Application of the Cyclic Coordinate Descent algorithm in the protein loop closure problem AndrzejW´ojtowicz andre@wmi.amu.edu.pl September 21, 2011 Abstract In 2003 Adrian Canutescu and Roland Dunbrack Jr. presented an algorithm used in protein structure prediction. In this case one protein element has to be fitted to a fixed fragment. Previous approaches relied on using Jacobian but they have two disadvan- tages: large computational overhead and need of matrix inversion. Fortunately protein loop closure and inverse kinematics problem are very similar. In robotics and computer animation the Cyclic Coordinate Descent (CCD) method is used to determine, accord- ingly, th movement of a robotic arm and animation of characters. In contrast to inverse Jacobian, CCD is extremely fast and has not the above-mentioned constraints. An appro- priate part of a protein may be considered as a robotic arm which is to e.g. catch a desired object, so we can benefit from CCD. This article presents the idea of the CCD algorithm. 1 Forward and inverse kinematics Before we focus on the biological aspect, let’s have a look at the origin of the problem in robotics. Consider a simple robotic arm in a 2D world presented on the picture below. x y l 1 l 2 l 3 (x E , y E ) α 3 α 1 α 2 α E (x' E , y' E ) Figure 1: An example configuration of the robotic arm. The red joint indicates, simplifying, the position of the end effector. The blue one is the object which the arm have to grab. The arm consists of three rotational joints, three links of length l 1 , l 2 , l 3 and the end effector. At the joints we have angles α 1 , α 2 and α 3 . 1