Quantifying allosteric effects in proteins

In allosteric regulation, protein activity is altered when ligand binding causes changes in the protein conformational distribution. Little is known about which aspects of protein design lead to effective allosteric regulation, however. To increase understanding of the relation between protein structure and allosteric effects, we have developed theoretical tools to quantify the influence of protein-ligand interactions on probability distributions of reaction rates and protein conformations. We define the rate divergence, Dk, and the allosteric potential, Dx, as the Kullback-Leibler divergence between either the reaction-rate distributions or protein conformational distributions with and without the ligand bound. We then define Dx as the change in the conformational distribution of the combined protein/ligand system, derive Dx in the harmonic approximation, and identify contributions from 3 separate terms: the first term, D[stackxomega], results from changes in the eigenvalue spectrum; the second term, D[stackxDeltax], results from changes in the mean conformation; and the third term, Dxv, corresponds to changes in the eigenvectors. Using normal modes analysis, we have calculated these terms for a natural interaction between lysozyme and the ligand tri-N-acetyl-D-glucosamine, and compared them with calculations for a large number of simulated random interactions. The comparison shows that interactions in the known binding-site are associated with large values of Dxv. The results motivate using allosteric potential calculations to predict functional binding sites on proteins, and suggest the possibility that, in Nature, effective ligand interactions occur at intrinsic control points at which binding induces a relatively large change in the protein conformational distribution.