Interaction nonbonded

There are two general forms of nonbonded interactions that need to be accoimted for by a classical force-field (1) the van der Waals (vdw) interactions and (2) the electrostatic interactions. 

Molecular Dynamics Simulation From Ab Initio to Coarse Grained  

In order to model the van der Waals interactions, we need a simple empirical expression that is not computationally intensive and that models both the dispersion and repulsive interactions that are known to act upon atoms and molecules. The most commonly used functional form of van der Waals energy ( vdw) in classical force-fields is the Lennard-Jones 12-6 function that has the form  

O Equation 7.36 models both the attractive part (the term) and the repulsive part (the term) of the nonbonded interaction. Other formulations of the Lennard-Jones nonbond potential commonly have the same power law description of the attractive part of the potential, but wUl have different power law dependence for the repulsive part of the interaction, such as the Lennard-Jones 9-6 function  

When the nonbond interactions of a system that contains multiple particle types and multiple molecules are modeled using a Lennard-Jones type nonbond potential, it is necessary to be able to define the values of a and e that apply to the interaction between particles of type I and /. The parameters for these cross interactions are generally found using one of the two following mixing rules. One common mixing rule is the Lorentz-Berthelot rule where the value of o// is found from the arithmetic mean of the two pure values and the value of ej/ is the geometric mean of the two pure values  

As a general rule, the best approach when developing a force field for a metal complex is to use, without modification, the nonbonded interaction terms developed for organic compounds. The best known and most popular is the MM2 force field and this has been extended and adapted for modeling metal complexes by a number of groups1113 1161. Force fields developed for organic molecules do not have parameters for some of the elements present in metal complexes but reasonable estimates are available for most of theset57 59,65 96 120 123 13 

One of the main reasons why nonbonded interactions involving metal ions have not been included in most force fields is a lack of good estimates for the parameters. As discussed in Section 3.2.5, values for the van der Waals radius and the polarizability (e) are required. In the case of metal complexes it is difficult to obtain estimates for the van der Waals radius because the metal ion is generally buried inside an organic sheath and does not make close contacts with atoms free to move away from it. In a few cases close contacts to M(II) metal ions have been observed and these are consistent with van der Waals radii in the range of 1.5-1.7 A. Where nonbonded interactions to the metal ions have been included in a force field the van der Waals radius used ranged from 1.7 to 2.44 At98,99,1421. 

In this section, we present a detailed discussion of methods relevant to counterion simulations of DNA. Electrostatic interactions dominate all energies involving the DNA and ions. The largest systems simulated to date typically require truncation of nonbonded interaction potentials at around ISA. Abrupt truncation of the potential has the adverse effect of creating an artificial boundary. In an MD simulation, the abrupt truncation produces a discontinuity in force because the first derivative of the interaction potential at the cutoff radius is infinity. Computer programs simply set the forces at the boundary to be zero instead of large values to represent infinity. 

The same type of argument can be applied to oppositely charged ions. Here the effect is in the reverse direction. When two such ions are farther apart than the cutoff, the sudden dip in the potential energy profile at the cutoff is favorable and the second ion will quickly roll down the dip, increasing the temperature again. It will continue to roll until contact with the other ion is established. Here, the truncation prevents the oppositely charged ions from ever separating beyond the cutoff distance. 

The main problem in the use of the hard-sphere potential is the selection of values of the van der Waals radii of the atoms. Much has been written on this subject, and the difficulty lies primarily in the fact that the concept of a van der Waals radius is itself a nebulous and inexact one. Also, since the radii are assigned to spherical atoms, no consideration is taken of the angle at which the two atoms approach each other. Thus, it is not surprising to find large ranges of values in the literature for the 

More refined treatments of the potential function have made use 

Selected Values of van der Waals Contact Distances (A) (Leach et al., 1966a) 

The Lennard-Jones potential is the one most commonly used at present since the Buckingham form has an additional parameter, exhibits a physically unrealistic maximum at very short intemuclear distances, and approaches —00 as the intemuclear distance approaches zero. While these artifacts can he eliminated by proper computer programming, they increase the time of computation. Nearly the same curves result in the region of r of interest if eqs. 5 and 6 are made to coincide at their minima. 

Hendrickson (1961) and Scott and Scheraga (1965, 1966a, b, c) have developed procedures for obtaining the constants of eqs. 5 and 6. The coefficients ci3- or ei3- of the attractive terms are obtained by using the Slater-Kirkwood equation 

Note that -y and y usually approach the Thomas-Fermi limit ( ) (Table 4.4) except for hydrogen, where y=2 because of the virial theorem, E=T + Vne, with E= - T. 

Recall that the core-valence separation in molecules is described in real space [83], as any atom-by atom or bond-by-bond partitioning of a molecule is inherently a real-space problem. Equation (10.6) does indeed refer to a partitioning in real space (as opposed to the usual Hartree-Fock orbital space), both for ground-state isolated atoms or ions and for atoms embedded in a molecule, with N = 2c for first-row elements. 

This concludes the enumeration of the concepts involved in our bond energy theory. 

Now we turn our attention to the intrinsic bond energies, s i. Two approaches will be considered (1) a derivation making use of Eq. (10.2) and (2) a derivation rooted in Eq. (4.47). But first, we deal with the nonbonded part, Eq. (10.3), and examine future possible simplifications regarding this energy contribution. 

The idea behind this survey of nonbonded interactions is to get rid of them elegantly as explicit terms requiring separate calculations by means of Eq. (10.3). We shall examine to what extent nonbonded Coulomb-type interactions are at least approximately additive. The formulation of additivity is presented here for C H2 +2-2m hydrocarbons [208], where m is the number of six-membered cycles. 

Generally the most important component of any molecular mechanics force field is the nonbonding potential function. The traditional form is the Lennard-Jones 6-12 potential (Eq. 2.47), where e and r are parameters that depend on the identities of the two interacting atoms and r is the distance between the atoms. 

The Lennard-Jones 6-12 potential function, with examples of both a hard and a soft potential. 

For hydrocarbons and simple organics, the force field we have defined so far is often sufficient. However, some force fields also include cross terms . For example, a stretch-bend term couples bond lengthening with angle bending. It could, for example, make it easier to stretch a bond if the bond is also involved in a distorted angle. Such terms usually make only small contributions to the total energy. 

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Guenot, J., Kollman, P.A. Conformational and energetic effects of truncating nonbonded interactions in an aqueous protein dynamics simulation. J. Comput. Chem. 14 (1993) 295-311. 

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

Formally, we describe the LN method with the above force splitting below for the triplet protocol At, Atm, At. The fast, medium, and slow force components are distinguished by subscripts we take the medium forces as those nonbonded interactions within a 6 A region. 

In an atomic level simulation, the bond stretch vibrations are usually the fastest motions in the molecular dynamics of biomolecules, so the evolution of the stretch vibration is taken as the reference propagator with the smallest time step. The nonbonded interactions, including van der Waals and electrostatic forces, are the slowest varying interactions, and a much larger time-step may be used. The bending, torsion and hydrogen-bonding forces are treated as intermediate time-scale interactions. 

Ding H-Q, N Karasawa and W A Goddard III 1992a. Atomic Level Simulations on a Milhon Particles The Cell Multipole Method for Coulomb and London Nonbonding Interactions. Journal of Chemical Physics 97 4309-4315. 

I J, J C Cole, J P M Lommerse, R S Rowland, R Taylor and M L Verdonk 1997. Isostar A Libraij )f Information about Nonbonded Interactions. Journal of Computer-Aided Molecular Design 11 525-531. g G, W C Guida and W C Still 1989. An Internal Coordinate Monte Carlo Method for Searching lonformational Space. Journal of the American Chemical Scociety 111 4379-4386. leld C and A J Collins 1980. Introduction to Multivariate Analysis. London, Chapman Hall, ig C-W, R M Cooke, A E I Proudfoot and T N C Wells 1995. The Three-dimensional Structure of 1 ANTES. Biochemistry 34 9307-9314. 

MOMEC is a force field for describing transition metal coordination compounds. It was originally parameterized to use four valence terms, but not an electrostatic term. The metal-ligand interactions consist of a bond-stretch term only. The coordination sphere is maintained by nonbond interactions between ligands. MOMEC generally works reasonably well for octahedrally coordinated compounds. 

YETI is a force held designed for the accurate representation of nonbonded interactions. It is most often used for modeling interactions between biomolecules and small substrate molecules. It is not designed for molecular geometry optimization so researchers often optimize the molecular geometry with some other force held, such as AMBER, then use YETI to model the docking process. Recent additions to YETI are support for metals and solvent effects. 

Most of the methods proposed include a van der Waals term for describing nonbonded interactions between atoms in the two regions. This is usually represented by a Leonard-Jones 6-12 potential of the form... 

If the QM and MM regions are separate molecules, having nonbonded interactions only might be sufficient. If the two regions are parts of the same molecule, it is necessary to describe the bond connecting the two sections. In most... 

First determine what parameters will be used for describing bond lengths and angles. Then determine torsional, inversion, and nonbonded interaction parameters. 

Calculating nonbonded interactions only to a certain distance imparts an error in the calculation. If the cutoff radius is fairly large, this error will be very minimal due to the small amount of interaction at long distances. This is why many bulk-liquid simulations incorporate 1000 molecules or more. As the cutoff radius is decreased, the associated error increases. In some simulations, a long-range correction is included in order to compensate for this error. 

Nonbonded interactions are the forces be tween atoms that aren t bonded to one another they may be either attractive or repulsive It often happens that the shape of a molecule may cause two atoms to be close in space even though they are sep arated from each other by many bonds Induced dipole/induced dipole interactions make van der Waals forces in alkanes weakly attractive at most distances but when two atoms are closer to each other than the sum of their van der Waals radii nuclear-nuclear and electron-electron repulsive forces between them dominate the fvan derwaais term The resulting destabilization is called van der Waals strain... 

United atom force fields (see United versus All Atom Force Fields on page 28) are sometimes used for biomolecules to decrease the number of nonbonded interactions and the computation time. Another reason for using a simplified potential is to reduce the dimensionality of the potential energy surface. This, in turn, allows for more samples of the surface. 

HyperChem also provides a shifting potential for terminating nonbonded interactions (equation 15). 

HyperChem supplements the standard MM2 force field (see References on page 106) by providing additional parameters (force constants) using two alternative schemes (see the second part of this book. Theory and Methods). This extends the range of chemical compounds that MM-t can accommodate. MM-t also provides cutoffs for calculating nonbonded interactions and periodic boundary conditions. 

You can choose to calculate all nonbonded interactions or to truncate (cut off) the nonbonded interaction calculations using a switched or shifted function. Computing time for molecular mechanics calculations is largely a function of the number of nonbonded interactions, so truncating nonbonded interactions reduces computing time. You must also truncate nonbonded interactions for periodic boundary conditions to prevent interaction problems between nearest neighbor images. 

Eor small and medium-sized molecules, calculate all nonbonded interactions. 

Usually, atoms with a vicinal relationship or more are considered to be nonbonded. Sometimes, however, only atoms with a 1-5 relationship are considered to be fully nonbonded and the atoms with a relationship have scaled down nonbonded interactions or are deleted completely from the nonbonded computations, or different parameters are used. 

The above potential describes the monopole-monopole interactions of atomic charges Q and Qj a distance Ry apart. Normally these charge interactions are computed only for nonbonded atoms and once again the interactions might be treated differently from the more normal nonbonded interactions (1-5 relationship or more). The dielectric constant 8 used in the calculation is sometimes scaled or made distance-dependent, as described in the next section. 

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