computer modeling in molecular biology.pdf

4 Molecular Dynamics and Free Energy Calculations 65Where c is the fixed distance between particles i and j, taken here as that observedin the starting conformation. The latter is the crystal structure of barnase after beingsubjected to energy minimization so as to regularize bond distances and angles andto relieve close contacts. To freeze bond stretching, the distance constraints aredirectly applied to the atoms defining the relevant bonds. To freeze the bending ofa valence angle between atoms i - j - k, they are applied to the distance betweenatoms i and k.Inserting the right hand side of Eq. (4-2) into the Verlet’s algorithm Eq. (4-1 a):At2ri (t + At) = 2ri (t)- ri (t- A t) + -[I;;.(t)- c lZ,ViG,] (4-4)mik=lwhere one readily recognizes the standard Verlet formulation of Eq. (4-1 a), plus acorrection term due to the constraints. This allows to “decouple” the integration ofthe positions into two steps:Step 1: the equation of motion are solved in absence of constraints using the Verletalgorithm to predict new “unconstrained” positions r’ (t + At).Step 2: the “unconstrained” position r’ is modified by an increment 6r to yield thenew “constrained” positions r (t + At):ri (t + At) = r; (t + At) + 6ri (4-5)This involves an iterative procedure which is applied until all the constraints aresatisfied to within a predefined tolerance.I4.2.1.2 The Force-FieldOne of the major challenges in simulations of large biological systems is the availabilityof force-fields that adequately represent their physical properties. The use ofclassical empirical force-fields offers the important advantage of testing differentfunctional forms and fitting parameters to a variety of experimental data and toresults from detailed quantum mechanical calculations. Obtaining an improved setof parameters and better potential functions is an ongoing effort in manylaboratories [37-421. Here, all the atoms of the system, including aliphatic and polarhydrogens, were considered explicitly. Forces and interaction energies between proteinatoms and between atoms of the protein and water, were calculated using arelatively recent version (version 19) of the CHARMM potentials [38] developed at