Computing Birth-Death Fixation Probabilities for Structured Populations

2013 ◽

Vol 469 (2153) ◽

pp. 20120248 ◽

Cited By ~ 17

Author(s):

Burton Voorhees

Keyword(s):

Complete Graph ◽

Present Model ◽

Automorphism Groups ◽

Linear Equations ◽

Structured Populations ◽

Edge Weight ◽

Simple Graphs ◽

Fixation Probabilities ◽

Fitness Parameter ◽

Birth Death

This paper presents an adaptation of the Moran birth–death model of evolutionary processes on graphs. The present model makes use of the full population state space consisting of 2 N binary-valued vectors, and a Markov process on this space with a transition matrix defined by the edge weight matrix for any given graph. While the general case involves solution of 2 N – 2 linear equations, symmetry considerations substantially reduce this for graphs with large automorphism groups, and a number of simple examples are considered. A parameter called graph determinacy is introduced, measuring the extent to which the fate of any randomly chosen population state is determined. Some simple graphs that suppress or enhance selection are analysed, and comparison of several examples to the Moran process on a complete graph indicates that in some cases a graph may enhance selection relative to a complete graph for only limited values of the fitness parameter.

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Evolution of cooperation in an epithelium

Journal of The Royal Society Interface ◽

10.1098/rsif.2018.0918 ◽

2019 ◽

Vol 16 (152) ◽

pp. 20180918 ◽

Cited By ~ 1

Author(s):

Jessie Renton ◽

Karen M. Page

Keyword(s):

Social Interactions ◽

Voronoi Tessellation ◽

Evolutionary Game ◽

Cellular Level ◽

Structured Populations ◽

Evolution Of Cooperation ◽

Animal Kingdom ◽

Population Structures ◽

Fixation Probabilities ◽

Evolutionary Graph Theory

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

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Fixation probabilities for simple digraphs

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2012.0676 ◽

2013 ◽

Vol 469 (2154) ◽

pp. 20120676 ◽

Cited By ~ 24

Author(s):

Burton Voorhees ◽

Alex Murray

Keyword(s):

Complete Bipartite Graph ◽

Directed Graphs ◽

Closed Form Solution ◽

Form Solution ◽

Upper And Lower Bounds ◽

Death Process ◽

Fixation Probability ◽

Single Vertex ◽

Fixation Probabilities ◽

Birth Death

The problem of finding birth–death fixation probabilities for configurations of normal and mutants on an N -vertex graph is formulated in terms of a Markov process on the 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth–death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the single-vertex fixation probabilities. This also yields a closed-form solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r >1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph.

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Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations

10.1101/082503 ◽

2016 ◽

Author(s):

F. Débarre

Keyword(s):

Social Behavior ◽

Life Cycles ◽

Structured Populations ◽

Altruistic Behavior ◽

Common Life ◽

Expected Frequency ◽

Phenotypic Differences ◽

Population Structures ◽

Birth Death

AbstractThe theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, i.e., for genetically transmitted traits, that mutation is very weak or absent. Here, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider populations of fixed size N, structured such that in the absence of selection, all individuals have the same probability of reproducing or dying (neutral reproductive values are the all same). Two types of individuals, A and B, corresponding to two types of social behavior, are competing; the fidelity of strategy transmission from parent to offspring is tuned by a parameter μ. Social interactions have a direct effect on individual fecundities. Under the assumption of small phenotypic differences (implyingweak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate our results with three common life-cycles (Wright-Fisher, Moran Birth-Death and Moran Death-Birth), and specific population structures (graph-structured populations). Qualitatively, we find that some life-cycles (Moran Birth-Death, Wright-Fisher) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. We also show that computing the expected frequency of altruists on a regular graph may require knowing more than just the graph’s size and degree.

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Improved multi-type birth-death phylodynamic inference in BEAST 2

10.1101/2020.01.06.895532 ◽

2020 ◽

Cited By ~ 1

Author(s):

Jérémie Scire ◽

Joëlle Barido-Sottani ◽

Denise Kühnert ◽

Timothy G. Vaughan ◽

Tanja Stadler

Keyword(s):

Influenza A ◽

Phylogenetic Trees ◽

Source Code ◽

Structured Populations ◽

Parameter Estimates ◽

Migration Patterns ◽

Global Migration ◽

Initial Release ◽

Past Population ◽

Birth Death

AbstractThe multi-type birth-death model with sampling is a phylodynamic model which enables quantification of past population dynamics in structured populations, based on phylogenetic trees. The BEAST 2 package bdmm implements an algorithm for numerically computing the probability density of a phylogenetic tree given the population dynamic parameters under this model. In the initial release of bdmm, analyses were limited to trees consisting of up to approximately 250 genetic samples for numerical reasons. We implemented important algorithmic changes to bdmm which dramatically increase the number of genetic samples that can be analyzed, and improve the numerical robustness and efficiency of the calculations. Being able to use bigger datasets leads to improved precision of parameter estimates. Furthermore, we report on several model extensions to bdmm, inspired by properties common to empirical datasets. We apply this improved algorithm to two partly overlapping datasets of Influenza A virus HA sequences sampled around the world, one with 500 samples, the other with only 175, for comparison. We report and compare the global migration patterns and seasonal dynamics inferred from each dataset.AvailabilityThe latest release with our updates, bdmm 0.3.5, is freely available as an open access package of BEAST 2. The source code can be accessed at https://github.com/denisekuehnert/bdmm.

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Selection and Drift in Subdivided Populations: A Straightforward Method for Deriving Diffusion Approximations and Applications Involving Dominance, Selfing and Local Extinctions

Genetics ◽

10.1093/genetics/165.4.2153 ◽

2003 ◽

Vol 165 (4) ◽

pp. 2153-2166 ◽

Cited By ~ 11

Author(s):

Denis Roze ◽

François Rousset

Keyword(s):

Fitness Function ◽

Structured Populations ◽

Allele Frequencies ◽

Frequency Change ◽

Diffusion Approximations ◽

Simple Method ◽

Straightforward Method ◽

Subdivided Populations ◽

Local Extinctions ◽

Fixation Probabilities

AbstractPopulation structure affects the relative influence of selection and drift on the change in allele frequencies. Several models have been proposed recently, using diffusion approximations to calculate fixation probabilities, fixation times, and equilibrium properties of subdivided populations. We propose here a simple method to construct diffusion approximations in structured populations; it relies on general expressions for the expectation and variance in allele frequency change over one generation, in terms of partial derivatives of a “fitness function” and probabilities of genetic identity evaluated in a neutral model. In the limit of a very large number of demes, these probabilities can be expressed as functions of average allele frequencies in the metapopulation, provided that coalescence occurs on two different timescales, which is the case in the island model. We then use the method to derive expressions for the probability of fixation of new mutations, as a function of their dominance coefficient, the rate of partial selfing, and the rate of deme extinction. We obtain more precise approximations than those derived by recent work, in particular (but not only) when deme sizes are small. Comparisons with simulations show that the method gives good results as long as migration is stronger than selection.

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Counterintuitive properties of the fixation time in network-structured populations

Journal of The Royal Society Interface ◽

10.1098/rsif.2014.0606 ◽

2014 ◽

Vol 11 (99) ◽

pp. 20140606 ◽

Cited By ~ 33

Author(s):

Laura Hindersin ◽

Arne Traulsen

Keyword(s):

Evolutionary Dynamics ◽

Transition Probabilities ◽

Structured Populations ◽

Death Process ◽

Fixation Time ◽

Fixation Probability ◽

Mutant Type ◽

Starting Conditions ◽

Fixation Probabilities ◽

Regular Networks

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.

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Fixation probabilities in graph-structured populations under weak selection

PLoS Computational Biology ◽

10.1371/journal.pcbi.1008695 ◽

2021 ◽

Vol 17 (2) ◽

pp. e1008695

Author(s):

Benjamin Allen ◽

Christine Sample ◽

Patricia Steinhagen ◽

Julia Shapiro ◽

Matthew King ◽

...

Keyword(s):

Natural Selection ◽

Search Algorithm ◽

Weighted Graph ◽

Structured Populations ◽

Death Process ◽

Fixation Probability ◽

Graph Structure ◽

Weak Selection ◽

Arbitrary Graph ◽

Fixation Probabilities

A population’s spatial structure affects the rate of genetic change and the outcome of natural selection. These effects can be modeled mathematically using the Birth-death process on graphs. Individuals occupy the vertices of a weighted graph, and reproduce into neighboring vertices based on fitness. A key quantity is the probability that a mutant type will sweep to fixation, as a function of the mutant’s fitness. Graphs that increase the fixation probability of beneficial mutations, and decrease that of deleterious mutations, are said to amplify selection. However, fixation probabilities are difficult to compute for an arbitrary graph. Here we derive an expression for the fixation probability, of a weakly-selected mutation, in terms of the time for two lineages to coalesce. This expression enables weak-selection fixation probabilities to be computed, for an arbitrary weighted graph, in polynomial time. Applying this method, we explore the range of possible effects of graph structure on natural selection, genetic drift, and the balance between the two. Using exhaustive analysis of small graphs and a genetic search algorithm, we identify families of graphs with striking effects on fixation probability, and we analyze these families mathematically. Our work reveals the nuanced effects of graph structure on natural selection and neutral drift. In particular, we show how these notions depend critically on the process by which mutations arise.

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