Principles of Protein Structure - G. Schultz 1982

Interactions Determining Protein Structure
Dispersion Forces and Electron Shell Repulsion

Dispersion attractive forces act between any pairs of atoms. Dispersion forces exist between any pair of atoms, even if they are completely nonpolar. If an atom is represented as an oscillating dipole, in a pair of atoms each dipole will polarize the neighboring atom. As a result, an attractive force arises between the atoms, the energy of which, in the first approximation, is inversely proportional to the sixth power of the distance between the atomic nuclei [45] and proportional to the polarizabilities of the atoms [46].

The polarizability of an atom incorporated into a molecule is anisotropic; therefore, dispersion forces depend on the relative orientation of molecular fragments. However, because The Effect of orientation is small and difficult to measure, it is usually neglected and dispersion forces are considered isotropic.

The electron shells of non-bonded atoms repel each other mutually. The dispersion attraction forces between a pair of non-bonded atoms are counterbalanced by the repulsion of their electron shells. This repulsion was approximated by Lennard-Jones by introducing a term inversely proportional to the m-th power of the distance [47]. Slater [48] described the repulsion using an exponential function, which was later modified by Buckingham [49] and bears his name. In protein calculations, the Lennard-Jones approximation with m = 12 is most frequently used. Taking into account the London term for dispersion forces [45], this leads to the computationally simple 6–12 potential, an example of which is shown in Fig. 3.1. Both parameters required to define this function are described in the figure legend. The potential in Fig. 3.1 has a minimum in the negative region at a distance of R_m. Therefore, the atoms are weakly "bound" at this distance (E_m being the attractive energy).

Energy parameters can be determined from crystal Structure data. The absolute magnitude of the 6–12 Lennard-Jones potential parameters can be determined from Contact distances and contact energies in crystals of small molecules, which are derived from X-Ray Diffraction data^a and, for example, heats of sublimation. These same data can also be obtained from atomic and molecular beam scattering experiments.

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Fig. 3.1. The 6–12 Lennard-Jones potential describing dispersion forces and electron repulsion (R_m = 3.24 Å, E_m = —0.13 kcal/mol).

The two-parameter formula is given below in a computationally convenient form (where A and B are parameters), as well as in its normalized form (where E_m and R_m are parameters); the quantities A and B are positive. With a decrease in B and an increase in A, E_m decreases, while the corresponding interatomic distance R_m increases. The potential barrier is asymmetric. Repulsion and attraction balance each other at a distance of 0.89∙R_m; at a distance of 1.5∙R_m, the attraction energy is only one-sixth of E_m.

Examples of such parameters are given in Table 3.2. The differences between the data characterize the magnitude of potential errors.

As can be seen from Table 3.2, the fraction of energy contributed by an atomic contact is very small. However, the number of contacts in a protein is large. For example, in a closest hexagonal packing of identical spheres, each sphere forms contacts with 12 neighbors. Since contact energies are additive, the total attractive energy per sphere in this case is six times greater than the energy contributed by a single contact.

Table 3.2 Parameters of the 6–12 Lennard-Jones potential for electron shell repulsion and dispersion forces in non-bonded contacts ^a

^a Only contacts between identical atom types are listed. For a contact between non-identical types i and j, we obtain