Smoothly Time-Varying Continuous Time Markov Chains in Phylogenetics

The dependence of evolutionary rate estimates on the timeframe of sampling poses a fundamental challenge for reconstructing evolutionary histories from molecular sequence data, which is central to evolutionary biology and infectious disease research. We present a novel and flexible approach to accommodate time-varying evolutionary rates by modeling the sequence substitution process using inhomogeneous continuous-time Markov chains (ICTMCs) acting along the branches of the phylogeny, and parameterizing the log transformed rate as a smooth function of time using a cubic B-spline basis expansion. Following the parlance of phylogenetics that refers to rates of molecular substitutions as molecular clocks, we call this a spline clock model. Integrals of the rate function over all branches, required for likelihood evaluation, are approximated efficiently using Gauss-Legendre quadrature, and smoothness is enforced by assigning a Gaussian Markov random field prior to the spline coefficients. Through a simulation study, we demonstrate that the spline clock model recovers the true time-varying rates more accurately and with tighter credible intervals than competing clock models. We apply the spline clock model to examine the evolutionary rate of foamy virus and the rate of spatial diffusion of SARS-CoV-2 across Europe, recovering strong time-varying signal in both settings.