  Naturally evolving proteins gradually accumulate mutations while continuing
to fold to thermodynamically stable native structures. This process of neutral
protein evolution is an important mode of genetic change, and forms the basis
for the molecular clock. Here we present a mathematical theory that predicts
the number of accumulated mutations, the index of dispersion, and the
distribution of stabilities in an evolving protein population from knowledge of
the stability effects (ddG values) for single mutations. Our theory
quantitatively describes how neutral evolution leads to marginally stable
proteins, and provides formulae for calculating how fluctuations in stability
cause an overdispersion of the molecular clock. It also shows that the
structural influences on the rate of sequence evolution that have been observed
in earlier simulations can be calculated using only the single-mutation ddG
values. We consider both the case when the product of the population size and
mutation rate is small and the case when this product is large, and show that
in the latter case proteins evolve excess mutational robustness that is
manifested by extra stability and increases the rate of sequence evolution. Our
basic method is to treat protein evolution as a Markov process constrained by a
minimal requirement for stable folding, enabling an evolutionary description of
the proteins solely in terms of the experimentally measureable ddG values. All
of our theoretical predictions are confirmed by simulations with model lattice
proteins. Our work provides a mathematical foundation for understanding how
protein biophysics helps shape the process of evolution.
