Energy Storage in a Hamiltonian System in Partial Contact with a Heat Bath

To understand the mechanism allowing for the long-term storage of excess energy in proteins, we study a Hamiltonian system consisting of several coupled pendula in partial contact with a heat bath. It is found that energy storage is possible when the motion of each pendulum switches between oscillatory (vibrational) and rotational (phase-slip) modes. The storage time increases almost exponentially to the square root of the injected energy. The relevance of our mechanism to protein motors is discussed. Proteins are among the most important biopolymers for living systems. They transform chemical energy to mechanical energy, and vice versa, and contribute to biological functions. However, the question of how proteins work dynamically remains unanswered. Recently, a noteworthy experiment concerning protein motors was performed [1]. In this experiment, the working process of a single molecule was directly investigated. The results suggest that proteins often store energy obtained from a reaction with ATP (adenosine triphosphate) and use it later (e.g., for enzymatic reactions with other proteins). The interval for energy storage was found to sometimes be very long, up to the order of seconds, while typical timescales for normal vibrations are several picoseconds. How can proteins store excess energy for such a long time, somehow overcoming the relaxation process toward thermal equilibrium? In order for a protein to store energy for a sufficiently long time, energy must be absorbed into a certain part of the protein, in accordance with its own dynamics. Furthermore, some characteristic type of dynamics is required to store the excess energy without losing it to the surrounding aqueous solution. Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki 310-8512 Department of Pure and Applied Sciences, College of Arts and Sciences, University of Tokyo, Tokyo 153