We study one of the simplest scenarios of polygenic selection that can be imagined: a subdivided population of diploid individuals expressing an additive trait under spatially homogeneous stabilizing selection. We are interested in the amounts of variation that can be maintained at mutation-selection-migration-drift equilibrium, at individual loci and at the level of the trait, within and among subpopulations. We derive analytical approximations for variance components and summary statistics such as FST and QST under the assumptions of the infinite-island model and compare these with individual-based simulations. We find that: (i) There is a critical migration threshold (which depends on effect sizes of trait loci) below which population structure strongly inflates genic variance in the subdivided population to levels well above those in a panmictic population. Variation within each subpopulation is maximized close to the critical migration rate. (ii) The genetic basis of trait variation across subpopulations is most similar close to this migration threshold and (counter-intuitively) becomes less similar for higher migration rates. This has consequences for the `portability' of Genome-Wide Association Studies (GWAS) between subpopulations, i.e, the extent to which loci with large contributions to variance in one subpopulation explain variance in other subpopulations. (iii) An analytical mean-field approach based on the single-locus diffusion approximation, together with effective migration and selection parameters (to account for associations between loci), very accurately predicts various quantities.

Assortative mating and sexual selection are widespread in nature and can play an important role in speciation by facilitating the buildup and maintenance of reproductive isolation (RI). However, their contribution to genome-wide suppression of gene flow during RI is rarely quantified. Here, we consider a polygenic "magic" trait that is divergently selected across two populations connected by migration, while also serving as the basis of assortative mating, thus generating sexual selection on one or both sexes. We obtain theoretical predictions for divergence at individual trait loci by assuming that the effect of all other loci on any locus can be encapsulated via an effective migration rate, which bears a simple relationship to measurable fitness components of migrants and various early-generation hybrids. Our analysis clarifies how "tipping points" (characterized by an abrupt collapse of adaptive divergence) arise, and when assortative mating can shift the critical level of migration beyond which divergence collapses. We quantify the relative contributions of viability and sexual selection to genome-wide barriers to gene flow and discuss how these depend on existing divergence levels. Our results suggest that effective migration rates provide a useful way of understanding genomic divergence, even in scenarios involving multiple, interacting mechanisms of RI.

Habitat fragmentation poses a significant risk to population survival, causing both demographic stochasticity and genetic drift within local populations to increase, thereby increasing genetic load. Higher load causes population numbers to decline, which reduces the efficiency of selection and further increases load, resulting in a positive feedback that may drive entire populations to extinction. Here, we investigate this eco-evolutionary feedback in a metapopulation consisting of local demes connected via migration, with individuals subject to deleterious mutation at a large number of loci. We first analyze the determinants of load under soft selection, where population sizes are fixed, and then build on this to understand hard selection, where population sizes and load coevolve. We show that under soft selection, very little gene flow (less than one migrant per generation) is enough to prevent fixation of deleterious alleles. By contrast, much higher levels of migration are required to mitigate load and prevent extinction when selection is hard, with critical migration thresholds for metapopulation persistence increasing sharply as the genome-wide deleterious mutation rate becomes comparable to the baseline population growth rate. Moreover, critical migration thresholds are highest if deleterious mutations have intermediate selection coefficients but lower if alleles are predominantly recessive rather than additive (due to more efficient purging of recessive load within local populations). Our analysis is based on a combination of analytical approximations and simulations, allowing for a more comprehensive understanding of the factors influencing load and extinction in fragmented populations.

We consider how a population of N haploid individuals responds to directional selection on standing variation, with no new variation from recombination or mutation. Individuals have trait values z1,…,zN, which are drawn from a distribution ψ; the fitness of individual i is proportional to ezi. For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance V0, and show that for large N, there is a scaling limit which depends on a single parameter NV0. When selection is weak relative to drift (NV0≪1), the variance decreases exponentially at rate 1/N, and the expected ultimate gain in log fitness (scaled by V0), is just NV0, which is the same as Robertson’s (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift (NV0≫1), the ultimate gain can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time t∼1/V0 of an allele with value z is P(z), with mean P¯, the winning allele is the fittest of NP¯ survivors drawn from a distribution ψP/P¯. The expected ultimate change is ∼2log(1.15NV0) for a Gaussian distribution, and ∼−12log0.36NV0−log−log0.36NV0 for a Laplace distribution. This approach also predicts the variability of the process, and its dynamics; we show that in the strong selection regime, the expected genetic variance decreases as ∼t−3 at large times. We discuss how these results may be related to selection on standing variation that is spread along a linear chromosome.

We consider how the genetic architecture underlying locally adaptive traits determines the strength of a barrier to gene flow in a mainland-island model. Assuming a general life cycle, we derive an expression for the effective migration rate when local adaptation is due to genetic variation at many loci under directional selection on the island, allowing for arbitrary fitness and dominance effects across loci. We show how the effective migration rate can be combined with classical single-locus diffusion theory to accurately predict multilocus differentiation between the mainland and island at migration-selection-drift equilibrium and determine the migration rate beyond which local adaptation collapses, while accounting for genetic drift and weak linkage. Using our efficient numerical tools, we then present a detailed study of the effects of dominance on barriers to gene flow, showing that when total selection is sufficiently strong, more recessive local adaptation generates stronger barriers to gene flow. We then study how heterogeneous genetic architectures of local adaptation affect barriers to gene flow, characterizing adaptive differentiation at migration-selection balance for different distributions of fitness effects. We find that a more heterogeneous genetic architecture generally yields a stronger genome-wide barrier to gene flow and that the detailed genetic architecture underlying locally adaptive traits can have an important effect on observable differentiation when divergence is not too large. Lastly, we study the limits of our approach as loci become more tightly linked, showing that our predictions remain accurate over a large biologically relevant domain.

We consider how the genetic architecture underlying locally adaptive traits determines the strength of a barrier to gene flow in a mainland-island model. Assuming a general life cycle, we derive an expression for the effective migration rate when local adaptation is due to a polygenic trait under directional selection on the island, allowing for arbitrary fitness and dominance effects across loci. We show how the effective migration rate can be combined with classical single-locus diffusion theory to accurately predict multilocus differentiation between the mainland and island at migration-selection-drift equilibrium and determine the migration rate beyond which local adaptation collapses, while accounting for genetic drift and weak linkage. Using our efficient numerical tools, we then present a detailed study of the effects of dominance on barriers to gene flow, showing that when total selection is sufficiently strong, more recessive local adaptation generates stronger barriers to gene flow. We show that details of the haplodiplontic life cycle can be captured using a set of effective parameters, and consider how the relative strength of selection in the two phases affects barriers to gene flow. We then study how heterogeneous genetic architectures of local adaptation affect barriers to gene flow, characterizing adaptive differentiation at migration-selection balance for different distributions of fitness effects. We find that a more heterogeneous genetic architecture generally yields a stronger genome-wide barrier to gene flow and that the detailed genetic architecture underlying locally adaptive traits can have an important effect on observable differentiation when divergence is not too large. Lastly, we study the limits of our approach as loci become more tightly linked, showing that our predictions remain accurate over a large biologically relevant domain.

Habitat fragmentation poses a significant threat to the persistence of populations, by generating increased genetic drift (and thus higher genetic load) as well as demographic stochasticity. Higher load causes population numbers to decline, which reduces the efficiency of selection and further increases load, resulting in a positive feedback which may drive entire populations to extinction. Here, we investigate this eco-evolutionary feedback in a metapopulation, focusing on how extinction thresholds depend on the total rate of deleterious mutations and their selection and dominance coefficients, the carrying capacity and growth rate of subpopulations, and the extent of migration between them. We first study the determinants of load under soft selection, where population sizes are constant, and then build upon this to analyse hard selection, where population sizes and load co-evolve. We show that in contrast to soft selection, high levels of migration are required to mitigate load and prevent extinction when selection is hard, with critical migration thresholds for metapopulation persistence increasing sharply with increasing ‘hardness’ of selection. Our analysis employs a combination of simulations and analytical approximations, allowing for a more comprehensive understanding of the factors influencing genetic load and extinction in fragmented landscapes.### Competing Interest StatementThe authors have declared no competing interest.

We analyse how migration from a large mainland influences genetic load and population numbers on an island, in a scenario where fitness-affecting variants are unconditionally deleterious, and where numbers decline with increasing load. Our analysis shows that migration can have qualitatively different effects, depending on the total mutation target and fitness effects of deleterious variants. In particular, we find that populations exhibit a genetic Allee effect across a wide range of parameter combinations, when variants are partially recessive, cycling between low-load (large-population) and high-load (sink) states. Increased migration reduces load in the sink state (by increasing heterozygosity) but further inflates load in the large-population state (by hindering purging). We identify various critical parameter thresholds at which one or other stable state collapses, and discuss how these thresholds are influenced by the genetic versus demographic effects of migration. Our analysis is based on a 'semi-deterministic' analysis, which accounts for genetic drift but neglects demographic stochasticity. We also compare against simulations which account for both demographic stochasticity and drift. Our results clarify the importance of gene flow as a key determinant of extinction risk in peripheral populations, even in the absence of ecological gradients. This article is part of the theme issue 'Species' ranges in the face of changing environments (part I)'.

This paper considers how polygenic local adaptation and reproductive isolation between hybridizing populations is influenced by linkage disequilibria (LD) between loci, in scenarios where both gene flow and genetic drift counteract selection. It shows that the combined effects of multi-locus LD and genetic drift on allele frequencies at selected loci and on heterozygosity at neutral loci are predicted accurately by incorporating (deterministic) effective migration rates into the diffusion approximation (for selected loci) and into the structured coalescent (for neutral loci). Theoretical approximations are tested against individual-based simulations and used to investigate conditions for the maintenance of local adaptation on an island subject to one-way migration from a differently adapted mainland, and in an infinite-island population with two habitats under divergent selection. The analysis clarifies the conditions under which LD between sets of locally deleterious alleles allows these to be collectively eliminated despite drift, causing sharper and (under certain conditions) shifted migration thresholds for loss of adaptation. Local adaptation also has counter-intuitive effects on neutral (relative) divergence: FST is highest for a pair of subpopulations belonging to the same (rare) habitat, despite the lack of reproductive isolation between them.

This article analyzes the conditions for local adaptation in a metapopulation with infinitely many islands under a model of hard selection, where population size depends on local fitness. Each island belongs to one of two distinct ecological niches or habitats. Fitness is influenced by an additive trait which is under habitat-dependent directional selection. Our analysis is based on the diffusion approximation and accounts for both genetic drift and demographic stochasticity. By neglecting linkage disequilibria, it yields the joint distribution of allele frequencies and population size on each island. We find that under hard selection, the conditions for local adaptation in a rare habitat are more restrictive for more polygenic traits: even moderate migration load per locus at very many loci is sufficient for population sizes to decline. This further reduces the efficacy of selection at individual loci due to increased drift and because smaller populations are more prone to swamping due to migration, causing a positive feedback between increasing maladaptation and declining population sizes. Our analysis also highlights the importance of demographic stochasticity, which exacerbates the decline in numbers of maladapted populations, leading to population collapse in the rare habitat at significantly lower migration than predicted by deterministic arguments.

Interspecific crossing experiments have shown that sex chromosomes play a major role in reproductive isolation between many pairs of species. However, their ability to act as reproductive barriers, which hamper interspecific genetic exchange, has rarely been evaluated quantitatively compared to Autosomes. This genome-wide limitation of gene flow is essential for understanding the complete separation of species, and thus speciation. Here, we develop a mainland-island model of secondary contact between hybridizing species of an XY (or ZW) sexual system. We obtain theoretical predictions for the frequency of introgressed alleles, and the strength of the barrier to neutral gene flow for the two types of chromosomes carrying multiple interspecific barrier loci. Theoretical predictions are obtained for scenarios where introgressed alleles are rare. We show that the same analytical expressions apply for sex chromosomes and autosomes, but with different sex-averaged effective parameters. The specific features of sex chromosomes (hemizygosity and absence of recombination in the heterogametic sex) lead to reduced levels of introgression on the X (or Z) compared to autosomes. This effect can be enhanced by certain types of sex-biased forces, but it remains overall small (except when alleles causing incompatibilities are recessive). We discuss these predictions in the light of empirical data comprising model-based tests of introgression and cline surveys in various biological systems.

This paper analyzes how partial selfing in a large source population influences its ability to colonize a new habitat via the introduction of a few founder individuals. Founders experience inbreeding depression due to partially recessive deleterious alleles as well as maladaptation to the new environment due to selection on a large number of additive loci. I first introduce a simplified version of the Inbreeding History Model (Kelly, 2007) in order to characterize mutation-selection balance in a large, partially selfing source population under selection involving multiple non-identical loci. I then use individual-based simulations to study the eco-evolutionary dynamics of founders establishing in the new habitat under a model of hard selection. The study explores how selfing rate shapes establishment probabilities of founders via effects on both inbreeding depression and adaptability to the new environment, and also distinguishes the effects of selfing on the initial fitness of founders from its effects on the long-term adaptive response of the populations they found. A high rate of (but not complete) selfing is found to aid establishment over a wide range of parameters, even in the absence of mate limitation. The sensitivity of the results to assumptions about the nature of polygenic selection are discussed.

Adaptive introgression is common in nature and can be driven by selection acting on multiple, linked genes. We explore the effects of polygenic selection on introgression under the infinitesimal model with linkage. This model assumes that the introgressing block has an effectively infinite number of loci, each with an infinitesimal effect on the trait under selection. The block is assumed to introgress under directional selection within a native population that is genetically homogeneous. We use individual-based simulations and a branching process framework to compute various statistics of the introgressing block, and explore how these depend on parameters such as the map length and initial trait value associated with the introgressing block, the genetic variability along the block, and the strength of selection. Our results show that the introgression dynamics of a block under infinitesimal selection are qualitatively different from the dynamics of neutral introgression. We also find that, in the long run, surviving descendant blocks are likely to have intermediate lengths, and clarify how their length is shaped by the interplay between linkage and infinitesimal selection. Our results suggest that it may be difficult to distinguish the long-term introgression of a block of genome with a single, strongly selected, locus from the introgression of a block with multiple, tightly linked and weakly selected loci.

We study how a block of genome with a large number of weakly selected loci introgresses under directional selection into a genetically homogeneous population. We derive exact expressions for the expected rate of growth of any fragment of the introduced block during the initial phase of introgression, and show that the growth rate of a single-locus variant is largely insensitive to its own additive effect, but depends instead on the combined effect of all loci within a characteristic linkage scale. The expected growth rate of a fragment is highly correlated with its long-term introgression probability in populations of moderate size, and can hence identify variants that are likely to introgress across replicate populations. We clarify how the introgression probability of an individual variant is determined by the interplay between hitchhiking with relatively large fragments during the early phase of introgression, and selection on fine scale variation within these, which at longer times results in differential introgression probabilities for beneficial and deleterious loci within successful fragments. By simulating individuals, we also investigate how introgression probabilities at individual loci depend on the variance of fitness effects, the net fitness of the introduced block, and the size of the recipient population, and how this shapes the net advance under selection. Our work suggests that even highly replicable substitutions may be associated with a range of selective effects, which makes it challenging to fine map the causal loci that underlie polygenic adaptation.

Assortative mating is an important driver of speciation in populations with gene flow and is predicted to evolve under certain conditions in few-locus models. However, the evolution of assortment is less understood for mating based on quantitative traits, which are often characterized by high genetic variability and extensive linkage disequilibrium between trait loci. We explore this scenario for a two-deme model with migration, by considering a single polygenic trait subject to divergent viability selection across demes, as well as assortative mating and sexual selection within demes, and investigate how trait divergence is shaped by various evolutionary forces. Our analysis reveals the existence of sharp thresholds of assortment strength, at which divergence increases dramatically. We also study the evolution of assortment via invasion of modifiers of mate discrimination and show that the ES assortment strength has an intermediate value under a range of migration-selection parameters, even in diverged populations, due to subtle effects which depend sensitively on the extent of phenotypic variation within these populations. The evolutionary dynamics of the polygenic trait is studied using the hypergeometric and infinitesimal models. We further investigate the sensitivity of our results to the assumptions of the hypergeometric model, using individual-based simulations.

A central issue in cell biology is the physico-chemical basis of organelle biogenesis in intracellular trafficking pathways, its most impressive manifestation being the biogenesis of Golgi cisternae. At a basic level, such morphologically and chemically distinct compartments should arise from an interplay between the molecular transport and chemical maturation. Here, we formulate analytically tractable, minimalist models, that incorporate this interplay between transport and chemical progression in physical space, and explore the conditions for de novo biogenesis of distinct cisternae. We propose new quantitative measures that can discriminate between the various models of transport in a qualitative manner- this includes measures of the dynamics in steady state and the dynamical response to perturbations of the kind amenable to live-cell imaging.

A central issue in cell biology is the physico-chemical basis of organelle biogenesis in intracellular trafficking pathways, its most impressive manifestation being the biogenesis of Golgi cisternae. At a basic level, such morphologically and chemically distinct compartments should arise from an interplay between the molecular transport and chemical maturation. Here, we formulate analytically tractable, minimalist models, that incorporate this interplay between transport and chemical progression in physical space, and explore the conditions for de novo biogenesis of distinct cisternae. We propose new quantitative measures that can discriminate between the various models of transport in a qualitative manner-this includes measures of the dynamics in steady state and the dynamical response to perturbations of the kind amenable to live-cell imaging.

We study analytically giant fluctuations and temporal intermittency in a stochastic one-dimensional model with diffusion and aggregation of masses in the bulk, along with influx of single particles and outflux of aggregates at the boundaries. We calculate various static and dynamical properties of the total mass in the system for both biased and unbiased movement of particles and different boundary conditions. These calculations show that (i) in the unbiased case, the total mass has a non-Gaussian distribution and shows giant fluctuations which scale as system size (ii) in all the cases, the system shows strong intermittency in time, which is manifested in the anomalous scaling of the dynamical structure functions of the total mass. The results are derived by taking a continuum limit in space and agree well with numerical simulations performed on the discrete lattice. The analytic results obtained here are typical of the full phase of a more general model with fragmentation, which was studied earlier using numerical simulations.

We study real space condensation in aggregation-fragmentation models where the total mass is not conserved, as in phenomena such as cloud formation and intracellular trafficking. We study the scaling properties of the system with influx and outflux of mass at the boundaries using numerical simulations, supplemented by analytical results in the absence of fragmentation. The system is found to undergo a phase transition to an unusual condensate phase, characterized by strong intermittency and giant fluctuations of the total mass. A related phase transition also occurs for biased movement of large masses, but with some crucial differences which we highlight.