisors: Prof. Paola Bonizzoni
Prof. Laxmi Parida
Tutor: Prof. Lucia Pomello
November 2015

To my parents Annamaria and Domenico
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Acknowledgements
This thesis is the result of three years of research efforts. Most of such work would not
have been possible without the support of many people that I wish to thank. First of all,
my gratitude goes to my two advisors.
I express my special appreciation to my advisor Professor Paola Bonizzoni, who dearly
supported me all the time, encouraging and guiding my work in the right direction. Her
advices on both research as well as personal issues have been priceless. I also want to
thank her for all the invaluable helpful suggestions about how to improve the quality of
the final work.
I am deeply grateful to my advisor Professor Laxmi Parida, who shared with me her
knowledge and work, who stimulated my scientific research allowing my professional
growth. Many thanks to Laxmi for giving me the opportunity to work with her and join
her team, as visiting research student at IBM T.J. Watson research center, making me
feel part of it.
The members of the computational genomics group have contributed immensely to my
personal and professional time at IBM. First of all, many thanks to Filippo Utro for
always being there ready to answer my questions and clarify my doubts. Thanks to him
also for reading the resulting dissertation and giving me insightful suggestions on how
to improve my draft. Thanks to Daniel Platt for all the math he wholeheartedly and
patiently taught me. Thanks to Niina Haiminen for her friendly and lovely help anytime
I needed it.
IexpressmygratitudetoRiccardoDondi. Hisfruitfulinteractionwithinmyresearchwork
stimulated and made possible the development of new contributions that are part of this
thesis. Thanks to Stefano Beretta and Yuri Pirola for the stimulating and enlightening
discussions. Stefano, Riccardo and Yuri constantly helped me and made pleasant, at the
same time, the working hours in the lab.
I would like also to thank Professor Lucia Pomello and Professor Gianluca Della Vedova
for the brilliant comments and advices they kindly gave me in the past three years.
Last but not least, I would like to thank my parents. They always supported me and
pushed me for higher education. I hope they will be proud.
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Contents
Acknowledgements iii
List of Figures vii
List of Tables ix
1 Introduction 1
2 The coalescent theory and the ancestral recombination graph 7
2.1 Forward and backward simulation algorithms . . . . . . . . . . . . . . . . 7
2.1.1 Forward simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The coalescent model and the backward approach. . . . . . . . . . . . . . 9
2.3 The Wright-Fisher Model of Evolving Population . . . . . . . . . . . . . . 10
2.4 The discrete-time coalescent process . . . . . . . . . . . . . . . . . . . . . 11
2.5 The continuous-time coalescent process. . . . . . . . . . . . . . . . . . . . 13
2.5.1 The exponential distribution . . . . . . . . . . . . . . . . . . . . . 13
2.6 Modeling recombination in the coalescent model . . . . . . . . . . . . . . 15
2.7 The Ancestral Recombination Graph . . . . . . . . . . . . . . . . . . . . . 17
2.8 Backward simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Hudson Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 SimRA: Sampling ARG of multiple populations 22
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Modeling multiple population evolution . . . . . . . . . . . . . . . . . . . 23
3.3 Modeling single population evolution . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Algorithm to generate the topology. . . . . . . . . . . . . . . . . . 30
3.3.2 Painting ARG edges with SNP & STR mutations . . . . . . . . . . 32
3.4 Difference from Hudson algorithm . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Comparison study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Time and space performance . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Compactness measure of ARG . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7.2 Reduced ARG (mdARG) is unique . . . . . . . . . . . . . . . . . . 42
3.7.3 Comparing the compactness of SimRA and Hudson . . . . . . . . 44
3.8 On the uniqueness of GMRCA . . . . . . . . . . . . . . . . . . . . . . . . 44
3.9 Four Quantitative Hallmarks of ARG network . . . . . . . . . . . . . . . . 45
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CONTENTS vi
3.9.1 Closed-Form approximations of the expected hallmark values . . . 46
3.9.2 Summary of closed-form formulations . . . . . . . . . . . . . . . . 54
3.10 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.11 Using closed-form approximations for complex scenarios parameter set-up 58
4 Topological signatures for population admixture 62
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Topology Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 The topological framework . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2 Persistent homology . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.3 Topological Signatures . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Experiments on simulated data . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Experiments on avocado germplasm . . . . . . . . . . . . . . . . . . . . . 70
5 Character based methods to reconstruct phylogenetic trees 76
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Maximum Parsimony Models and the perfect phylogeny . . . . . . . . . . 77
5.3 Camin Sokal Parsimony, Dollo Parsimony and its variants . . . . . . . . . 80
5.4 The persistent perfect phylogeny . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1 The Persistent Perfect Phylogeny (PPP) Problem . . . . . . . . . 83
5.4.2 The extended matrix, the red black graph and the conflict graph . 85
5.4.3 An algorithm to compute a persistent perfect phylogeny . . . . . . 90
6 Explaining evolution via constrained persistent perfect phylogeny 92
6.1 The constrained persistent perfect phylogeny . . . . . . . . . . . . . . . . 93
6.2 The c-reduction and the persistent perfect phylogeny . . . . . . . . . . . . 94
6.3 Solving CPPP on matrices with edgeless conflict graphs . . . . . . . . . . 98
6.4 An algorithm for CPPP problem . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 Experimental analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6 A characterization of matrices without a persistent phylogeny . . . . . . . 105
7 Filling incomplete genomic sequences 108
7.1 Background and motivations . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3 An FPT algorithm for One-sided SF-MNSA . . . . . . . . . . . . . . . . . 115
7.4 An FPT algorithm for Two-sided SF-MNSA . . . . . . . . . . . . . . . . . 120
8 Conclusions and future research 131
8.1 SimRA: Sampling ARG of multiple populations - Ch. 3 . . . . . . . . . . 131
8.2 Topological Signatures for Population Admixture - Ch. 4. . . . . . . . . . 133
8.3 Explaining evolution via Constrained Persistent Phylogeny - Ch. 6 . . . . 133
8.4 Filling incomplete genomic sequences - Ch. 7 . . . . . . . . . . . . . . . . 135
Bibliography 136
List of Figures
2.1 The genealogy of three sample sequences . . . . . . . . . . . . . . . . . . . 9
2.2 Hudson model of recombination . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Pedigree graph and ARG of Wright-Fisher population . . . . . . . . . . . 17
3.1 An example of scaffold with four contemporary populations . . . . . . . . 24
3.2 A detailed example of scaffold with four extant populations . . . . . . . . 25
3.3 Merge and split nodes in a scaffold . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Density and length of a chromosomal segment . . . . . . . . . . . . . . . . 26
3.5 Time and space performance of SimRA and Hudson . . . . . . . . . . . . 36
3.6 An ARG and its marginal or embedded trees . . . . . . . . . . . . . . . . 37
3.7 Removal of a non t-coalescent node . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Portions of an ARG with non t-coalescent nodes . . . . . . . . . . . . . . 43
3.9 Compactness of ARGs produced by SimRA and Hudson . . . . . . . . . . 43
3.10 Genetic flow of three non-mixing units in an ARG . . . . . . . . . . . . . 47
3.11 Decomposition of an ARG in four trees . . . . . . . . . . . . . . . . . . . 48
3.12 Height of the ARG H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.13 Number of recombinations Z . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.14 Number of mutations Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.15 Diversity of population D . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.16 Height of the truncated ARG . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.17 A complex and detailed example of scaffold with three extant populations 59
4.1 An example of scaffold and the corresponding ARG. . . . . . . . . . . . . 63
4.2 Examples of CPs in scaffolds versus CPs in the respective ARGs . . . . . 64
4.3 Topology signatures embedded in the ARGs, on simulated data . . . . . . 71
4.4 Analysis of the persistent cycles of avocado germplasm data . . . . . . . . 72
4.5 Results of experiments with different simulation parameters . . . . . . . . 73
4.6 Six simulations with effective population size N = 10K and recombination
(r = 0.3×10−8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.7 Haplotypes from three groups of avocado germplasm data . . . . . . . . . 75
5.1 Example of perfect phylogeny over a binary matrix M . . . . . . . . . . . 78
5.2 Example of Camin-Sokal phylogeny . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Example of Dollo phylogeny . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Example of Persistent Perfect Phylogeny . . . . . . . . . . . . . . . . . . . 84
5.5 A binary matrix M and its associated M . . . . . . . . . . . . . . . . . . 86
e
5.6 A conflict graph G for a binary matrix . . . . . . . . . . . . . . . . . . . 86
c
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LIST OF FIGURES viii
5.7 Realization of a character in the red black graph and corresponding
completion of an extended matrix . . . . . . . . . . . . . . . . . . . . . . . 88
5.8 A red Σ-graph and the corresponding forbidden matrix. . . . . . . . . . . 89
6.1 Binary matrix M of size 5×5 . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 A c-reduction and a persistent perfect phylogeny . . . . . . . . . . . . . . 97
6.3 A chordal conflict graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4 A red black graph which is a simple cycle . . . . . . . . . . . . . . . . . . 107
6.5 A matrix without a persistent perfect phylogeny . . . . . . . . . . . . . . 107
7.1 An instance for the One-sided SF-MNSA problem. . . . . . . . . . . . . . 112
7.2 An instance of the Two-sided SF-MNSA problem . . . . . . . . . . . . . . 122
List of Tables
3.1 SimRA input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 An example of parameter-set up for a scaffold with 3 extant populations 60
6.1 Running times on unconstrained simulated instances . . . . . . . . . . . . 104
6.2 Improvements of constrained simulated instances over unconstrained in-
stances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
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