Approximate Free Energy Calculations

One should assume that at least 200 ps of equilibration+sampling will be required for any reliable simulation in explicit water solvent. Since each simulation should be run at least twice (or forwards and backwards) to ensure a reproducible result, this means a floor of 400 ps simulation time will be required. Note that 200 ps (400 ps) is a lower bound, and that many simulations will need to be run considerably longer. It is not unusual to run protein-based simulations for a nanosecond or more to achieve convergence. For a large (protein based) system, this requires a substantial investment of computer time on today s computers. 

Aerr is the proportionality constant from Equation 23, Aerr= ( r2(Xi)(l + 2r))1/2 which depends on variance and correlation in 

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. 

The linear interaction energy method allows the approximation of the free energy of binding using the relationships64 

Finally, surprisingly good initial results have been obtained with a simple approach that makes use of the Poisson-Boltzmann approach to estimating solvation free energies. In this method,74 76 

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Radmer, R.. 1., Kollman, P. A. Approximate free energy calculation methods and structiire based ligand design. J. Comp. Aid. Mol. Desgn (in press)... 

Radmer, R J. Kollman, P.A., The application of three approximate free energy calculations methods to structure based ligand design trypsin and its complex with inhibitors., J. Comput. Aided Mol. Des. 1998,12, 215-227... 

There now exist several methods for predicting the free energy associated with a compositional or conformational change.7 These can be crudely classified into two types "exact" and "approximate" free energy calculations. The former type, which we shall discuss in the following sections, is based directly on rigorous equations from classical statistical mechanics. The latter type, to be discussed later in this chapter, starts with statistical mechanics, but then combines these equations with assumptions and approximations to allow simulations to be carried out more rapidly. 

R. Radmer and P. Kollman, The application of three approximate free energy calculations... 

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

The titration coordinates evolve along with the dynamics of the conformational degrees of freedom, r, in simulations with GB implicit solvent models [37, 57], An extended Hamiltonian formalism, in analogy to the A dynamics technique developed for free energy calculations [50], is used to propagate the titration coordinates. The deprotonated and protonated states are those, for which the A value is approximately 1 or 0 (end-point states), respectively. Thus, in contrast to the acidostat method, where A represents the extent of deprotonation, is estimated from the relative occupancy of the states with A 1 (see later discussions). The extended Hamiltonian in the CPHMD method is a sum of the following terms [42],... 

The quality of the mean-field approximation can be tested in simulations of the same lattice model [13]. Ideally, direct free-energy calculations of the liquid and solid phases would allow us to locate the point where the two phases coexist. However, in the present studies we followed a less accurate, but simpler approach we observed the onset of freezing in a simulation where the system was slowly cooled. To diminish the effect of supercooling at the freezing point, we introduced a terraced substrate into the system to act as a crystallization seed [14]. We verified that this seed had little effect on the phase coexistence temperature. For details, see Sect. A.3. At freezing, we have... 

Verkhivker, G. Elber, R. Nowak, W., Locally enhanced sampling in free energy calculations application of mean held approximation to accurate calculation of free energy differences, J. Chem. Phys. 1992, 97, 7838-7841... 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

The Feynman-Hibbs and QFH models perform quite well in free energy calculations as long as the quantum corrections are modest. The conditions for validity of the approximations are given above. 

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]. 

Free Energy Calculations Approximate Methods for Biological Macromolecules... 

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. 

Overall, free energy calculations continue to evolve-they have gotten more reliable, faster, and (with the approximate methods) more universally applicable. As such, they remain, and will continue to remain, a vital part tool in the modeler s arsenal. 

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. 

The results of the free energy calculations for electron transfer in bulk water show that the full free energy curves are well approximated by paraboli. The calculations for electron transfer at the solution/metal interface are also, in general, in agreement with the linear response assumption. 

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