The coupon collector and the suppressor mutation: Estimating the number of compensatory mutations by maximum likelihood

Compensatory mutation occurs when a loss of fitness caused by a deleterious mutation is restored by its epistatic interaction with a second mutation at a different site in the genome. How many different compensatory mutations can act on a given deleterious mutation? Although this quantity is fundamentally important to understanding the evolutionary consequence of mutation and the genetic complexity of adaptation, it remains poorly understood. To determine the shape of the statistical distribution for the number of compensatory mutations per deleterious mutation, we have performed a maxim Lim-likelihood analysis of experimental data collected from the suppressor mutation literature. Suppressor mutations are used widely to assess protein interactions and are under certain conditions equivalent to compensatory mutations. By comparing the maximum likelihood of a variety of candidate distribution functions, we established that an L-shaped gamma distribution (alpha = 0.564, theta = 21.01) is the most successful at explaining the collected data. This distribution predicts an average of 11.8 compensatory mutations per deleterious mutation. Furthermore, the success of the L-shaped gamma distribution is robust to variation in mutation rates among sites. We have detected significant differences among viral, prokaryotic, and eukaryotic data subsets in the number of compensatory mutations and also in the proportion of compensatory mutations that are intragenic. This is the first attempt to characterize the overall diversity of compensatory mutations, identifying a consistent and accurate prior distribution of compensatory mutation diversity for theoretical evolutionary models.