A semigroup representation and asymptotic behavior of certain statistics of the fisher-wright-moran coalescent

Abstract

We derive new results giving mathematical properties of functions of allele frequencies under the time-continuous Fisher-Wright-Moran model with mutations of the general Markov-chain form. The matrix R(t) (possibly infinite) of the joint distributions of the types of a pair of alleles sampled from the population at time t , satisfies a matrix differential equation of the form d R(t) /d t = [ Q * R(t) + R(t)Q ] − [1 / (2 N ] R(t) + [1/(2 N )] Π(t) where Q is the intensity matrix of the Markov chain, II(t) is its diagonalized probability distribution, and N is the effective population size. This is the Lyapunov differential equation, known in control theory. Investigation of behavior of its solutions leads to consideration of tensor products of transition (Markov) semigroups. Semigroup theory methods allow proofs of asymptotic results for the model, also in the cases when the population size does not stay constant. If population is composed of a number of disjoint subpopulations, the asymptotics depend on the growth rate of the population. Special cases of the model include stepwise mutation models with and without allele size constraints, and with directional bias of mutations. Allele state changes caused by recombinatorial misalignment and more complex sequence conversion patterns also can be incorporated in this model. The methodology developed can also be applied to model coevolution of disease and marker loci, of further use for linkage disequilibrium mapping of disease genes.