Free energy perturbation

Free energy perturbation (FEP) theory is a method based on statistical mechanics that is used in computational chemistry for computing free energydifferences from molecular dynamics or Metropolis Monte Carlosimulations. The FEP method was introduced by R. W. Zwanzig in 1954.^[1] According to free-energy perturbation theory, the free energy difference for going from state A to state B is obtained from the following equation, known as the Zwanzig equation:

http://upload.wikimedia.org/wikipedia/en/math/8/5/5/85545b0b56fcd83c2a850f807eb06b26.png

where T is the temperature, k_B is Boltzmann's constant, and the triangular brackets denote an average over a simulation run for state A. In practice, one runs a normal simulation for state A, but each time a new configuration is accepted, the energy for state B is also computed. The difference between states A and B may be in the atom types involved, in which case the ΔG obtained is for "mutating" one molecule onto another, or it may be a difference of geometry, in which case one obtains a free energy map along one or more reaction coordinates. This free energy map is also known as a potential of mean force or PMF. Free energy perturbation calculations only converge properly when the difference between the two states is small enough; therefore it is usually necessary to divide a perturbation into a series of smaller “windows”, which are computed independently. Since there is no need for constant communication between the simulation for one window and the next, the process can be trivially parallelized by running each window in a different CPU, in what is known as an “embarrassingly parallel” setup.

FEP calculations have been used for studying host-guest binding energetics, pKapredictions, solvent effects on reactions, and enzymatic reactions. For the study of reactions it is often necessary to involve a quantum-mechanical representation of the reaction center because the molecular mechanics force fields used for FEP simulations can't handle breaking bonds. A hybrid method that has the advantages of both QM and MM calculations is called QM/MM.

Umbrella sampling is another free-energy calculation technique that is typically used for calculating the free-energy change associated with a change in "position" coordinates as opposed to "chemical" coordinates, although Umbrella sampling can also be used for a chemical transformation when the "chemical" coordinate is treated as a dynamic variable (as in the case of the Lambda dynamics approach of Kong and Brooks). An alternative to free energy perturbation for computing potentials of mean force in chemical space is thermodynamic integration. Another alternative, which is probably more efficient, is the Bennett acceptance ratio method.

1.       Zwanzig, R. W. J. Chem. Phys. 1954, 22, 1420-1426. doi:10.1063/1.1740409

Thermodynamic integration

Thermodynamic integration is a method used to compare the thermal quantity difference of two given phases in molecular dynamics simulation. Free energydifferences are one quantity commonly computed in this way; since they are not simply functions of the phase space coordinates of the system, but are related to the canonical partition function Q(N,V,T), they cannot be directly measured in a simulation. These differences are usually calculated by designing a thermodynamic cycle and integrating along the relevant paths. Such paths can either be real chemical processes or alchemical processes. A good example of the alchemical process is the Kirkwood's coupling parameter method.^[1]

As free energy can be expressed by:

F = − k_BTln Q(N,V,T),

where in the partition function take the derivate of F, we will get that it equals the derivative of potential energy. thus the free energy difference of different state can be used to calculate the difference of potential energy.

http://upload.wikimedia.org/wikipedia/en/math/c/a/f/caf4207bf2d387dcc508f9c70d387a03.png

Umbrella sampling is a related free energy method. It adds a bias to the potential energy. In the limit of an infinite strong bias it is equivalent to thermodynamic integration.^[2]

1. J. G. Kirkwood. Statistical mechanics of fluid mixtures, J. Chem. Phys., 3:300-313,1935

2. J Kästner et al. (2006). "QM/MM Free-Energy Perturbation Compared to Thermodynamic Integration and Umbrella Sampling: Application to an Enzymatic Reaction". JCTC 2 (2): 452–461. doi:10.1021/ct050252w