Forcefields enable the potential energy of a molecular system to be calculated rapidly and fairly accurately. A typical forcefield represents each atom in the sytem as a single point and energies as a sum of two-, three-, and four-particle interactions. The potential energy of a particular interaction is described by an equation which involves the positions of the particles and a small number of parameters which have been determined experimentally or by quantum mechanical calculations. For large systems with many particles, the equations are usually quite simple in order to allow for a rapid calculation of the total energy. There is often a trade-off between simplicity and accuracy. For instance, the bond between two particles and can be described by a harmonic potential,

or a Morse potential:

Here, is the distance between the two particles; , , and are force constant, equilibrium geometry, and bond energy parameters, respectively; and . The Morse potential is more accurate, especially when is significantly larger than the equilibrium bond distance. The harmonic potential, however, is calculated very fast and gives reasonable answers for bonds near their equilibrium geometries. Forcefields designed for proteins and nucleic acids almost always use the simpler harmonic form.

During the past ten years, several forcefields have been developed for protein simulations. Those having the most widespread usage are AMBER [11] and CHARMm [12]. Recently, the DREIDING forcefield was enhanced and published[13]. Although this forcefield is much more general than either AMBER or CHARMm, it is equally effective for protein simulations. All three are ``united atom'' forcefields: hydrogens bonded to carbons are not treated specially but are treated as a unit with the carbon atom. The three forcefields also share many of the same potential functions. However, specific parameters, such as force constants and equilibrium geometries, and atom type assignments, i.e., which parameters should be used for which atoms in a simulation, are different. Most of the calculations presented here used the DREIDING forcefield. However, in some instances results are reported for simulations using AMBER for the sake of comparison.

**Valence interactions.** The overall potential energy of a
molecular system is typically described by a forcefield like

where the energy potential terms are either between bonded atoms (``valence interactions'') or through-space (``nonbonded interactions''). Valence interactions include bonds(), angles(), torsions(), and inversions(). The nonbonded interactions are electrostatic(), van der Waals(), and hydrogen bonds(). The valence interactions are generally quite simple, like the bond energy terms in Equation (). The total bond energy is a sum over all bonds in the system, which is typically very close to the number of atoms:

The angle term is very similar and is also the same in both DREIDING and AMBER:

Here, is the angle formed by the atoms , and . Torsion terms are also treated identically in DREIDING and AMBER, but the equation is very different from the equations for bonds and angles:

Here, each four-body torsion is itself a sum of up to six terms, each of which can have its own periodicity. The periodicity is determined by , while determines whether the term has a maximum at or at .

The most complex term in a typical protein forcefield is the inversion term, which is added to ensure that a particular atom , which is bonded to three other atoms ,, and , remains planar or non-planar. AMBER and DREIDING treat this term differently. AMBER uses the angle between the and planes and the equation:

Planarity is enforced by and a tetrahedral geometry is enforced by . The DREIDING forcefield uses a different angle between the and planes and a simpler harmonic term:

Note that and are unrelated to the important and backbone dihedrals of proteins. For more details, see the AMBER[11] and DREIDING[13] papers.

**Nonbonded interactions.** The number of valence interactions that
must be calculated for a molecule is usually proportional to the
number of atoms, . The number of nonbonded terms, however is
roughly proportional to , because they involve almost all
possible pairs of atoms. It is slightly less than because two
atoms involved in a particular bond or angle are not considered to
have a through-space interaction. Also, it is very common to ignore
interactions between atoms too far apart in space (typically, more
than 9 Å), even though this technique can be quite inaccurate (N.
Karasawa and H.Q. Ding, unpublished data). Nevertheless, for large
systems, the bulk of computational time is spent calculating the
nonbonded interactions, so a great deal of work has been done to
optimize these calculations for vector and parallel processors. There
has also been considerable work in developing new methods of accurate
nonbond calculations which are proportional to
(see Reference [25]). Nevertheless, most
calculations are done using the techniques.

Both van der Waals and electrostatic interactions are calculated over pairs of atoms, so they are usually done concurrently:

If all atoms in a system are explicitly included in a calculation, the vacuum dielectric constant () should be used. However, is often set proportional to , ostensibly to represent the electrostatic screening effect of solvent atoms when they not present, but more practically to make the electrostatic term proportional to rather than . This speeds calculations substantially because can be directly calculated from the Cartesian coordinates of and without requiring a lengthy square-root calculation. Each forcefield includes van der Waals radii and well depths for each atom type. The equilibrium bond strength in Equation () is the geometric mean of the van der Waal's well depths of the individual atoms and . The equilibrium bond length is the arithmetic mean of the two van der Waals radii.

Hydrogen atoms are treated specially. The AMBER forcefield assigns charges to hydrogens but does not give them van der Waals parameters. Instead, it uses ``off-diagonal'' van der Waals terms. In other words, these are special terms for interactions when , rather than simply using the averages of the individual atomic terms. These interactions do not use the Lennard-Jones 12-6 potential, but rather a Lennard-Jones 12-10 potential, which goes to zero much more quickly:

DREIDING treats hydrogens even more unusually. Hydrogens are not given charges or van der Waals parameters, so Equation () does not apply at all. Rather, the DREIDING forcefield has a special hydrogen bond term for D-H-A interactions, where D is the hydrogen bond donor, H is the hydrogen bonded to it covalently, and A is the hydrogen bond acceptor, non-covalently attached. The DREIDING hydrogen bond uses both a radial and an angular part:

Both the radial and angular parts are set to zero beyond certain cutoff values and switching functions are used to make the transition to smooth. See Reference [15] for details.

Sat Jun 18 14:06:11 PDT 1994