Stabilizing selection of protein function and distribution of selection coefficient among sites

In this study, I take a new approach to modeling the evolutionary constraint of protein sequence, introducing the stabilizing selection of protein function into the nearly-neutral theory. In other words, protein function under stabilizing selection generates the evolutionary conservation at the sequence level. With the help of random mutational effects of nucleotides on protein function, I have derived the distribution of selection coefficient among sites, called the S-distribution whose parameters have clear biological interpretations. Moreover, I have studied the inverse relationship between the evolutionary rate and the effective population size, showing that the number of molecular phenotypes of protein function, i.e., independent components in the fitness of the organism, may play a key role for the molecular clock under the nearly-neutral theory. These results are helpful for having a better understanding of the underlying evolutionary mechanism of protein sequences, as well as human disease-related mutations.