The conditional Markov chain in a genetic context

Abstract

A theory of the stochastic change in the frequency of a gene in finite populations conditional to its eventual fixation has been developed employing a Condiditional Markov Chain. The probability generating function of the distribution of time until fixation of a particular allele as well as the eigen-roots and eigenvectors of the conditional process with binomial transition probabilities have been studied. The theory has been applied to investigate the effect of linkage on the mean and standard deviation of time until fixation of a gamete in populations practising self-fertilization. It has been found that linkage decreases or increases the average and standard deviation of time to fixation of a coupled gamete according as the initial population consists of a coupling or repulsion heterozygote respectively.