Free energy perturbation equilibration

The free energy perturbation calculations on mutation of the central statine residue of pepstatin to its dehydroxy and other derivatives were carried out using the window method. The crystal structure reported by Suguna et al.l4 l5was used for these calculations. In most simulations, the mutations were achieved either in 101 or 51 windows with 0.4 ps of equilibration and 0.4 ps of data collection at each window. The calculation for each mutation was repeated in water to determine the difference in the free energies of solvation and to complete the thermodynamic cycle. 

The protein systems were equilibrated by a 20 ps stepwise heating scheme and thereafter 50 ps simulation at a constant temperature of 300 K. The water systems were equilibrated by directly simulating them for 50 ps at 300 K. The MD trajectories were run using a time step of 1 fs and energy data were collected every fifth step. The free energy perturbations were sampled using 47-83. -points and 5 ps simulation for each value of X,. Data from the first 2 ps of each step were discarded for equilibration. 

UU is the Hamiltonian difference (the perturbation) the angle brackets represent a canonical ensemble average performed on an equilibrated system designated by the subscript. Usually the kinetic component of the Hamiltonian is not included in the free energy calculation, and A- //, n can be replaced by the potential energy difference AU = Ui - U0. 

Rao and Singh32 calculated relative solvation free energies for normal alkanes, tetra-alkylmethanes, amines and aromatic compounds using AMBER 3.1. Each system was solvated with 216 TIP3P water molecules. The atomic charges were uniformly scaled down by a factor of 0.87 to correct the overestimation of dipole moment by 6-31G basis set. During the perturbation runs, the periodic boundary conditions were applied only for solute-solvent and solvent-solvent interactions with a non-bonded interaction cutoff of 8.5 A. All solute-solute non-bonded interactions were included. Electrostatic decoupling was applied where electrostatic run was completed in 21 windows. Each window included 1 ps of equilibration and 1 ps of data... 

Internally equilibrated subsystems, which act as free energy reservoirs, are already as random as possible given their boundary conditions, even if they are not in equilibrium with one another because of some bottleneck. Tlius, the only kinds of perturbation that can arise and be stabilized when they are coupled are those that make the joint system less constrained than the subsystems originally were. (This is Boltzmann s H-theorem [9] only a less constrained joint system has a liigher maximal entropy than die sum of entropies from the subsystems independently and can stably adopt a different form.) The flows that relax reservoir constraints are thermochemical relaxation processes toward the equilibrium state for tlte joint ensemble. The processes by wliich such equilibration takes place are by assumption not reachable within the equilibrium distribution of either subsystem. As the nature of the relaxation phenomenon often depends on aspects of the crosssystem coupling that are much more specific than the constraints that define either reservoir, they are often correspondingly more complex than the typical processes... 

It is assumed that the system free energy is again given by (8.1) where Us = Us 0, e) depends on the elastic strain field and on the local orientation of the material surface in a characteristic way for a given material. The extensional strain of the surface is related to the bulk strain field through (8.124). The process of elastic equilibration is expected to be very rapid compared to shape equilibration, which requires diffusive transport of material. Consequently, it is assumed from the outset that the elastic field is always in mechanical equilibrium for any surface shape. This expectation is enforced by requiring the variation in free energy due to a kinematically admissible small perturbation 6ui in the displacement field from its equilibrium distribution to be stationary, that is, to vanish to first order in the perturbation. 

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