Toroidal diffusions and protein structure evolution

TL;DR

This paper introduces toroidal diffusions as a probabilistic tool for modeling protein evolution, develops an ergodic diffusion process with a wrapped normal stationary distribution, and applies it within a Bayesian network to analyze protein structure changes.

Contribution

It presents a novel toroidal diffusion process with a wrapped normal stationary distribution and integrates it into a Bayesian network for protein evolution analysis.

Findings

The wrapped normal approximation outperforms other methods in empirical tests.

ETDBN effectively models both smooth and abrupt conformational changes.

Case study demonstrates insights into sequence-structure evolution relationships.

Abstract

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.