LINKAGE DISEQUILIBRIUM AND THE INFINITESIMAL LIMIT

Under the classical Fisher-Bulmer infinitesimal model of quantitative genetics, the within family distribution for an additive trait with no environmental component is Gaussian with mean at the mid-parent value and a variance which is the same for all families. When an additive t rait is determined by unlinked loci, the Fisher-Bulmer model can arise in the limit as the number of loci contributing to variation in the t rait increases. However, a counterexample is presented where the Fishe r-Bulmer model fails to arise in the infinite locus limit because ther e is too much linkage disequilibrium. An example is also presented whe re a degenerate form of the Fisher-Bulmer model arises. Under what con ditions does the Fisher-Bulmer model arise in the infinite locus limit ? It follows from the central limit theorem that the within-family dis tribution is Gaussian. But, under what conditions is the within-family distribution the same for almost all families in the population? An a lternative population genetic derivation of the Fisher-Bulmer model is presented for a population at linkage equilibrium. This approach is t hen extended to allow many patterns of linkage disequilibrium. Diallel ic models are used to illustrate the type of linkage disequilibrium al lowed. The results on the limiting behaviour of population genetic mod els with many unlinked loci can be regarded as special cases of a more general limiting property of sequences of random variables. A possibl e application of this more general result to models of cultural inheri tance is suggested. (C) 1997 Academic Press.