Damien Pitman | Searcharxiv

arXiv2007

Splitting pairs and the number of clusters generated by random pair incompatibilities

We consider a random fitness landscape on the space of haploid diallelic genotypes with n genetic loci, where each genotype is considered either inviable or viable depending on whether or not there are any incompatibilities among its allele pairs. We suppose that each allele pair in the set of all possible allele pairs on the n loci is independently incompatible with probability p=c/(2n). We examine the connectivity of the viable genotypes under single locus mutations and show that, for 0 1, there are no viable genotypes with probability converging to one. The genotype space is equivalent to the n-dimensional hypercube and the viable genotypes are solutions to a random 2-SAT problem, so the same result holds for the connectivity of solutions in the hypercube to a random 2-SAT problem.

math.PR↗

arXiv2006

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes. We consider three classes of fitness models. The first is a continuous phenotype space in which individuals are characterized by a large number of continuously varying traits such as size, weight, color, or concentrations of gene products which directly affect fitness. The second is a simple model that explicitly describes genotype-to-phenotype and phenotype-to-fitness maps allowing for neutrality at both phenotype and fitness levels and resulting in a fitness landscape with tunable correlation length. The third is a class of models in which particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have reduced (or zero) fitness. This class of models can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation. We also demonstrate that the discrete NK model shares some signature properties of models with high correlations. Throughout the paper, our focus is on the percolation threshold, on the number, size and structure of connected clusters, and on the number of viable genotypes.

q-bio.PE↗