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

Hathcock, David; Strogatz, Steven H.

doi:10.1101/496380

Cited by 5 publications

(21 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While our focus was on mutant fixation probabilities, our model can be employed to investigate fixation times and evolution rate [15,[32][33][34][35][36][37][38]. It can also address more complex population structures [17,39], e.g.…”

Section: Discussionmentioning

confidence: 99%

Toward a universal model for spatially structured populations

Marrec

Lamberti

Bitbol

2020

Preprint

View full text Add to dashboard Cite

Microbial populations often have complex spatial structures, with homogeneous competition holding only at a local scale. Population structure can strongly impact evolution, in particular by affecting the fixation probability of mutants. Here, we propose a model of structured microbial populations on graphs, where each node of the graph contains a well-mixed deme whose size can fluctuate, and where migrations are independent from birth and death events. We study analytically and numerically the mutant fixation probabilities in different structures, in the rare migration regime. In particular, we demonstrate that the star graph continuously transitions between amplifying and suppressing natural selection as migration rate asymmetry is varied. This elucidates the apparent paradox in existing constant-size models on graphs, where the star is an amplifier or a suppressor depending on the details of the dynamics or update rule chosen, e.g. whether each birth event precedes or follows a death event. The celebrated amplification property of the star graph for large populations is preserved in our model, for specific migration asymmetry. We further demonstrate a general mapping between our model and constant-size models on graphs, under a constraint on migration rates, which directly stems from assuming constant size. By lifting this constraint, our model reconciles and generalizes previous results, showing that migration rate asymmetry is key to determining whether a given population structure amplifies or suppresses natural selection.SummaryA key question in evolution is how likely a mutant is to take over. This depends on natural selection and on stochastic fluctuations. Compared to a well-mixed population, some spatial structures have been predicted to amplify or suppress the impact of natural selection. This can have a major impact on evolution, as well as applications in biotechnology, e.g. by facilitating selection in directed evolution of biomolecules. We introduce a model for complex structured populations that generalizes over existing ones by allowing population size to fluctuate. We demonstrate that whether a given population structure amplifies or suppresses selection strongly depends on the asymmetry of migration rates. Our results are universal in the sense that they do not depend on update rules.

show abstract

Section: Discussionmentioning

confidence: 99%

Toward a universal model for spatially structured populations

Marrec

Lamberti

Bitbol

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The transition probabilities of Y t are independent and identical [4,6]. Therefore Wald's 80 Eliminating time steps where X t = 0 does not affect the fixation probability of the Moran process.…”

mentioning

confidence: 99%

“…approximations to the problem [4]. Second, we can settle for closed-form but intractable 234 expressions for those distributions [3] and numerically evaluate them.…”

mentioning

confidence: 99%

“…When they are applicable, martingales have several advantages compared to other 240 popular approaches to analyzing evolutionary stochastic processes. Markov chain 241 analysis seems to be the predominant method for studying these kinds of problems in 242 the literature [4,[27][28][29][30][31][32][33][34][35][36]. The Markov chain approach constructs a matrix of transition 243 probabilities and calculates e.g.…”

mentioning

confidence: 99%

“…We can calculate them by brute force if the size of state space is 247 small enough [5,31,32,37]. But quite often they are written using sums of products over 248 all state space [4,5,[33][34][35][36]. If we consider more complicated evolutionary stochastic 249 processes, e.g.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Wald’s martingale and the Moran process

Monk

Schaik

2020

Preprint

View full text Add to dashboard Cite

Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moran birth-death process, where that quantity is the number of mutants in a population. In such processes we are often interested in their absorption (i.e. fixation) probabilities, and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth-death model. Instead of considering the time to absorption, we consider a closely-related quantity: the number of mutant population size changes before absorption. We use Wald's martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical, and exact. The parameter dependence of the characteristic functions is explicit, so it is easy to explore their properties in parameter space. We also use them to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so we can quickly and tractably obtain absorption probabilities and times of evolutionary stochastic processes.February 18, 2020 1/21The Moran process is a probabilistic birth-death model of evolution. A mutant is introduced to an indigenous population, and we randomly choose organisms to live or die on subsequent time steps. Our goals are to calculate the probabilities that the mutant eventually dominates the population or goes extinct, and the distribution of time it requires to do so. The conditional distributions of time are difficult to obtain for the Moran process, so we consider a slightly different but related problem. We instead calculate the conditional distributions of the number of times that the mutant population size changes before it dominates the population or goes extinct. We use a martingale identified by Abraham Wald to obtain elegant and exact expressions for those distributions. We then use them to approximate conditional time distributions, and we show when that approximation is accurate. Our analysis outlines the basic concepts martingales and demonstrates why they are a formidable tool for studying probabilistic evolutionary models such as the Moran process. Introduction 1 The Moran birth-death process is a classic model of evolution [1, 2]. It considers a 2 population of two species that we call mutants and indigenous. On sequential time 3 steps, one organism is chosen to reproduce from either species. The newborn is the 4 same species as its parent, and it replaces another organism in the population uniformly 5 at random. The total population size then remains constant, but the numbers of mutant 6 and indigenous organisms in it are stochastic in time. The mutants will drive the 7 indigenous organisms to extinctio...

show abstract

Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Monk

Schaik

2022

R. Soc. open sci.

View full text Add to dashboard Cite

Evolutionary graph theory (EGT) investigates the Moran birth–death process constrained by graphs. Its two principal goals are to find the fixation probability and time for some initial population of mutants on the graph. The fixation probability of graphs has received considerable attention. Less is known about the distribution of fixation time. We derive clean, exact expressions for the full conditional characteristic functions (CCFs) of a close proxy to fixation and extinction times. That proxy is the number of times that the mutant population size changes before fixation or extinction. We derive these CCFs from a product martingale that we identify for an evolutionary graph with any number of partitions. The existence of that martingale only requires that the connections between those partitions are of a certain type. Our results are the first expressions for the CCFs of any proxy to fixation time on a graph with any number of partitions. The parameter dependence of our CCFs is explicit, so we can explore how they depend on graph structure. Martingales are a powerful approach to study principal problems of EGT. Their applicability is invariant to the number of partitions in a graph, so we can study entire families of graphs simultaneously.

show abstract

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

Cited by 5 publications

References 48 publications

Toward a universal model for spatially structured populations

Toward a universal model for spatially structured populations

Wald’s martingale and the Moran process

Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Contact Info

Product

Resources

About