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

Abstract

Loop closure in proteins requires computing the values of the inverse kinematics (IK) map for a backbone fragment with 2n ≥ 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 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 and a slightly idealized kinematic model, called the canonical model. This model, which assumes that every two consecutive torsional bonds C α -C and N-C α 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) and 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 × SO(3) where the IK of the 6-dof fragment has 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 non-canonical fragment. Computational results for fragments ranging from 8 to 30 dofs are presented.

Backbone flexibility in computational protein design

“…Commonly referred to as 'loop closure', Go and Scheraga [43] first applied these concepts to biomolecules with a numerical formulation for determining allowable values for 6 torsions of peptide chainsspecifically, tripeptides and cyclic peptides. Subsequent closure techniques [44][45][46][47][48] find conformations for longer chains using related ideas from inverse kinematics, a subfield of robotics. Finding accessible conformations of linked objects subject to constraints has been wellstudied in inverse kinematics, such as determining the possible rotations of the internal joints of a robotic arm that satisfy fixed positions for the shoulder and hand (Figure 1c).…”

Modeling Structures and Motions of Loops in Protein Molecules

“…Other methods subject open loop conformations to attractive mechanical forces formulated to pull the mobile anchor to its target pose in the stationary anchor [33]. Yet other methods incorporate IK techniques in a probabilistic sampling framework to close open loop conformations [32,[68][69][70][130][131][132][133][134][135]. In [130], for instance, the loop is broken into an active part, for which open conformations are generated disregarding the constraints, and a passive part of exactly three amino acids that is closed through exact IK methods [130].…”

Computational Models of Protein Kinematics and Dynamics: Beyond Simulation

“…Another polynomial formulation is proposed in Reference 55, but the polynomial equations are solved with a subdivision algorithm, which yields approximate solutions. Along a related line of research, the structure of the IK map over R 3 × SO(3) is studied in Reference 56, which shows that the critical poses of Ω 2 relative to Ω 1 decompose R 3 × SO(3) into regular regions, such that over each such region the number of IK solutions is constant. This decomposition leads to a constructive proof of the existence of a region in which the theoretical maximum of 16 solutions is attained.…”