Protein evolution can be viewed as a repeated mutation-fixation process. At each step, one amino acid is randomly mutated. The mutant will eventually be either discarded or fixed, replacing the parent. The fixation probability varies from site to site, thus different sites evolve at different rates. This variation of rates among sites is due to thermodynamic and/or functional constraints.

Almost all biophysical models of protein evolution developed so far consider only selection for stability, disregarding functional constraints, which were thought to affect only a handful of residues: the active site and its immediate neighbours. However, a recent study showed that site-specific evolutionary rates increase rather smoothly with increasing distance to the protein’s active residues. Such long-range rate-distance dependence cannot be explained with current stability-based models and is at odds with the localized fast exponential decrease of coupling strength one would expect on physics grounds.

To understand whether and why functional constraints have long-range effects, we need new models of protein evolution that go beyond selection for stability and consider protein activity explicitly. Here, I will describe a model of protein evolution that considers explicitly both stability and activity constraints and I will discuss its predictions. The stability-activity model predicts that fitness cost is localized (short-range) yet rate variation is delocalized (long-range); short-range fitness effects are consistent with long-range rate effects. Yet, such long-range coupling is not universal but range varies among proteins. Such range variation among proteins does not depend on intrinsic protein properties but on external functional selection pressure (e.g. the role of an enzyme in the metabolic network).