Selection originating from protein stability/foldability: Relationships between protein folding free energy, sequence ensemble, and fitness

Assuming that mutation and fixation processes are reversible Markov processes, we prove that the equilibrium ensemble of sequences obeys a Boltzmann distribution with $\exp (4N_{e}m(1-1\mathrm{/}(2N)))$, 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 $\exp(-\Delta G_{ND}/k_B T_s)$orby$\exp(- G_{N}/k_B T_s)$ if an amino acid composition is kept constant, where $\mathrm{\Delta }{G}_{ND}\equiv G_N-G_D$, $G_N$ and $G_D$ are the native and denatured free energies, and $T_s$ is selective temperature. Thus, $4N_{e}m(1-1\mathrm{/}(2N))$, $-\mathrm{\Delta }{\psi }_{ND}$, and $-\mathrm{\Delta }{G}_{ND}\mathrm{/}{k}_B{T}_s$ must be equivalent to each other. Based on the analysis of the changes ($\Delta \psi_N$)of$\psi_N$duetosinglenucleotidenonsynonymoussubstitutions,$T_s$,andthenglasstransitiontemperature$T_g$,and$\Delta G_{ND}$ are estimated with reasonable values for 14 protein domains. In addition, approximating the probability density function (PDF) of $\mathrm{\Delta }{\psi }_N$ by a log-normal distribution, PDFs of $\mathrm{\Delta }{\psi }_N$ and $K_a\mathrm{/}{K}_s$, 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 $K_a\mathrm{/}{K}_s$. 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.