  This chapter shows how toroidal diffusions are convenient methodological
tools for modelling protein evolution in a probabilistic framework. The chapter
addresses the construction of ergodic diffusions with stationary distributions
equal to well-known directional distributions, which can be regarded as
toroidal analogues of the Ornstein-Uhlenbeck process. The important challenges
that arise in the estimation of the diffusion parameters require the
consideration of tractable approximate likelihoods and, among the several
approaches introduced, the one yielding a specific approximation to the
transition density of the wrapped normal process is shown to give the best
empirical performance on average. This provides the methodological building
block for Evolutionary Torus Dynamic Bayesian Network (ETDBN), a hidden Markov
model for protein evolution that emits a wrapped normal process and two
continuous-time Markov chains per hidden state. The chapter describes the main
features of ETDBN, which allows for both "smooth" conformational changes and
"catastrophic" conformational jumps, and several empirical benchmarks. The
insights into the relationship between sequence and structure evolution that
ETDBN provides are illustrated in a case study.
