EFFECTS OF LINKAGE AND EPISTASIS ON THE EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. NUMERICAL SOLUTIONS

Abstract

(1) The effects of linkage and epistasis on the equilibrium frequencies of lethal genes were studied by numerical means, making use of an electronic computer. The changes in gamete and gene frequencies, linkage disequilibrium, and population fitness under selection were also examined. The selective values of A1A1B1B1, A1A2B1B1, A1A1B1B2, and A1A2B1B2 are denoted by a, b, c, and d respectively, those of the remaining genotypes being all 0, where A2 and B2 represent lethal genes.(2) If a=b=c=d, there are no stable equilibria for two lethal genes even if there is linkage disequilibrium. If a is smaller than b and c or both, there occur non-trivial equilibria for the two genes jointly. The equilibium frequencies of lethal genes increases the recombination value between the two loci decreases, unless ad-bc=0 and the original population is in linkage equilibrium. If ad-bc=0 and the original population is not in linkage equilibrium the equilibrium gene frequencies depend on the sign of the original linkage disequilibrium as well as the recombination value. It does not appear that the initial gene frequencies affect the equilibrium values.(3) Gene frequencies do not necessarily approach the equilibrium values straightly but in some situations the lethal gene frequencies first increase considerably and then begin to decrease, depending on the initial gamete frequencies. If the recombination value is large, the rate of change in gene frequencies is usually larger in early generations than in later generations, but if the recombination value is small, there occur situations where little changes of gene frequencies are observed in early generations.(4) The equilibrium value of linkage disequilibrium has the same sign as that of ad-bc, if ad-bc≠0. If ad-bc=0, the sign of the equilibrium value conforms to the sign of the original linkage disequilibrium. The equilibrium values of linkage disequilibrium increases as linkage becomes tight for a given set of selective values. The forms of changes in gamete frequencies and linkage disequilibrium are again affected considerably by the recombination value and initial gamete frequencies.(5) In some situations the population fitness decreases appreciably. This decrease is usually associated with the breakdown of linkage disequilibrium, but not always so.(6) It has been indicated that the frequency of lethal chromosomes and the allelic rate of lethal genes are differentially affected by linkage disequilibrium. The effect of linkage on the genetic load manifested under inbreeding is also discussed. Further, the applicability of the results obtained to the artificial selection for quantitative characters is pointed out.