Thermodynamic integration methods

Gouda, H. Kuntz, I.D. Case, D.A. Kollman, P.A., Free energy calculations for theophylline binding to an RNA aptamer Comparison of MM-PBSA and thermodynamic integration methods, Biopolymers 2003, 68,16-34. 

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]. 

The evaluation of the free energy is essential to quantitatively treat a chemical process in condensed phase. In this section, we review methods of free-energy calculation within the context of classical statistical mechanics. We start with the standard free-energy perturbation and thermodynamic integration methods. We then introduce the method of distribution functions in solution. The method of energy representation is described in its classical form in this section, and is combined with the QM/MM methodology in the next section. 

Free-Energy Perturbation and Thermodynamic Integration Methods... 

The standard and often used methods to circumvent the difficulty associated with the form of Eq. (17-30) are the free-energy perturbation and thermodynamic integration methods [13,42,43], These methods are generally applicable to free-energy... 

In the thermodynamic integration method, the intermediate states are introduced with respect to the coupling parameter A(0 < k < 1) The potential function at the coupling parameter of A. is denoted by Ux and satisfies Ux = Uo and Ux = U at the initial and final states (A. = 0 and 1), respectively. The intermediate states correspond to 0 < A < 1. The form of averaging-the-exponential is then avoided by rewriting Eq. (17-28) as... 

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 ... 

With the thermodynamic integration method the difference is evaluated with a 12-point Gaussian quadrature. 

The anbarmonic contribution to the free energy is evaluated by a thermodynamic integration method with a reference system of harmonic oscillators. This free energy difference between the real and the reference system A - is given by... 

The GCMC is successfully applied to many adsorption systems. For temperatures greater than the pore critical temperature, the adsorption isotherms exhibit a smooth behaviour. However, for temperatures less than the pore critical temperature, there is a possibility of transition. > en there is transition, the equilibrium point may be obtained by applying the thermodynamic integration method [12]. 

Solid-state phase transitions of salts have been studied by fitting the pressure and internal energy of each phase to an equation of state and determining the temperature for which AG = 0 at each pressure. Simulations for each solid-state phase are performed separately in the NVE ensemble.[158] In general, the thermodynamic -integration method discussed in Sec. 4.1 can be used to study solid-state phase transitions as well. 

Recently, Kollman and coworkers calculated the effect of a methyl substitution on the a versus P carbon atom of the unnatural amino acid (S-2-amino-3-cyclopentylpropanoic acid) in position 133 on T4 lysozyme stability. The authors used AMBER 4.1 and compared the results obtained by thermodynamic integration method with the results obtained by newer methods. 2 5 The study determined that the methyl group substitution on the a carbon stabilizes T4 lysozyme by 1.8 kcal/mol, whereas the methyl substitution on the P carbon destabilizes the protein by 0.4 kcal/mol. The result is in good agreement with the less CPU-intensive method, pictorial representation of free energy changes (PROFEC). 

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