Multilocus models of selection and genetic drift in subdivided populations

Many species are distributed in space with local subpopulations connected by migration. If environmental conditions vary in space, such species experience spatially heterogeneous selection which often acts on multiple genes. Local adaptation of a subpopulation to its environment may be counteracted by maladaptive gene flow from other subpopulations that are genetically different. Strong gene flow or small population size can lead to loss of variation and to genetic homogeneity. Lack of genetic diversity may hamper future adaptation to changing environmental conditions and endanger population survival. Of particular interest are quantitative insights into the dependence of the degree of local adaptation and the degree of genetic divergence among subpopulations on the pattern and rate of migration, the form and intensity of selection, the genetic architecture of the traits under selection, and on population size and random genetic drift. From a genomics or data-oriented point of view, it is important to quantify the combined consequences of selection and gene flow on neutral (marker) sites linked to the selected loci. Using mathematical models, we shall explore the consequences of genetic architecture on the maintenance of genetic variation, on the evolution of local adaptation, and on the advance of genetic divergence (an important factor in speciation). In particular, we will examine the role of linkage, of the relative magnitude of locus effects, and of epistatic interactions between loci. Of equal interest, and intimately related, is the study of the evolution of genetic architecture caused by divergent selection. We shall investigate the evolution of recombination, of chromosomal inversions, and of genetic incompatibilities between subpopulations. A proper mathematical treatment is very challenging because it requires the analysis of genuine multilocus models, both in a deterministic and a stochastic context. Mathematical analyses will be complemented by extensive computer simulations. One major line of research will focus on stochastic models that describe evolution of a finite population inhabiting discrete demes. For the scenarios outlined above, we will analyze the invasion and sojourn properties of locally beneficial mutations in a genomic context and the concomitant signatures induced at linked neutral sites. The corresponding tools are provided by the theory of branching processes, Markov chains, and diffusion processes. A second major line of research will focus on (infinitely large) populations distributed continuously in space. Their evolution is modeled by systems of semilinear parabolic differential equations. We strive to establish general existence, convergence, and perturbation results for multilocus clines, i.e., of non-constant stationary solutions that describe the distribution of genotype frequencies in space. We shall study the invasion of locally advantageous alleles at linked loci and the consequences of linkage for the existence and properties of clines.

We developed and explored mathematical models that advance our understanding of the evolution of local adaptation and of genetic differentiation in biological species inhabiting a heterogeneous environment. Most species, animals and plants, are distributed in space and exchange migrants among local subpopulations. If environmental conditions vary geographically, such species experience spatially heterogeneous selection, which often acts on multiple genes. Local adaptation of a subpopulation to its environment may be counteracted by maladaptive gene flow from other subpopulations that are genetically different. Strong gene flow or small population size can lead to loss of variation and to genetic homogeneity. Lack of genetic diversity may hamper future adaptation to changing environmental conditions and endanger population survival. By contrast, increasing local adaptation and divergence between populations may eventually lead to speciation through the evolution of reproductive barriers. Biologists are striving for a quantitative understanding of the dependence of the degree of local adaptation and of genetic divergence on the patterns and rates of migration, the form and intensity of natural and sexual selection, the genetic architecture of the traits under selection, and on population size and random genetic drift. From a genomics or data-oriented point of view, it is important to quantify the combined consequences of selection and gene flow on genetic markers linked to the selected loci. Such a quantitative understanding requires the analysis of mathematical models of the processes of interest. This was the main goal of this project. We developed models to explore the consequences of genetic architecture on the evolution of local adaptation and on the emergence and maintenance of genetic differentiation. In particular, we examined the roles of genetic linkage between loci, of the migrational patterns, and of sexual selection. A proper mathematical treatment is very challenging because of the complexity of the models, which are formulated in terms of systems of recursive equations, differential equations, or stochastic processes. The mathematical analyses were complemented by extensive computer simulations. Our results provide a better understanding of the ecological and evolutionary processes leading to or preventing local adaptation and speciation. Two of the main achievements are the following. (i) We showed how well established measures of neutral genetic diversity change qualitatively and quantitatively along chromosomes if local adaptation is counteracted by migration. (ii) We showed that, in contrast to widespread opinion, sexual selection does in general not build barriers to gene flow between populations. Instead, it may weaken or even destroy them, thus degrading incipient speciation.