0.2 Protein inverse kinematics and the loop closure problem

• Background Material
• Inverse Kinematics and its Relevance to Proteins
• Solving Inverse Kinematics
  • Inverse Kinematics Methods
    • Classical Methods
    • Optimization-Based Methods
  • Cyclic Coordinate Descent and Its Application to Proteins

Background material

The math involved in solving the Inverse Kinematics problem requires some background in linearalgebra, specifically in the anatomy and application of transformation matrices. Please refer to Forward Kinematics for an introduction to transformation matrices. It is very important thatyou understand how to apply transformations for the Forward Kinematics of a chain.

Inverse kinematics and its relevance to proteins

Inverse kinematics (IK) is the problem of finding the right values for the underlying degrees offreedom of a chain, in the case of a protein polypeptide chain, of the dihedral angles, so that the chain satisfies certain spatialconstraints. For example, in some applications, it is necessary to find rotations that can steer certain atoms to desired locations inspace. To achieve a particular function, protein regions sometimes have to undergo concerted motion where atoms move together in orderto locate themselves near another protein or molecule. The motion of atoms is spatially constrained because they have to assume specifictarget locations in space. However, since atoms must move together in order not to break bonds by their motion, it is easier to modeltheir motion in dihedral angle space , where bond lengths and bond angles are fixed. This parameterization of proteinmotion, called the idealized or rigid geometry model , is discussed in Representing Proteins in silico: Data Structures andKinematics .

Solving the Inverse Kinematics problem in the context of proteins, i.e., finding what values of the dihedral angles of aprotein polypeptide chain yield configurations of the chain where the endpoints satisfy spatial constraints, is a very importantproblem in structural biology. The relevance of Inverse Kinematics for proteins can be seen in three main applications:

• Finding a missing loop (Loop Closure Problem)
• Characterizing the Flexibility of a fragment of the protein polypeptide chain
• Generating ensembles of protein structures