Free energy calculations partitioning

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

To illustrate how stratification works in the context of free energy calculations, let us consider the transformation of state 0 into state 1 described by the parameter A. We further assume that these two states are separated by a high-energy barrier that corresponds to a value of A between Ao and Ai. Transitions between 0 and 1 are then rare and the free energy estimated from unstratified computer simulations would converge very slowly to its limiting value, irrespective of the initial conditions. If, however, the full range of A is partitioned into a number of smaller intervals, and... 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

Lombardo F, Blake JF, Curatolo WJ (1996) Computation of brain-blood partitioning of organic solutes via free energy calculations. J Med Chem 39 4750-4755. 

The standard free energy of partition was calculated from water-hexane partition coefficients ... 

Lombardo, F., Blake, J.F. and Curatolo, W.J. (1996). Computation of Brain-Blood Partitioning of Organic Solutes via Free Energy Calculations. J.Med.Chem., 39, 4750-4755. 

In the study of Li et u/. good agreement was found between experimental and theoretical hydration energies, with somewhat better agreement for the full 18-water cluster compared to the 6-water cluster. The authors carefully partition the components of the redox energy (free energy) calculation to include geometry relaxation and relativistic corrections ... 

Solvation is an important parameter that influences macromolecular properties such as solubility, reaction rate equilibria, partition coefficients, enzyme-substrate, and ligand-receptor binding. In free energy calculations, solvent is represented either explicitly or implicitly. The decision of whether to use explicit or implicit solvent models in a given simulation typically depends on the available computer resources. Inclusion of explicit solvent molecules is more realistic than continuum models. 

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