Linkage disequilibrium at steady state determined by random genetic drift and recurrent mutation.

order to understand the genetic architecture of Mendelian populations, we '2ave to know, in addition to gene frequencies, the degree of linkage disequilibrium between loci, namely, the extent to which combinations of genes at linked loci deviate from randomness. In panmictic populations, two factors are mainly responsible for the production of linkage disequilibrium. They are epistatic interaction in fitness and random sampling of gametes in reproduction. Although a number of papers have been published since the work of KIMURA (1956), treating the problem of epistasis and linkage in an infinite population, it was only last year that the first systematic treatment of linkage disequilibrium due to random drift was presented by HILL and ROBERTSON (1968). Using the method of moment-generating matrix, they obtained the variance of the linkage disequilibrium coefficient as a function of time and initial gametic frequencies for the case of no recombination, assuming that mutation and selection are absent. In addition, they studied numerically several cases of recombination by multiplying the moment-generating matrix and also by carrying out simulation experiments. They demonstrated that significant linkage disequilibrium may result from random drift under tight linkage and small population number. Their results were extended by OHTA and KIMURA (1969) who obtained, by the method of Kolmogorov backward equation, the formula for the variance of linkage disequilibrium in which recombination is incorporated. In natural populations, however, it is expected that random drift and recurrent mutation balance each other so that a steady state is reached with respect to linkage disequilibrium. SVED (1968) studied this problem assuming that all gene frequencies are held at 50% by strong overdominance while mutation is so rare as to be negligible. Presumably, his treatment is applicable to a certain transient state but not necessarily to the equilibrium state. In the present paper we intend to present a theoretical foundation for the treatment of linkage disequilibrium at steady state determined by random drift and mutation (or more generally, linear evolutionary pressure). The problem of linkage disequilibrium will become particularly important when we consider two neighboring nucleotide sites within a cistron for which the recombination fraction may be much smaller than the reciprocal of the population number. It will also