In an effort to reduce the number of degrees of freedom necessary to describe a polypeptide chain we analyze the statistical behavior of polypeptide chains when represented as Cα chains, Cα chains with Cβ atoms attached, and Cα chains with rotational ellipsoids as models of side chains. A statistical analysis on a restricted data set of 75 unrelated protein structures is performed. The comparison of the database distributions with those obtained by model calculation on very short polypeptide stretches allows the dissection of local versus nonlocal features of the distributions. The database distribution of the bend angles of polypeptide chains of pseudo bonded Cα atoms spans a restricted range of values and shows a bimodal structure. On the other hand, the torsion angles of the Cα chain may assume almost all possible values. The distribution is bimodal, but with a much broader probability distribution than for bend angles. The Cα-Cβ vectors may be taken as representative of the orientation of the lateral chain, as the direction of the bond is close to the direction of the vector joining Cα to the ad hoc defined center of the "steric mass" of the side chain. Interestingly, both the bend angle defined by Cαi-Cαi+1-Cβi+1 and the torsional angle offset of the pseudo-dihedral Cαi-Cαi+1-Cαi-2-C βi+2 with respect to Cαi-Cαi+1-Cαi-2-C αi+3 span a limited range of values. The latter results show that it is possible to give a more realistic representation of polypeptide chains without introducing additional degrees of freedom, i.e., by just adding to the Cα chain a Cβ with given side-chain properties. However, a more realistic description of side chains may be attained by modeling side chains as rotational ellipsoids that have roughly the same orientation and steric hindrance. To this end, we define the steric mass of an atom as proportional to its van der Waals volume and we calculate the side-chain inertia ellipsoid assuming that the steric mass of each atom is uniformly distributed within its van der Waals volume. Finally, we define the rotational ellipsoid representing the side chain as the uniform density ellipsoid possessing the same rotationally averaged inertia tensor of the side chain. The statistics of ellipsoid parameters support the possibility of representing a side chain via an ellipsoid, independently of the local conformation. To make this description useful for molecular modeling we describe ellipsoid-ellipsoid interactions via a Lennard-Jones potential that preserves the repulsive core of the interacting ellipsoids and takes into account their mutual orientation. Tests are performed for two different forms of the interaction potential on a set of high-resolution protein structures. Results are encouraging, in view of the drastic simplifications that were introduced.
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