The Interaction of Selection and Linkage. I. General Considerations; Heterotic Models.

HILE the theory of the genetic changes in a population due to selection is quite well understood for single loci, our theory for multiple-gene characters is in a rudimentary stage. Most of the formulations for multiple-gene characters are simply extensions of single-locus models, extensions which ignore the problem of linkage. There are, however, a few papers in which the role of linkage has been investigated for more or less special cases of selection (KIMURA 1956; LEWONTIN and KOJIMA 1960; BODMER and PARSONS 1962). The results of these investigations were sufficient to show that even for relatively simple cases (two loci, simple symmetrical selective values) linkage might have profound effects on the course of natural selection and, pari passu, natural selection may have major effects on the distribution of coupling and repulsion linkage in a population. The results of the investigations of LEWONTIN and KOJIMA (1960) of the twolocus model can be summarized as follows: (1) If the fitnesses are additive between loci (no epistasis), linkage does not effect the final equilibrium state of the population. (2) If linkage is tighter than the value demanded by the magnitude of the epistasis (the greater the epistasis the greater the value) there may be permanent linkage disequilibrium and alteration of equilibrium gene frequencies. (3) The rate of genetic change with time is affected by the tightness of the linkage. (4) In some cases stable gene frequency equilibria are possible only if linkage is tight enough. Although these conclusions were based only on two-locus model and for selective values of a fairly restricted sort, they point clearly to the importance of taking linkage into account in understanding the changes of gene frequencies in populations. In fact, some experimental results (an example of which will be given below) can be understood only if the interaction of selection and linkage is taken into account, The equations describing the interaction between selection and linkage (see below) do not usually have general literal solutions. It is for this reason that the authors cited above have restricted themselves to relatively simple cases. In view of the interesting findings of those previous papers, however, it is worthwhile to explore the subject more intensively. To do so requires the numerical rather than general literal solutions to the equations, but such numerical solutions apply, obviously, only to the particular parameter values chosen. To make such a nu-