Counterintuitive properties of the fixation time in network-structured populations

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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Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

10.1101/496380 ◽

2018 ◽

Author(s):

David Hathcock ◽

Steven H. Strogatz

Keyword(s):

Complete Graph ◽

Time Distribution ◽

Evolutionary Dynamics ◽

Structured Populations ◽

Relative Fitness ◽

Death Process ◽

Fixation Time ◽

Large Network ◽

Moran Process ◽

Fitness Dependence

Evolutionary graph theory models the effects of natural selection and random drift on structured populations of mutant and non-mutant individuals. Recent studies have shown that fixation times, which determine the rate of evolution, often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, each of which admits an exact solution in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a weighted convolution of two Gumbel distributions, with a weight depending on the relative fitness. When fitness is neutral, however, the Moran process has a highly skewed fixation-time distribution on both the complete graph and the ring. In this sense, the case of neutral fitness is singular. Even on these simple network structures, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Applications of our methods to a multi-fitness Moran model, times to partial fixation, and evolution on random networks are discussed.

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Fixation probabilities in network structured meta-populations

Scientific Reports ◽

10.1038/s41598-021-97187-6 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Sedigheh Yagoobi ◽

Arne Traulsen

Keyword(s):

Evolutionary Dynamics ◽

Evolutionary Ecology ◽

Research Topic ◽

Fixation Probability ◽

Popular Approach ◽

Fixation Probabilities ◽

Evolutionary Graph Theory ◽

Selection For ◽

Carry Over ◽

Regular Networks

AbstractThe effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

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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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Fixation probabilities in network structured meta-populations

10.1101/2021.07.21.453198 ◽

2021 ◽

Author(s):

Sedigheh Yagoobi ◽

Arne Traulsen

Keyword(s):

Evolutionary Dynamics ◽

Evolutionary Ecology ◽

Research Topic ◽

Fixation Probability ◽

Popular Approach ◽

Fixation Probabilities ◽

Evolutionary Graph Theory ◽

Selection For ◽

Carry Over ◽

Regular Networks

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two-patch structure with unequal population size suppresses selection for low migration probabilities.

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The effect of population structure on the rate of evolution

Proceedings of The Royal Society B Biological Sciences ◽

10.1098/rspb.2013.0211 ◽

2013 ◽

Vol 280 (1762) ◽

pp. 20130211 ◽

Cited By ~ 56

Author(s):

Marcus Frean ◽

Paul B. Rainey ◽

Arne Traulsen

Keyword(s):

Population Structure ◽

Evolutionary Process ◽

Ecological Factors ◽

Structured Populations ◽

Fixation Time ◽

Fixation Probability ◽

Star Graphs ◽

Rate Of Evolution ◽

Population Structures ◽

Evolution Proceeds

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by ‘star graphs’, that amplify the effects of selection by further increasing the fixation probability of advantageous mutants and decreasing the fixation probability of disadvantageous mutants. The fact that star graphs increase the fixation probability of beneficial mutations has lead to the conclusion that evolution proceeds more rapidly in star-structured populations, compared with mixed (unstructured) populations. Here, we show that the effects of population structure on the rate of evolution are more complex and subtle than previously recognized and draw attention to the importance of fixation time. By comparing population structures that amplify selection with other population structures, both analytically and numerically, we show that evolution can slow down substantially even in populations where selection is amplified.

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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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The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population

PLoS Computational Biology ◽

10.1371/journal.pcbi.1009537 ◽

2021 ◽

Vol 17 (10) ◽

pp. e1009537

Author(s):

Mohammad Ali Dehghani ◽

Amir Hossein Darooneh ◽

Mohammad Kohandel

Keyword(s):

Network Structure ◽

Evolutionary Dynamics ◽

Finite Population ◽

Small World ◽

Fixation Time ◽

Fixation Probability ◽

Neutral Drift ◽

Scale Free ◽

Wide Range ◽

And Mathematics

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

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A theory of evolutionary dynamics on any complex spatial structure

10.1101/2021.02.07.430151 ◽

2021 ◽

Author(s):

Yang Ping Kuo ◽

César Nombela Arrieta ◽

Oana Carja

Keyword(s):

Bone Marrow ◽

Stem Cell ◽

Spatial Structure ◽

Evolutionary Dynamics ◽

Natural Populations ◽

Spatial Arrangement ◽

Death Process ◽

Stem Cell Population ◽

Evolutionary Simulations ◽

Regular Networks

AbstractUnderstanding how the spatial arrangement of a population shapes its evolutionary dynamics has been of long-standing interest in population genetics. Most previous studies assume a small number of demes connected by migration corridors, symmetrical structures that most often act as well-mixed populations. Other studies use networks to model the more complex topologies of natural populations and to study the structures that suppress or amplify selection. However, they usually assume very small, regular networks, with strong constraints on the strength of selection considered. Here we build network generation algorithms, evolutionary simulations and derive general analytic approximations for probabilities of fixation in populations with complex spatial structure. By tuning network parameters and properties independent of each other, we systematically span across network families and show that both a network’s degree distribution, as well as its node mixing pattern shape the evolutionary dynamics of new mutations. We analytically write the relevant selective parameter, predictive of evolutionary dynamics, as a combination of network statistics. As one application, we use recent imaging datasets and build the cellular spatial networks of the stem cell niches of the bone marrow. Across a wide variety of parameters and regardless of the birth-death process used, we find these networks to be strong suppressors of selection, delaying mutation accumulation in this tissue. We also find that decreases in stem cell population size decrease the suppression strength of the tissue spatial structure, hinting at a potential diminishing spatial suppression in the bone marrow tissue as individuals age.

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Fixation Probabilities of Evolutionary Graphs Based on the Positions of New Appearing Mutants

Journal of Applied Mathematics ◽

10.1155/2014/901363 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Pei-ai Zhang

Keyword(s):

Markov Chain ◽

Graph Theory ◽

Evolutionary Dynamics ◽

Chain Process ◽

Fixation Probability ◽

Spatial Structures ◽

Single Level ◽

Fixation Probabilities ◽

Evolutionary Graph Theory

Evolutionary graph theory is a nice measure to implement evolutionary dynamics on spatial structures of populations. To calculate the fixation probability is usually regarded as a Markov chain process, which is affected by the number of the individuals, the fitness of the mutant, the game strategy, and the structure of the population. However the position of the new mutant is important to its fixation probability. Here the position of the new mutant is laid emphasis on. The method is put forward to calculate the fixation probability of an evolutionary graph (EG) of single level. Then for a class of bilevel EGs, their fixation probabilities are calculated and some propositions are discussed. The conclusion is obtained showing that the bilevel EG is more stable than the corresponding one-rooted EG.

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