Dynamic linear response theory for conformational relaxation of proteins

Dynamic linear response theory is adapted to the problem of computing the time evolution of the atomic coordinates of a protein in response to the unbinding of a ligand molecule from a binding pocket within the protein. When the ligand dissociates from the molecule, the protein molecule finds itself out of equilibrium and its configuration begins to change, ultimately coming to a new stable configuration corresponding to equilibrium in a force field that lacks the ligand-protein interaction terms. Dynamic linear response theory (LRT) relates the nonequilibrium motion of the protein atoms that ensues after the ligand molecule dissociates to equilibrium dynamics in the force field, or equivalently, on the potential energy surface (PES) relevant to the unliganded protein. In general, the connection implied by linear response theory holds only when the ligand-protein force field is small. However, in the case where the PES of the unliganded protein system is a quadratic (harmonic oscillator) function of the coordinates, and the force of the ligand upon the protein molecule in the ligand-bound conformation is constant (the force on each atom in the protein is independent of the location of the atom), dynamic LRT is exact for any ligand-protein force field strength. An analogous statement can be made for the case where the atoms in the protein are subjected to frictional and random noise forces in accord with the Langevin equation (to account for interaction of the protein with solvent, for example). We numerically illustrate the application of dynamic LRT for a simple harmonic oscillator model of the ferric binding protein, and for an analogous model of T4 lysozyme. Using a physically appropriate value of the viscosity of water to guide the choice of friction parameters, we find relaxation time scales of residue-residue distances on the order of several hundred ps. Comparison is made to relevant experimental measurements.