Evolution of sparsity and modularity in a model of protein allostery

The sequence of a protein is not only constrained by its physical and biochemical properties under current selection, but also by features of its past evolutionary history. Understanding the extent and the form that these evolutionary constraints may take is important to interpret the information in protein sequences. To study this problem, we introduce a simple but physical model of protein evolution where selection targets allostery, the functional coupling of distal sites on protein surfaces. This model shows how the geometrical organization of couplings between amino acids within a protein structure can depend crucially on its evolutionary history. In particular, two scenarios are found to generate a spatial concentration of functional constraints: high mutation rates and fluctuating selective pressures. This second scenario offers a plausible explanation for the high tolerance of natural proteins to mutations and for the spatial organization of their least tolerant amino acids, as revealed by sequence analysis and mutagenesis experiments. It also implies a faculty to adapt to new selective pressures that is consistent with observations. The model illustrates how several independent functional modules may emerge within the same protein structure, depending on the nature of past environmental fluctuations. Our model thus relates the evolutionary history of proteins to the geometry of their functional constraints, with implications for decoding and engineering protein sequences.