Principles for optimal cooperativity in allosteric materials

TL;DR

This paper investigates the architectures of allosteric materials that optimize cooperative energy transmission, revealing universal principles and optimal frequency scaling, with implications for understanding natural allosteric proteins.

Contribution

It introduces an in-silico evolution scheme and theoretical analysis to identify architectures that optimize cooperativity, differing from strain-propagating materials, and reveals universal mechanisms and scaling laws.

Findings

Optimal cooperative architectures display a soft mode mechanism.

Optimal frequency decreases as system size increases, following a power law.

Cooperativity decays logarithmically in 2D and remains constant in 3D with size.

Abstract

Allosteric proteins transmit a mechanical signal induced by binding a ligand. However, understanding the nature of the information transmitted and the architectures optimizing such transmission remains a challenge. Here we show using an {\it in-silico} evolution scheme and theoretical arguments that architectures optimized to be cooperative, which propagate efficiently energy, {qualitatively} differ from previously investigated materials optimized 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 {\it mechanism}, i.e. an extended {soft} mode}. We show that its optimal frequency decreases with the spatial extension $L$ of the system as $L^{-d/2}$, where $d$ is the spatial dimension. For these optimal designs, cooperativity decays logarithmically with $L$ for $d=2$ and does not decay for $d=3$. Overall our approach leads to a natural explanation for several observations in allosteric proteins, and { indicates an experimental path to test if allosteric proteins lie close to optimality}.