Exact analytical loop closure in proteins using polynomial equations

Loop closure in proteins has been studied actively for over 25 years. Using spherical geometry and polynomial equations, several loop-closure problems in proteins are solved exactly by reducing them to the determination of the real roots of a polynomial. Loops of seven, eight, and nine atoms are treated explicitly, including the tripeptide and disulfide-bonded loop-closure problems. The number of valid loop closures can be evaluated by the method of Sturm chains, which counts the number of real roots of a polynomial. Longer loops can be treated by three methods: by sampling enough dihedral angles to reduce the problem to a soluble loop-closure problem; by applying the loop-closure algorithm hierarchically; or by decimating the chain into independently moving rigid elements that can be reconnected using loop-closure algorithms. Applications of the methods to docking, homology modeling and NMR problems are discussed. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 819-844, 1999.