On the rotational operators in protein structure simulations

The reduction of the computational complexity of the algorithms dealing with protein structure analysis and conformation predictions is of prime importance. One common element in most of these algorithms is the process of transforming geometrical information between dihedral angles and Cartesian coordinates of the atoms in the protein using rotational operators. In the literature, the operators used in protein structures are rotation matrices, quaternions in vector and matrix forms and the Rodrigues-Gibbs formula. In the protein structure-related literature, the most widely promoted rotational operator is the quaternions operator. In this work, we studied the computational efficiency of the mathematical operations of the above rotational operators applied to protein structures. A similar study applied to protein structures has not been reported previously. We concluded that the computational efficiency of these rotational operators applied to protein chains is different from those reported for other applications (such as mechanical machinery) and the conclusions are not analogous. Rotation matrices are the most efficient mathematical operators in the protein chains. We examined our findings in two protein molecules: Ab1 tyrosine kinase and heparin-binding growth factor 2. We found that the rotation matrix operator has between 2 and 187% fewer mathematical operations than the other rotational operators.