Principles of Protein Structure - G. Schultz 1982

Interactions Determining Protein Structure
Van der Waals Potentials

The potentials include repulsion of the electron shells, dispersion forces of Structure/103.html">Van der Waals attraction, and Electrostatic Interactions. For computational convenience, it is practical to combine all three non-valent forces into a single simple potential function (or force field), traditionally referred to as the van der Waals potential. This requires a further simplification of the electrostatic interaction model. First, it is assumed that contacts occur only between nearest neighbors, and the electrostatic interactions are averaged over all relative mutual orientations that are sterically permissible for the two interacting groups:

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and so forth. Thus, the resulting contribution depends solely on the distance between the interacting groups. With this simplification, the van der Waals potential becomes isotropic:

It incorporates three parameters: A and B, as illustrated in Fig. 3.1, and the product of the effective charges q_iq_j of the interacting atoms.

Effective charges can be determined from the partial charges of individual atoms (Table 3.3). Parameters A and B, which describe electron cloud repulsion and dispersion forces, can be derived from crystal structure data [52]. Because partial charges can only be calculated with low precision [53], Lifson et al. [54] attempted to extract them directly from crystal structure data. This was achieved by simultaneously varying all three parameters of the 1–6–12 potential to obtain the best possible agreement with experimental observations.

6–12 Potentials. When refining protein structures using an energy Minimization Procedure, Levitt [55] omitted the R^-1 term for computational convenience. This led to a 6–12 van der Waals potential that differs significantly from the 6–12 potentials describing solely electron shell repulsion and attractive dispersion forces (Table 3.2). Examples of such potentials are listed in Table 3.5. Note that when contacts are formed by amide nitrogen atoms, purely repulsive forces operate. In this case, electrostatic repulsion between the negative charges on the nitrogen atoms outweighs the attraction due to the dispersion forces given in Table 3.2.

Table 3.5 Parameters of the 6–12 van der Waals potential obtained from known Cell/13.html">Protein Structure Data [55]^a

^a A and B are defined in Fig. 3.1.

Van der Waals radii correspond approximately to Contact distances. Van der Waals potentials make it possible to determine the "distances corresponding to van der Waals contacts" between specific atoms. The lower limits of these distances, derived from crystal structures, were established by Ramachandran and Sasisekharan [29]. They amount to about 75% of the equilibrium distances R_m in Table 3.2 and correspond to a repulsion energy of roughly 1 kcal/mol. These contact distances were employed in the hard-sphere model to estimate steric constraints at the C_α atom of the polypeptide chain [28] (Fig. 2.3).

Observed van der Waals contact distances for pairs of atoms can be converted into more general "van der Waals radii" by assuming that each distance is simply the sum of the radii of two specific atoms. This assumption is only partially valid; the resulting radii represent an average over many types of contacts between various atoms. Examples of van der Waals radii are presented in Table 3.6. The "lower normal limits" of contact distances given in Table 2.1 are approximately 10% smaller than the sums of the corresponding van der Waals radii.

Table 3.6 Van der Waals radii according to Bondi [73]