Free-energy calculations thermodynamic integration

Within a particle-based model, there is no well-defined reference state for the self-assembled structure. However, one can try to relate the seF-assembled structure to a disordered melt (or a different self-assembled morphology) via a reversible path and calculate the change of the free energy by thermodynamic integration. Typically, transitions between disordered and ordered morphologies or between different self-assembled structures are of first order. Thus, in an analogy to crystallization of hard condensed matter, there is no path in the space of physical intensive variables - for example, temperature, incompatibility, or composition - that reversibly cormects disordered and ordered structures. 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

From (9.27), we see that this approach will work nicely if the variance is always small Taylor s theorem with remainder tells us that the error of the first-derivative - mean-field - contribution is proportional to the second derivative evaluated at an intermediate A. That second derivative can be identified with the variance as in (9.27). If that variance is never large, then this approach should be particularly effective. For further discussion, see Chap. 4 on thermodynamic integration, and Chap. 6 on error analysis in free energy calculations. 

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. 

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. 

Although historically less common, free energy calculations based on a different equation from classical statistical mechanics have grown in popularity in recent years. These calculations, termed Thermodynamic Integration (TI), are based on the integral... 

An alternative approach to free energy calculations is the thermodynamic integration (TI) method,18 20 which considers the ensemble average of the first derivative of the hybrid potential with respect to A at various values of A... 

Blondel. A. 2004. Ensemble Variance in Free Energy Calculations by Thermodynamic Integration Theoiy. Optimal Alchemical Path, and Practical Solution. J. Compul. Chem., 25, 985. 

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. 

Molecular dynamics and minimization of proteins locally enhanced sampling and free energy calculations along reaction paths by perturbation or thermodynamic integration. Stardent, Silicon Graphics, IBM, and HP workstations, moil-view for visualization of shaded spheres and sticks on Silicon Graphics. Available by anonymous ftp from 128.248.186.70. 

Our free energy calculations, using the thermodynamic integration technique [37], show different results for solvation in the bulk phases and for the ion transfer across the interface in the two-phase system, which may be understood by hypothesizing the existence of an electrostatic potential at the ice/water interface. Calculations of the free energy profiles across the ice/water interface show opposite tendencies of the ions in the bulk crystal phase, which may also can be explained by that same hypothesis, namely the existence of an interface potential attractive to Cl- ions and repulsive to Na+ ions. 

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