The Mathematics of Darwin’s Theory of Evolution: 1859 and 150 Years Later

Abstract

A mathematical formulation of Darwin’s theory of evolutionary optimization through variation and selection is derived in terms of conventional ODEs that can be interpreted as chemical kinetics of evolution. Variation in form of mutation and recombination operates on genotypes being DNA or RNA sequences, whereas phenotypes, which are represented by organisms or molecular structures, are the target of selection. The impact of recombination on optimization is briefly sketched. Differential equations modelling selection in populations with correct replication and mutation are derived from the molecular mechanisms of polynucleotide replication. The analysis of these ODEs reveals restrictions of the optimization principle caused by mutation. Error propagation over generations sets a limit to mutation rates in evolution, which manifests itself in the form of a phase transition-like phenomenon characterized as error threshold. Conditions on fitness landscapes for the occurrence of error thresholds derived from numerical investigations are presented: Smooth fitness landscapes show no error thresholds but gradual transitions, sufficiently steep landscapes and rugged landscapes sustain error thresholds. Sharp transitions are also found with realistic landscapes combining ruggedness and neutrality. Lethal mutants may lead to extinction of populations and set another upper limit to mutation rates in form of an extinction threshold through lethal mutagenesis.