Does Sex Induce a Phase Transition?

Abstract

We discovered a dynamic phase transition induced by sexual reproduction. The dynamics is a pure Darwinian rule with both fundamental ingredients to drive evolution: 1) random mutations and crossings which act in the sense of increasing the entropy (or diversity); and 2) selection which acts in the opposite sense by limiting the entropy explosion. Selection wins this competition if mutations performed at birth are few enough. By slowly increasing the average number m of mutations, however, the population suddenly undergoes a mutational degradation precisely at a transition point mc. Above this point, the "bad" alleles spread over the genetic pool of the population, overcoming the selection pressure. Individuals become selectively alike, and evolution stops. Only below this point, m < mc, evolutionary life is possible. The finite-size-scaling behaviour of this transition is exhibited for large enough "chromosome" lengths L. One important and surprising observation is the L-independence of the transition curves, for large L. They are also independent on the population size. Another is that mc is near unity, i.e. life cannot be stable with much more than one mutation per diploid genome, independent of the chromosome length, in agreement with reality. One possible consequence is that an eventual evolutionary jump towards larger L enabling the storage of more genetic information would demand an improved DNA copying machinery in order to keep the same total number of mutations per offspring.