Free energy methods

2 Basic Formulation of Conventional Free Energy Methods 

The free energy difference between a state with the value of X and the initial state (A=0) is given by the FEP method17 

The angle bracket denotes that the configurational integral is taken over the initial state. The conformational sampling indicated by Equation 4 is generated according to the Boltzmann probability associated with the initial state potential. As discussed in Section 2.1, convergence of conformational 

An alternative approach to free energy calculations is the thermodynamic integration (TI) method,18 20 which considers the ensemble average of the first derivative of the hybrid potential with respect to A at various values of A 

In many applications, a single biasing potential is not sufficient to cover the whole range of and simultaneously produce good sampling. Thus a set of restraining potentials, C/ ( ), are used to shift the local minima in the desired direction. In this windowing approach, the potential of mean force, ), in each window takes the form 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accuracy within a finite simulation time. It is. however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natural thermodynamical quantity under normal experimental conditions (constant temperature and pressure. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

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Lyubartsev A P, MartsInovskI A A, Shevkunov S V and Vorontsov-Velyamlnov P N 1992 New approach to Monte Carlo calculation of the free-energy—method of expanded ensembles J. Chem. Phys. 96... 

This equation relates the free energy difference between two systems to the individual perturbations x and the / and g distribution functions. The relationship (6.15) is important in both the characterization of free energy error and the development of improved free energy methods. 

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

We can expect to see future research directed at QM/MM and ab initio simulation methods to handle these electronic structure effects coupled with path integral or approximate quantum free energy methods to treat nuclear quantum effects. These topics are broadly reviewed in [32], Nuclear quantum effects for the proton in water have already received some attention [30, 76, 77]. Utilizing the various methods briefly described above (and other related approaches), free energy calculations have been performed for a wide range of problems involving proton motion [30, 67-69, 71, 72, 78-80]. 

The list of fluids which exhibit important quantum effects is not large. Getting back to the original question of this chapter, it is clear that for liquids like helium and hydrogen, a full quantum treatment is necessary. Liquids such as neon and water, however, show modest quantum effects which can be modeled with approximate free energy methods. The quantum correction to the free energy of water is roughly 10%... 

We present and analyze the most important simplified free energy methods, emphasizing their connection to more-rigorous methods and the underlying theoretical framework. The simplified methods can all be superficially defined by their use of just one or two simulations to compare two systems, as opposed to many simulations along a complete connecting pathway. More importantly, the use of just one or two simulations implies a common approximation of a near-linear response of the system to a perturbation. Another important theme for simplified methods is the use, in many cases, of an implicit description of solvent usually a continuum dielectric model, often supplemented by a simple description of hydrophobic effects [11]. 

We now turn to the problem of proton binding to proteins, an important area for simplified free energy methods. The linear response formalism earlier underlies most of the methods used today. It leads directly to one of the more useful practical methods, the so-called LRA, or linear response approximation method [59], presented here. 

A key element of many simplified free energy methods is the use of an implicit description of the solvent. Implicit solvent models are based on the concept of a PMF, presented briefly in this section see [11] for a detailed review see Chap. 4 for applications of the PMF concept that are not related to implicit solvation. 

Free Energy Methods Using an Implicit Solvent PBFE, MM/PBSA, and Other Acronyms... 

The most important model parameter in PBFE and MM/PBSA is the dielectric constant used for the solutes. Most studies have taken an empirical approach, viewing the dielectric constant as an adjustable parameter. While this seems plausible, it is prudent to analyze the physical problem in more detail, because, in some cases, the experimental data can be fit by models that are distinctly unphysical, despite some plausible features. We therefore come back to the simplest possible PBFE calculation the important problem of proton binding, or pKa shifts. We discuss a nonem-pirical model that attempts to avoid parameter fitting and that gives insights into the limitations of simplified continuum electrostatic free energy methods. 

Another apparent difference between the various free energy methods lies in the treatment of order parameters. In the original formulation of a number of methods, order parameters were dynamical variables - i.e., variables that can be expressed in terms of the Cartesian coordinates of the particles - whereas in others, they were parameters in the Hamiltonian. This implies a different treatment of the order parameter in the equations of motion. If one, however, applies the formalism of metadynamics, or extended dynamics, in which any parameter can be treated as a dynamical variable, most conceptual differences between these two cases vanish. 

Since there is no information about the site, a scoring function based on free energy methods cannot be used. 

Other variations on these basic free energy methods have been published, although for various reasons they have not yet been widely adopted. These methods include MD/MC methods,38 the acceptance ratio method,39, 40 the weighted histogram method,41 the particle insertion method,42 43 and the energy distribution method.39 The reader is referred to the original publications for additional discussion of these approaches. 

To address these needs, a variety of methods have been developed allow approximate free energies to be calculated. These methods are based, in one way or another, on the precise free energy methods described above. But they make various assumptions or simplifications that allow them to be carried out much more quickly. All of these methods have shown promise on limited data, but as of yet, all are still in the development stage. 

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