Counterintuitive properties of fixation probability and fixation time in population structures with spatially periodic resource distribution
Hossein Nemati, Mohammad Reza Ejtehadi, Kamran Kaveh
备注：
28 pages, 14 figures (10 in main text and 4 in appendices). Corresponding authors: Kamran Kaveh(kkaveh@uw.edu, kkavehma@gmail.com) and Mohammad Reza Ejtehadi(ejtehadi@sharif.edu)
Resource are often not uniformly distributed within a population. Spatial variations of concentration of a resource, change the fitness of competing strategies locally. The notion of fitness varying with respect to both genotype and environment is important in modeling cancer initiation, microbial evolution and evolution of drug resistance. Environmental interactions can be asymmetric, that is, they affect the fitness of one type more than the other. The question is how local environmental variations in network population structures change the selection dynamics in a finite population setting. We consider one-dimensional lattice population structures with spatial fitness distributions with a periodic pattern. Heterogeneity is determined by standard deviation of fitnesses and period. The model covers biologically relevant limits of two-habitat subdivided populations and randomly-distributed resources in high- and low-periods. We numerically calculate fixation probability and fixation times for a constant population birth-death process as fitness heterogeneity and period vary. We identify levels of heterogeneity for which a previously deleterious mutant, in a uniform environment, becomes beneficial. In other regimes of the problem we observe unexpected behavior where the fixation probability of both types are larger than their neutral value at the same time. This coincides with an exponential increase in time to fixation as a function of population size, which points to significant slow-down in selection process and the potential for coexistence between types in realistic time scales. We also discuss `fitness shift' model where the fitness function of one type is identical to the other up to a constant spatial shift. This leads to significant increase (or decrease) in the fixation probability of the mutant depending the value of the shift.
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 o…
The Moran process on graphs is a popular model to study the dynamics of evolution in a spatially structured population. Exact analytical solutions for the fixation probability and time of a new mutant have been found for only a few classes of graphs so far. Simulations are time-expensive and many realizations are necessary, as the variance of the fixation times is high. We present an algorithm that numerically computes these quantities for arbitrary small graphs by an approach based on the transition matrix. The advantage over simulations is that the calculation has to be executed only once. B…
A key question in evolution is how likely a mutant is to take over. This depends on natural selection and on stochastic fluctuations. Population spatial structure can impact mutant fixation probabilities. We introduce a model for structured populations on graphs that generalizes previous ones by making migrations independent of birth and death. We demonstrate that by tuning migration asymmetry, the star graph transitions from amplifying to suppressing natural selection. The results from our model are universal in the sense that they do not hinge on a modeling choice of microscopic dynamics or …
Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant in…
For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2\log(α)/α$ for a large selection coefficient $α$. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $μ$ for which the fixation times have different asymptotics as $α\to \infty$. If $μ$ is of order $α$, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log(α)/α$. If $μ$ is of order $α^γ, 0\leq γ\leq 1$, the fixatio…
How should dispersal strategies be chosen to increase the likelihood of survival of a species? We obtain the answer for the spatially extended versions of three well-known models of two competing species with unequal diffusivities. Though identical at the mean-field level, the three models exhibit drastically different behaviour leading to different optimal strategies for survival, with or without a selective advantage for one species. With conserved total particle number, dispersal has no effect on survival probability. With a fluctuating number, faster dispersal is advantageous if intra-spec…
We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially-structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model) and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the the degree distribution of the network. H…
We study the stochastic dynamics of evolutionary games, and focus on the so-called `stochastic slowdown' effect, previously observed in (Altrock et. al, 2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the prisoner's dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the micr…
Environment heterogeneity creates fast amplifiers of natural selection in graph-structured populations
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；SIB Swiss Institute of Bioinformatics（瑞士生物信息学研究所）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；Inserm（法国国立健康与医学研究院）
Predicting success of cooperators across arbitrary heterogeneous environmental landscapes
School of Biological Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran（基础科学研究所生物科学学院）；University of Minnesota, Minneapolis, MN, USA（明尼苏达大学）；Institute of Science and Technology Austria, Klosterneuburg, Austria（奥地利科学技术研究所）
The dynamics of well-mixed biological populations is usually studied by mean-field methods and weak-noise expansions. Similar methods have been applied also in spatially extended problems, relying on the fact that these populations are organized in colonies with a large local density of individuals. We provide a counterexample discussing a one-dimensional neutral population with negative frequency-dependent selection. The system exhibits a continuous phase transition between genetic fixation and coexistence unexpected from weak-noise arguments. We show that the behavior is a non-perturbative e…
We study a spatial, one-shot prisoner's dilemma (PD) model in which selection operates on both an organism's behavioral strategy (cooperate or defect) and its choice of when to implement that strategy across a set of discrete time slots. Cooperators evolve to fixation regularly in the model when we add time slots to lattices and small-world networks, and their portion of the population grows, albeit slowly, when organisms interact in a scale-free network. This selection for cooperators occurs across a wide variety of time slots and it does so even when a crucial condition for the evolution of …
The growth and evolution of microbial populations is often subjected to advection by fluid flows in spatially extended environments, with immediate consequences for questions of spatial population genetics in marine ecology, planktonic diversity and origin of life scenarios. Here, we review recent progress made in understanding this rich problem in the simplified setting of two competing genetic microbial strains subjected to fluid flows. As a pedagogical example we focus on antagonsim, i.e., two killer microorganism strains, each secreting toxins that impede the growth of their competitors (c…
We review and extend results for mutation, selection, genetic drift, and migration in a one-dimensional continuous population. The population is described by a continuous limit of the stepping stone model, which leads to the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation with additional terms describing mutations. Although the stepping stone model was first proposed for population genetics, it is closely related to "voter models" of interest in nonequilibrium statistical mechanics. The stepping stone model can also be regarded as an approximation to the dynamics of a thin layer of a…
