Architecture of Allosteric Materials and Principles for Optimal Cooperativity

Allostery, a long-range mediated interaction, is a key feature in the functionality of several proteins, resulting to be crucial for life. Understanding the nature of the information transmitted and the architectures optimising such transmission remains a challenge. We recently introduced a numerical scheme to evolve functional materials that can accomplish a specific task. Architectures optimised to be cooperative, which propagate efficiently energy, strongly differ from previously investigated materials optimised to propagate strain. Although we observe a large diversity of functioning cooperative architectures —including shear, hinge and twist designs, they all obey the same principle of displaying a mechanism, i.e. an extended and nearly zero energy mode. We compute its optimal frequency, and show that —for such designs— cooperativity decays logarithmically with the system size L for d=2 and does not decay for d=3 where d is the spatial dimension, in great contrast with undesigned materials for which cooperativity decays as L^-d. Overall, our approach leads to a natural explanation for several observations in allosteric proteins, and suggests a path to discover new ones, also viewing allostery as a consequence of design in disordered media.