Fixation probabilities in graph-structured populations under weak selection

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.Author summary: When a new mutation appears in a population, it may ultimately spread to all individuals, or it may go extinct. Which outcome occurs depends on how the mutation affects the organism’s fitness (i.e., natural selection), but also on random chance. The spatial arrangement of organisms in the population can alter the balance between selection and random chance: amplifying one, suppressing the other. However, these effects can be difficult to predict or compute, even in simple, idealized mathematical models. We develop a method to efficiently calculate the effects of spatial structure on natural selection, in the case that mutations have only a weak effect on fitness. We use this method to comb through millions of distinct spatial structures, identifying those with the most extreme effects on natural selection. The question we study has applications to cancer research, with an individual’s cells considered as a population, of which cancer cells are an invading mutant type.