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. Combining this finding with the knowledge of protein folding, we derive a correspondence between protein fitness and folding free energy, i.e., 4N_e m (1 - 1/(2N)) and -\Delta G_{ND}/k_B T_s  , 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 the effective temperatureof natural selection. In the preceding manuscript, we analyzed dimensionless free energy changes, \Delta \psi_N (= \Delta G_N /  k_B T_s ), due to single nucleotide mutations, for which the total interaction \psi_N of a sequence was estimated from a multiple sequence alignment . Based on this analyis, we study the equilibrium state of protein stability in a monoclonal approximation, in which protein evolution proceeds with single amino acid substitutions fixed at a time in a population. The ensemble of proteins attains the equilibrium, \psi_N= \psi_N ^{eq, where the average of \Delta \psi}  _{ N } over fixed mutants is equal to zero. Approximating the probability density function (PDF) of \Delta \psi_N by a log-normal distribution, the PDF of \Delta \psi _{N (\simeq \Delta\Delta \psi_{ND}) in fixed mutants is estimated. \psi_N ^{eq} for 14 protein domains agree well with \psi_N averaged over homologous sequences. The PDFs of }  K_a/K_s, which is the ratio of nonsynonymous to synonymous substitution rate per site, at equilibrium confirm that T_s negatively correlates with the amino acid substitution rate. Contrary to the neutral theory,  neutral selection is not significant even in fixed mutants .
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 , 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 G_{ND k_B T_s , and -\Delta \psi_{ND . Based on , where the average of \Delta \psi} 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 (\simeq \Delta\Delta \psi_{ND}) in fixed mutants is estimated. \psi_N ^{eq} for 14 protein domains agree well with \psi_N averaged over homologous sequences. The PDFs of } _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 .
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