POPULATION-GENETICS OF MULTIGENE FAMILY WITH SPECIAL REFERENCE TO DECREASE OF GENETIC CORRELATION WITH DISTANCE BETWEEN GENE MEMBERS ON A CHROMOSOME

A mathematical method is developed which enables us to treat exactly the process of coincidental evolution under mutation, unequal intrachromosomal crossing-over as well as ordinary crossing-over between homologous chromosomes in a finite population of the effective size N. It makes use of finite difference equations involving two quantities denoted by f(i) and Φ(i), in which f(i) is the identity coefficient of two gene members that are i steps apart on the same chromosome and Φ(i) is that of two members i steps apart on two homologous chromosomes. When the number of genes (n) per family is large, the finite difference equations can be approximated by ordinary second-order differential equations which can then be solved analytically. Results obtained by the present method are compared with the corresponding results previously obtained by one of us (T.O.) using conventional diffusion models of gene frequency changes in population genetics. It is shown that the previous results obtained by T.O. regarding second-order statistics are essentially valid, and they give good approximations particularly when Nβ is small, where β is the rate of ordinary interchromosomal crossing-over within the multigene family.