computer modeling in molecular biology.pdf

4 Molecular Dynamics and Free Energy Calculations 87pling parameter, varied from 0 to 1. rN are the atomic coordinates of the system encompassinghere the protein and solvent molecules. With this simple form of thehybrid potential problems of convergence are not uncommon and can be overcomeusing more complex forms, non-linear in 1 [85].The free energy difference AG between the two states A (wild type) and B (mutant)can be obtained from either of two formally exact expressions. One is the socalled "exponential formula" (EF) [91, 921 :(4-13)where AV= VB (r") - V, (r"), dAi = Ai+l - Ai, kB the Boltzmann constant and 7the absolute temperature; the angle brackets represent an ensemble average obtainedusing the potential V(r", Ai) from Eq. (4-12) to represent the system. To implementEq. (4-13), a series of simulations is set up corresponding to a succession of discrete1 values between 0 and 1. From each simulation at a given 1, ensemble averages ofthe exponential factor are evaluated. Summation of the natural logarithm of theseaverages then leads to the required free energy value. Equation (4-13) is formallyexact provided that the configuration ensemble generated at a given Ai is representativeof the potential corresponding to Ai+l. This implies conserving reversibility ofthe perturbation at each step along the pathway and requires taking small enough1 intervals. It is also the main reason why the method is restricted to small size perturbations.The reversibility condition also stipulates that the EF expression, whenused in both the forward and reverse directions along a given perturbation, shouldyield the same result in absolute value. This is often done to monitor convergenceand as a consistency check, and involves using the mirror expression of Eq. (4-13)which gives the free energy difference evaluated from an ensemble obtained from asimulation performed at Ai+l :AG = kBT c Ini(4-14)The other equivalent formulation, is called thermodynamic integration (TI). It hasoften been applied to compute the free energy change of systems where a parameterspecifying the thermodynamic state, such as the volume or temperature, is variedslowly. It uses the folIowing expression [93]: