Algorithms for Molecular and Stochastic Dynamics

A variety of algorithms have been used for integrating the equations of motion in molecular dynamics simulations of macromolecules. Most widely employed are the algorithms due to Gear91 and Verlet.90 The algorithm introduced by Verlet in his initial studies of the dynamics of Lennard-Jones fluids is derived from the two Taylor expansions, 

In this equation x, is the Cartesian coordinate x for atom i, /3, is the frictional drag on atom i and/, is the Langevin random force on atom i obtained from a Gaussian random distribution of zero mean and variance 

This algorithm has been used in the integration of the Langevin equation applied to the buffer zone atoms in the stochastic boundary molecular dynamics method (Chapt. IV.C), as well as in other stochastic dynamics calculations.102 

For some problems, such as the motion of heavy particles in aqueous solvent (e.g., conformational transitions of exposed amino acid sidechains, the diffusional encounter of an enzyme-substrate pair), either inertial effects are unimportant or specific details of the dynamics are not of interest e.g., the solvent damping is so large that inertial memory is lost in a very short time. The relevant approximate equation of motion that is applicable to these cases is called the Brownian equation of motion, 

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