The consequences of gene flow for local adaptation and differentiation: A two-locus two-deme model

We consider a population subdivided into two demes connected by migration in which selection acts in opposite direction. We explore the effects of recombination and migration on the maintenance of multilocus polymorphism, on local adaptation, and on differentiation by employing a deterministic model with genic selection on two linked diallelic loci (i.e., no dominance or epistasis). For the following cases, we characterize explicitly the possible equilibrium configurations: weak, strong, highly asymmetric, and super-symmetric migration, no or weak recombination, and independent or strongly recombining loci. For independent loci (linkage equilibrium) and for completely linked loci, we derive the possible bifurcation patterns as functions of the total migration rate, assuming all other parameters are fixed but arbitrary. For these and other cases, we determine analytically the maximum migration rate below which a stable fully polymorphic equilibrium exists. In this case, differentiation and local adaptation are maintained. Their degree is quantified by a new multilocus version of \Fst and by the migration load, respectively. In addition, we investigate the invasion conditions of locally beneficial mutants and show that linkage to a locus that is already in migration-selection balance facilitates invasion. Hence, loci of much smaller effect can invade than predicted by one-locus theory if linkage is sufficiently tight. We study how this minimum amount of linkage admitting invasion depends on the migration pattern. This suggests the emergence of clusters of locally beneficial mutations, which may form `genomic islands of divergence'. Finally, the influence of linkage and two-way migration on the effective migration rate at a linked neutral locus is explored. Numerical work complements our analytical results

Muller's ratchet and mutational meltdowns

We extend our earlier work on the role of deleterious mutations in the extinction of obligately asexual populations. First, we develop analytical models for mutation accumulation that obviate the need for time-consuming computer simulations in certain ranges of the parameter space. When the number of mutations entering the population each generation is fairly high, the number of mutations per individual and the mean time to extinction can be predicted using classical approaches in quantitative genetics. However, when the mutation rate is very low, a fixation-probability approach is quite effective. Second, we show that an intermediate selection coefficient (s) minimizes the time to extinction. The critical value of s can be quite low, and we discuss the evolutionary implications of this, showing that increased sensitivity to mutation and loss of capacity for DNA repair can be selectively advantageous in asexual organisms. Finally, we consider the consequences of the mutational meltdown for the extinction of mitochondrial lineages in sexual species

The mutational meltdown in asexual populations

Loss of fitness due to the accumulation of deleterious mutations appears to be inevitable in small, obligately asexual populations, as these are incapable of reconstituting highly fit genotypes by recombination or back mutation. The cumulative buildup of such mutations is expected to lead to an eventual reduction in population size, and this facilitates the chance accumulation of future mutations. This synergistic interaction between population size reduction and mutation accumulation leads to an extinction process known as the mutational meltdown, and provides a powerful explanation for the rarity of obligate asexuality. We give an overview of the theory of the mutational meltdown, showing how the process depends on the demographic properties of a population, the properties of mutations, and the relationship between fitness and number of mutations incurred

Polygenic dynamics underlying the response of quantitative traits to directional selection

We study the response of a quantitative trait to exponential directional selection in a finite haploid population at the genetic and the phenotypic level. We assume an infinite sites model, in which the number of new mutations per generation in the population follows a Poisson distribution (with mean $\Theta$) and each mutation occurs at a new, previously monomorphic site. Mutation effects are beneficial and drawn from a distribution. Sites are unlinked and contribute additively to the trait. Assuming that selection is stronger than random genetic drift, we model the initial phase of the dynamics by a supercritical Galton-Watson process. This enables us to obtain time-dependent results. We show that the copy-number distribution of the mutant in generation n, conditioned on non-extinction until n, is described accurately by the deterministic increase from an initial distribution with mean 1. This distribution is related to the absolutely continuous part $W^+$ of the random variable, typically denoted $W$, that characterizes the stochasticity accumulating during the mutant's sweep. On this basis, we derive explicitly the (approximate) time dependence of the mutant frequency distribution, of the expected mean and variance of the trait and of the expected number of segregating sites. Unexpectedly, we obtain highly accurate approximations for all times, even for the quasi-stationary phase where we refine classical results. In addition, we find that $\Theta$ is the main determinant of the pattern of adaptation at the genetic level, i.e., whether the initial allele-frequency dynamics are best described by sweep-like patterns at few loci or small allele-frequency shifts at many. The selection strength determines primarily the rate of adaptation. The accuracy of our results is tested by comprehensive simulations in a Wright-Fisher framework

Phenotypic Mutation Rates and the Abundance of Abnormal Proteins in Yeast

Phenotypic mutations are errors that occur during protein synthesis. These errors lead to amino acid substitutions that give rise to abnormal proteins. Experiments suggest that such errors are quite common. We present a model to study the effect of phenotypic mutation rates on the amount of abnormal proteins in a cell. In our model, genes are regulated to synthesize a certain number of functional proteins. During this process, depending on the phenotypic mutation rate, abnormal proteins are generated. We use data on protein length and abundance in Saccharomyces cerevisiae to parametrize our model. We calculate that for small phenotypic mutation rates most abnormal proteins originate from highly expressed genes that are on average nearly twice as large as the average yeast protein. For phenotypic mutation rates much above 5 × 10−4, the error-free synthesis of large proteins is nearly impossible and lowly expressed, very large proteins contribute more and more to the amount of abnormal proteins in a cell. This fact leads to a steep increase of the amount of abnormal proteins for phenotypic mutation rates above 5 × 10−4. Simulations show that this property leads to an upper limit for the phenotypic mutation rate of approximately 2 × 10−3 even if the costs for abnormal proteins are extremely low. We also consider the adaptation of individual proteins. Individual genes/proteins can decrease their phenotypic mutation rate by using preferred codons or by increasing their robustness against amino acid substitutions. We discuss the similarities and differences between the two mechanisms and show that they can only slow down but not prevent the rapid increase of the amount of abnormal proteins. Our work allows us to estimate the phenotypic mutation rate based on data on the fraction of abnormal proteins. For S. cerevisiae, we predict that the value for the phenotypic mutation rate is between 2 × 10−4 and 6 × 10−4