Topology of the Interconversion Pathway Networks of Cycloheptane Conformations and Those of Related n-Membered Rings

We, a collaboration group of mathematician and chemists, previously reported that the configuration space of n-membered alicyclic compounds (n = 5, 6, and 7) is diffeomorphic to (n - 4)-dimensional surface Sn-4 (S. Goto, K. Komatsu, Hiroshima Mathematical Journal 2012 42, 115-126). In the present work, we spread concrete conformations and their interconversion pathway networks over a conformational scattering manifold, stochastically computing the optimized conformations of the alicyclic compounds (cyclopentane, cyclohexane, and cycloheptane). The conformational scattering manifold, namely world map of conformation, of these compounds depends on the degree of freedom in the endocyclic torsion angles. The present computational study confirmed that the world map of cyclohexane consists of a compact two-dimensional surface S-2 equivalent to a traditional ring-puckering coordinate space. We then claim the world map of cycloheptane exists in a compact three-dimensional surface S-3 (3-manifold). Since these series of studies rationalizes the world map of transformation of the cyclic compound containing small-, medium-, and large-size ring portions and their various connections, it might give an opportunity to improve such medicinal resources.