1.2 Assignment 3: inverse kinematics

Motivation for inverse kinematics in proteins

One important application of inverse kinematics is in determining missing portions of protein structure. We traditionally depend on experimentaltechniques to provide us with the picture of an average structure for a protein. X-ray crystallography for example, relies on crystallizing proteinsand reporting the structure of the protein crystal within a certain resolution. One of the inherent problems with X-ray crystallography is thatmobile protein regions such as loops cause disorder in the crystal and as a consequence, coordinates for the atoms of these mobile regions cannot bereported. Often, in the PDB, crystallographically determined proteins are partially resolved, i.e. a portion of the structure may be missing due to itsintrinsic mobility. Even when experimental techniques such as NMR and cryo-EM can report an average picture of the fully resolved protein, the averagestructure reported is not indicative of the different conformations mobile regions can assume inside our cells at room temperature.

The specific problem of completing a partially resolved protein structure by finding conformations for its missing loop is known as the fragmentcompletion or the loop closure problem. Note that the loop closure problem is actually an inverse kinematics problem. Using sequence information alone, i.e.knowing the aminoacid sequence of the missing loop, one can generate starting loop conformations. The loop closure problem requires these loopconformations to be geometrically constrained by attaching them to the portion of the protein structure that is experimentally determined.Note that, as the picture below indicates, one can generate many loop conformations in space through forward kinematics. One end of the loop can beattached to its counterpart in the protein through translation alone. The other end however, needs to be attached without breaking bonds or stretchingbond angles. One way to do this is through inverse kinematics; that is, knowing the goal position in space for the end of the loop, can we solve forthe dihedral DOFs of the loop conformation? This question can be answered by Inverse Kinematics techniques.

Inverse kinematics for a polypeptide chain

Cyclic coordinate descent (ccd)