Aneuploidy Can Be an Evolutionary Diversion on the Path to Adaptation.

Aneuploidy is common in eukaryotes, often leading to decreased fitness. However, evidence from fungi and human tumur cells suggests that specific aneuploidies can be beneficial under stressful conditions and facilitate adaptation. In a previous evolutionary experiment with yeast, populations evolving under heat stress became aneuploid, only to later revert to euploidy after beneficial mutations accumulated. It was therefore suggested that aneuploidy is a "stepping stone" on the path to adaptation. Here, we test this hypothesis. We use Bayesian inference to fit an evolutionary model with both aneuploidy and mutation to the experimental results. We then predict the genotype frequency dynamics during the experiment, demonstrating that most of the evolved euploid population likely did not descend from aneuploid cells, but rather from the euploid wild-type population. Our model shows how the beneficial mutation supply-the product of population size and beneficial mutation rate-determines the evolutionary dynamics: with low supply, much of the evolved population descends from aneuploid cells; but with high supply, beneficial mutations are generated fast enough to outcompete aneuploidy due to its inherent fitness cost. Our results suggest that despite its potential fitness benefits under stress, aneuploidy can be an evolutionary "diversion" rather than a "stepping stone": it can delay, rather than facilitate, the adaptation of the population, and cells that become aneuploid may leave less descendants compared to cells that remain diploid.

The effects of epistasis and linkage on the invasion of locally beneficial mutations and the evolution of genomic islands.

We study local adaptation of a peripheral population by investigating the fate of new mutations using a haploid two-locus two-allele continent-island migration model. We explore how linkage, epistasis, and maladaptive gene flow affect the invasion probability of weakly beneficial de-novo mutations that arise on the island at an arbitrary physical distance to a locus that already maintains a stable migration-selection polymorphism. By assuming a slightly supercritical branching process, we deduce explicit conditions on the parameters that permit a positive invasion probability and we derive approximations for it. They show how the invasion probability depends on the additive and epistatic effects of the mutant, on its linkage to the polymorphism, and on the migration rate. We use these approximations together with empirically motivated distributions of epistatic effects to analyze the influence of epistasis on the expected invasion probability if mutants are drawn randomly from such a distribution and occur at a random physical distance to the existing polymorphism. We find that the invasion probability generally increases as the epistasis parameter increases or the migration rate decreases, but not necessarily as the recombination rate decreases. Finally, we shed light on the size of emerging genomic islands of divergence by exploring the size of the chromosomal neighborhood of the already established polymorphism in which 50% or 90% of the successfully invading mutations become established. These 'window sizes' always decrease in a reverse sigmoidal way with stronger migration and typically increase with increasing epistatic effect.

Loss of genetic variation in the two-locus multiallelic haploid model.

In the evolutionary biology literature, it is generally assumed that for deterministic frequency-independent haploid selection models, no polymorphic equilibrium can be stable in the absence of variation-generating mechanisms such as mutation. However, mathematical analyses that corroborate this claim are scarce and almost always depend upon additional assumptions. Using ideas from game theory, we show that a monomorphism is a global attractor if one of its alleles dominates all other alleles at its locus. Further, we show that no isolated equilibrium exists, at which an unequal number of alleles from two loci is present. Under the assumption of convergence of trajectories to equilibrium points, we resolve the two-locus three-allele case for a fitness scheme formally equivalent to the classical symmetric viability model. We also provide an alternative proof for the two-locus two-allele case.

Evolutionary dynamics in the two-locus two-allele model with weak selection.

Two-locus two-allele models are among the most studied models in population genetics. The reason is that they are the simplest models to explore the role of epistasis for a variety of important evolutionary problems, including the maintenance of polymorphism and the evolution of genetic incompatibilities. Many specific types of models have been explored. However, due to the mathematical complexity arising from the fact that epistasis generates linkage disequilibrium, few general insights have emerged. Here, we study a simpler problem by assuming that linkage disequilibrium can be ignored. This is a valid approximation if selection is sufficiently weak relative to recombination. The goal of our paper is to characterize all possible equilibrium structures, or more precisely and general, all robust phase portraits or evolutionary flows arising from this weak-selection dynamics. For general fitness matrices, we have not fully accomplished this goal, because some cases remain undecided. However, for many specific classes of fitness schemes, including additive fitnesses, purely additive-by-additive epistasis, haploid selection, multilinear epistasis, marginal overdominance or underdominance, and the symmetric viability model, we obtain complete characterizations of the possible equilibrium structures and, in several cases, even of all possible phase portraits. A central point in our analysis is the inference of the number and stability of fully polymorphic equilibria from the boundary flow, i.e., from the dynamics at the four marginal single-locus subsystems. The key mathematical ingredient for this is index theory. The specific form of epistasis has both a big influence on the possible boundary flows as well as on the internal equilibrium structure admitted by a given boundary flow.