Representing Rod-Shaped Protein 3D Structures in Cylindrical Coordinates

Based on overall 3D structure, proteins may be grouped into two broad categories, namely, globular proteins (spheroproteins), and elongated or rod-shaped proteins (RSP). The former comprises a significant majority of proteins. This work concerns the second category. Unlike a spheroprotein, an RSP possesses a conspicuous axis along its longest dimension. To take advantage of this symmetry element, we decided to represent RSPs using cylindrical coordinates, (rho, theta, z), with the z-axis as the main axis, and one tip of the protein at the origin. A "tip" is one of two extreme points in the protein lying along the protein axis along its longest dimension. We first identify the two tips, T1 and T2, of the RSP using a protein graphics software, then determine their (Cartesian) coordinates, (h, k, l) and (m, n, o), respectively. Arbitrarily selecting T1 as the tip at the origin, we translate the protein by subtracting (h, k, l) from all structural coordinates. We then find the angle alpha between vector T1-T2 and the positive z-axis by computing the scalar product of vectors T1- T2 and OP where P is an arbitrary point along the positive z-axis. We typically use (0, 0, p) where p is a suitable positive number. Then we compute the cross product of the two vectors to determine the axis about which we should rotate vector T1-T2 to make it coincide with the positive z-axis. We use a matrix form of Rodrigues' formula to do the rotation. We then apply the Cartesian to cylindrical coordinate transformation equations to the system. We have applied the above transformation to 15 RSPs: 1QCE, 2JJ7, 2KPE, 3K2A, 3LHP, 2LOE, 2L3H, 2L1P, 1KSG, 1KSJ, 1KSH, 2KOL, 2KZG, 2KPF and 3MQC. We have also created a web server that can take the PDB coordinate file of an RSP and output its cylindrical coordinates. The URL of our web server will be announced publicly in due course.
Comment: 40 pages, 14 figures, 1 table