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The effect of graph structure on the dynamics of a stochastic evolutionary process

Hindersin, L. (2018). The effect of graph structure on the dynamics of a stochastic evolutionary process. PhD Thesis, University of Lübeck, Lübeck.

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Creators:
Hindersin, Laura1, Author
Traulsen, Arne1, Referee
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1Department Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Max Planck Society, ou_1445641

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Abstract: Evolutionary graph theory is the study of how spatial population structure affects evolutionary processes. The nodes of the graph are inhabited by individuals, e.g. cells. The links between nodes represent possibilities for these individuals to spread. We study the Moran process, where there is one birth and death event per time step. In the most commonly used updating mechanism, birth happens with probability proportional to the individual’s fitness. The individual giving birth then randomly replaces one of its neighbors by an identical copy of itself. Initially, the network is inhabited entirely by wild-type individuals with fitness 1 and one mutant with relative fitness r > 0. One interesting property of this system is the probability with which the mutant will give rise to a lineage that takes over the whole network, the so-called fixation probability. There are certain networks that can increase or decrease this probability compared to the unstructured case, called amplifiers and suppressors of selection, respectively. We find that most small undirected random graphs are amplifiers of selection. If we however change the updating rule to remove a random individual first and subsequently let its neighbors compete for the empty slot according to their fitness, this completely changes the result. Under death-birth updating, almost all undirected random graphs are suppressors of selection. Another evolutionary outcome of interest is the expected time this process takes until the mutants fixate in the population. Since it is known that certain amplifier graphs also increase the time to fixation, we are interested in the specific effect of graph structure on fixation time. We show that this fixation time can both increase or decrease when removing a link from a graph. Often, the fixation probability and time are either calculated analytically for simple cases or simulated for larger or more complicated graphs. We use standard Markov chain methods to numerically solve the system which has advantages over both analytical calculations and simulations. For this, we provide code to automate the part of creating the transition matrix for arbitrary graph structure. Lastly, we apply this abstract model to a conceptual question in biology, namely to cancer initiation. We are interested to find a graph which would provide an optimal tissue structure to prevent cancer mutations from spreading through the whole graph. Surprisingly, we conclude that it is not always the strongest suppressor of selection that works best at preventing this. But instead it highly depends on the fitness distribution of newly arising mutations and on the detailed update mechanism.

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Language(s): eng - English
Dates: 2018-02-202018-02-20
Publication Status: Published in print
Pages: 105
Publishing info: Lübeck : University of Lübeck
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Evolutionary graph theory . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The Moran process . . . . . . . . . . . . . . . . . . . . 3
1.2.2 The Moran process on graphs . . . . . . . . . . . . . . 5
1.2.3 Amplification and suppression of selection . . . . . . . 6
1.3 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . 9
2 Counterintuitive properties of the fixation time 11
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 The Moran process in well-mixed populations . . . . . 13
2.2.2 The Moran process in structured populations . . . . . 14
2.2.3 A general approach to calculate probabilities and times
of fixation . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Analytical results for small networks . . . . . . . . . . 17
2.3.2 Numerical simulations for larger networks . . . . . . . 28
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Amplifiers and suppressors of selection 33
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Fixation probabilities in well-mixed populations . . . . 36
3.3.2 Numerical procedure . . . . . . . . . . . . . . . . . . . 38
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Birth-death . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2 death-Birth . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.3 Directed graphs . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Numerical method and algorithm 51
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Software description . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 Computing the transition matrix from the adjacency
matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Fixation probability . . . . . . . . . . . . . . . . . . . 55
4.3.3 Fixation time . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.4 Computational limitations and performance . . . . . . 58
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Application to a question in cancer initiation 63
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Fixation of novel mutations . . . . . . . . . . . . . . . . . . . 66
5.4 The distribution of fitness effects of cancer mutations . . . . . 69
5.5 Population structures and their effect on fixation probabilities 72
5.6 Double mutations . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Discussion 81
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bibliography 91
Rev. Method: -
Identifiers: Other: Diss/12914
Degree: PhD

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