Potential free energy calculations

Amadei, A., Apol, M. E. F., Di Nola, A., Berendsen, H. J. C. The quasi-Gaussian entropy theory Free energy calculations based on the potential energy distribution function. J. Chem. Phys. 104 (1996) 1560-1574... 

Potential Pitfalls with Free Energy Calculations... 

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... 

Figure 4.4 Schematic diagram of the free energy calculated from (4.4), Fftee. versus potential cf> for the generic electrocatalytic reaction A —> B. Points indicated hy squares and circles are for specific external charges (various q) for the systems A and B, respectively. Solid and dashed lines indicate the best-fit curves for the free energy versus potential relationship for systems A and B, respectively.

Figure 4.12 The upper plots (a, c, e, g) show the free energies (calculated by (4.4) from DFT) versus the estimated potential for reactants and products involved in the first, second, third, and fourth consecutive methanol dehydrogenation steps, as indicated, over Pt(lll) from Cao et al. [2005]. Filled symbols in (a) refer to the energy and potential for the system tq = Q. The lower plots (b, d, f, h) show the corresponding reaction energies for the first, second, third, and fourth consecutive methanol dehydrogenation steps, as indicated.

Zhang Y, Liu H, Yang W (2000) Free energy calculation on enzyme reactions with an efficient iterative procedure to determine minimum energy paths on a combined ab initio QM/MM potential energy surface. J Chem Phys 112 3483-3492... 

Almost 20 years later, Robert Zwanzig [4] followed a perturbative route to free energy calculations, showing how physical properties of a hard-core molecule change upon adding a rudimentary form of an attractive potential. The high-temperature... 

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. 

Four years later, Christian Bartels and Martin Karplus [55] used the WHAM equations as the core of their adaptive US approach, in which the efficiency of free energy calculations was improved through refinement of the biasing potentials as the simulation progressed. Efforts to develop adaptive US techniques had, however, started even before WHAM was developed. They were pioneered by Mihaly Mezei [56], who used a self-consistent procedure to refine non-Boltzmann biases. 

Fig. 2.5. Possible applications of a coupling parameter, A, in free energy calculations, (a) and (b) correspond, respectively, to simple and coupled modifications of torsional degrees of freedom, involved in the study of conformational equilibria (c) represents an intramolecular, end-to-end reaction coordinate that may be used, for instance, to model the folding of a short peptide (d) symbolizes the alteration of selected nonbonded interactions to estimate relative free energies, in the spirit of site-directed mutagenesis experiments (e) is a simple distance separating chemical species that can be employed in potential of mean force (PMF) calculations and (f) corresponds to the annihilation of selected nonbonded interactions for the estimation of e.g., free energies of solvation. In the examples (a), (b), and (e), the coupling parameter, A, is not independent of the Cartesian coordinates, x. Appropriate metric tensor correction should be considered through a relevant transformation into generalized coordinates...

In the following, we will briefly illustrate the application of nonequilibrium free energy calculations for a simple ID model system. Shown in Fig. 5.1 are the potential energy surfaces... 

One of the objectives in free energy calculations is to obtain potentials of mean force C(r) along a chosen coordinate r = r(z) defined as... 

UU is the Hamiltonian difference (the perturbation) the angle brackets represent a canonical ensemble average performed on an equilibrated system designated by the subscript. Usually the kinetic component of the Hamiltonian is not included in the free energy calculation, and A- //, n can be replaced by the potential energy difference AU = Ui - U0. 

As discussed in Sect. 6.1, the bias due to finite sampling is usually the dominant error in free energy calculations using FEP or NEW. In extreme cases, the simulation result can be precise (small variance) but inaccurate (large bias) [24, 32], In contrast to precision, assessing the systematic part (accuracy) of finite sampling error in FEP or NEW calculations is less straightforward, since these errors may be due to choices of boundary conditions or potential functions that limit the results systematically. 

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 ... 

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