Evolution in heterogeneous populations: from migration models to fixation probabilities

The role of spatial structure in multi-deme models of evolutionary rescue

Genetic variation and population sizes are critical factors for successful adaptation to novel environmental conditions. Gene flow between sub-populations is a potent mechanism to provide such variation and can hence facilitate adaptation, for instance by increasing genetic variation or via the introduction of beneficial variants. On the other hand, if gene flow between different habitats is too strong, locally beneficial alleles may not be able to establish permanently. In the context of evolutionary rescue, intermediate levels of gene flow are therefore often optimal for maximizing a species chance for survival in metapopulations without spatial structure. To which extent and under which conditions gene flow facilitates or hinders evolutionary rescue in spatially structured populations remains unresolved. We address this question by studying the differences between evolutionary rescue in the island model and in the stepping stone model in a gradually deteriorating habitat. We show that evolutionary rescue is modulated by the rate of gene flow between different habitats, which in turn depends strongly on the spatial structure and the pattern of environmental deterioration. We use these insights to show that in many cases spatially structured models can be translated into a simpler island model using an appropriately scaled effective migration rate.

The Effect of Habitat Choice on Evolutionary Rescue in Subdivided Populations

AbstractEvolutionary rescue is the process by which a population, in response to an environmental change, successfully avoids extinction through adaptation. In spatially structured environments, dispersal can affect the probability of rescue. Here, we model an environment consisting of patches that degrade one after another, and we investigate the probability of rescue by a mutant adapted to the degraded habitat. We focus on the effects of dispersal and of immigration biases. We identify up to three regions delimiting the effect of dispersal on the probability of evolutionary rescue: (i) starting from low dispersal rates, the probability of rescue increases with dispersal; (ii) at intermediate dispersal rates, it decreases; and (iii) at large dispersal rates, it increases again with dispersal, except if mutants are too counterselected in not-yet-degraded patches. The probability of rescue is generally highest when mutant and wild-type individuals preferentially immigrate into patches that have already undergone environmental change. Additionally, we find that mutants that will eventually rescue the population most likely first appear in nondegraded patches. Overall, our results show that habitat choice, compared with the often-studied unbiased immigration scheme, can substantially alter the dynamics of population survival and adaptation to new environments.

Drivers of diversity in individual life courses: Sensitivity of the population entropy of a Markov chain

Individuals differ in their life courses, but how this diversity is generated, how it has evolved and how it is maintained is less understood. However, this understanding is crucial to comprehend evolutionary and ecological population dynamics. In structured populations, individual life courses represent sequences of stages that end in death. These life course trajectories or sequences can be described by a Markov chain and individuals diversify over the course of their lives by transitioning through diverse discrete stages. The rate at which stage sequences diversify with age can be quantified by the population entropy of a Markov chain. Here, we derive sensitivities of the population entropy of a Markov chain to identify which stage transitions generate - or contribute - most to diversification in stage sequences, i.e. life courses. We then use these sensitivities to reveal potential selective forces on the dynamics of life courses. To do so we correlated the sensitivity of each matrix element (stage transition) with respect to the population entropy, to its sensitivity with respect to fitness λ, the population growth rate. Positive correlation between the two sensitivities would suggest that the stage transitions that selection has acted most strongly on (high sensitivities with respect to λ) are also those that contributed most to the diversification of life courses. Using an illustrative example on a seabird population, the Thick-billed Murres on Coats Island, that is structured by reproductive stages, we show that the most influential stage transitions for diversification of life courses are not correlated with the most influential transitions for population growth. Our finding suggests that observed diversification in life courses is neutral rather than adaptive, note this does not imply that the life histories themselves are not adaptive. We are at an early stage of understanding how individual level dynamics shape ecological and evolutionary dynamics, and many discoveries await.

Environmental fitness heterogeneity in the Moran process

Many mathematical models of evolution assume that all individuals experience the same environment. Here, we study the Moran process in heterogeneous environments. The population is of finite size with two competing types, which are exposed to a fixed number of environmental conditions. Reproductive rate is determined by both the type and the environment. We first calculate the condition for selection to favour the mutant relative to the resident wild-type. In large populations, the mutant is favoured if and only if the mutant's spatial average reproductive rate exceeds that of the resident. But environmental heterogeneity elucidates an interesting asymmetry between the mutant and the resident. Specifically, mutant heterogeneity suppresses its fixation probability; if this heterogeneity is strong enough, it can even completely offset the effects of selection (including in large populations). By contrast, resident heterogeneity has no effect on a mutant's fixation probability in large populations and can amplify it in small populations.

Transitions from Single- to Multi-Locus Processes during Speciation with Gene Flow

During speciation-with-gene-flow, a transition from single-locus to multi-locus processes can occur, as strong coupling of multiple loci creates a barrier to gene flow. Testing predictions about such transitions with empirical data requires building upon past theoretical work and the continued development of quantitative approaches. We simulated genomes under several evolutionary scenarios of gene flow and divergent selection, extending previous work with the additions of neutral sites and coupling statistics. We used these simulations to investigate, in a preliminary way, if and how selected and neutral sites differ in the conditions they require for transitions during speciation. For the parameter combinations we explored, as the per-locus strength of selection grew and/or migration decreased, it became easier for selected sites to show divergence-and thus to rise in linkage disequilibrium (LD) with each other as a statistical consequence-farther in advance of the conditions under which neutral sites could diverge. Indeed, even very low rates of effective gene flow were sufficient to prevent differentiation at neutral sites. However, once strong enough, coupling among selected sites eventually reduced gene flow at neutral sites as well. To explore whether similar transitions might be detectable in empirical data, we used published genome resequencing data from three taxa of Heliconius butterflies. We found that fixation index ( F S T ) outliers and allele-frequency outliers exhibited stronger patterns of within-deme LD than the genomic background, as expected. The statistical characteristics of within-deme LD-likely indicative of the strength of coupling of barrier loci-varied between chromosomes and taxonomic comparisons. Qualitatively, the patterns we observed in the empirical data and in our simulations suggest that selection drives rapid genome-wide transitions to multi-locus coupling, illustrating how divergence and gene flow interact along the speciation continuum.

Establishment of Locally Adapted Mutations Under Divergent Selection

We study the establishment probabilities of locally adapted mutations using a multi-type branching process framework. We find a surprisingly simple and intuitive analytical approximation for the establishment probabilities in a symmetric two-deme model under the assumption of weak (positive) selection. This is the first analytical closed-form approximation for arbitrary migration rate to appear in the literature. We find that the establishment probability lies between the weak and the strong migration limits if we condition the origin of the mutation to the deme where it is advantageous. This is not the case when we condition the mutation to first occur in a deme where it is disadvantageous. In this case we find that an intermediate migration rate maximizes the probability of establishment. We extend our results to the cases of multiple demes, two demes with asymmetric rates of gene flow, and asymmetric carrying capacities. The latter case allows us to illustrate how density regulation can affect establishment probabilities. Finally, we use our results to investigate the role of gene flow on the rate of local adaptation and identify cases in which intermediate amounts of gene flow facilitate the rate of local adaptation as compared to two populations without gene flow.

Effects of motion in structured populations

In evolutionary processes, population structure has a substantial effect on natural selection. Here, we analyse how motion of individuals affects constant selection in structured populations. Motion is relevant because it leads to changes in the distribution of types as mutations march towards fixation or extinction. We describe motion as the swapping of individuals on graphs, and more generally as the shuffling of individuals between reproductive updates. Beginning with a one-dimensional graph, the cycle, we prove that motion suppresses natural selection for death-birth (DB) updating or for any process that combines birth-death (BD) and DB updating. If the rule is purely BD updating, no change in fixation probability appears in the presence of motion. We further investigate how motion affects evolution on the square lattice and weighted graphs. In the case of weighted graphs, we find that motion can be either an amplifier or a suppressor of natural selection. In some cases, whether it is one or the other can be a function of the relative reproductive rate, indicating that motion is a subtle and complex attribute of evolving populations. As a first step towards understanding less restricted types of motion in evolutionary graph theory, we consider a similar rule on dynamic graphs induced by a spatial flow and find qualitatively similar results, indicating that continuous motion also suppresses natural selection.

Genetic hitchhiking and the dynamic buildup of genomic divergence during speciation with gene flow

A major issue in evolutionary biology is explaining patterns of differentiation observed in population genomic data, as divergence can be due to both direct selection on a locus and genetic hitchhiking. "Divergence hitchhiking" (DH) theory postulates that divergent selection on a locus reduces gene flow at physically linked sites, facilitating the formation of localized clusters of tightly linked, diverged loci. "Genome hitchhiking" (GH) theory emphasizes genome-wide effects of divergent selection. Past theoretical investigations of DH and GH focused on static snapshots of divergence. Here, we used simulations assessing a variety of strengths of selection, migration rates, population sizes, and mutation rates to investigate the relative importance of direct selection, GH, and DH in facilitating the dynamic buildup of genomic divergence as speciation proceeds through time. When divergently selected mutations were limiting, GH promoted divergence, but DH had little measurable effect. When populations were small and divergently selected mutations were common, DH enhanced the accumulation of weakly selected mutations, but this contributed little to reproductive isolation. In general, GH promoted reproductive isolation by reducing effective migration rates below that due to direct selection alone, and was important for genome-wide "congealing" or "coupling" of differentiation (F(ST)) across loci as speciation progressed.

Species Differentiation on a Dynamic Landscape: Shifts in Metapopulation Genetic Structure Using the Chronology of the Hawaiian Archipelago

Species formation during adaptive radiation often occurs in the context of a changing environment. The establishment and arrangement of populations, in space and time, sets up ecological and genetic processes that dictate the rate and pattern of differentiation. Here, we focus on how a dynamic habitat can affect genetic structure, and ultimately, differentiation among populations. We make use of the chronology and geographical history provided by the Hawaiian archipelago to examine the initial stages of population establishment and genetic divergence. We use data from a set of 6 spider lineages that differ in habitat affinities, some preferring low elevation habitats with a longer history of connection, others being more specialized for high elevation and/or wet forest, some with more general habitat affinities. We show that habitat preferences associated with lineages are important in ecological and genetic structuring. Lineages that have more restricted habitat preferences are subject to repeated episodes of isolation and fragmentation as a result of lava flows and vegetation succession. The initial dynamic set up by the landscape translates over time into discrete lineages. Further work is needed to understand how genetic changes interact with a changing set of ecological interactions amongst a shifting mosaic of landscapes to achieve species formation.