Energy-minimization methods

Energy minimization methods that exploit information about the second derivative of the potential are quite effective in the structural refinement of proteins. That is, in the process of X-ray structural determination one sometimes obtains bad steric interactions that can easily be relaxed by a small number of energy minimization cycles. The type of relaxation that can be obtained by energy minimization procedures is illustrated in Fig. 4.4. In fact, one can combine the potential U r) with the function which is usually optimized in X-ray structure determination (the R factor ) and minimize the sum of these functions (Ref. 4) by a conjugated gradient method, thus satisfying both the X-ray electron density constraints and steric constraint dictated by the molecular potential surface. 

Chymotrypsin, 170,171, 172, 173 Classical partition functions, 42,44,77 Classical trajectories, 78, 81 Cobalt, as cofactor for carboxypeptidase A, 204-205. See also Enzyme cofactors Condensed-phase reactions, 42-46, 215 Configuration interaction treatment, 14,30 Conformational analysis, 111-117,209 Conjugated gradient methods, 115-116. See also Energy minimization methods Consistent force field approach, 113 Coulomb integrals, 16, 27 Coulomb interactions, in macromolecules, 109, 123-126... 

Electrostatic stabilization, 181, 195,225-228 Empirical valence bond model, see Valence bond model, empirical Energy minimization methods, 114-117 computer programs for, 128-132 convergence of, 115 local vr. overall minima, 116-117 use in protein structure determination,... 

See also Energy minimization methods computer program for, 130-132 Nonbonded interactions, 56,61 Normal modes analysis, 117-119 computer program for, 132-134... 

In the case of crystals, both intramolecular (conformational) and packing energies should be taken into account simultaneously. Such a total energy minimization method, with suitable crystallographic constraints, has been applied in different steps of the analysis of crystalline structures of three different synthetic polymers. Structures of these molecules, namely, isotactic trans-1,4-poly-penta-1,3-diene (ITPP), poly-pivalolactone (PPVL), and isotactic cis-1,4-poly(2-methyl-penta-1,3-diene)(PMPD), do not have troublesome features such as charged groups, counterions, and solvent molecules. 

Combined Monte Carlo/Energy Minimization Methods... 

The Gibbs energy minimization method allows for calculations of the formation conditions for any phase (including the hydrate). It also allows for the calculation of phases present at any T and P (whether hydrates are present or not). Therefore, included are the options to perform all thermodynamic calculations with every phase and not just the hydrate. The types of calculations, combined with plotting capability, included in CSMGem are... 

In this chapter we consider in detail how one goes about performing molecular mechanics calculations. There are a number of important considerations such as having an adequate starting model, choosing an appropriate energy minimization method, and the probability that there are many possible energy minima. There are also pitfalls that need to be avoided such as false minima, saddle points and unstable refinements. 

Figure 6. Application of packing energy minimization method to poly(ethylene oxide) (16). (a) Starting model of uniform helix (b) stable crystal structure model obtained by energy minimization calculations and (c) the structure determined by x-ray analysis.

The study of chemical equilibrium can detect thermodynamic constraints on the achievable conversion and selectivity. In this section we make use of the Gibbs free-energy minimization method available in Aspen Plus [9], We assume that both cyclohexanone and cyclohexanol are products. The curves in Figure 5.2 show the evolution of the phenol equilibrium conversion, yield and selectivity with the ratio hydrogen/phenol at temperatures of 180, 200, 220 °C and a pressure of 3 bar. 

Thus, if we restrict ourselves to those simulations that have a clear quantum-chemical component, in the simplest case one is basing the procedure on perturbation theory, with a model in which the protein is represented by a collection of point particles (i.e., the atoms), with some distance constraints in an attempt to account for the existence of bonds, using an energy minimization method with appropriate computational techniques, the final... 

In order to calculate Ug(r) and Ud r) from the specified operating parameters Ug and Gs, it is necessary to know the dominant factor defining radial heterogeneity. Extending the energy minimization method to overall hydrodynamics, a stable radial profile calls for not only Model LR for local hydrodynamics at every point, but also the minimization of the cross-sectional average Nst for overall stability, which is defined by... 

In typical environmental situations, the liquid phase may contain a number of ions which can form complexes with the multiple charge cations present. In order to be able to calculate the thermodynamic driving force for the precipitation of a particular solid phase, it is necessary to determine the concentrations of free lattice ions in the solutions. The commonly used procedures for the calculations of solute components in a homogeneous solution are the "equilibrium constant and the "free energy minimization" methods. The former utilizes an approach wherein first, a "basis" is selected from the species, usually that having the highest concentration at equilibriiam. 

The other solution species are formed from this "basis" by a series of chemical reactions, and their concentrations can be expressed through the use of equilibrium constants, in terms of the concentration of the chosen "basis". The resulting set of nonlinear simultaneous equations consists of as many unknowns as there are elements and can be solved by conventional niamerical methods. The "free energy minimization" method utilizes only free energy criteria for chemical equilibria making no distinction among the constituent species and is essentially a constraint non-linear minimization problem. A number of search methods have... 

Two approaches to the treatment of large systems have been developed the Gibbs energy minimization method and the equilibrium constant approach. Both are based on a knowledge of the chemical potentials of species under standard conditions and under actual conditions of composition of conditions in the chemical system. In the Gibbs energy minimization approach (Clason,... 

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