Approximation of Indel Evolution by Differential Calculus of Finite State Automata

Abstract

We introduce a systematic method of approximating finite-time transition probabilities for continuous-time insertion-deletion models on sequences. The method uses automata theory to describe the action of an infinitesimal evolutionary generator on a probability distribution over alignments, where both the generator and the alignment distribution can be represented by Pair Hidden Markov Models (Pair HMMs). In general, combining HMMs in this way induces a multiplication of their state spaces; to control this, we introduce a coarse-graining operation to keep the state space at a constant size. This leads naturally to ordinary differential equations for the evolution of the transition probabilities of the approximating Pair HMM. The TKF model emerges as an exact solution to these equations for the special case of single-residue indels. For the general case, the equations can be solved by numerical integration. Using simulated data we show that the resulting distribution over alignments, when compared to previous approximations, is a better fit over a broader range of parameters. We also propose a related approach to develop differential equations for sufficient statistics to estimate the underlying instantaneous indel rates by Expectation-Maximization. Our code and data are available at https://github.com/ihh/trajectory-likelihood.