0.2 Protein inverse kinematics and the loop closure problem  (Page 2/5)

When this happens, the result is a partially resolved protein structure, with fragments of the protein chain,such as mobile loops, missing. The only information available for the missing fragment is its amino acid sequence and where its twoendpoints need to be spatially located in order to connect with the known, resolved, part of the protein structure. Given the spatialconstraints on the endpoints of the missing fragment, one needs to find values to the dihedral angles of the fragment in order toobtain configurations of the fragment consistent with the constraints. This problem is known as the Loop Closure problem inthe structural biology community. It is easy to note that even though this problem is cast in the context of finding atomicpositions of a missing fragment such as a mobile loop, it is nothing new but a statement of the Inverse Kinematics problem for proteins.

Solving the Inverse Kinematics problem in the context of a missing fragment in proteins is not limited tofinding mobile loops. More generally, through the Inverse Kinematics problem, one can search for alternative configurations of anyfragment of a protein polypeptide chain (even fragments containing secondary structural elements) that satisfy the spatial constraintson their endpoints. Very recently, a third application has emerged, where alternative configurations of consecutive fragments that covera polypeptide chain are generated to obtain an ensemble of alternative protein structures.

Solving inverse kinematics

In applying inverse kinematics algorithms to proteins, we are taking advantage of a striking similarity between organic molecules androbotic manipulators (robot arms) in terms of how they move. As robot manipulators have joints, proteins have atoms. As robotmanipulators have links that connect their joints, proteins have bonds that connect their atoms. The similarity between proteins androbots makes it possible for us to apply to proteins a large existing literature of solutions to the Inverse Kinematics problem,developed in the context of robot manipulators (robotic arms).

Before we proceed with some simple inverse kinematics examples, note that inverse kinematics isthe inverse of the forward kinematics problem. Therefore, an immediate attempt to solve the inverse kinematics problem would beby inverting forward kinematics equations.

Let's illustrate how to solve the inverse kinematics problem for robot manipulators on a simple example. Thefigure below shows a simple planar robot with two arms. The underlying degrees of freedom of this robot are the two anglesdictating the rotation of the arms. These are labeled in the figure below as θ1 and θ2. The inverse kinematics question in this case wouldbe: What are the values for the degrees of freedom so that the endeffector of this robot (the tip of the last arm) lies at position (x,y) in the two-dimensional Cartesian space? One straightforward approach to solving the problem is to try to write down theforward kinematics equations that relate (x,y) to the two rotational degrees of freedom (see Forward Kinematics for details on how to do so), then try to solve these equations. The solutions will give you an answer to the inverse kinematics problem for thisrobot.