Diallelic multilocus model of neutral genes

The evolution of a completely linked diallelic multilocus system of neutral genes in a finite population is studied. A diffusion model incorporating random genetic drift and mutation is used. We neglect the recombination. To begin with, the spectral analysis of the Kolmogorov backward equation for this model is investigated. We apply this to two extreme situations when the number of sites approaches to infinity. One is a DeMoivre-Laplace type approximation and the other is a Poisson type approximation. The former is applied to the study of the simultaneous distribution and evolution of a large number of neutral genes. It is applicable to the distribution of a polygenic character controlled by clustered loci on a chromosome, and we show that it differs from the normal distribution on account of random genetic drift and linkage disequilibrium. The latter is applied to the distribution of the number of segregating sites in DNA nucleotide sequences, and the rate of evolution is obtained.