Title: Principles for optimal cooperativity in allosteric materials,
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}.