  Neutral evolution is the simplest model of molecular evolution and thus it is
most amenable to a comprehensive theoretical investigation. In this paper, we
characterize the statistical properties of neutral evolution of proteins under
the requirement that the native state remains thermodynamically stable, and
compare them to the ones of Kimura's model of neutral evolution. Our study is
based on the Structurally Constrained Neutral (SCN) model which we recently
proposed. We show that, in the SCN model, the substitution rate decreases as
longer time intervals are considered, and fluctuates strongly from one branch
of the evolutionary tree to another, leading to a non-Poissonian statistics for
the substitution process. Such strong fluctuations are also due to the fact
that neutral substitution rates for individual residues are strongly correlated
for most residue pairs. Interestingly, structurally conserved residues,
characterized by a much below average substitution rate, are also much less
correlated to other residues and evolve in a much more regular way. Our results
could improve methods aimed at distinguishing between neutral and adaptive
substitutions as well as methods for computing the expected number of
substitutions occurred since the divergence of two protein sequences.
