Empirical potential functions

An ideal empirical potential function on the residue level is a function V that assigns to each sequence-coordinates pair s,x) an energy V s,x) such that... 

While this is disappointing, the nonuniqueness theorem also shows that if some empirical potential is able to predict correct protein folds then many other empirical potentials will do so, too. Thus, the construction of empirical potentials for fold prediction is much less constrained than one might think initially, and one is justified in using additional qualitative theoretical assumptions in the derivation of an appropriate empirical potential function. 

Finally some other developments are briefly mentioned for the sake of completeness. These include the work of Sposito and Babcock 184> which bears close resemblance to the partitioned potential methods. Their idea is to solve the quantum mechanical energy spectrum of the complex using only the empirical potential function as the potential operator in the Hamiltonian. This spectrum then leads to calculations of temperature-dependent energies of formation. 

A. T. Hagler, A. Lapicirella, Spatial Electron Distribution and Population Analysis of Amides, Carboxylic Acid, and Peptides, and Their Relation to Empirical Potential Functions , Biopolymers 1976, 1167-1200 A. T. Hagler, L. Leiserowitz, M. Tuval, Experimental and Theoretical Studies of the Barrier to Rotation about the N-C° and Ca-C Bonds ([Pg.369]

The Gordon-Kim interaction functions may be compared with empirical potential functions derived by energy- or net-force minimization methods using known crystal structures. The O—O Gordon-Kim potentials are more repulsive, as illustrated in Fig. 9.2. Spackman points out that the empirical potentials likely contain a significant attractive component because of the inadequate allowance for electrostatic interactions in their derivation. This attractive component is included in the electrostatic interaction in the density functional model. 

Vedani, A., Dobler, M., and Dunitz, J. D. (1986). An empirical potential function for metal centers Application to molecular mechanics calculations on metallo proteins.y. Corn-put. Chem. 7, 701-710. 

Beardmore etal. [Ill] have presented a realistic empirical potential function to model the head-group interaction for SAMs of alkanethiols on Au(lll). The main result of these calculations is that the barriers within the surface corrugation potential are too small to pin S atoms at any particular site. 

Two groups of workers121,122 have, independently, performed calculations of the free energies of these molecules, using semi-empirical, potential functions. The calculated compositions agreed well with those found experimentally. The change of composition on acetylation, and on protonation, of the amino group appears to be caused by electrostatic interactions. 

Vibrational frequency (We) the vibrational frequency is determined by fitting a harmonic or other empirical potential function to the calculated potential-energy curves. 

Calculations have been performed on atom clusters by Hoare and Pal (55) and Burton (39) to determine their geometrical and thermodynamic properties using empirical potential functions. In this technique the pairwise potentials V(Rjj) between atoms, a distance Ry apart, are assumed additive to give a potential energy... 

There are other empirical potential functions which are especially useful to describe intermolecular potentials, i.e., the potentials between atoms, which are not connected by chemical bonds. Two of these (the Buckingham and the Lennard-Jones potential) are discussed in Secs. 2.5.4 and 5.2. 

Empirical Potential Functions. There is one more type of treatment which is properly classified as an empirical model of the H bond. An explicit form is assumed for the potential function associated with the movement of the H atom within the H bond. Nordman and Lipscomb have proposed such a model based upon a Morse function (1521). This proposal will not be explored, however, in favor of the similar and more completely developed potential Junction model of Lippincott and Schroeder (1242, 1815). The potential function is written as a sum of four terms, the first of which has the form... 

The usefulness of this function, like that of the Morse function for diatomic molecules, depends upon its applicability to observational data and upon its convenience. Unfortunately it does not possess the virtue of simplicity, a substantial deficiency from the standpoint of convenience. Whether its applicability to data will stimulate general acceptance of the function remains to be seen. Nevertheless, the goal of discovering an empirical potential function which expresses the energetics of the vibrational degrees of freedom of the H bond is one of considerable interest and value. (See also 989, 2070, 1792, 112, 1951, 1078.)... 

Figure 4. Comparison of Mott-Littleton, periodic empirical potential functions and periodic DFT quantum mechanics. The horizontal scale is not significant. Transition state energies are shown as open symbols...

A variant of the combined QM/MM approach introduces a hybrid description of the solute. The main motivation for the introduction of this additional approximation lies in computational costs. Combined QM/MM calculations are quite costly, even when all the possible simplifications are introduced in the QM part and in the MM interaction potentials. On the other hand, QM formulation is more reliable than an empirical potential function to describe chemical reactions which involve bond-formation and disruption processes. To temperate contrasting factors, i.e. the need for a QM description and the computational costs, one may resort to the well established fact that, in chemical reactions, the quantum bond-breaking and bond-forming processes are limited to a restricted portion of the molecular system, with the remainder playing an auxiliary role. Hence, it may be convenient to resort to hybrid descriptions, where the active part of the molecule is described at the QM level and the remainder via MM potentials. 

Finally we note in brief some approaches to solve a one-dimensional Schrodinger equation with a given empirical diatomic potential function. The list of empirical potential functions known from the literature is very... 

To obtain the accuracy required for a realistic analysis of the structure and dynamics of macromolecules it is necessary to use a relatively complex form for the empirical potential function and to optimize the values of the parameters that determine the magnitudes of the different contributing terms. In general, the function will have terms that depend not only on the relative position of all pairs of atoms but certain triples and quadruples of atoms as well. Usually, one does not need to go beyond four-body terms in the model potential function. This approach to calculating energies is often referred to as molecular mechanics.6061... 

It has also been demonstrated that molecular dynamics can play a useful role in the refinement of protein structures against X-ray data.420a By adding an effective potential that represents the difference between the observed and calculated structure factors (Eq. 99) to the standard empirical potential function (Eq. 6), simulated annealing4206 can be used to automatically refine a crude X-ray structure. In this way much of the manual rebuilding of the model structure, that is, the most time-consuming part of standard structure refinement,421 can be avoided. 

T. Oie, G. M. Maggiora, R. E. Christoffersen, and D. J. Duchamp, Int. J. Quantum Chem., Quantum Biol. Symp., 8, 1 (1981). Development of a Flexible Intra- and Intermolecular Empirical Potential Function for Large Molecular Systems. 

Organometallic systems necessarily exhibit greater diversity in the forms of bonding that must be accounted for by empirical potential functions than do their organic counterparts and Molecular Mechanics calculations have accordingly played a smaller role for these materials. However, a number of studies have now been reported which extend the isolated description (Allured et al., 1991 Castonguay and Rappe, 1992 Rappe et al., 1992) of such compounds to the solid state (Talamo et al., 1995). 

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