  Recently described stochastic models of protein evolution have demonstrated
that the inclusion of structural information in addition to amino acid
sequences leads to a more reliable estimation of evolutionary parameters. We
present a generative, evolutionary model of protein structure and sequence that
is valid on a local length scale. The model concerns the local dependencies
between sequence and structure evolution in a pair of homologous proteins. The
evolutionary trajectory between the two structures in the protein pair is
treated as a random walk in dihedral angle space, which is modelled using a
novel angular diffusion process on the two-dimensional torus. Coupling sequence
and structure evolution in our model allows for modelling both "smooth"
conformational changes and "catastrophic" conformational jumps, conditioned on
the amino acid changes. The model has interpretable parameters and is
comparatively more realistic than previous stochastic models, providing new
insights into the relationship between sequence and structure evolution. For
example, using the trained model we were able to identify an apparent
sequence-structure evolutionary motif present in a large number of homologous
protein pairs. The generative nature of our model enables us to evaluate its
validity and its ability to simulate aspects of protein evolution conditioned
on an amino acid sequence, a related amino acid sequence, a related structure
or any combination thereof.
