Title:
The properties of bio-energy transport and influence of structure nonuniformity and temperature of systems on energy transport along polypeptide chains.

Abstract:
A new theory of bio-energy transport along protein molecules in living systems, where the energy is released by hydrolysis of adenosine triphosphate (ATP), is proposed based on some physical and biological reasons. In the new theory, the Davydov's Hamiltonian and wave function of the systems are simultaneously modified and extended. A new interaction has been added into Davydov's Hamiltonian. The wave function of the excitation state of single particles for the excitons in the Davydov model is replaced by a new wave function of two-quanta quasicoherent state. In such a case, the bio-energy is transported by the new soliton, which differs from the Davydov's soliton. The soliton is formed through self- trapping of two excitons interacting amino acid residues. The exciton is generated by vibrations of amide-I (CO stretching) arising from the energy of hydrolysis of ATP. The properties of the new soliton are extensively studied by analytical method and its lifetime is calculated using the nonlinear quantum perturbation theory and a wide ranges of parameter values relevant to protein molecules. The lifetime of the new soliton at the biological temperature 300 K is enough large and belongs to the order of 10⁻¹⁰ s, or τ/τ₀≥700, in which the soliton can transports over several hundreds amino acid residues. These studied results show clearly that the new soliton is thermally stable and has so larger lifetime that it can play an important role in biological processes. Thus the new model is a candidate of the bio-energy transport mechanism in protein molecules. In the meanwhile, the influences of structure nonuniformity in protein molecules and temperature of the systems on the states and properties of the soliton transport of bio-energy are numerically simulated and studied by the fourth-order Runge-Kutta method. The structure nonuniformity arises from the disorder distributions of masses of amino acid residues, side groups and impurities, which results also in the fluctuations of the spring constant of protein molecules, dipole-dipole interaction between the neighboring amides, exciton-phonon (vibration of amino acids)interaction, chain-chain interaction among the three channels and ground state energy of the systems. We investigated the behaviors and states of the new solitons in a single protein molecular chain and α-Helix protein molecules with three channels under influences of the structure nonuniformity. We prove first that the bio-energy is transported by a soliton, which can move without dispersion, retaining its shape, velocity and energy in a uniform and periodic protein molecule. When the structure nonuniformity exists, although the fluctuations of the spring constant, dipole-dipole interaction constant, exciton-phonon coupling constant and ground state energy and the nonuniformity distributions of masses of amino acid residues can change the states and properties of motion of new soliton, they are still quite stable and very robust against these structure nonuniformities, i.e., even there are a larger structure nonuniformity in the protein molecules, the new solitons cannot be still dispersed. If the effects of thermal perturbation of medium on the soliton in nonuniform proteins is considered again, the new soliton can transport also over a larger spacing of 400 amino acids and has a longer time period of 300 ps, it is still thermally stable up to 320 K under the influence of the above structure nonuniformities. However, the new soliton disperses in the case of a higher temperature of 325 K and in more large structure nonuniformity. Thus, we determine that the new soliton's lifetime and critical temperature are 300 ps and 320 K, respectively. These results are also consistent with analytical data obtained via quantum perturbed theory. For α-Helix protein molecules with three channels, the results obtained show that the structure nonuniformity and quantum fluctuation can change the states and features of the new solitons, for example, the amplitudes, energies and velocities of the new soliton are decreased, but the solitons have been not destroyed, they can still transport steadily along the molecular chains retaining energy and momentum. When the quantum fluctuations are larger, such as, structure disorders and quantum fluctuations of 0.67<α(K)<2, ΔW=±8%W¯, ΔJ=±1%J¯, Δ(χ₁+χ₂)=±3%(χ¯₁+χ¯₂) and ΔL=±1%L¯ and Δɛ₀=ɛ|β(n)|, ɛ=0.1 meV, |β(n)|<0.5, the new soliton is still stable. Therefore, the new solitons are quite robust against these nonuniform effects. However, they will be dispersed or disrupted in cases of very large structure nonuniformity. When the influence of temperature on solitons is considered, we find that the new solitons can transport steadily over 333 amino acid residues in the case of a long time period of 120 ps, in which the soliton can retain its shape and energy to travel forward along protein molecules after their mutual collision at the biological temperature of 300 K. However, the soliton disperses in cases of higher temperatures 325 K under action of a larger structure disorder. Thus, its critical temperature is about 320 K. When the effects of structure nonuniformity and temperature are considered simultaneously, then the new soliton has still high thermal stability and can transport also along the protein molecular chains retaining its amplitude, energy and velocity, they will disperses in the larger fluctuations, for example, 0.67 M¯<M(k)<2M¯, ΔW=±6%W¯, ΔJ=±1.3%J¯, Δ(χ₁+χ₂)=±2%(χ¯₁+χ¯₂), ΔL=±1.5%L¯ and Δɛ₀=ɛ|β(n)|, ɛ=0.82 meV, |β(n)|≤0.5 at T=300 K, or at higher temperatures 320 K and the fluctuations of 0.67M¯<M(k)<2M¯, Δ(χ₁+χ₂=±1%(χ¯₁+χ¯₂), ΔJ=±0.7%J¯, ΔW=±7%W¯, ΔL=±0.8%L¯ and Δɛ₀=ɛ|β(n)|, ɛ=0.4 meV, |β(n)|≤0.5. These results mean that the critical temperature of the new soliton is 315K in this condition. Thus, we can conclude from these investigations that the new soliton is quite robust against the structure nonuniformities and thermal perturbation of proteins at 300 K in α-helix protein molecules, then it is a carrier of bio-energy transport and the improved model is a candidate for the mechanism of the bio-energy transport in the protein molecules. Finally we gave some experimental evidences for real existence of the soliton and validity of the theory of bio-energy transport in proteins.