Free Energy Perturbation in MacroModel

The fundamental expression for free energy calculations as implemented in MacroModel is:

(1)

where H_A and H_B are the Hamiltonians for the two systems, A and B, and the <> notation represents an ensemble average over system A. This expression is valid only when there is a very small difference between the two systems, A and B (i.e., H_B−H_A ≈ RT). To perform free energy calculations on meaningful systems, you generally perform a series of smaller simulations (windows), which can be summed to obtain a total free energy difference.

The coupling parameter <λ> is used to define each window in terms of the two endpoints, A and B, so that at any stage the Hamiltonian over which the ensemble average will be generated for the system is described in terms of the Hamiltonians for the endpoints:

(2)

Then the overall expression for performing a simulation becomes:

(3)

where n is the number of windows to be used in the simulation. At any window the “perturbation” is between a state at λ and that at λ + dλ. The method for performing this simulation as implemented in MacroModel is described as “single topology.” That is, at any value of λ, a set of interactions is generated by mixing the interactions of the endpoints. For example, if an atom has a charge x in the starting point of the simulation and a charge y in the end point, at any given value of λ (i.e., at any window), the charge on the atom will be z = yλ + (1-λ)x. This process will be repeated for all the interactions in the system and the simulation will be performed with this new set of interactions. At points during the simulation, the energy, H(λ), will be evaluated. Then the interactions corresponding to the “next” value of lambda (λ+dλ) will be generated and H(λ+dλ) will be calculated. These terms are used in the exponential of expression (3). In MacroModel we usually take advantage of the fact that for most values of λ, the energy at λ-dλ can also be evaluated—this procedure is known as “double wide sampling,” and effectively allows a “forward” and “reverse” simulation to be performed in one simulation.