Cyclic coordinate descent: A robotics algorithm for protein loop closure

Abstract

In protein structure prediction, it is often the case that a protein segment must be adjusted to connect two fixed segments. This occurs during loop structure prediction in homology modeling as well as in ab initio structure prediction. Several algorithms for this purpose are based on the inverse Jacobian of the distance constraints with respect to dihedral angle degrees of freedom. These algorithms are sometimes unstable and fail to converge. We present an algorithm developed originally for inverse kinematics applications in robotics. In robotics, an end effector in the form of a robot hand must reach for an object in space by altering adjustable joint angles and arm lengths. In loop prediction, dihedral angles must be adjusted to move the C-terminal residue of a segment to superimpose on a fixed anchor residue in the protein structure. The algorithm, referred to as cyclic coordinate descent or CCD, involves adjusting one dihedral angle at a time to minimize the sum of the squared distances between three backbone atoms of the moving C-terminal anchor and the corresponding atoms in the fixed C-terminal anchor. The result is an equation in one variable for the proposed change in each dihedral. The algorithm proceeds iteratively through all of the adjustable dihedral angles from the N-terminal to the C-terminal end of the loop. CCD is suitable as a component of loop prediction methods that generate large numbers of trial structures. It succeeds in closing loops in a large test set 99.79% of the time, and fails occasionally only for short, highly extended loops. It is very fast, closing loops of length 8 in 0.037 sec on average.

Figure 1.

(A) C_α trace of a loop before (red) and after (green) closure with the flanking secondary structures (blue). The moving C-terminal anchor and the fixed C-terminal anchor are indicated. The loop closure problem is to adjust the dihedral angle degrees of freedom of the loop so that the moving C-terminal anchor is superimposed on the fixed C-terminal anchor. (B) Schematic of the CCD algorithm. Variables are defined in the text.

Figure 2.

Histograms of RMS values for unclosed loops at loop lengths of 4, 8, and 12 for the CCD No Constraint and the CCD Ramachandran Map algorithms. Note that all loops of length 12 closed for the CCD Ramachandran Map algorithm, so there is no plot.

Figure 3.

Histograms of the number of steps for loop closure at loop lengths 4, 8, and 12 for the CCD No Constraint and the CCD Ramachandran Map algorithms.

Figure 4.

C_α renderings of the lowest RMS loop generated from 5000 trials of the CCD Ramachandran Map method for loops of 4, 8, and 12 amino acids, compared with the X-ray structures (dark figures). (A) Loop 1ej0A_74–77, (B) loop 1ctqA_144–151, (C) loop 1eguA_508–519.

Figure 5.

Comparison of distribution of RMS among conformations generated from the same initial structure (light bars) and different initial structures (dark bars). PDB entry 1egu, residues 508–519, was used as the test case.