Selection maintaining protein stability at equilibrium

The common understanding of protein evolution has been that neutral mutations are fixed by random drift, and a proportion of neutral mutations depending on the strength of structural and functional constraints primarily determines evolutionary rate. Recently it was indicated that fitness costs due to misfolded proteins are a determinant of evolutionary rate and selection originating in protein stability is a driving force of protein evolution. Here we examine protein evolution under the selection maintaining protein stability. Protein fitness is a generic form of fitness costs due to misfolded proteins; s=κexp(ΔG/kT)(1-exp(ΔΔG/kT)), where s and ΔΔG are selective advantage and stability change of a mutant protein, ΔG is the folding free energy of the wildtype protein, and κ is a parameter representing protein abundance and indispensability. The distribution of ΔΔG is approximated to be a bi-Gaussian distribution, which represents structurally slightly- or highly-constrained sites. Also, the mean of the distribution is negatively proportional to ΔG. The evolution of this gene has an equilibrium point (ΔGe) of protein stability, the range of which is consistent with observed values in the ProTherm database. The probability distribution of Ka/Ks, the ratio of nonsynonymous to synonymous substitution rate per site, over fixed mutants in the vicinity of the equilibrium shows that nearly neutral selection is predominant only in low-abundant, non-essential proteins of ΔGe>-2.5 kcal/mol. In the other proteins, positive selection on stabilizing mutations is significant to maintain protein stability at equilibrium as well as random drift on slightly negative mutations, although the average 〈Ka/Ks〉 is less than 1. Slow evolutionary rates can be caused by both high protein abundance/indispensability and large effective population size, which produces positive shifts of ΔΔG through decreasing ΔGe, and strong structural constraints, which directly make ΔΔG more positive. Protein abundance/indispensability more affect evolutionary rate for less constrained proteins, and structural constraint for less abundant, less essential proteins. The effect of protein indispensability on evolutionary rate may be hidden by the variation of protein abundance and detected only in low-abundant proteins. Also, protein stability (-ΔGe/kT) and 〈Ka/Ks〉 are predicted to decrease as growth temperature increases.