Solving the Geometric Docking Problem for Planar and Spatial Sets

Motivation. A docking algorithm working without charge calculations is needed for molecular modeling studies. Two sets of n points in the d–dimensional Euclidean space are considered. The optimal translation and/or rotation minimizing the variance of the sum of the n squared distances between the fixed and the moving set is computed. An analytical solution is provided for d–dimensional translations and for planar rotations. The use of the quaternion representation of spatial rotations leads to the solving of a quadratically constrained non– linear system. When both spatial translations and rotations are considered, the system is solved using a projected Lagrangian method requiring only 4–dimensional initial starting tuples. Method. The projected Lagrangian method was used in the docking algorithm. Results. The automatic positioning of the moving set is performed without any a priori information about the initial orientation. Conclusions. Minimizing the variance of the squared distances is an original and simple geometric docking criterion, which avoids any charge calculation. Availability. The FORTRAN source is available within framework of scientific collaborations. Contact: petitjean@itodys.jussieu.fr.