Geometric constraints within tripeptides and the existence of tripeptide reconstructions

Designing movesets providing high quality protein conformations remains a hard problem, especially when it comes to deform a long protein backbone segment, and a key building block to do so is the so-called tripeptide loop closure (TLC). Consider a tripeptide whose first and last bonds ( $$ and $$ ) are fixed, and so are all internal coordinates except the six $$ dihedral angles associated to the three $$ carbons. Under these conditions, the TLC algorithm provides all possible values for these six dihedral angles-there exists at most 16 solutions. TLC moves atoms up to $$ in one step and retains low energy conformations, whence its pivotal role to design move sets sampling protein loop conformations. In this work, we relax the previous constraints, allowing the last bond ( $$ ) to freely move in 3D space-or equivalently in a 5D configuration space. We exhibit necessary geometric constraints in this 5D space for TLC to admit solutions. Our analysis provides key insights on the geometry of solutions for TLC. Most importantly, when using TLC to sample loop conformations based on $$ consecutive tripeptides along a protein backbone, we obtain an exponential gain in the volume of the $$ -dimensional configuration space to be explored.