Free Energy Calculations in Practice

Both factors must be considered when applying any method to enhance sampling in free energy calculations. In practice it is necessary to constrain the system very close to 3D Cartesian space. 

Chipot, C., Free energy calculations in biological systems. How useful are they in practice in New Algorithms for Maeromolecular Simulation, Leimkuhler, B. Chipot, C. Elber, R. Laaksonen, A. Mark, A. E. Schlick, T. Schutte, C. Skeel, R., Eds., vol. 49. Springer Berlin, Heidelberg, New York, 2005, pp. 183-209... 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Realizing that practical application of free energy calculations outside the purely academic environment, in particular in the pharmaceutical industry, required significant cost reductions, much effort was invested towards developing faster and cheaper methods for estimating free energy differences in complex systems. The goal... 

Pearlman, D. A. Charifson, P. S., Are free energy calculations useful in practice A comparison with rapid scoring functions for the p38 MAP kinase protein system, J. Med. Chem. 2001, 44, 3417-3423... 

In practice it is helpful to know the order of magnitude of the sample size N needed to reach a reasonably accurate free energy. The inaccuracy model described above presents an effective way to relate the sample size N and the finite sampling error through perturbation distribution functions. Alternatively, one can develop a heuristic that does not involve distribution functions and is determined by exploring the common behavior of free energy calculations for different systems [25]. Although only FEP calculations are considered in this section, the analysis extends to NEW calculations. 

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2,..., r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes ... 

To exploit the concept of PMF to represent solvent in free energy calculations, practical approximations must be constructed. A common approach is to treat the two components Z H/"P(X) and Z lYelec(X) separately. Approximations for the nonpolar term are usually derived from geometric considerations, as in scaled particle theory, for example [62], The electrostatic contribution is usually derived from continuum electrostatics. We consider these two contributions in turn. 

The theory is capable of describing both the regimes of equilibrium and nonequilibrium solvation for the latter we have developed a framework of natural solvent coordinates which greatly helps the analysis of the reaction system along the ESP, and displays the ability to reduce considerably the burden of the calculation of the free energy surface in the nonequilibrium solvation regime. While much remains to be done in practical implementations for various reactions, the theory should prove to be a very useful and practical description of reactions in solution. 

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