On the structure of the inverse kinematics map of a fragment of protein backbone

Loop closure in proteins requires computing the values of the inverse kinematics (IK) map for a backbone fragment with 2n > or = 6 torsional degrees of freedom (dofs). It occurs in a variety of contexts, e.g., structure determination from electron-density maps, loop insertion in homology-based structure prediction, backbone tweaking for protein energy minimization, and the study of protein mobility in folded states. The first part of this paper analyzes the global structure of the IK map for a fragment of protein backbone with 6 torsional dofs for a slightly idealized kinematic model, called the canonical model. This model, which assumes that every two consecutive torsional bonds C(alpha)--C and N--C(alpha) are exactly parallel, makes it possible to separately compute the inverse orientation map and the inverse position map. The singularities of both maps and their images, the critical sets, respectively, decompose SO(3) x R(3) into open regions where the number of IK solutions is constant. This decomposition leads to a constructive proof of the existence of a region in R(3) x SO(3) where the IK of the 6-dof fragment attains its theoretical maximum of 16 solutions. The second part of this paper extends this analysis to study fragments with more than 6 torsional dofs. It describes an efficient recursive algorithm to sample IK solutions for such fragments, by identifying the feasible range of each successive torsional dof. A numerical homotopy algorithm is then used to deform the IK solutions for a canonical fragment into solutions for a noncanonical fragment. Computational results for fragments ranging from 8 to 30 dofs are presented.