Substitution processes in molecular evolution. III. Deleterious alleles.

The substitution processes for various models of deleterious alleles are examined using computer simulations and mathematical analyses. Most of the work focuses on the house-of-cards model, which is a popular model of deleterious allele evolution. The rate of substitution is shown to be a concave function of the strength of selection as measured by alpha = 2N sigma, where N is the population size and sigma is the standard deviation of fitness. For alpha < 1, the house-of-cards model is essentially a neutral model; for alpha > 4, the model ceases to evolve. The stagnation for large alpha may be understood by appealing to the theory of records. The house-of-cards model evolves to a state where the vast majority of all mutations are deleterious, but precisely one-half of those mutations that fix are deleterious (the other half are advantageous). Thus, the model is not a model of exclusively deleterious evolution as is frequently claimed. It is argued that there are no biologically reasonable models of molecular evolution where the vast majority of all substitutions are deleterious. Other models examined include the exponential and gamma shift models, the Hartl-Dykhuizen-Dean (HDD) model, and the optimum model. Of all those examined, only the optimum and HDD models appear to be reasonable candidates for silent evolution. None of the models are viewed as good candidates for protein evolution, as none are both biologically reasonable and exhibit the variability in substitutions commonly observed in protein sequence data.