How proteins' negative cooperativity emerges from entropic optimisation of versatile collective fluctuations.

The fact that allostery, a nonlocal signaling between distant binding sites, can arise mainly from the entropy balance of collective thermal modes, without conformational changes, is by now well known. However, the propensity to generate negative cooperativity is still unclear. Starting from an elastic-network picture of small protein complexes, in which effector binding is modeled by locally altering interaction strengths in lieu of adding a node-spring pair, we elucidate mechanisms particularly for such negative cooperativity. The approach via a few coupled harmonic oscillators with internal elastic strengths allows us to trace individual eigenmodes, their frequencies, and their statistical weights through successive bindings. We find that the alteration of the oscillators' couplings is paramount to covering both signs of allostery. Binding-modified couplings create a rich set of eigenmodes individually for each binding state, modes inaccessible to an ensemble of noninteracting units. The associated shifts of collective-mode frequencies, nonuniform with respect to modes and binding states, result in an enhanced optimizability, reflected by a subtle phase map of allosteric behaviors.