Variation of an inherited trait across a population cannot be explained by additive contributions of relevant genes, due to epigenetic effects and biochemical interactions (epistasis). Detecting epistasis in genomic data still represents a significant challenge that requires a better understanding of epistasis from the mechanistic point of view. Using a standard Wright-Fisher model of bi-allelic asexual population, we study how compensatory epistasis affects the process of adaptation. The main result is a universal relationship between four haplotype frequencies of a single site pair in a genome, which depends only on the epistasis strength of the pair defined regarding Darwinian fitness. We demonstrate the existence, at any time point, of a quasi-equilibrium between epistasis and disorder (entropy) caused by random genetic drift and mutation. We verify the accuracy of these analytic results by Monte-Carlo simulation over a broad range of parameters, including the topology of the interacting network. Thus, epistasis assists the evolutionary transit through evolutionary hurdles leaving marks at the level of haplotype disequilibrium. The method allows determining selection coefficient for each site and the epistasis strength of each pair from a sequence set. The resulting ability to detect clusters of deleterious mutation close to full compensation is essential for biomedical applications. These findings help to understand the role of epistasis in multiple compensatory mutations in viral resistance to antivirals and immune response.
An intriguing fact long defying explanation is the observation of a universal exponential distribution of beneficial mutations in fitness effect for different microorganisms. To explain this effect, we use a population model including mutation, directional selection, linkage, and genetic drift. The multiple-mutation regime of adaptation at large population sizes (traveling wave regime) is considered. We demonstrate analytically and by simulation that, regardless of the inherent distribution of mutation fitness effect across genomic sites, an exponential distribution of fitness effects emerges in the long term. This result follows from the exponential statistics of the frequency of the less-fit alleles, f, that we predict to evolve, in the long term, for both polymorphic and monomorphic sites. We map the logarithmic slope of the distribution onto the previously derived fixation probability and demonstrate that it increases linearly in time. Our results demonstrate a striking difference between the distribution of fitness effects observed experimentally for naturally occurring mutations, and the "inherent" distribution obtained in a directed-mutagenesis experiment, which can have any shape depending on the organism. Based on these results, we develop a new method to measure the fitness effect of mutations for each variable residue using DNA sequences sampled from adapting populations. This new method is not sensitive to linkage effects and does not require the one-site model assumptions.
10An intriguing fact long defying explanation is the observation of a universal exponential distribution of 11 beneficial mutations in fitness effect for different microorganisms. Here we use a general and 12 straightforward analytic model to demonstrate that, regardless of the inherent distribution of mutation 13 fitness effect across genomic sites, an observed exponential distribution of fitness effects emerges 14 naturally, as a consequence of the evolutionary process. Using this result, we develop a technique to 15 measure the mutation fitness effects for specific genomic sites from a single-time sequence set and apply 16 it to influenza A H1N1 hemagglutinin protein. Our results demonstrate the difference between the 17 distribution of fitness effects experimentally observed for naturally occurring mutations and the inherent 18 distribution obtained in directed-mutagenesis experiments. The technique will enable researchers to 19 measure fitness effects of mutations across the genome from a single DNA sample, which is important 20 for predicting the evolution of a population. 21 22 Recently, selection coefficients across the sites of the hemagglutinin gene of human influenza 48 A/H3N2 were estimated by fitting the deterministic one-locus model and its approximate 49 extension for two-loci (Illingworth and Mustonen 2012). The authors fit the model to time-series 50 data on allele frequencies of hemagglutinin (HA) gene of human influenza A H3N2. (Keightley and 51 Eyre-Walker 2007) proposed a method of DFE estimation in mutation-selection-drift equilibrium 52based on the assumption that DFE has the shape of the gamma distribution. They estimate 53 parameters of gamma distribution from maximization of the likelihood under the assumption that 54 the derived sites are binomially distributed. Thus, the two modeling papers used strong 55 assumptions about the dynamics of the system. 56
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