A proof about molecular bearings

Abstract

Molecular bearings are likely to be as ubiquitous in future molecular machines as conventional bearings are in today's macroscopic machines. The ability to design molecular bearings will therefore be crucial. In molecular bearings made from a shaft with m-fold symmetry and a sleeve with n-fold symmetry, it is proved that the potential energy of the bearing as a function of the rotational position of the shaft within the sleeve will be periodic, with a period of GCD(m,n)/mn. This result continues to hold true even when the shaft and sleeve are jointly minimized, so that the abstract perfect symmetries of the shaft and sleeve are marred by the perturbations in structure each induces in the other. If the period is sufficiently short, e.g. if a small rotational change drives the bearing from one minimum to the adjacent minimum in the potential energy function, then the barrier height will be small and the bearing will be able to rotate without difficulty.