Stationary multi-wave resonant ensembles in a microtubule

Stationary energy distribution between nonlinearity coupled modes of oscillations in microtubules is investigated using standard asymptotic approaches in the framework of classical mechanics. The first-order approximation analysis reveals three various types of triple-mode resonant ensembles consisting of one longitudinal wave and a pair of transverse modes being in phase. In turn, the nonlinear coupling between triads of various types and spectral states creates stationary multi-mode resonant ensembles. The corresponding stationary multi-mode solutions exhibit Lyapunov-stable quasiperiodic orbits in contrast to chaotic motions inherent in the same system. It is found that the energy distribution in microtubules being in thermal equilibrium is described by a recursive function that cannot be reduced to well-known classical statistical laws. Also, multi-frequency envelope solitons, taking place far from the thermal equilibrium of a microtubule, are possible when getting some portion of energy from the environment. The mechanical momentum of such solitons can propagate along microtubules without dispersion in long distances that can help for efficient heat exchange as well as in providing coherent movement of motor organelles.