Free energy methods thermodynamic perturbation

Simulations, Time-dependent Methods and Solvation Models 16.1 Simulation Methods 16.1.1 Free Energy Methods 16.1.2 Thermodynamic Perturbation Methods 16.1.3 Thermodynamic Integration Methods 16.2 Time-dependent Methods ill 373 380 380 381 383 ... 

Fne Enenv Methods Thermodynamic Perturbation or Free Energy Perturbation (FEP). 

From the free energy all thermodynamic properties of the system can be calculated. For example, the entropy is S = -3A/dTand the energy is E = A + TS. For more details we refer to the review by Werthamer (1976). One important point should be mentioned. Expanding the potential VppiQp,. Qp ) as a Taylor series in the displacement coordinates Qp, we observe, using the analog of Eq. (87) for the force constants (88), that the odd power terms do not contribute to the effective force constants the SCP method neglects these terms. Their relative importance can, of course, be estimated by perturbation techniques as described in Section III,B. 

This chapter reviews some theoretical aspects of the two most popular free energy difference methods, thermodynamic perturbation and thermodynamic integration, as well as assumptions and approximations made in the implementation. Advantages and disadvantages of certain implementations are discussed, and general recommendations are given for the practical application of these methods. 

What has been developed within the last 20 years is the computation of thermodynamic properties including free energy and entropy [12, 13, 14]. But the ground work for free energy perturbation was done by Valleau and Torrie in 1977 [15], for particle insertion by Widom in 1963 and 1982 [16, 17] and for umbrella sampling by Torrie and Valleau in 1974 and 1977 [18, 19]. These methods were primarily developed for use with Monte Carlo simulations continuous thermodynamic integration in MD was first described in 1986 [20]. 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy 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 natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. 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. 

Tobias, D. J. Brooks III, C. L., Calculation of free energy surfaces using the methods of thermodynamic perturbation theory, Chem. Phys. Lett. 1987,142, 472-476... 

Thermodynamic perturbation theory represents a powerful tool for evaluating free energy differences in complex molecular assemblies. Like any method, however, FEP has limitations of its own, and particular care should be taken not only when carrying out this type of statistical simulations, but also when interpreting their results. We summarize in a number of guidelines the important concepts and features of FEP calculations developed in this chapter ... 

The next three chapters deal with the most widely used classes of methods free energy perturbation (FEP) [3], methods based on probability distributions and histograms, and thermodynamic integration (TI) [1, 2], These chapters represent a mix of traditional material that has already been well covered, as well as the description of new techniques that have been developed only recendy. The common thread followed here is that different methods share the same underlying principles. Chapter 5 is dedicated to a relatively new class of methods, based on calculating free energies from nonequilibrium dynamics. In Chap. 6, we discuss an important topic that has not received, so far, sufficient attention - the analysis of errors in free energy calculations, especially those based on perturbative and nonequilibrium approaches. 

Whilst this Chapter is primarily concerned with the methods of determining the free energies of tautomeric or ionisation equilibria via computer simulation of free energy differences, many of the issues raised relate also to the determination of other molecular properties upon which behaviour of the molecule within the body may depend, such as the redox potential or the partition coefficient.6 In the next section, we shall give a brief explanation of the methods used to calculate these free energy differences -namely the use of a thermodynamic cycle in conjunction with ab initio and free energy perturbation (FEP) methods. This enables an explicit representation of the solvent environment to be used. In depth descriptions of the various simulation protocols, or the accuracy limiting factors of the simulations and methods of validation, have not been included. These are... 

The free energy perturbation calculations on mutation of the central statine residue of pepstatin to its dehydroxy and other derivatives were carried out using the window method. The crystal structure reported by Suguna et al.l4 l5was used for these calculations. In most simulations, the mutations were achieved either in 101 or 51 windows with 0.4 ps of equilibration and 0.4 ps of data collection at each window. The calculation for each mutation was repeated in water to determine the difference in the free energies of solvation and to complete the thermodynamic cycle. 

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