Shared Information between Residues Is Sufficient to Detect Pairwise Epistasis in a Protein

In a comment [1] on our manuscript “Strong Selection Significantly Increases Epistatic Interactions in the Long-Term Evolution of a Protein” [2], Dr. Crona challenges our assertion that shared entropy (that is, information) between two residues implies epistasis between those residues by constructing an explicit example of three loci (say A, B, and C), in which A and B are epistatically linked (leading to shared entropy between A and B) and A and C are also epistatically dependent (leading to shared entropy between A and C) so that loci B and C are correlated (share entropy). She goes on to assert that (as per her examples) even though there will be correlations (and thus shared entropy) between the residues at loci B and C, there is no pairwise epistasis between loci B and C, contradicting our assertion in [1] that shared entropy implies epistasis. The disagreement is based on two different interpretations of the meaning of pairwise epistasis, and the comment gives us an opportunity to discuss those. We do not disagree that epistasis refers to mutational effects that are conditional on the states of other alleles. In our paper, we are in particular interested in pairwise epistasis—that is, how the fitness effects of mutations at two loci depend on each other. Of course, the dependence between those two loci could depend on the state of many other alleles in the genome. The different interpretations of epistasis hinge upon whether a quantitative assessment of the epistasis between two loci should be conditional on the state of other loci in the genome, or whether instead we should consider the state of these loci averaged over what their state would be in a population at mutation–selection balance. Crona’s example helps us illustrate that distinction. A fitness landscape constructed in the supplementary information of [1] (example 1) has three loci: A, B, and C. The values given are (note that in [1], the log-transformed relative fitnesses are given instead):