Gain-loss-duplication models on a phylogeny: exact algorithms for computing the likelihood and its gradient

TL;DR

This paper introduces exact algorithms for efficiently computing the likelihood and its gradient in gene gain-loss-duplication models on phylogenies, facilitating probabilistic inference of gene content evolution.

Contribution

It presents a novel probabilistic framework with exact, quadratic-time algorithms for likelihood and posterior calculations in gene duplication models, including a linear-time gradient computation.

Findings

Exact likelihood computation in quadratic time.

Linear-time algorithm for likelihood gradient.

Framework based on dependent random variables with known distributions.

Abstract

Gene gain-loss-duplication models are commonly based on continuous-time birth-death processes. Employed in a phylogenetic context, such models have been increasingly popular in studies of gene content evolution across multiple genomes. While the applications are becoming more varied and demanding, bioinformatics methods for probabilistic inference on copy numbers (or integer-valued evolutionary characters, in general) are scarce. We describe a flexible probabilistic framework for phylogenetic gene-loss-duplication models. The framework is based on a novel elementary representation by dependent random variables with well-characterized conditional distributions: binomial, P\'olya (negative binomial), and Poisson. The corresponding graphical model yields exact numerical procedures for computing the likelihood and the posterior distribution of ancestral copy numbers. The resulting algorithms take quadratic time in the total number of copies. In addition, we show how the likelihood gradient can be computed by a linear-time algorithm.