  The amino acid sequences of proteins provide rich information for inferring
distant phylogenetic relationships and for predicting protein functions.
Estimating the rate matrix of residue substitutions from amino acid sequences
is also important because the rate matrix can be used to develop scoring
matrices for sequence alignment. Here we use a continuous time Markov process
to model the substitution rates of residues and develop a Bayesian Markov chain
Monte Carlo method for rate estimation. We validate our method using simulated
artificial protein sequences. Because different local regions such as binding
surfaces and the protein interior core experience different selection pressures
due to functional or stability constraints, we use our method to estimate the
substitution rates of local regions. Our results show that the substitution
rates are very different for residues in the buried core and residues on the
solvent exposed surfaces. In addition, the rest of the proteins on the binding
surfaces also have very different substitution rates from residues. Based on
these findings, we further develop a method for protein function prediction by
surface matching using scoring matrices derived from estimated substitution
rates for residues located on the binding surfaces. We show with examples that
our method is effective in identifying functionally related proteins that have
overall low sequence identity, a task known to be very challenging.
