Equations of Evolutionary Dynamics in High Dimensions

We study quasi-species and closely related evolutionary dynamics like the replicator-mutator equation in high dimensions. In particular, we show that under certain conditions the fitness of almost all quasi-species becomes independent of mutational probabilities and the initial frequency distributions of the sequences in high dimensional sequence spaces. This result is the consequence of the concentration of measure on a high dimensional hypersphere and its extension to Lipschitz functions known as the Levy’s Lemma. Therefore, evolutionary dynamics almost always yields the same value for fitness of the quasi-species, independent of the mutational process and initial conditions, and is quite robust to mutational changes and fluctuations in initial conditions. Our results naturally extend to any Lipschitz function whose input parameters are the frequencies of individual constituents of the quasi-species. This suggests that the functional capabilities of high dimensional quasi-species are robust to fluctuations in the mutational probabilities and initial conditions. We discuss the consequences of our study for the replicator-mutator equation.