Transferability of Force Field Parameters

The range of systems that have been studied by force field methods is extremely varied. Some force fields have been developed to study just one atomic or molecular species under a wider range of conditions. For example, the chlorine model of Rodger, Stone and Tildesley [Rodger etal. 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules depends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with respect to the bond vectors of the two molecules. The model includes an electrostatic component which contains dipole-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

The effective atomic charges are either obtained by fitting to data on diatomic molecules (where it exists) or by interpolation or extrapolation from this fit. 

In Equation (4.103) is the bond order about the central bond i-J of the torsion angle calculated for a torsion angle of zero and is the resonance integral from the molecular orbital calculation. The parameter A has a value of —009 and so the V2 term is lower for those conjugated bonds with a lower bond order. In Equation (4.104) p y is now the bond order for the bond i-j calculated for the torsion angle uj Ky, equals 1.25 and so V3 increases with decreasing bond order. A bond with a lower bond order (and so a lower V2 and a higher V3) is thus more likely to deviate from planarity. 

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Such results are interesting for force field development as they clearly establish the existence of conserved electrostatic blocks within amino acids, an encouraging step for transferability of force field parameters. 

We may now begin to explore the transferability and combination rules of force constants, partial charges, van der Waals parameters, etc., using the methods described above. The issue of the transferability of force field parameters is crucial to the practitioner of molecular mechanics applications. Is the carbonyl oxygen in an amide the same as in an acid —Are the intermolecular interactions between peptides the same as in model amide compounds Clearly, our ability to derive force fields from well-characterized model compounds and to transfer them to polymeric and biomolecular systems of interest depends substantially on our ability to answer such questions. 

Force-field calculations could be simple energy minimization or advanced monte-carlo and molecular dynamics calculations. The major assumption here is the transferability of force-field parameters among the related materials. These calculations can provide wealth of information such as the relative ordering of adsorption sites on surface, diffusion mechanism of molecules particularly inside zeolites, energy barrier for difihision, diffusion coefficients, heats of adsorption and more importantly, the effect of temperature on all these properties. 

Geerke DP, van Gunsteren WF (2007) Calculation of the fiee energy of polarization quantifying the effect of explicitly treating electrrauc polarization rai the transferability of force-field parameters. J Phys Chem B 111 6425-6436... 

D. P. Geerke and W. F. van Gunsteren,/. Phys. Chem. B, 111(23), 6425-6436 (2007). Calculation of the Free Energy of Polarization Quantifying the Effect of Explicitly Treating Electronic Polarization on the Transferability of Force-Field Parameters. 

The parameterization process may be done sequentially or in a combined fashion. In the sequential method a certain class of compound, such as hydrocarbons, is parameterized first. These parameters are held fixed, and a new class of compound, for example alcohols and ethers, is then parameterized. Tins method is in line with the basic assumption of force fields parameters are transferable. The advantage is that only a fairly small number of parameters are fitted at a time. The ErrF is therefore a relatively low-dimensional function, and one can be reasonably certain that a good minimum has been found (although it may not be the global minimum). The disadvantage is that the final set of parameters necessarily provides a poorer fit (as defined from the value of the ErrF) than if all the parameters are fitted simultaneously. 

The usefulness of quantum-chemical methods varies considerably depending on what sort of force field parameter is to be calculated (for a detailed discussion, see [46]). There are relatively few molecular properties which quantum chemistry can provide in such a way that they can be used directly and profitably in the construction of a force field. Quantum chemistry does very well for molecular bond lengths and bond angles. Even semiempirical methods can do a good job for standard organic molecules. However, in many cases, these are known with sufficient accuracy a C-C single bond is 1.53 A except under exotic circumstances. Similarly, vibrational force constants can often be transferred from similar molecules and need not be recalculated. 

The force fields have to be defined through their functional form as well as the parameters. Different force fields can use the same functionality with very different parameters (or force fields with different functional forms) and still provide results of comparable accuracy. This highlights the problem of transferability of force fields and parameters. Force fields are often developed to study certain characteristics and properties of materials and that information forms part of the force field itself. Force fields that are derived to reproduce the structural properties of solids and molecules do not always provide good results when calculating vibrational spectra. This does not mean that the force field fails it is simply being used outside its range of applicability. 

While simulations reach into larger time spans, the inaccuracies of force fields become more apparent on the one hand properties based on free energies, which were never used for parametrization, are computed more accurately and discrepancies show up on the other hand longer simulations, particularly of proteins, show more subtle discrepancies that only appear after nanoseconds. Thus force fields are under constant revision as far as their parameters are concerned, and this process will continue. Unfortunately the form of the potentials is hardly considered and the refinement leads to an increasing number of distinct atom types with a proliferating number of parameters and a severe detoriation of transferability. The increased use of quantum mechanics to derive potentials will not really improve this situation ab initio quantum mechanics is not reliable enough on the level of kT, and on-the-fly use of quantum methods to derive forces, as in the Car-Parrinello method, is not likely to be applicable to very large systems in the foreseeable future. 

The final part is devoted to a survey of molecular properties of special interest to the medicinal chemist. The Theory of Atoms in Molecules by R. F.W. Bader et al., presented in Chapter 7, enables the quantitative use of chemical concepts, for example those of the functional group in organic chemistry or molecular similarity in medicinal chemistry, for prediction and understanding of chemical processes. This contribution also discusses possible applications of the theory to QSAR. Another important property that can be derived by use of QC calculations is the molecular electrostatic potential. J.S. Murray and P. Politzer describe the use of this property for description of noncovalent interactions between ligand and receptor, and the design of new compounds with specific features (Chapter 8). In Chapter 9, H.D. and M. Holtje describe the use of QC methods to parameterize force-field parameters, and applications to a pharmacophore search of enzyme inhibitors. The authors also show the use of QC methods for investigation of charge-transfer complexes. 

The molecular mechanics method is extremely parameter dependent. A force field equation that has been empirically parameterized for calculating peptides must be used for peptides it cannot be applied to nucleic acids without being re-parameterized for that particular class of molecules. Thankfully, most small organic molecules, with molecular weights less than 800, share similar properties. Therefore, a force field that has been parameterized for one class of drug molecules can usually be transferred to another class of drug molecules. In medicinal chemistry and quantum pharmacology, a number of force fields currently enjoy widespread use. The MM2/MM3/MMX force fields are currently widely used for small molecules, while AMBER and CHARMM are used for macromolecules such as peptides and nucleic acids. 

The authors finish by exploring the transferability of their force field parameters to a different zeolite, namely, silicalite. In this instance, a Fourier transform of the total dipole correlation function provides another model infrared (IR) spectrum for comparison to experiment, and again excellent agreement is obtained. Dominant computed bands appear at 1099, 806, 545, and464 cm while experimental bands are observed at 1100, 800,550, and 420 cm A Some errors in band intensity are observed in the lower energy region of tlie spectrum. 

Therefore, two parts of any molecular mechanics package that have a direct influence on a particular optimized structure, i. e., on the nuclear coordinates of a specific energy minimum on the calculated potential energy surface, are the mathematical functions and the corresponding parameters. The potential energy functions and the force field parameters are interrelated and, therefore, the parameters may not, in general, be transferred from one force field into another. 

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. 

To finalize the development of the aqueous CO2 force field parameters, the C02 model was used in free energy perturbation Monte Carlo (FEP/MC) simulations to determine the solubility of C02 in water. The solubility of C02 in water is calculated as a function of temperature in the development process to maintain transferability of the C02 model to different simulation techniques and to quantify the robustness of the technique used in the solubility calculations. It is also noted that the calculated solubility is based upon the change in the Gibbs energy of the system and that parameter development must account for the entropy/enthalpy balance that contributes to the overall structure of the solute and solvent over the temperature range being modeled [17]. 

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