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Abstract Phylogenetic and discrete-trait evolutionary inference depend heavily on an appropriate characterization of the underlying character substitution process. In this paper, we present random-effects substitution models that extend common continuous-time Markov chain models into a richer class of processes capable of capturing a wider variety of substitution dynamics. As these random-effects substitution models often require many more parameters than their usual counterparts, inference can be both statistically and computationally challenging. Thus, we also propose an efficient approach to compute an approximation to the gradient of the data likelihood with respect to all unknown substitution model parameters. We demonstrate that this approximate gradient enables scaling of sampling-based inference, namely Bayesian inference via Hamiltonian Monte Carlo, under random-effects substitution models across large trees and state-spaces. Applied to a dataset of 583 SARS-CoV-2 sequences, an HKY model with random-effects shows strong signals of nonreversibility in the substitution process, and posterior predictive model checks clearly show that it is a more adequate model than a reversible model. When analyzing the pattern of phylogeographic spread of 1441 influenza A virus (H3N2) sequences between 14 regions, a random-effects phylogeographic substitution model infers that air travel volume adequately predicts almost all dispersal rates. A random-effects state-dependent substitution model reveals no evidence for an effect of arboreality on the swimming mode in the tree frog subfamily Hylinae. Simulations reveal that random-effects substitution models can accommodate both negligible and radical departures from the underlying base substitution model. We show that our gradient-based inference approach is over an order of magnitude more time efficient than conventional approaches. 

ABSTRACT Comparative phylogenetic studies of adaptation are uncommon in biomechanics and physiology. Such studies require data collection from many species, a challenge when this is experimentally intensive. Moreover, researchers struggle to employ the most biologically appropriate phylogenetic tools for identifying adaptive evolution. Here, we detail an established but greatly underutilized phylogenetic comparative framework – the Ornstein–Uhlenbeck process – that explicitly models long-term adaptation. We discuss challenges in implementing and interpreting the model, and we outline potential solutions. We demonstrate use of the model through studying the evolution of thermal physiology in treefrogs. Frogs of the family Hylidae have twice colonized the temperate zone from the tropics, and such colonization likely involved a fundamental change in physiology due to colder and more seasonal temperatures. However, which traits changed to allow colonization is unclear. We measured cold tolerance and characterized thermal performance curves in jumping for 12 species of treefrogs distributed from the Neotropics to temperate North America. We then conducted phylogenetic comparative analyses to examine how tolerances and performance curves evolved and to test whether that evolution was adaptive. We found that tolerance to low temperatures increased with the transition to the temperate zone. In contrast, jumping well at colder temperatures was unrelated to biogeography and thus did not adapt during dispersal. Overall, our study shows how comparative phylogenetic methods can be leveraged in biomechanics and physiology to test the evolutionary drivers of variation among species.