Conditional free energy

In Eq. (17-63), 8v = dist - E is imposed and 8jl can be computed by using an approximate functional introduced in the next subsection. As a result, the total excess chemical potential Afi in Eq. (17-59) can be rewritten as... 

The quantity U , i , x is computed by summing over the ECEPP energies of all pairs of nonbonded atoms of the whole molecule. Equations [11] and [12] are used to compute conditional free energy (4>,i]i) contour maps, similar to standard conformational energy (< >, ) contour maps, for each residue of the polypeptide. The probable conformation of the /th residue is taken as the one of lowest conditional free energy, and the probable conformation of the whole polypeptide chain is assumed to be the combination of the probable conformations of the individual residues. 

The class composite-operator contains procedures that transform a set of chemical species of predisposed behaviors into a set of products, provided that an optional set of prespecified conditions is satisfied. These conditions can be specified by the user or imposed by the system. They may encompass virtually any symbolically encodable concept structural character, chemical behavior, spatial orientation, reaction conditions, free-energy requirements, enthalpy considerations, toxicity, etc. An instance of the composite-operator (e.g., k) calls on the following procedures to carry out the corresponding tasks ... 

Now we have two functions that can be used to predict spontaneity the entropy of the universe, which applies to all processes, and free energy, which can be used for processes carried out at constant temperature and pressure. Since so many chemical reactions occur under the latter conditions, free energy is the more useful to chemists. 

For spontaneous processes at constant temperature and pressure it is the Gibbs free energy G that decreases, while at equilibrium under such conditions dG = 0. 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

The alternative to direct simulation of two-phase coexistence is the calculation of free energies or chemical potentials together with solution of the themiodynamic coexistence conditions. Thus, we must solve (say) pj (P) = PjjCT ) at constant T. A reasonable approach [173. 174. 175 and 176] is to conduct constant-AT J simulations, measure p by test-particle insertion, and also to note that the simulations give the derivative 3p/3 7 =(F)/A directly. Thus, conducting... 

Free energy is related to two other energy quantities, the enthalpy (the heat of reaction measured at constant pressure) and the entropy. S. an energy term most simply visualised as a measure of the disorder of the system, the relationship for a reaction taking place under standard conditions being... 

Wood, R.H. Continuum electrostatics in a computational universe with finite cut-off radii and periodic boundary conditions Correction to computed free energies of ionic solvation. J. Chem. Phys. 103 (1995) 6177-6187. 

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

Equilibrium constants for protein-small molecule association usually are easily measured with good accuracy it is normal for standard free energies to be known to within 0.5 kcal/mol. Standard conditions define temperature, pressure and unit concentration of each of the three reacting species. It is to be expected that the standard free energy difference depends on temperature, pressure and solvent composition AA°a also depends on an arbitrary choice of standard unit concentrations. 

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