Fast and exact single and double mutation-response scanning of proteins

Studying the effect of perturbations on protein structure is a basic approach in protein research. Important problems, such as predicting pathological mutations and understanding patterns structural evolution, have been addressed by computational simulations based on modelling mutations as forces and predicting deformations using the Linear Response Approximation. In single mutation-response scanning simulations, a sensitivity matrix is obtained by averaging deformations over point mutations. In double mutation-response scanning simulations, a compensation matrix is obtained by minimizing deformations over pairs of mutations. These very useful simulation-based methods may be too slow to deal with large supra-molecular complexes, such as a ribosome or a virus capsid, or large number of proteins, such as the human proteome, which limits their applicability. To address this issue, I derived analytical closed formulas to calculate the sensitivity and compensation matrices directly, without simulations. Here, I present these derivations and show that the resulting analytical methods are much faster than their simulation counterparts, and that where the simulation methods are approximate, the analytical methods are exact by design.