Scaling and fractal behaviour underlying meiotic recombination

In this paper we investigate some of the mathematical properties of meiotic recombination. Working within the framework of a genetic model with n loci, where alpha alleles are possible at each locus, we find that the proportion of all possible diploid parental genotypes that can produce a particular haploid gamete is exp[-n log(alpha(2)/[2alpha-1])]. We show that this proportion connects recombination with a fractal geometry of dimension log(2alpha-1)/log(alpha). The fractal dimension of a geometric object manifests itself when it is measured at increasingly smaller length scales. Decreasing the length scale of a geometric object is found to be directly analogous, in a genetics problem, to specifying a multilocus haplotype at a larger number of loci, and it is here that the fractal dimension reveals itself.