Smoothly Time-Varying Continuous Time Markov Chains in Phylogenetics

2026-06-24

AI Summary

This study addresses the challenge that molecular evolutionary rates vary across timescales by introducing the “spline clock” model, which for the first time incorporates smoothly time-varying rates into phylogenetic inference. Built upon a non-homogeneous continuous-time Markov chain, the method represents the log-rate function using cubic B-spline basis expansions, computes branch lengths via Gauss–Legendre numerical integration, and enforces smoothness through a Gaussian Markov random field prior. Simulation experiments demonstrate that this approach recovers true time-varying rates more accurately and yields tighter credible intervals than existing molecular clock models. Analyses of foamy virus evolution and SARS-CoV-2 spread in Europe both reveal significant time-dependent rate variation, underscoring the model’s practical utility.

Abstract

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.

Problem

• time-varying evolutionary rates
• molecular clock
• phylogenetics
• sequence substitution
• evolutionary history

Innovation

• inhomogeneous continuous-time Markov chains
• spline clock model
• time-varying evolutionary rates
• Gaussian Markov random field
• phylogenetics