Structure of the Singular Manifold of Protein Backbone and Its Applications

This paper aims to study the structure of the singular manifold of a fragment of protein backbone using the method of robot kinematics. By modeling chemical bonds as rigid links, and torsional motion around chemical bonds as revolute joints, a fragment of protein backbone turns into a kinematic linkage. The singular manifold of the forward kinematic map of this linkage contains important information about the structure of its inverse kinematics map, and the results also apply to the study of the conformation space of protein loops. By splitting the forward kinematic map into an orientation and position map, we derive equations of the singular manifolds for both maps and analyze their geometric and combinatorial structures. We discuss the application of singular manifolds in several problems of computational biology, including minimal parametrization of protein conformation space and conformation space sampling of protein loops.