Constraint energy function, distance constraints

Consider now the behaviour of the HF wave function 0 (eq. (4.18)) as the distance between the two nuclei is increased toward infinity. Since the HF wave function is an equal mixture of ionic and covalent terms, the dissociation limit is 50% H+H " and 50% H H. In the gas phase all bonds dissociate homolytically, and the ionic contribution should be 0%. The HF dissociation energy is therefore much too high. This is a general problem of RHF type wave functions, the constraint of doubly occupied MOs is inconsistent with breaking bonds to produce radicals. In order for an RHF wave function to dissociate correctly, an even-electron molecule must break into two even-electron fragments, each being in the lowest electronic state. Furthermore, the orbital symmetries must match. There are only a few covalently bonded systems which obey these requirements (the simplest example is HHe+). The wrong dissociation limit for RHF wave functions has several consequences. 

The first step of the structure refinement is the appHcation of distance geometry (DG) calculations which do not use an energy function but only experimentally derived distances and restraints which follow directly from the constitution, the so-caUed holonomic constraints. Those constraints are, for example, distances between geminal protons, which normally are in the range between 1.7 and 1.8 A, or the distance between vicinal protons, which can not exceed 3.1 A when protons are in anti-periplanar orientation. 

The role of the potential energy is taken by the Dyana target function [8, 28] that is defined such that it is zero if and only if all experimental distance constraints and torsion angle constraints are fulfilled and all nonbonded atom pairs satisfy a check for the absence of steric overlap. A conformation that satisfies the constraints more closely than another one will lead to a lower target function value. The exact definition of the Dyana target function is ... 

To generate the distance constraints to be used in the constraint energy function, E(NOE), the spatial contacts were established on the basis of a 2D-N0ESY experiment with a 400 ms mixing time performed on a Bruker VTH-400 NMR spectrometer. The following... 

As described in the Introduction, it is usually possible to consider the modeling of experimental data separately from the scheme actually used to move atoms about. Ideally, the different models should be able to be used in the different minimization or dynamics schemes. Thus, the subsequent sections describe the kind of data offered by NMR and the kinds of penalty functions or pseudo-energy terms that can be used to represent them. For convenience, we use nomenclature common to force field-based approaches where one refers to a distance constraint potential Vdc r) as a function of intemudear distance. 

Equation (2.2.12) may be directly obtained from minimizing the elastic free energy under the constraint that the mean-square radius of gyration has a fixed value [see Eqs. (2.1.63) and (2.1.39), C q) -+ ot q)C q) [10]. The physical meaning of this result is that under chain compression the free energy due to the interatomic contacts is basically a function of only, no matter what are the individual values of the a (q). As a consequence, all the mean-square distances (r (k)) may be expressed under a general form [53]. Defining... 

The potential is the same as the square-well potential when R < dup + rsw sw is the switching distance, a and b are chosen so that Enoe is a smooth function at the point R = dup + rgw and c is the asymptote. When tiie violation is very large, the constraint energy will only increase linearly with distance, so that the potential is effective at long range. 

Restraint A restraint biases or forces a target function such as the energy function in molecular mechanics toward a specific value for a degree of freedom. Various restraints are in common use torsional restraints, distance restraints, and tethering. A constraint is the most restrictive version of restraint. 

Blommers et al. performed a conformational search on small peptides in an attempt to satisfy a set of NOE distance constraints. They used an SGA, the only variant on which was the use of a sharing operator to slow down convergence. At the end of the initial search phase, several interesting conformations were gradient minimized using an MM energy function. 

Recently, MD simulation has been applied as a tool in the refinement of three-dimensional biomolecular structures from X-ray diffraction and two-dimensional NMR data. In the refinement process, MD trajectories are run at elevated temperatures (perhaps several thousand degrees Kelvin) to enhance conformational sampling. The molecular systems are cooled periodically to permit the trajectories to settle into local minimum energy conformations. Constraint terms based on X-ray structure factors or NMR NOE distances are added to the standard potential energy functions, so that the MD trajectories relax to conformations that satisfy the experimental data as the systems are cooled. Thus, the complete potential function in a refinement simulation has the form... 

Set counter, c = 1. Perform local minimization using NPSOL with dihedral angle box constraints to implicitly enforce bounds. The objective function is a weighted combination of forcefield energy and distance restraint terms ... 

The symbol (, j] indicates that the sums are only over those pairs of atoms for which explicit distance constraints exist. Since in most problems the distance constraints are very sparse, such an error function can be computed much more rapidly than semi-empirical energy functions. The parameters (, Su are made large enough to avoid having wy single term become much bigger than the others when the distance or upper bound is very small, respectively. Each term in this error function is called a distance restraint. 

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