Potential energy function refinement

Sensitivity analysis is also a tool that can help to refine potential energy functions for (bio)molecular simulations. Sensitivity analysis can help one decide whether a specific feature needs to be included in a potential function for describing a specified set of properties of a given class of molecules. For example, because point charge models are commonly used in bio(molecular) modeling, it is useful to inquire whether a dispersed charge representation would improve the description of intra- and intermolecular electrostatic interactions. One study of this type was carried out by Zhu and Wong, ° who included in the force field a squared Lorentzian function f r - f/ ) of the form 

Correlations among parameters and observables can complicate a parameter refinement process. There may be insufficient data to determine N parameters of a force field given N experimental or theoretical data points if there exist significant correlations among the potential constants or among the data used for determining these parameters. A principal component analysis can 

Therefore, from a sensitivity analysis and a principal component analysis, one can gain insights into how many useful relations can be derived from a given set of experimental/theoretical data for refining force field parameters. These analyses can also be useful in the selection of suitable experimental/theoretical data to use for force field parameterization. Ideally, one would like to include the smallest amount of data containing the largest amount of information the judicious choice of experimental/theoretical data needed to accomplish this can help reduce the computational costs of refining a set of potential parameters. 

Different force fields with a similar functional form commonly have different values of analogous parameters. Even so, similar values of certain selected properties can often be obtained by those different force fields. The possibility exists of having many possible combinations of parameters that can describe certain properties with comparable accuracies, and it is consistent with 

Force fields must be relatively simple and computationally efficient for studying complex macromolecules such as proteins and DNAs. The force fields usually describe properties of certain types better than others, depending on how the force fields were developed. We have already learned from the sensitivity analysis studies of liquid water and a two-dimensional square lattice model of protein folding that different system properties can be determined by different features of a potential model. 

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A particularly important application of molecular dynamics, often in conjunction with the simulated annealing method, is in the refinement of X-ray and NMR data to determine the three-dimensional structures of large biological molecules such as proteins. The aim of such refinement is to determine the conformation (or conformations) that best explain the experimental data. A modified form of molecular dynamics called restrained moleculai dynarrdcs is usually used in which additional terms, called penalty functions, are added tc the potential energy function. These extra terms have the effect of penalising conformations... 

The additional penalty function that is added to the empirical potential energy function in restrained dynamics X-ray refinement has the form ... 

Most force fields used in coordination chemistry, in respect of the organic part of the molecules, are based on or are at least similar to the MM2 11 or AMBER 11 parameterization schemes, or mixtures thereof. However, it is of importance to stress again that transferring parameters from one force field to another without appropriate checks is not valid. This is not only a question of the different potential energy functions that may be used, but it is also a consequence of the interrelatedness of the entire set of parameters. Force field parameters imported from any source, whether a well-established force field or experimental data, should only be used as a starting point for further parameter refinement. 

There is a strong correlation between the parameters of different potential energy functions so that they should not be developed or refined in isolation. For example, the barrier to rotation about a bond can be modified by changing the explicit torsion angle term or by changing the nonbonded interactions. Thus, the effect of any change on a force field parameter needs to be tested extensively, i.e.,... 

Displacements derived from temperature factors have been compared with those obtained from molecular dynamics. In these calculations, an empirical potential energy function is expressed as a function of the positional co-ordinates of the atoms. This function is then used to o,.. ain the force on each atom (energy is a generalised force X a generalised displacement) and the Newtonian equations of motion are solved for a small time interval, usually a fraction of a picosecond. Good agreement has been obtained for BPTI [195] and cytochrome c [196]. There are likely to be significant developments in this field as the sophistication of both refinement and simulation methods is increased. 

Although the energy calculations described here are of interest, they have a number of limitations. The first of these is inherent in the inaccuracies of the empirical potential energy functions that are being used. These are known to be significant, as indicated by the sizable difference found between the minimum-energy structure obtained from the potential functions and the observed crystallographic structure, even when the calculations are done for the full crystal system.161,1613 Such errors can be reduced, in principle, by further refinements of the form of the potential function and the associated parameters. 

Once a successful set of rules for the folding of all a proteins is available, we can turn our attention to the known mainly a, a/jS and mainly p proteins and find, from a statistical analysis of their structures, which pairs of side chains destabilize helix-helix interactions and result in jS-sheet formation (previous investigations indicate that Phe-Phe, Phe-Tyr, Glu-Arg and Glu-Lys pairs are probable candidates for helix-helix destabilizers ). The rules thus derived from all the known protein structures can then be applied to the prediction of the three dimensional fold and packing of the secondary structures of other amino acid sequences whose structures are not known, that is, to the prediction of a cursory but full three-dimensional structure of a protein, which may be further refined by the application of standard potential energy functions. 

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... 

In general, atomic-level computer simulations of channels in membranes or membrane-mimetic systems have proven to be quite useful in refining model structures, often postulated somewhat ad hoc. However, full assessments of channel stability from simulations on the currently accessible timescales is not nearly as reliable. Owing to the long relaxation times required to equilibrate membrane-bound systems, it is not currently feasible to arrive at the correct, equilibrium structure starting from an arbitrary initial conditions. Even if the potential energy functions used in simulations were accurate, correct results could be expected only if the initial and equilibrium structures were not too different. 

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