  Despite the importance of a thermodynamically stable structure with a
conserved fold for protein function, almost all evolutionary models neglect
site-site correlations that arise from physical interactions between
neighboring amino acid sites. This is mainly due to the difficulty in
formulating a computationally tractable model since rate matrices can no longer
be used. Here we introduce a general framework, based on factor graphs, for
constructing probabilistic models of protein evolution with site
interdependence. Conveniently, efficient approximate inference algorithms, like
Belief Propagation, can be used to calculate likelihoods for these models. We
fit an amino acid substitution model of this type that accounts for both
solvent accessibility and site-site correlations. Comparisons of the new model
with rate matrix models and a model accounting only for solvent accessibility
demonstrate that it better fits the sequence data. We also examine evolution
within a family of homohexameric enzymes and find that site-site correlations
between most contacting subunits contribute to a higher likelihood. In
addition, we show that the new substitution model has a similar mathematical
form to the one introduced in (Rodrigue et al. 2005), although with different
parameter interpretations and values. We also perform a statistical analysis of
the effects of amino acids at neighboring sites on substitution probabilities
and find a significant perturbation of most probabilities, further supporting
the significant role of site-site interactions in protein evolution and
motivating the development of new evolutionary models like the one described
here. Finally, we discuss possible extensions and applications of the new
substitution models.
