Thermodynamic Perturbation Methods

Bash et al. (1987) applied the thermodynamic perturbation method to complexes of thermolysin with a phosphonamidate [Cbz-Gly -(NH)-Leu-Leu] and the corresponding phosphonate inhibitor [Cbz-Gly -(0)-Leu-Leu]. The perturbation was carried out by using 20 windows, with 2-psec molecular dynamics simulations in each window. Computations were for the ligand in solution and bound to the enzyme. The solvation of the enzyme was represented by a spherical cap of 168 water molecules about the bound inhibitor. The difference in free energy of binding of the two inhibitors was calculated to be 4.38 kcal/mol, to be compared with the experimental value, 4.10 kcal/mol. These calculations point out the importance of solvation effects, which are seen in the 3.4 kcal/mol difference between the NH and O forms of the inhibitor. 

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 ... 

An extension to the pseudo-receptor approach is to utilize the thermodynamic perturbation method (8) to calculate approximate values for the free energy of binding of various... 

The implementation of the thermodynamic perturbation methods is relatively straightforward. An ensemble generated by a Monte Carlo simulation or the time trajectory generated by a molecular dynamics simulation for a system described by Hamiltonian "Kq is used to evaluate the ensemble average of expi-AM/k T). The free energy difference between the reference system described by Hamiltonian o> for which the ensemble is generated, and the perturbed system with Hamiltonian + A% is found using Eq, [17],... 

Figure 1 Schematic illustration of the forward, reverse, and double-wide sampling thermodynamic perturbation methods.

M. R. Reddy and M. D. Erion, J. Comput. Chem., 20,1018 (1999), Calculation of Relative Solvation Free Energy Differences by Thermodynamic Perturbation Method Dependence of the Free Energy Results on the Simulation Length. 

Kovalenko A, Hirata F, Kinoshita M Hydration structure and stabdity of Met-enkephalin studied by a three-dimensional reference interaction site model with a repulsive bridge correction and a thermodynamic perturbation method, J Chem Phys 113(21) 9830—9836, 2000. 

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 ... 

Considerable use has been made of the thermodynamic perturbation and thermodynamic integration methods in biochemical modelling, calculating the relative Gibbs energies of binding of inhibitors of biological macromolecules (e.g. proteins) with the aid of suitable thermodynamic cycles. Some applications to materials are described by Alfe et al. [11]. 

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

Because of the long-range and reduced symmetry of the dipole-dipole interaction, analytical methods such as the thermodynamic perturbation theory presented in Section II.B.l. will be applicable only for weak interaction. Numerical simulation techniques are therefore indispensable for the study of interacting nanoparticle systems, beyond the weak coupling regime. 

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