MacroModel Reference Manual - ISP

Appendix G: MINTA BackgroundFEP establishes a link between free-energy difference ∆G and potential energy difference ∆Eutilizing Zwanzig’s famous equation (Zwanzig, 1954):∆G AB = – RTln expE B – E〈 ( A– ------------------- )〉 RT A(3)∆G AB is the free-energy difference between two systems, “A” and “B,” which can, e.g., representtwo different ligands in a situation described by the thermodynamic cycle in Equation (1).E A and E B are the potential energy (molecular mechanics energy including solvation energy) ofsystem “A” and “B,” respectively. Note that E A and E B are functions of the coordinates of thetwo different systems. The bracket 〈〉 A refers to an ensemble average over system “A,” which isdetermined at a particular temperature T (R is the gas constant).It should be noted that Equation (3) is exact. It will be discussed later why is FEP a perturbationtheory. For the sake of argument, let us assume that we want to use Equation (3) directly tocalculate the binding free-energy difference between two ligands L A and L B with respect to amacromolecular host H. What does it mean to compute an ensemble average over system “A”?The answer depends on what kind of simulation is applied. If it is MD, the answer is that theMD simulation is running using L A ,L A is replaced by L B after every time step, and the exponentialin Equation (3) is accumulated during the full course of the MD simulation. MD(stochastic dynamics, to be correct) mimics thermal motion in a thermal bath and, therefore,generates the so-called canonical ensemble, i.e., samples the configuration space with the Boltzmannprobability. Thus, the simple arithmetic mean of the accumulated exponentials exp(–(E B –E A )/RT) gives the ensemble average in Equation (3).In case of MMC simulation, the Metropolis algorithm guaranties that the canonical ensembleis generated (Metropolis et al., 1953). With MMC, therefore, the well-known Metropolis criterionis applied to L A to drive the simulation. Similar to MD, the ensemble average is calculatedas the arithmetic mean of the accumulated exponentials exp(–(E B –E A )/RT) where L A is temporarilyreplaced by L B after each accepted MMC move. It is important to note that FEP isalways carried out in an explicit solvent box.Thermodynamic integration provides an alternative way for free-energy perturbation and isbased on the following formula (Kollman, 1993):∆G AB=ρ = B∂E 〈 -------- ρ∂ρ〉dρ∫ρρ = A(4)where the ensemble average 〈〉 ρ of the derivative of the energy with respect to ρ is evaluated atvarious values of ρ, and the outer integral is solved numerically. ρ represents a “reaction coordinate,”which is in most cases non-physical, but delineates a computationally accessible230MacroModel 9.7 Reference Manual