Toward a universal model for spatially structured populations

Abstract

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.

Experimental evidence that network topology can accelerate the spread of beneficial mutations

“…Recent theoretical work, which considers nodes as subpopulations rather than individuals as in our experiments, shows that asymmetry in migration between leaves and the hub in star networks can potentially amplify probabilities of fixation relative to the well-mixed case. Specifically, star networks with net outward or inward migration have been predicted to be suppressors or amplifiers of selection, respectively, while those with no net inward or outward, or balanced, migration have been shown to have no advantage over a well-mixed network in fixing a beneficial mutant (9). Our experimental set-up and modelling framework allow us to test these predictions by manipulating the relative amount of migration between the hub and the three leaves of the star network.…”

“…Metapopulations were transferred daily following dispersal among subpopulations (see below) by taking an aliquot corresponding to ∼10 7 CFUs per mL and inoculating into fresh medium. The population density in each subpopulation reached~10 9 CFUs, so this transfer regime corresponds to~6.67 daily generations of growth.…”

“…The fact that EGT rests on a stochastic model of evolution in finite populations (8) where individuals occupy the nodes of the graph means, moreover, that we do not know how robust its predictions are to more realistic situations involving nodes composed of subpopulations or variable rates of dispersal. While recent work relaxes some of the more restrictive assumptions of EGT (9) to show that migration asymmetry (10) can better account for amplification when migration is sufficiently rare to allow each node to be composed of either all wild-type or all mutant allele, the relevance of this work to real-world situations remains speculative.…”

“…We accounted for asymmetries in the direction of migration between leaves and hub generated by bidirectional dispersal by reducing inward migration rates from the leaves to the hub by ⅓ (Fig S3). Daily transfers involved 100-fold dilutions, resulting in a bottleneck population size of ∼10 7 CFUs and a maximum population size of~10 9 CFUs, corresponding tõ 6.67 generations per day. Since the theory makes predictions about fixation of a single beneficial mutation, we focus on the first 5 to 6 days (~35 to~40 generations) to minimize the opportunity for de novo mutations rising to high frequency.…”

Hydrodynamic flow and concentration gradients in the gut enhance neutral bacterial diversity

“…While complex population structures with asymmetric migrations can impact the fixation probabilities of beneficial and deleterious mutants [13,14,21], that of neutral mutants appearing uniformly in the population, e.g. upon division, is unaffected [14,21]. Similarly, chaotic hydrodynamic flow has been predicted to impact non-neutral mutant fixation probabilities, but not neutral ones [49].…”