Force field (chemistry)

Functional form

The basic functional form of potential energy in molecular mechanics includes bonded terms for interactions of atoms that are linked by covalent bonds, and nonbonded (also termed noncovalent) terms that describe the long-range electrostatic and van der Waals forces. The specific decomposition of the terms depends on the force field, but a general form for the total energy in an additive force field can be written as   E total = E bonded + E nonbonded where the components of the covalent and noncovalent contributions are given by the following summations:

  E bonded = E bond + E angle + E dihedral

  E nonbonded = E electrostatic + E van der Waals

The bond and angle terms are usually modeled by quadratic energy functions that do not allow bond breaking. A more realistic description of a covalent bond at higher stretching is provided by the more expensive Morse potential. The functional form for dihedral energy is highly variable. Additional, "improper torsional" terms may be added to enforce the planarity of aromatic rings and other conjugated systems, and "cross-terms" that describe coupling of different internal variables, such as angles and bond lengths. Some force fields also include explicit terms for hydrogen bonds.

The nonbonded terms are most computationally intensive. A popular choice is to limit interactions to pairwise energies. The van der Waals term is usually computed with a Lennard-Jones potential and the electrostatic term with Coulomb's law, although both can be buffered or scaled by a constant factor to account for electronic polarizability and produce better agreement with experimental observations.

Parameterization

In addition to the functional form of the potentials, force fields define a set of parameters for different types of atoms, chemical bonds, dihedral angles and so on. The parameter sets are usually empirical. A force field would include distinct parameters for an oxygen atom in a carbonyl functional group and in a hydroxyl group. The typical parameter set includes values for atomic mass, van der Waals radius, and partial charge for individual atoms, and equilibrium values of bond lengths, bond angles, and dihedral angles for pairs, triplets, and quadruplets of bonded atoms, and values corresponding to the effective spring constant for each potential. Most current force fields parameters use a fixed-charge model by which each atom is assigned one value for the atomic charge that is not affected by the local electrostatic environment; proposed developments in next-generation force fields incorporate models for polarizability, in which a particle's charge is influenced by electrostatic interactions with its neighbors. For example, polarizability can be approximated by the introduction of induced dipoles; it can also be represented by Drude particles, massless, charge-carrying virtual sites attached by a springlike harmonic oscillator potential to each polarizable atom. The introduction of polarizability into force fields in common use has been inhibited by the high computational expense associated with calculating the local electrostatic field.

Although many molecular simulations involve biological macromolecules such as proteins, DNA, and RNA, the parameters for given atom types are generally derived from observations on small organic molecules that are more tractable for experimental studies and quantum calculations. Different force field parameters can be derived from dissimilar types of experimental data, such as enthalpy of vaporization (OPLS), enthalpy of sublimation, dipole moments, or various spectroscopic parameters.

Parameter sets and functional forms are defined by interatomic potentials developers to be self-consistent. Because the functional forms of the potential terms vary extensively between even closely related interatomic potentials (or successive versions of the same interatomic potential), the parameters from one interatomic potential function should clearly never be used together with another interatomic potential function.

Deficiencies

All interatomic potentials are based on many approximations and derived from different types of experimental data. Thus, they are termed empirical. Some existing energy functions do not account for electronic polarization of the environment, an effect that can significantly reduce electrostatic interactions of partial atomic charges. This problem was addressed by developing polarizable force fields or using macroscopic dielectric constant. However, application of one value of dielectric constant is questionable in the highly heterogeneous environments of proteins or biological membranes, and the nature of the dielectric depends on the model used.

All types of van der Waals forces are also strongly environment-dependent, because these forces originate from interactions of induced and "instantaneous" dipoles (see Intermolecular force). The original Fritz London theory of these forces can only be applied in vacuum. A more general theory of van der Waals forces in condensed media was developed by A. D. McLachlan in 1963 (this theory includes the original London's approach as a special case). The McLachlan theory predicts that van der Waals attractions in media are weaker than in vacuum and follow the like dissolves like rule, which means that different types of atoms interact more weakly than identical types of atoms. This is in contrast to combinatorial rules or Slater-Kirkwood equation applied for development of the classical force fields. The combinatorial rules state that interaction energy of two dissimilar atoms (e.g., C…N) is an average of the interaction energies of corresponding identical atom pairs (i.e., C…C and N…N). According to McLachlan theory, the interactions of particles in a media can even be fully repulsive, as observed for liquid helium. The conclusions of McLachlan theory are supported by direct measurements of attraction forces between different materials (Hamaker constant), as explained by Jacob Israelachvili in his book Intermolecular and surface forces. It was concluded that "the interaction between hydrocarbons across water is about 10% of that across vacuum". Such effects are unaccounted in standard molecular mechanics.

Another round of criticism came from practical applications, such as protein structure refinement. It was noted that Critical Assessment of protein Structure Prediction (CASP) participants did not try to refine their models to avoid "a central embarrassment of molecular mechanics, namely that energy minimization or molecular dynamics generally leads to a model that is less like the experimental structure". The force fields have been applied successfully for protein structure refinement in different X-ray crystallography and NMR spectroscopy applications, especially using program XPLOR. However, such refinement is driven mainly by a set of experimental constraints, whereas the interatomic potentials serve merely to remove interatomic hindrances. The results of calculations are practically the same with rigid sphere potentials implemented in program DYANA (calculations from NMR data), or with programs for crystallographic refinement that do not use any energy functions. The deficiencies of the interatomic potentials remain a major bottleneck in homology modeling of proteins. Such situation gave rise to development of alternative empirical scoring functions specifically for ligand docking, protein folding, homology model refinement, computational protein design, and modeling of proteins in membranes.

There is also an opinion that molecular mechanics may operate with energy which is irrelevant to protein folding or ligand binding. The parameters of typical force fields reproduce enthalpy of sublimation, i.e., energy of evaporation of molecular crystals. However, it was recognized that protein folding and ligand binding are thermodynamically very similar to crystallization, or liquid-solid transitions, because all these processes represent freezing of mobile molecules in condensed media. Thus, free energy changes during protein folding or ligand binding are expected to represent a combination of an energy similar to heat of fusion (energy absorbed during melting of molecular crystals), a conformational entropy contribution, and solvation free energy. The heat of fusion is significantly smaller than enthalpy of sublimation. Hence, the potentials describing protein folding or ligand binding must be weaker than potentials in molecular mechanics. Indeed, the energies of H-bonds in proteins are ~ -1.5 kcal/mol when estimated from protein engineering or alpha helix to coil transition data, but the same energies estimated from sublimation enthalpy of molecular crystals were -4 to -6 kcal/mol. The depths of modified Lennard-Jones potentials derived from protein engineering data were also smaller than in typical potential parameters and followed the like dissolves like rule, as predicted by McLachlan theory.

Future perspectives

The use of interatomic potentials in chemistry was first introduced in 1949, apparently independently by Hill and by Westheimer, applied mainly to organic chemistry to estimate properties such as strain energies among others. The functional form of the interatomic potential, focused in this article applied to biological systems, was established by Lifson in the 1960s. For over a half century, interatomic potentials have served us well, providing useful insights into and interpretation of biomolecular structure and function. Undoubtedly, it will continue to be widely used, thanks to its computational efficiency, while its reliability will continue to be improved. Yet, there are many well-known deficiencies as noted above. Further, the number of energy terms used in a given interatomic potential cannot be uniquely determined and a highly redundant number of degrees of freedom are typically used. Consequently, the "parameters" in different interatomic potentials can be vastly different. Of course, the emphasis to incorporate polarization into the standard pair-wise potentials can be very useful; however, there is no unique way of treating polarization in molecular mechanics because it is of quantum mechanical origin. Furthermore, often we are more interested in the properties derived from the dynamic dependence of the interatomic potential itself on molecular fluctuations.

One possibility is that the future development of interatomic potential ought to move beyond the current molecular mechanics approach, by using quantum mechanics explicitly to construct the interatomic potential. A number of the polarizable interatomic potentials listed below, such as density fitting and bond-polarization, already included some of the key ingredients towards this goal. The explicit polarization (X-Pol) method appears to have established the fundamental theoretical framework for a quantal force field; the next step is to develop the necessary parameters to achieve more accurate results than classical mechanics can offer.

Popular force fields

Different force fields are designed for different purposes. All are implemented in various computer software.

MM2 was developed by Norman Allinger mainly for conformational analysis of hydrocarbons and other small organic molecules. It is designed to reproduce the equilibrium covalent geometry of molecules as precisely as possible. It implements a large set of parameters that is continuously refined and updated for many different classes of organic compounds (MM3 and MM4).

CFF was developed by Arieh Warshel, Lifson and coworkers as a general method for unifying studies of energies, structures and vibration of general molecules and molecular crystals. The CFF program, developed by Levitt and Warshel, is based on the Cartesian representation of all the atoms, and it served as the basis for many subsequent simulation programs.

ECEPP was developed specifically for modeling of peptides and proteins. It uses fixed geometries of amino acid residues to simplify the potential energy surface. Thus, the energy minimization is conducted in the space of protein torsion angles. Both MM2 and ECEPP include potentials for H-bonds and torsion potentials for describing rotations around single bonds. ECEPP/3 was implemented (with some modifications) in Internal Coordinate Mechanics and FANTOM.

AMBER, CHARMM, and GROMOS have been developed mainly for molecular dynamics of macromolecules, although they are also commonly used for energy minimizing. Thus, the coordinates of all atoms are considered as free variables.

Classical force fields

Polarizable force fields

Reactive force fields

Coarse-grained force fields

Water models

The set of parameters used to model water or aqueous solutions (basically a force field for water) is called a water model. Water has attracted a great deal of attention due to its unusual properties and its importance as a solvent. Many water models have been proposed; some examples are TIP3P, TIP4P, SPC, flexible simple point charge water model (flexible SPC), and ST2.

Post-translational modifications and unnatural amino acids

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