Mathematics of Genetic Diversity (CBMS-NSF Regional by J. F. C. Kingman

By J. F. C. Kingman

Attracts jointly a few mathematical rules which are worthwhile in inhabitants genetics, focusing on a couple of facets that are either biologically appropriate and mathematically attention-grabbing.

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Example text

However, it should be stressed that this argument is a very crude and approximate one. Starting from rather different assumptions, Li [2] is led to different conclusions on the basis of extensive computations. 2. The Wright-Fisher model. 2 can be modified to allow for selective differences. Thus consider a diploid population of fixed size N, and a particular locus at which the possible alleles are At (i G S). In a particular generation Gt, list the genes present in the population, in random order, as We shall assume here that selection acts at the genie level so that the fitness of the genotype AtAj is of product form wtWj for positive constants wt (i e S).

Convergence of probability measures onV is weak convergence relative to this topology (Billingsley [1]). Then if a sequence of probability measures pr converges to another measure p, the corresponding partition structures converge pointwise: for all n ^ 1, a G wn. Conversely, suppose that (P ( n r) ; n ^ 1) is for each r a partition structure, and that the limit 46 CHAPTER 3 exists for all n ^ 1 and a G wn. Then (Pn) is a partition structure, and the corresponding representing measures converge in the sense described above.

Thus for instance x^n) = j/n, where 7 is the largest integer with Uj ^ 1. Each xr(n) is likewise a function of the partition a G wn, and is a random element of V, and has a distribution pn which can be computed in terms of Pn. It turns out that pn —» p as n —»• °°, in_the usual sense of "weak" convergence (relative to the product topology on V). ) Applying this analysis to the genetical context, we of course read "colours" as alleles (or as equivalence classes of alleles which can be discriminated experimentally).

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