Davydov model: The quantum, mixed quantum-classical, and full classical systems

The need for a mechanism for energy transfer in proteins, such as the Davydov model, is emphasized. Here we concentrate on the finite-temperature properties of the Davydov model in three regimes: the quantum regime, in which both the excitation and the lattice are treated quantum mechanically; the mixed quantum-classical regime, in which the excitation is treated quantum mechanically but the lattice is considered classical; and the classical regime, in which both the excitation and the lattice are treated classically. The equilibrium behavior can be determined exactly in the three regimes and thus provides a way to evaluate the validity of the latter two regimes as well as a reference point for the nonequilibrium studies. Our results indicate that while at low temperature both the classical and the semiclassical regimes differ from the full quantum Davydov system, at biological temperatures the mixed quantum-classical regime leads to the same equilibrium behavior as the full quantum Davydov system. The nonequilibrium properties in the mixed quantum-classical regime are studied with a different set of equations of motion for finite temperature, which are derived in great detail in Sec. VI. At biological temperatures, these equations predict that the Davydov soliton is unstable. However, the states populated at biological temperatures preserve one of the features of the Davydov soliton, namely, the localization of the amide I excitation. The nonequilibrium equations in Sec. VI lead to a Brownian-like motion of the amide I excitation from the active site to other regions of the protein. This stochastic mechanism for energy transfer may constitute a first step in many biological processes.