Assuming that mutation and fixation processes are reversible Markov processes, we prove that the equilibrium ensemble of sequences obeys a Boltzmann distribution with \exp(4N_e m(1 - 1/(2N))), where m is a Malthusian fitness and N_e and N are the effective and actual population sizes. On the other hand, the probability distribution of sequences with maximum entropy that satisfies a given amino acid composition at each site and a given pairwise amino acid frequency at each site pair is a Boltzmann distribution with \exp(-\psi_N), where \psi_N is represented as the sum of one body and pairwise potentials. A protein folding theory indicates that homologous sequences obey a canonical ensemble characterized by \exp(-\Delta G_{ND}/k_B T_s) or by \exp(- G_{N}/k_B T_s) if an amino acid composition is kept constant, where \Delta G_{ND} \equiv G_N - G_D, G_N and G_D are the native and denatured free energies, and T_s is selective temperature. Thus, 4N_e m (1 - 1/(2N)), -\Delta , and -\Delta } G_{ND}/k_B T_s , and -\Delta \psi_{ND must be equivalent to each other. Based on the analysis of the changes (\Delta \psi_N) of \psi_N due to single nucleotide nonsynonymous substitutions, T_s, and then glass transition temperature T_g, and \Delta G_{ND} are estimated with reasonable values for 14 protein domains. In addition, approximating the probability density function (PDF) of \Delta \psi_N by a log-normal distribution, PDFs of \Delta \psi_N and K_a/K_s, which is the ratio of nonsynonymous to synonymous substitution rate per site, in all and in fixed mutants are estimated. It is confirmed that T_s negatively correlates with the average of K_a/K_s. Stabilizing mutations are significantly fixed by positive selection, and balance with destabilizing mutations fixed by random drift. Contrary to the neutral theory, the proportion of neutral selection is not large .
Assuming that mutation and fixation processes are reversible Markov processes, we prove that the equilibrium ensemble of sequences obeys a Boltzmann distribution with \exp(4N_e m(1 - 1/(2N))), where m is Malthusian fitness and N_e and N are effective and actual population sizes. On the other hand, the probability distribution of sequences with maximum entropy that satisfies a given amino acid composition at each site and a given pairwise amino acid frequency at each site pair is a Boltzmann distribution with \exp(-\psi_N), where \psi_N is represented as the sum of one body and pairwise potentials. A protein folding theory indicates that homologous sequences obey a canonical ensemble characterized by \exp(-\Delta G_{ND}/k_B T_s) or by \exp(- G_{N}/k_B T_s) if an amino acid composition is kept constant, where \Delta G_{ND} \equiv G_N - G_D, G_N and G_D are the native and denatured free energies, and T_s is selective temperature. Thus, 4N_e m (1 - 1/(2N)), -\Delta \psi_{ND, and -\Delta } G_{ND}/k_B T_s must be equivalent to each other. Based on the analysis of the changes (\Delta \psi_N) of \psi_N due to single nucleotide nonsynonymous substitutions, T_s, and then glass transition temperature T_g, and \Delta G_{ND} are estimated with reasonable values for 14 protein domains. In addition, approximating the probability density function (PDF) of \Delta \psi_N by a log-normal distribution, PDFs of \Delta \psi_N and K_a/K_s, which is the ratio of nonsynonymous to synonymous substitution rate per site, in all and in fixed mutants are estimated. It is confirmed that T_s negatively correlates with the mean of K_a/K_s. Stabilizing mutations are significantly fixed by positive selection, and balance with destabilizing mutations fixed by random drift. Supporting the nearly neutral theory, neutral selection is not significant .
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