Protein folding, protein dynamics and the topology of self-motions

It has long been recognized that segments of the protein main chain are like robotic manipulators and inverse kinematics methods from robotics have been applied to model loops to bridge gaps in protein comparative modelling. The complex internal motion of a redundant manipulator with fixed ends is called a self-motion and its character is determined by the relative position of its ends. Self-motions that are topologically equivalent (homotopic) occupy the same continous region of the configuration space. Topologically inequivalent (non-homotopic) regions are separated by co-regular surfaces and crossing a co-regular surface can result in a sudden dramatic change in the character of the self-motion. It is shown, using a five-residue type I β-turn, that these concepts apply to protein segments and that as the ends of the five-residue segment come closer together, a co-regular surface is crossed, and the structure is locked in to becoming either a type I or type I' turn. It is also shown that the type II turn is topologically equivalent to the type I' turn, not the type I turn. These results have implications for both native-state protein dynamics and protein folding.