Free energy perturbations in enzyme kinetic models reveal cryptic epistasis

Epistasis-the context-dependence of mutational effects-is a key driver of protein evolution, influencing adaptive pathways and functional diversity. While specific epistasis arises from direct physical interactions between mutations, non-specific epistasis emerges when a non-linear mapping links a protein's biophysical properties to its function. Enzyme kinetic parameters map directly to free energies, enabling researchers to connect epistasis in these parameters to an enzyme's structural features. Here, we show that this approach is incorrect: enzyme catalytic parameters like kcat and KM inherently exhibit non-specific epistasis due to the multi-state nature of the catalytic cycle. Using enzyme catalytic cycle models, parameterized by free energies of ground and transition states, we simulated 1000 "mutations" or perturbations to the sub-state free energies within the kinetic ensemble. We then combined these mutations, creating one million double mutants with strictly additive free energy effects. Despite the absence of explicit mutational interactions, we observed substantial epistasis in catalytic parameters; its prevalence and complexity increasing with the number of kinetic states in the mechanism. We derived analytical conditions for the emergence of this form of epistasis in a simple kinetic model, demonstrating that non-specific epistasis depends on the relative values of key microscopic rate constants. Finally, we validated our framework by reanalyzing kinetic data for double mutants in Bacillus cereus β-lactamase I and found that reported specific epistasis in catalytic efficiency was substantially stronger than previously inferred, altering mechanistic interpretations. Our results identify an intrinsic, previously unknown source of epistasis that can distort both the magnitude and sign of mutational effects in enzyme kinetics. We provide theoretical and computational tools for recognizing and correcting for this form of non-specific epistasis, enabling accurate mechanistic inference from kinetic data and improving our understanding of the links between epistasis, structure-function relationships, enzyme evolution, and protein design.