Markov Chains with Absorbing States: A Genetic Example

Abstract

If a finite Markov chain (discrete time, discrete states) has a number of absorbing states, one of these will eventually be reached. In this paper are given theoretical formulae for the probability distribution, its generating function and moments of the time taken to first reach an absorbing state, and these formulae are applied to an example taken from genetics. While first passage time problems and their solutions are known for a wide variety of Markov chain processes (e.g., [2], [7], [4]), the theory seems not to have been used in population genetics. Suppose a genetic population consists of a constant number of individuals and the state of the population is defined by the numbers of the various genotypes existing at a given time. Then if mutation is absent, all individuals will eventually become of the same genotype because of random influences such as births, deaths, mating, selection, chromosome breakages and recombinations. The population behavior may in some circumstances be approximated by a Markov chain with absorbing states. In Section 2, two alternative approaches are given for the theoretical determination of absorption time properties, using well known techniques. In Section 3, the consequences of the theoretical results are investigated for a particular population model introduced by Moran [9], [10], and explicit expressions for the distribution of the gene fixation time are obtained in terms of Chebyshev's orthogonal polynomials. The derivation requires finding the pre- and post-eigenvectors of the matrix of transition probabilities, and an incidental by-product is the proof of certain identities for the orthogonal polynomials. The material presented in Section 2 and Section 3 is obtained by exact methods. In Section 4, the Fokker-Planck diffusion equation is used to obtain approximate results, and these are compared with those of the exact theory to ascertain the accuracy of the diffusion approximation.