Free perturbation

The results, described in the following section, test the efficacies of the spin-free perturbative and CEPA approximants of the SS-MRCC method, along with the APl-SSMRPT versions. 

The first important dynamic CyD studies published in 1987 and 1988 were the result of cooperation by the theoreticians van Gunsteren and Koehler with the X-ray specialist Saenger [98, 99, 100, 101, 102]. The aim of these studies was twofold on the one hand, they served the development of the GROMOS force field [103] while on the other they had to show that DSs for such complicated systems as CyDs were feasible. Starting from the experimental structures, these simulations of 15 or 20 ps, very short by today s standards, are of historical interest only. Similarly, the work by Mark et al. [104] of 1994 on free perturbation calculations was mainly devoted to the development of the method. 

It follows that the change in Ftelmholtz free energy due to the perturbation is... 

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

Perturbation theory is also used to calculate free energy differences between distinct systems by computer simulation. This computational alchemy is accomplished by the use of a switching parameter X, ranging from zero to one, that transfonns tire Hamiltonian of one system to the other. The linear relation... 

Free energy perturbation (FEP) theory is now widely used as a tool in computational chemistry and biochemistry [91]. It has been applied to detennine differences in the free energies of solvation of two solutes, free energy differences in confonnational or tautomeric fonns of the same solute by mutating one molecule or fonn into the other. Figure A2.3.20 illustrates this for the mutation of CFt OFl CFt CFt [92]. 

Wesolowski T A and Warshel A 1994 Ab initio free energy perturbation calculations of solvation free energy using the frozen density functional approach J. Phys. Chem. 98 5183... 

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]. 

Bash, P.A., Field, M.J.,Karplus, M. Free energy perturbation method for chemical reactions in the condensed phase A dynamical approach baaed on a combined quantum and molecular dynamics potential. J. Am. Chem. Soc. 109 (1987) 8092-8094. 

S. Miyamoto and P. A. Kollman. Absolute and relative binding free energy calculations of the interaction of biotin and its analogs with streptavidin using molecular dynamics/free energy perturbation approaches. Proteins, 16 226-245, 1993. 

Such a free energy is called a potential of mean force. Average values of Fs can be computed in dynamics simulations (which sample a Boltzmann distribution), and the integral can be estimated from a series of calculations at several values of s. A third method computes the free energy for perturbing the system by a finite step in s, for example, from si to S2, with... 

This principle has been applied in a contribution by Mark, Schafer, Liu and van Gunsteren to this volume, and in section 6 of this article. For a review of free energy perturbation methods see [8]. 

The problems that occur when one tries to estimate affinity in terms of component terms do not arise when perturbation methods are used with simulations in order to compute potentials of mean force or free energies for molecular transformations simulations use a simple physical force field and thereby implicitly include all component terms discussed earlier. We have used the molecular transformation approach to compute binding affinities from these first principles [14]. The basic approach had been introduced in early work, in which we studied the affinity of xenon for myoglobin [11]. The procedure was to gradually decrease the interactions between xenon atom and protein, and compute the free energy change by standard perturbation methods, cf. (10). An (issential component is to impose a restraint on the... 

Fig. 5. To generate an ensemble using Molecular Dynamics or Monte-Carlo simulation techniques the interaction between all pairs of atoms within a given cutoff radius must be considered. In contrast, to estimate changes in free energy using a stored trajectory only those interactions which are perturbed need be determined making the approach highly efficient.

Mark, A.E. Free energy perturbation calculations. Encyclopaedia of Computational Chemistry, Wiley, New York, (1998) (in press). 

T.P. Lybrand, Computer simulations of biomolecular systems using molecular dynamics and free energy perturbation methods, in Reviews in Computational Chemistry, Vol. 1, K.B. Lipkowitz, D.B. Boyd (Eds.), VCH, New York, 1990, pp. 295-320. 

It is often the case that the solvent acts as a bulk medium, which affects the solute mainly by its dielectric properties. Therefore, as in the case of electrostatic shielding presented above, explicitly defined solvent molecules do not have to be present. In fact, the bulk can be considered as perturbing the molecule in the gas phase , leading to so-called continuum solvent models [14, 15]. To represent the electrostatic contribution to the free energy of solvation, the generalized Bom (GB) method is widely used. Wilhin the GB equation, AG equals the difference between and the vacuum Coulomb energy (Eq. (38)) ... 

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