Scaling and Fractal Behaviour Underlying Meiotic Recombination

D. Waxman and N. Stoletski

Biosystems 99: 42-49 (2010)

Centre for the Study of Evolution, School of Life Sciences, University of Sussex, Brighton BN1 9QG, Sussex UK

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 a 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(a^2/[2a-1])). We show that this proportion connects recombination with a fractal geometry of dimension log(2a-1)/log(a). 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.