Quantifying how constraints limit the diversity of viable routes to adaptation.

Convergent adaptation occurs at the genome scale when independently evolving lineages use the same genes to respond to similar selection pressures. These patterns of genetic repeatability provide insights into the factors that facilitate or constrain the diversity of genetic responses that contribute to adaptive evolution. A first step in studying such factors is to quantify the observed amount of repeatability relative to expectations under a null hypothesis. Here, we formulate a novel index to quantify the constraints driving the observed amount of repeated adaptation in pairwise contrasts based on the hypergeometric distribution, and then generalize this for simultaneous analysis of multiple lineages. This index is explicitly based on the probability of observing a given amount of repeatability by chance under a given null hypothesis and is readily compared among different species and types of trait. We also formulate an index to quantify the effective proportion of genes in the genome that have the potential to contribute to adaptation. As an example of how these indices can be used to draw inferences, we assess the amount of repeatability observed in existing datasets on adaptation to stress in yeast and climate in conifers. This approach provides a method to test a wide range of hypotheses about how different kinds of factors can facilitate or constrain the diversity of genetic responses observed during adaptive evolution.