  The common understanding of protein evolution has been that neutral or
slightly deleterious mutations are fixed by random drift, and evolutionary rate
is determined primarily by the proportion of neutral mutations. However, recent
studies have revealed that highly expressed genes evolve slowly because of
fitness costs due to misfolded proteins. Here we study selection maintaining
protein stability.
  Protein fitness is taken to be $s = \kappa \exp(\beta\Delta G) (1 -
\exp(\beta\Delta\Delta G))$, where $s$ and $\Delta\Delta G$ are selective
advantage and stability change of a mutant protein, $\Delta G$ is the folding
free energy of the wild-type protein, and $\kappa$ represents protein abundance
and indispensability. The distribution of $\Delta\Delta G$ is approximated to
be a bi-Gaussian function, which represents structurally slightly- or
highly-constrained sites. Also, the mean of the distribution is negatively
proportional to $\Delta G$.
  The evolution of this gene has an equilibrium ($\Delta G_e$) of protein
stability, the range of which is consistent with experimental values. The
probability distribution of $K_a/K_s$, 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 $\Delta G_e > -2.5$ 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 $\langle K_a/K_s \rangle$ is less than
1. Slow evolutionary rates can be caused by high protein
abundance/indispensability, which produces positive shifts of $\Delta\Delta G$
through decreasing $\Delta G_e$, and by strong structural constraints, which
directly make $\Delta\Delta G$ more positive.
