Testing for Independence between Evolutionary Processes.

Evolutionary events co-occurring along phylogenetic trees usually point to complex adaptive phenomena, sometimes implicating epistasis. While a number of methods have been developed to account for co-occurrence of events on the same internal or external branch of an evolutionary tree, there is a need to account for the larger diversity of possible relative positions of events in a tree. Here we propose a method to quantify to what extent two or more evolutionary events are associated on a phylogenetic tree. The method is applicable to any discrete character, like substitutions within a coding sequence or gains/losses of a biological function. Our method uses a general approach to statistically test for significant associations between events along the tree, which encompasses both events inseparable on the same branch, and events genealogically ordered on different branches. It assumes that the phylogeny and themapping of branches is known without errors. We address this problem from the statistical viewpoint by a linear algebra representation of the localization of the evolutionary events on the tree.We compute the full probability distribution of the number of paired events occurring in the same branch or in different branches of the tree, under a null model of independence where each type of event occurs at a constant rate uniformly inthephylogenetic tree. The strengths andweaknesses of themethodare assessed via simulations;we then apply the method to explore the loss of cell motility in intracellular pathogens.