Free energy derivatives method

D. A. Pearlman, A. free energy derivatives A new method for probing the convergence... 

B. G. Rao, and U. C. Singh, Studies on the binding of pepstatin and its derivatives to Rhizopus pepsin by quantum methanics, molecular mechanics, and free energy perturbation methods, J. Am. Chem. Soc. 113 6735 (1991). 

Also, because such derivatives are to be evaluated at the equilibrium geometry, a key point is the determination of that geometry on the solvated PES, which leads to the so-called indirect solvent effects , which still requires a viable method to calculate free energy gradients (and possibly hessians). The problem of the formulation of free energy derivatives within continuum solvation models is treated elsewhere in this book and for this reason it will not considered here. Instead, it is worth remarking in this context another implication of such a formulation, i.e. that a choice between a complete equilibrium scheme or the account for vibrational and/or electronic nonequilibrium solvent effects [42, 43] should be done (see below). 

The standard potentials of bromine and iodine have been determined by somewhat similar methods with bromine the results are expressed in terms of two alternative standard states, viz., the gas at 1 atm. pressure or the pure liquid. The standard state adopted for iodine is the solid state, so that the solution is saturated with respect to the solid phase. The acandard potential of fluorine has not been determined by direct experiment, but its value has been calculated from free energies derived from thermal and entropy data. ... 

The advantages of LIA over the classical (TP and TI) free energy simulation methods are (i) the absolute binding free energies are computed (AF instead of AAF), (ii) the intermediate hybrid states are eliminated and only the end-points are sampled, and (iii) a physical path is used between the initial and the final states which prevents the X goes to end-points catastrophe problem. However, a certain degree of empiricism is associated with the model, and in fact, somewhat different values for the scaling parameters a and P have been derived by various authors [274-276]. 

The study of reaction mechanisms by means of theoretical methods heavily depends on the availability of analytical expressions of the free energy derivatives with respect to nuclear coordinates. We shall summarize here the status of the art for the PCM method developed in Pisa. Other approaches will be considered later in Section 8. 

As it will be discussed later, PCM-QINTn also gives good derivatives of the free energy. The method has been thus far implemented for standard PCM only. 

L. Wang, D. L. Veenstra, R. J. Radmer, and P, A, Kollman, Proteins, 32,438 (1998). Can One Predict Protein Stability An Attempt to Do So for Residue 133 of T4 Lysozyme Using a Combination of Free Energy Derivatives, PROFEC, and Free Energy Perturbation Methods. 

The theory required for the determination of analytical free energy derivatives was recently developed by Kantorovich and applied to alkali halide crystals (Kantorovich 1995). Subsequently the method has been refined by Taylor et al. (1997) who have discussed many of the details of its implementation. However, as the approach is relatively new, a summary of the main features will be given here. 

While simulations reach into larger time spans, the inaccuracies of force fields become more apparent on the one hand properties based on free energies, which were never used for parametrization, are computed more accurately and discrepancies show up on the other hand longer simulations, particularly of proteins, show more subtle discrepancies that only appear after nanoseconds. Thus force fields are under constant revision as far as their parameters are concerned, and this process will continue. Unfortunately the form of the potentials is hardly considered and the refinement leads to an increasing number of distinct atom types with a proliferating number of parameters and a severe detoriation of transferability. The increased use of quantum mechanics to derive potentials will not really improve this situation ab initio quantum mechanics is not reliable enough on the level of kT, and on-the-fly use of quantum methods to derive forces, as in the Car-Parrinello method, is not likely to be applicable to very large systems in the foreseeable future. 

Let us illustrate this with the example of the bromination of monosubstituted benzene derivatives. Observations on the product distributions and relative reaction rates compared with unsubstituted benzene led chemists to conceive the notion of inductive and resonance effects that made it possible to explain" the experimental observations. On an even more quantitative basis, linear free energy relationships of the form of the Hammett equation allowed the estimation of relative rates. It has to be emphasized that inductive and resonance effects were conceived, not from theoretical calculations, but as constructs to order observations. The explanation" is built on analogy, not on any theoretical method. 

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