Computing Free-energy Differences

As noted previously in Chapters 3 and 10, statistical thermodynamics relates all thermodynamic observables to the partition function Q. For ease of reference, the definition of Q and the equations defining various thermodynamic variables as a function of Q, some of which have appeared previously, are as follows 

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer 

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If we are interested (as is almost always the case) in computing free energy differences, the factor y/nj2/3 can be ignored. The factor Ze r represents a contribution to the free energy, in addition to the mechanical forces given by A, due to entropic effects. 

Hendrix, D.A. Jarzynski, C., A fast growth method of computing free energy differences, J. Chem. Phys. 2001,114, 5974-5981... 

In this chapter, we will examine in depth the characteristic errors of two free energy techniques and present improved methods based on a better understanding of their behavior. The two techniques examined are free energy perturbation (FEP) [2] and nonequilibrium work (NEW) based on Jarzynski s equality [3-6]. These techniques are discussed in Chaps. 2 and 5. The FEP method is one of the most popular approaches for computing free energy differences in molecular simulation see, e.g., [1, 7-10]. The recently developed NEW method, which is closely related to FEP, is gaining popularity in both simulation [11-18] and experimental applications [19-21],... 

Errors of this magnitude make the useful prediction of free energies a difficult task, when differences of only one to three kcal/mol are involved. Nevertheless, within the error limits of the computed free energy differences, the trend is that relative to 8-methyl-N5-deazapterin or 8-methyl-pterin, the compounds methyl substituted in the 5, 6 or 7 positions are thermodynamically more stable when bound to DHFR largely by virtue of a hydrophobic effect, i.e. methyl substitution reduces the affinity of the ligand for the solvent more than it reduces affinity for the DHFR active-site. The stability of ligand binding to DHFR appears to be optimal with a 6-methyl substituent additional 5-methyl and/or 7-methyl substitution has little effect... 

While our focus has been primarily on thermodynamic quantities, like free energy, it should be borne in mind that the ultimate motivation for computing free energy differences is usually to pennit calculation of chemical concentrations in actual systems. To accomplish this for a generic equilibrium is straightforward. For example, consider the following reaction (chosen in a completely arbitrary fashion)... 

Mooney, S. D., Huang, G. C., Kollman, P. A., and Klein, T. E. (2001). Computed free energy differences between point mutations in a collagen-like peptide. Biopolymers 58, 347-353. 

To compute free energy differences between configurations or potentials of mean force, umbrella potentials are used to sample phase space at increments of X (see Figure 8). Protocols such as multi-step thermodynamic perturbation (MSTP) or multi-configuration thermodynamic integration (MCTl), are used to compute free energies [86] ... 

To compute free energy differences between states with little overlap, it will usually be more efficient to compute the free energy along a pathway of intermediate states. Free energy calculations can be made significantly more efficient by optimizing the choice of intermediate states for increased phase space overlap [36, 59-61],... 

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