Physical principle that governs the protein's conformational conversion between different conformers, which are far apart in a phase space, remains unknown. Recent development in NMR techniques enabled us to observe the slow dynamics on the time scale of micro- to milliseconds of nuclei in a protein. However, the nature of such a slow motion is generally not well understood. For example, at room temperature the time constant of the refolding of prion protein is about ?100 mseconds, but that of slow dynamics is between sub-millisecond to milliseconds. During such a slow motion, the native structure has a chance to unfold completely. This kind of slow motion occurs in the critical region where fluctuation interferes with the thermal stability of the conformation. In order to describe the protein dynamics in a large phase space, here we have constructed a novel representation theory, termed "the thermodynamics of quantum cryptography", which is essentially based on the number theory. We could represent the conformational rearrangement with an extremely low probability in a critical region. This formalism includes the Dirichlet series to represent the partition function, which consists of periodic trajectories with a prime number step. These theories have been employed to interpret the slow dynamics of prion protein as well as its pathogenicity.
Keywords:slow dynamics; conformational rearrangement; perturbation theory; trace formula; number theory