Population evolution in a single peak fitness landscape how high are the clouds?

Abstract

A theory for evolution of molecular sequences must take into account that a population consists of a finite number of individuals with related sequences. Such a population will not behave in the deterministic way expected for an infinite population, nor will it behave as in adaptive walk models, where the whole of the population is represented by a single sequence. Here we study a model for evolution of population in a fitness landscape with a single fitness peak. This landscape is simple enough for finite size population effects to be studied in detail. Each of the N individuals in the population is represented by a sequence of L genes which may either be advantageous or disadvantageous. The fitness of an individual with k disadvantageous genes is w_k=(1−s)^k, where s determines the strength of selection. In the limit L tends to infinity the model reduces to the problem of Muller's Ratchet: the population moves away from the fitness peak at a constant rate due to the accumulation of disadvantageous mutations. For finite length sequences, a population placed initially at the fitness peak will evolve away from the peak until a balance is reached between mutation and selection. From then on the population will wander through a spherical shell in sequence space at a constant mean Hamming distance <k> from the optimum sequence. This has been likened to the idea of a cloud layer hanging below the mountain peak. We give an approximate theory for the way <k> depends on N, L, s, and the mutation rate u. This is found to agree well with numerical simulation.