| The Rotation and Translation block (RTB) method of Durand et al. [1] [Biopolymers 34, 759 (1994)] and Tama et al. [2] [Proteins 41, 1 (2000)], which is an appealing way to calculate low frequency normal modes of biomolecules by restricting the space of motions to exclude internal motions of pre-selected rigid fragments within the molecule, is extended to a method for computing the Newtonian dynamics of a biomolecule, or any large molecule, with effective rigid-body constraints applied to a pre-chosen set of internal molecular fragments. This method, to be termed RTB-dynamics does not require the construction of the matrix of second spatial derivatives of the potential energy function, and can be used to compute the classical dynamics of a system moving in an arbitrary anharmonic force field with an efficient way of freezing out the high frequency motions within rigid fragments. Moreover, an approximation scheme is developed to compute Brownian motion according to the Langevin Equation for a molecular system moving in a harmonic force field and characterized by one or more rigid internal fragments by using the RTB methodology. To illustrate these methods elementary numerical applications to signal propagation in the small membrane-bound polypeptide gramicidin-A are presented.;Finally, Dynamic Linear Response Theory (DLRT) 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. DLRT relates the non-equilibrium motion of the protein atoms which 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. We numerically illustrate the application of DLRT for a simple harmonic oscillator model of the ferric binding protein, and for an analogous model of T4 lysozyme including the solvent effects on the motion using the Langevin prescription. 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. |
