The regulation of several processes inside and outside the cell depends on the action of a particular class of enzymes, called allosteric. In allosteric macromolecules, binding a ligand at one site affects the binding activity at a distal functional site, providing a reliable tool to regulate the corresponding function. The physical mechanisms underpinning allostery and its long-range communication are not yet fully understood, despite a great number of advances were made possible by significant works spanning 60 years, in between biology, bioinformatics and physics.

In physics terms, proteins can be viewed as amorphous materials that however underwent billions of years of evolution to be functional as observed today. The framework introduced in this dissertation allows to explore how the structural organisation of an allosteric system is constrained by the function that it has evolved to perform. It allows a classification of allosteric architectures and suggests a physical explanation behind the emergence of such long-range allosteric coupling. Furthermore, it is also apt to build in silico a large amount of allosteric architectures that share the same evolutionary history. The constraints imprinted by evolution on sequences that share a common ancestor motivate the exploration of inference methods that try to predict the fitness of a protein solely from the knowledge of sequences.

The strategy used to build this framework is to resort to a coarse-grained model of a protein, on the line of elastic network models resulted successful in the description of the large-scale dynamics of proteins. Ideas on how to pursue these research directions further are discussed throughout the chapters.

Firstly, we introduce an in-silico model for the evolution of allosteric behaviour in discrete lattices of harmonic springs. The in-silico evolution is performed for two different allosteric tasks: one optimising for the transmission of strain between the allosteric and active site, while the other maximising the cooperative binding energy between the two. To optimise the transmission of strain, the network develops a lever that amplifies the response at the active site. In such a way, our model proposes a novel allosteric architecture, potentially in use in proteins as well. Cooperative architectures show, among others, hinge and shear motions and rationalise the observation of a low energy mode that describes conformational changes in proteins. Indeed, to achieve proper function, the mode is predicted to get softer as the size of the system increases. This prediction is tested by collecting a database of 34 high resolution structures of allosteric proteins and is proven valid even when elastic nonlinearities are introduced.

Secondly, the sequences generated with the in-silico model serve to benchmark existing methods that infer co-evolutionary couplings between amino acids, proven to be successful in predicting local structural constraints, but with unclear performance in the presence of global allosteric constraints. These models do predict local features reflecting structure, but fail in the prediction of long-range functional dependencies and are not able to generate synthetic sequences that function as native ones. Thus, the exploration of new directions is needed.