Free energy calculating-example

Figure 6 Thermodynamic cycle for multi-substate free energy calculation. System A has n substates system B has m. The free energy difference between A and B is related to the substate free energy differences through Eq. (41). A numerical example is shown in the graph (from Ref. 39), where A and B are two isomers of a surface loop of staphylococcal nuclease, related by cis-trans isomerization of proline 117. The cis trans free energy calculation took into account 20 substates for each isomer only the six or seven most stable are included in the plot.

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...

Consider the forward calculation as an example. Earlier in this chapter, we concluded that the most important contribution to a forward free energy calculation comes from the low-x tail of f(x)9 and its poor sampling results in the major systematic error of the calculation. To simplify further the analysis we assume that there is a limit-perturbation x f such that x > Xf is well sampled and x < Xf is never sampled. This is illustrated in Fig. 6.5. 

As discussed in the example from the work of Hodel et al. [3], one of the most efficient ways to improve the accuracy of free energy calculations with a given force field is to enhance the conformational sampling. Thus, it is important to assess the extent to which phase space is covered. 

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. 

It is not uncommon for protons to be taken up or released upon formation of a biomolecular complex. Experimental data on such processes can be compared to computational results based on, for example, Poisson-Boltzmann calculations.25 There is a need for methods that automatically probe for the correct protonation state in free energy calculations. This problem is complicated by the fact that proteins adapt to and stabilize whatever protonation state is assigned to them during the course of a molecular dynamics simulation.19 When the change in protonation state is known, equations are available to account for the addition or removal of protons from the solvent in the overall calculation of the free energy change.11... 

The examples discussed here show that the new LIE parametrization of Ref. 26, while reliable for a number of systems, could not be the final word in the development of this type of approximate binding free energy calculations. As we will see below there may be more examples of ligand-receptor systems that don t fit the simple picture of Figure 1. 

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