Population evolution on a multiplicative single-peak fitness landscape.

A theory for evolution of either gene sequences or 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 Wk = (1-s)k, where s determines the strength of selection. In the limit L-->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. 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. Selection is less effective on small populations, so <k> increases as N decreases. Our simulations also show that the mean overlap between gene sequences separated by a time of t generations is of the form Q(t) = Q infinity + (Q0-Q infinity)exp(-2ut), which means that the rate of evolution within the spherical shell is independent of the selection strength. We give a simplified model which can be solved exactly for which Q(t) has precisely this form. We then consider the limit L-->infinity keeping U = uL constant. We suppose that each mutation may be favourable with probability p, or unfavourable with probability 1-p. We show that for p less than a critical value pc, the population decreases in fitness for all values of U, whereas for pc < p < 1/2, the population increases in fitness for small U and decreases in fitness for large U. In this case there is an optimum non-zero value of U at which the fitness increases most rapidly, and natural selection will favour species with non-zero mutation rates.