The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equation
s framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient.
We investigate the properties of a Wright-Fisher diffusion process started from frequency x at time 0 and conditioned to be at frequency y at time T. Such a process is called a bridge. Bridges arise naturally in the analysis of selection acting on st
anding variation and in the inference of selection from allele frequency time series. We establish a number of results about the distribution of neutral Wright-Fisher bridges and develop a novel rejection sampling scheme for bridges under selection that we use to study their behavior.
One of the outstanding challenges in comparative genomics is to interpret the evolutionary importance of regulatory variation between species. Rigorous molecular evolution-based methods to infer evidence for natural selection from expression data are
at a premium in the field, and to date, phylogenetic approaches have not been well-suited to address the question in the small sets of taxa profiled in standard surveys of gene expression. We have developed a strategy to infer evolutionary histories from expression profiles by analyzing suites of genes of common function. In a manner conceptually similar to molecular evolution models in which the evolutionary rates of DNA sequence at multiple loci follow a gamma distribution, we modeled expression of the genes of an emph{a priori}-defined pathway with rates drawn from an inverse gamma distribution. We then developed a fitting strategy to infer the parameters of this distribution from expression measurements, and to identify gene groups whose expression patterns were consistent with evolutionary constraint or rapid evolution in particular species. Simulations confirmed the power and accuracy of our inference method. As an experimental testbed for our approach, we generated and analyzed transcriptional profiles of four emph{Saccharomyces} yeasts. The results revealed pathways with signatures of constrained and accelerated regulatory evolution in individual yeasts and across the phylogeny, highlighting the prevalence of pathway-level expression change during the divergence of yeast species. We anticipate that our pathway-based phylogenetic approach will be of broad utility in the search to understand the evolutionary relevance of regulatory change.
We examine the distribution of heterozygous sites in nine European and nine Yoruban individuals whose genomic sequences were made publicly available by Complete Genomics. We show that it is possible to obtain detailed information about inbreeding whe
n a relatively small set of whole-genome sequences is available. Rather than focus on testing for deviations from Hardy-Weinberg genotype frequencies at each site, we analyze the entire distribution of heterozygotes conditioned on the number of copies of the derived (non-chimpanzee) allele. Using Levenes exact test, we reject Hardy-Weinberg in both populations. We generalized Levenes distribution to obtain the exact distribution of the number of heterozygous individuals given that every individual has the same inbreeding coefficient, F. We estimated F to be 0.0026 in Europeans and 0.0005 in Yorubans, but we could also reject the hypothesis that F was the same in each individual. We used a composite likelihood method to estimate F in each individual and within each chromosome. Variation in F across chromosomes within individuals was too large to be consistent with sampling effects alone. Furthermore, estimates of F for each chromosome in different populations were not correlated. Our results show how detailed comparisons of population genomic data can be made to theoretical predictions. The application of methods to the Complete Genomics data set shows that the extent of apparent inbreeding varies across chromosomes and across individuals, and estimates of inbreeding coefficients are subject to unexpected levels of variation which might be partly accounted for by selection.
