Nonbonded atoms

When using XOH signals as restrain ts, choose the distance between the nonbonded atoms. A typical distance is about 4 to 5 Angstroms. A typical force constant is 0..5 to. 5.0 kcal/A-, depending on the flexibdity ofthe rest of the molecule and the strength of the NOE signal. 

This term describes the repulsive forces keeping two nonbonded atoms apart at close range and the attractive force drawing them together at long range. 

Note Restraints apply to distances, angles and dihedrals between bonded or nonbonded atoms. You can also restrain atoms to points in space. 

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. 

To understand the function of a protein at the molecular level, it is important to know its three-dimensional stmcture. The diversity in protein stmcture, as in many other macromolecules, results from the flexibiUty of rotation about single bonds between atoms. Each peptide unit is planar, ie, oJ = 180°, and has two rotational degrees of freedom, specified by the torsion angles ( ) and /, along the polypeptide backbone. The number of torsion angles associated with the side chains, R, varies from residue to residue. The allowed conformations of a protein are those that avoid atomic coUisions between nonbonded atoms. 

Fig. 3.2. Energy as a function of intemuclear distance for nonbonded atoms.

Both attractive forces and repulsive forces are included in van der Waals interactions. The attractive forces are due primarily to instantaneous dipole-induced dipole interactions that arise because of fluctuations in the electron charge distributions of adjacent nonbonded atoms. Individual van der Waals interactions are weak ones (with stabilization energies of 4.0 to 1.2 kj/mol), but many such interactions occur in a typical protein, and, by sheer force of numbers, they can represent a significant contribution to the stability of a protein. Peter Privalov and George Makhatadze have shown that, for pancreatic ribonuclease A, hen egg white lysozyme, horse heart cytochrome c, and sperm whale myoglobin, van der Waals interactions between tightly packed groups in the interior of the protein are a major contribution to protein stability. 

The intermolecular forces operative in nonpolar compounds are also electrostatic-in nature. These weak van der Waals forces involve attraction between nonbonded atoms and are effective over short ranges only. 

Inspection of models show that the interaction between the non-bonded atoms is different in an isotactic and a syndiotactic sequence. A somewhat naive calculation based on van der Waals repulsions between the nonbonded atoms, suggests that for nonpolar vinyl monomers the syndiotactic sequence may be favored by a few tenths of a kcal/mole, and this difference might increase to... 

The first two terms on the right-hand side of Eq. (83) are usually assumed to be harmonic, as given for example by Eq. (6-74). The third term is often developed in a Fourier series, as given by Eq. (82). The potential function appropriate to the interaction between nonbonded atoms is taken to be of the Lennard-Jones type (Section 6.7.3). In all of these cases the necessary force constants are estimated by comparing the results obtained from a large number of similar molecules. If electrostatic interactions are to be considered, effective atomic charges must be suggested and Coulomb s law applied directly [see Eq. (6-81)]. 

While nonbonded atom pairs will typically not come within 1A of each other, it is possible for covalently bound pairs, either directly bounds, as in 1-2 pairs, or at the vertices of an angle, as in 1-3 pairs. Accordingly it may be considered desirable to omit the 1-2 and 1-3 dipole-dipole interactions as is commonly performed on additive force fields for the Coulombic and van der Waals terms. However, it has been shown that inclusion of the 1-2 and 1-3 dipole-dipole interactions is required to achieve anistropic molecular polarizabilites when using isotropic atomic polariz-abilites [50], For example, in a Drude model of benzene in which isotropic polarization was included on the carbons only inclusion of the 1-2 and 1-3 dipole-dipole interactions along with the appropriate damping of those interactions allowed for reproduction of the anisotropic molecular polarizability of the molecule [64], Thus, it may be considered desirable to include these short range interactions in a polarizable force field. 

The discussion thus far has focused on the forces between an array of atoms connected together through covalent bonds and their angles. Important interactions occur between atoms not directly bonded together. The theoretical explanation for attractive and repulsive forces for nonbonded atoms i and j is based on electron distributions. The motion of electrons about a nucleus creates instantaneous dipoles. The instantaneous dipoles on atom i induce dipoles of opposite polarity on atom j. The interactions between the instantaneous dipole on atom i with the induced instantaneous dipole on atom j of the two electron clouds of nonbonded atoms are responsible for attractive interactions. The attractive interactions are know as London Dispersion forces,70 which are related to r 6, where r is the distance between nonbonded atoms i and j. As the two electron clouds of nonbonded atoms i and j approach one another, they start to overlap. There is a point where electron-electron and nuclear-nuclear repulsion of like charges overwhelms the London Dispersion forces.33 The repulsive... 

In MM3, the two types of nonbonded interactions which are included in the program are handled separately. The first includes attractive and repulsive forces between nonbonded atoms, their origins are described above, and the second is hydrogen bonding. 

Tables 22.1 and 22.2 show how the general principles sketched above are manifested in real systems. The C-C and C-H bond order indices and the C and H valence indices were calculated for ethane, ethene, ethyne, and benzene at the HF/6-31G geometry with various basis sets. The bond order of the C-H bond is close to unity in all cases. The carbon-carbon bonds have bond orders close to one, two and three in ethane, ethene, and ethyne, respectively. In benzene, all C-C bonds have the same bond order, which is close to 1.5. Note that definition (Equation 22.1) yields nonzero bond orders between nonbonded atoms also, and in certain cases,...

In Section 3.1., we shall show that the dynamic model leads to an unambiguous determination of the type of nonbonded interactions involved while the static model may lead to erroneous predictions as a result of an ambiguous definition of the nature of a nonbonded interaction. The superiority of the dynamic model is due to the fact that nonbonded interactions affect bonded interactions and, thus, the change in an overall overlap population rather than the change of a specific overlap population between nonbonded atoms or groups is the most appropriate index of a nonbonded interaction. Accordingly, we shall employ the dynamic model in all subsequent discussions of molecular structure, unless otherwise stated. 

It should be noted that in this example the sign of AN is positive while the sign of the F—F pi overlap population is negative and the static model fails or becomes ambiguous. Obviously, both models will lead to indentical predictions when ANj and a given pi overlap population between nonbonded atoms or groups have the same sign. 

---