Optimality of cooperativity in allosteric materials and proteins

Allostery is responsible for the activity regulation of many proteins essential for life. Many efforts on understanding this long-range communication have been made, but the physical picture of allosteric mechanics is not yet clear. Recent progress employing in-silico evolutions studied mechanical networks of harmonic springs with allosteric behaviors. These networks are found to share common principles for the long-range communication to occur. Specifically, the stiffness of the allosteric response scales with the system size with a nontrivial power law. In this work, we test these principles in real systems, using a large set of X-ray structural data of allosteric proteins. Overall, we find that the functional allosteric response of each protein is related to a “mechanism”, a soft and extended mode with strong strain. By extending the theory of allosteric materials to include nonlinearities, we identify two scalings of stiffness setting a regime where the allosteric cooperative binding is optimal. A new scaling exponent appears, in addition to the one from the linear theory. The stiffness from the X-ray structural data falls in this predicted range, suggesting that proteins actually work at optimal cooperativity.