Ensemble epistasis: thermodynamic origins of nonadditivity between mutations.

Epistasis-when mutations combine nonadditively-is a profoundly important aspect of biology. It is often difficult to understand its mechanistic origins. Here, we show that epistasis can arise from the thermodynamic ensemble, or the set of interchanging conformations a protein adopts. Ensemble epistasis occurs because mutations can have different effects on different conformations of the same protein, leading to nonadditive effects on its average, observable properties. Using a simple analytical model, we found that ensemble epistasis arises when two conditions are met: (1) a protein populates at least three conformations and (2) mutations have differential effects on at least two conformations. To explore the relative magnitude of ensemble epistasis, we performed a virtual deep-mutational scan of the allosteric Ca2+ signaling protein S100A4. We found that 47% of mutation pairs exhibited ensemble epistasis with a magnitude on the order of thermal fluctuations. We observed many forms of epistasis: magnitude, sign, and reciprocal sign epistasis. The same mutation pair could even exhibit different forms of epistasis under different environmental conditions. The ubiquity of thermodynamic ensembles in biology and the pervasiveness of ensemble epistasis in our dataset suggests that it may be a common mechanism of epistasis in proteins and other macromolecules.