Concerted Fragment Motion

Residues must move in concert as a mathematical necessity: there are not enough degrees of freedom for a single residue to flex independently of its neighbors. Consider the segment of the backbone that lies between the C atom on residue i and the N atom on residue i +n + 1 for some integer number of residues, n. This segment of

the backbone, regardless of its length, has 6 degrees of freedom – I describe these degrees of freedom in the paragraphs below. If these residues are restricted to ideal bond geometry, then each residue may flex at itsφ and ψ dihedrals only. That is each residue contributes two degrees of freedom to the segment. Suppose we are interested in moving a single residue independently of its neighbors: that is, n = 1. There are six degrees of freedom between the end of residue i and the beginning of residuei+ 2, however residue i+ 1 has only two degrees of freedom. The system of equations for moving residue i+ 1 independently of its neighbors is over-constrained.

With three residues (n = 3), the backbone segment has six degrees of freedom (sixφ

and ψ angles), matching the number of degrees of freedom in the gap between residues

i and i+ 4. Therefore the system of equations to solve for theφ and ψ angles for the segment has a finite number of solutions: in fact, it has at most 16 solutions (Raghavan and Roth, 1989). If residue i is held fixed as is residue i+ 4, then the 16 possible solutions for the dihedral angles of residuesi+ 1, i+ 2 andi+ 3 may be computed. If a seventh degree of freedom can be given to the system, then motion can be simulated – the seventh degree of freedom can be sampled, and the values for the other six degrees of freedom can be computed. Small changes to the seventh degree of freedom simulate backbone motion (Figure 7.1).

Raghavan and Roth (1989) provide the robotics formulation of this “loop closure” problem, a robot arm has some number of degrees of freedom (torsional or otherwise), and at the end of the arm it has a hand. The robot must position its hand at a particular point and at a particular orientation. If the base of the arm is at the origin, then it takes three degrees of freedom to describe the point in space for the hand’s final position; it takes another three degrees of freedom to describe the hand’s orientation (which can be thought of as a point on the unit sphere, which takes two degrees of freedom to specify, and then a twist angle for the hand – the direction to point the thumb). If the robot’s arm has six degrees of freedom, then given the desired hand location and orientation, a finite number of solutions for positioning the arm can be found that place the hand correctly.

Raghavan and Roth observed that if the dihedral angles are given for a robot arm with six “revolute” (dihedral) degrees of freedom (often called an R6 arm), then the kinematics computation for the location and orientation of the hand position can be made with a series of linear operations, and in particular, by multiplying a set of Denavit-Hartenburg coordinate frames (Denavit and Hartenberg, 1955). Raghavan and Roth derived a set of linear equations to describe the relationship between the

Figure 7.1: PROBIK: Protein Backbone Motion By Inverse Kinematics. An illus- tration of the kinds of “motion” that can be generated by inverse kinematics. These are residues 212 through 215 of ribose binding protein (PDB ID: 1RBP), the scaffold protein on which Homme Hellinga’s lab built both a receptor protein (Looger et al., 2003) and an enzyme (Dwyer et al., 2004). By varying the N-Cα-C bond angle, τ, for residue 214 away from ideality, and solving for the other six dihedral angles pictured, inverse kinematics creates a series of conformations. Backbone carbonyl oxygen atoms are in red, Cα-Cβ bonds are in turquoise.

dihedral angles and the position of the robot’s hand. They proved that computing the valid dihedral angles given the location of the hand was equivalent to computing the roots of a degree-16 polynomial. There are at most sixteen roots to this polynomial, and thus there are a finite number of solutions for the dihedral angle solutions for the revolute joints. In practice, most of the roots for this polynomial are complex so that the typical number of realizable angular assignments to the revolute angles is 2, 4 or 6 (there are always an even number of realizable solutions since the complex solutions exist as conjugate pairs). Manocha and Canny expanded upon this solution by proving that the task of finding the roots of this degree-16 polynomial, which is numerically unstable, could be performed by finding the eigenvalues of a matrix, which is numerically stable (Manocha and Canny, 1994).

With Manocha’s implementation of the the inverse kinematics solution to computing the six revolute angles for an R6 robot arm, Noonan, O’Brien and Snoeyink (Noonan et al., 2004) proposed a method for flexing backbones by sampling a seventh degree of freedom and using inverse kinematics to solve for the other six degrees of freedom. A series of solutions produced by sampling the seventh degree of freedom appears to computer scientists and biochemist alike as smooth backbone motion (Figure 7.1).