A kinematic view of loop closure

Abstract

We consider the problem of loop closure, i.e., of finding the ensemble of possible backbone structures of a chain segment of a protein molecule that is geometrically consistent with preceding and following parts of the chain whose structures are given. We reduce this problem of determining the loop conformations of six torsions to finding the real roots of a 16th degree polynomial in one variable, based on the robotics literature on the kinematics of the equivalent rotator linkage in the most general case of oblique rotators. We provide a simple intuitive view and derivation of the polynomial for the case in which each of the three pair of torsional axes has a common point. Our method generalizes previous work on analytical loop closure in that the torsion angles need not be consecutive, and any rigid intervening segments are allowed between the free torsions. Our approach also allows for a small degree of flexibility in the bond angles and the peptide torsion angles; this substantially enlarges the space of solvable configurations as is demonstrated by an application of the method to the modeling of cyclic pentapeptides. We give further applications to two important problems. First, we show that this analytical loop closure algorithm can be efficiently combined with an existing loop-construction algorithm to sample loops longer than three residues. Second, we show that Monte Carlo minimization is made severalfold more efficient by employing the local moves generated by the loop closure algorithm, when applied to the global minimization of an eight-residue loop. Our loop closure algorithm is freely available at http://dillgroup. ucsf.edu/loop_closure/.

On the structure of the inverse kinematics map of a fragment of protein backbone

“…For any given X ∈ R 3 , when t varies, the left-hand side of Equation (6), X − r(t), spans a unit circle centered at X . When u and w vary, the right-hand side, q(u, w), spans a bounded and connected quartic surface Q (see Figure 4) that is the Minkowski sum of the first and the last circles in Equation (5). Appendix B gives a detailed description of Q.…”

“…Hence, the backbone of a protein formed by r amino-acids has 2r dofs. The study of the kinematics of a protein backbone -i.e., the study of its intrinsic mobility without regard to the forces needed to actually create motion -has attracted considerable interest (e.g., [5,12,13,14,25]), in particular to compute protein conformations subject to geometric constraints. Of special interest is the inverse kinematics (IK) problem for a backbone fragment.…”

“…Of special interest is the inverse kinematics (IK) problem for a backbone fragment. This problem is also known as the loop closure problem [2,5,12,24].…”

“…Case where n = 3: It is well known that for a 6-dof fragment F the number of values of f −1 (c) is finite and upper bounded by 16 (as they are the zeros of a monovariate polynomial of degree 16) [5,13]. However, it has recently been questioned whether this bound is tight [5,8,24].…”

“…However, it has recently been questioned whether this bound is tight [5,8,24]. Algorithms are available to compute the values of f −1 when n = 3, for instance the elegant analytical method developed in [5], but they only return the values of f −1 for given poses of T .…”