Research Article

The production of Bio-energy and its properties of Transport in the Living Systems

Pang Xiao Feng*

Institutes of Physical electron and Life Science and Technology, University of Electronic Science and Technology of Chengdu 610054, China

*Address for Correspondence: Pang xiao feng, Institutes of Physical electron and Life Science and Technology, University of Electronic Science and Technology of Chengdu 610054, China, E-mail: pangxf2006@aliyun.com

Dates: 23 October 2017; Approved: 06 November 2017; Published: 09 November 2017

Citation this article: Feng PX. The production of Bio-energy and its properties of Transport in the Living Systems. Sci J Biomed Eng Biomed Sci. 2017;1(1): 030-052.

Copyright: © 2017 Feng PX. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Keywords: Form; Living system; Bio – energy; Protein; Biological energy; Soliton; ATP hydrolysis; Amide; Exciton; Life time; Amino acid; Quasi - coherent state

Abstract

We here introduced the form of bio - energy in living system and eluciduted again the new theory of bio-energy transport along protein molecules in living systems based on the changes of structure and conformation of molecules arising from the energy, which is released by hydrolysis of Adenosine Triphosphate (ATP). In this theory, the Davydov’s Hamiltonian and wave function of the systems are simultaneously improved and extended. A new interaction has been added into the original Hamiltonian. The original wave functions of the excitation state of single particles have been replaced by a new wave function of two-quanta quasicoherent state. In such a case, bio-energy is carried and transported by the new soliton along protein molecular chains. The soliton is formed through self - trapping of two excitons interacting amino acid residues. The exciton is generated by vibrations of amide - I (C = O stretching) arising from the energy of hydrolysis of ATP. The properties of the soliton are extensively studied by analytical method and its lifetime for a wide ranges of parameter values relevant to protein molecules is calculated using the nonlinear quantum perturbation theory. The lifetime of the new soliton at the biological temperature 300K is enough large and belongs to the order of 10-10 second orτ/τ0 ≥ 700. The different properties of the new soliton are further studied. The results show that the new soliton in the new model is a better carrier of bio - energy transport and it can play an important role in biological processes. This model is a candidate of the bio-energy transport mechanism in protein molecules.

The Phosphorylation and De - Phosphorylation Reactions in the cell and the features of energy released in hydrolysis of ATP molecules

As it is known, Kal’kar first proposedthe idea of aerobic phosphorylation, which is carried out by the phosphorylationcoupled to the respiration. Belitser studied in detail the stoichometric ratios between the conjugated bound phosphate and the absorption of oxygen and gave further the ratio of the number of ionorganic phosphate molecules to the number of oxygen atoms absorbed during the respiration, which is not less than two. He thought also that the transfer of electrons from the substrate to the oxygen is a possible source of energy for the formation of two or more ATP molecules per atom of absorbed oxygen. Therefore Belitser and Kal’kar’s research results are foundations establishing modern theory of oxidative phosphorylation of ATP molecules in the cell [1-3]. In such a case we must know clearly the mechanism and properties of the oxidation process, which involves the transfer of hydrogen atoms from the oxdised molecule to another molecule, in while there are always protons present in water and in the aqueous medium of the cell, thus we may only consider the transfer of electrons in this process. The necessary number of protons to form hydrogen atoms is taken from the aqueous medium. The oxidation reaction is usually proceded inside the cell under the action of special enzymes, in which two electrons are transferred from the food substance to some kind of initial acceptor, another enzymes transfer them further along the electron transfer chain to the second acceptor etc. Thus a water molecule is formed in which each oxygen atom requires two electrons and two protons. The main initial acceptors of electrons in cells are the oxidised forms NAD+ and NADP+ of NAD (nicotine amide adenine dinucleotide or pyridine nucleotide with two phosphate groups) molecules and NADP(nicotine amide adenine nucleotide phosphate or pyridine nucleotide with three phosphate groups) as well as FAD (flavin adenine dinucleotide or flavoquinone) and FMN (flavin mononucleotide).The above oxidised forms of these molecules servefor primary acceptors of electrons and hydrogen atoms through attaching two hydrogen atoms [3], which is expressed by

$N A D P^{+} + 2 H^{+} + 2 e^{-} \to N A D P \cdot H + H^{+}$

Where, NADP molecule becomes the reduced molecule NADP H. The NAD molecule has also the same active center as the NADP molecule; it can be converted to the reduced molecule NAD. H under combining with two atoms of hydrogen according to the reaction [3]:

$N A D^{+} + 2 H^{+} + 2 e^{-} \to N A D P^{+} + H^{+}$

The NAD and NADP are the enzymes, which can perform the reaction of dehydrogenation on compounds containing the group of atoms through removing two hydrogen atoms.

In the presence of enzymes, such as pyridine-dependent hydrogenases，and with the participation NAD and NADP molecules two hydrogen atoms, including two protons and twoo electrons, are removed from this group of atoms. One proton and two electrons combine with the NAD or NADP molecule converting them to the reduced forms NADP H or NAD H and the second proton is released. This mechanism can be also used to oxidise lactic acid (lactate) with the formation of pyruvic acid (pyruvate) and NAD H, in which the reduced molecules NADP H and NAD H serve as electron donors (reducing agents) in other reactions. They are involved in a large number of biosynthetic processes, such as in the synthesis of fatty acids and cholesterol. Therefore, the molecule NAD • H can serve as an electron donor in the process of oxidative phosphorylation, then the phosphorylation reaction is of [3].

$H^{+} + N A D \cdot H + 3 H_{3} P O_{4} + 3 A D P + 1 / 2 O_{2} \to N A D^{+} + 4 H_{2} O + 3 A T P$

Where, ADP is called the adenosine diphosphate. The abbreviated form of this reaction can be written as

$A D P + P_{i} \to A T P + H_{2} 0$

Thus three ATP molecules are formed in the reaction, in which the synthesis of ATP molecule are carried out through the transfer of two electrons from the NAD • H molecule along the electron transport chain to the oxygen molecule in the mitochondria. In this way the energy of each electron is reduced by 1.14 eV. The reaction is called the phosphorylation of ADP molecules. However, an ATP molecule can reacts with water in an aqueous medium, which results in the energy release of about 0.43eV under normal physiological conditions by virtue of some special enzymes. The reaction can be represented by

$A T P^{4 -} + H_{2} O \to A D P^{3 -} + H P O_{4}^{2 -} + H^{+} + 0.43 e V$

Its abbreviated form is of

$A T P + H_{2} O \to A D P + P_{i}$

In this process ATP molecules are transformed as ADP molecules and the bio-energy of about 0.43eV is also released. Then it is referred to as de-phosphorylation reaction of ATP molecules. We know from the above representations that an increase in free energy G in reaction and its decrease in reaction depend on their temperatures, concentrations of the ions Mg2+ and Ca2+ and on the pH value of the medium. Under the standard conditions $Δ G_{0}$ = 0.32 eV (～7.3 kcal/mole). If the appropriate corrections are made taking into consideration the physiological pH values and the concentration of Mg2+ and Ca2* inside the cell as well as the normal values for the concentrations of ATP and ADP molecules and inorganic phosphate in the cytoplasm we can obtain a value of ～ 0.54 eV (～12.5 kcal/mole) for the free energy in the hydrolysis of ATP molecu. Hence the free energy for the hydrolysis of ATP molecules is not constant. But it is impossibly the same at different sites of the same cell if these sites have different concentrations of ATP, ADP, Pi, Mg2+ and Ca .

On the other hand, cells contain a number of phosphorylated compounds the hydrolysis of which in the cytoplasm is associated with the release of free energy. Then the values for the standard free energy of hydrolysis for some of these compounds are also different. The enzymes carrying out the above synthesis of ATP molecules from ADP molecules and inorganic phosphate in the coupling membranes of mitochondria are the same as in the cytoplasmic membranes of bacteria, which are mainly composed of F₁ and F₀ , which are joined to each other by the small proteins F₅ and F₆. These proteins form the F₁ - F₀ complex or the enzyme ATP - ase, in which F₁ is composed of five protein subunits and has the shape of a sphere with a diameter of about 9nm which projects above the surface of the membrane in the form of a protuberance. In the coupling membrane of mitochondria and the cytoplasmic membrane of bacteria the complex F₁ - F₀ is positioned so that the enzyme F₁ is on the inside of the membrane [1-3]. The enzyme F₀ can extend from one side of the membrane to the other and has a channel which lets protons through. When two protons pass through the complex F₁ - F₀ in the coupling mitochondrial membrane one ATP molecule is synthesized inside the matrix from an ADP molecule and inorganic phosphate. This reaction is reversible. Under certain condition the enzyme transports protons from the matrix to the outside using the energy of dissociation of ATP molecules, which may be observed in a solution containing isolated molecules of enzyme F₁ and ATP. The largest two proteins in F₁, which is composed of fiveprotein molecules, take part in the synthesis and dissociation of ATP molecules, the other three are apparently inhibitors controlling these reactions. After removing enzyme F₁ molecules from mitochondria the remaining F₀ enzymes increase greatly the permeability of protons in the coupling membranes, which confirms that the enzyme F₀ has really a channel for the passage of protons which is constructed by the enzyme F₁. However, the complete mechanism for the synthesis of ATP molecules by the enzyme ATP - ase is still not clearly known up to now.

The Physical and Biological Foundations of Construction of New Theory

As it is known, many biological processes, such as muscle contraction, DNA reduplication, neuroelectric pulse transfer on the neurolemma and work of calcium pump and sodium pump, and so on, are associated with bioenergy transport through protein molecules, where the energy is released by the hydrolysis of Adenosine Triphosphate (ATP) in the living systems. Thus there here are always a biological process of energy transport from production place to absorption place in the living systems. In general, the bioenergy transport is carried out by virtue of protein molecules. Therefore, the study of the bioenergy transport along protein molecules is a very interesting subject in biology and has important significance in life science. However, understanding the mechanism of bioenergy transport in biomacromolecular systems has been a long-standing problem that remains of great interest today. As an alternative to electronic mechanisms [1], one can assume that the energy is stored as vibrational energy in the C = 0 stretching mode (amide-I) of a protein molecular chain of polypeptide. Following Davydov’s idea [2], ones take into account the coupling between the amide-I vibrarional quantum (exciton ) and the acoustic phonon (molecular displacements) in the amino acid residues; Through the coupling, nonlinear interaction appears in the motion of the vibrartional quanta, which could lead to a self-trapped state of the vibrational quantum.The latter plus the deformational amino acid lattice together can travel over macroscopic distances along the molecular chains, retaining the wave shape, energy, momentum and other properties of the quasiparticle. In this way, the bioenergy can be transported as a localized “wave packet” or soliton.This is just the Davydov’s model of bioenergy transport in proteins, which was proposed in the 1970s [2,3]. Davydov model of bioenergy transport work at $α -$ helical proteins as shown in figure1.

Following Davydov idea [3], the Hamiltonian describing such system has in the form of

$\begin{array}{l} H_{D} = \sum_{n} [ε_{0} B_{n}^{+} B_{n} - J (B_{n}^{+} B_{n - 1} + B_{n} B_{n + 1}^{+})] + \sum_{n} [\frac{P_{n}^{2}}{2 M} + \frac{1}{2} w {(u_{n} - u_{n - 1})}^{2}] \\ + \sum_{n} [χ_{1} (u_{n + 1} - u_{n - 1}) B_{n}^{+} B_{n}] = H_{e x} + H_{p h} + H_{int} (1) \end{array}$

where $ε_{0}$ = 0.205ev is the amide-I quantum energy, - J is the dipole-dipole interaction energy between neighbouring sites, $B_{n}^{+} (B_{n})$ is the creation (annihilation) operator for an amide - I quantum excitation (exciton) in the site n, un is the displacement operator of amino acid residues at site n, Pn is its conjugate momentum operator, M is the mass of an amino acid molecule, w is the elasticity constant of the protein molecular chains, and $χ_{1}$ is an nonlinear coupling parameter and represents the coupling size of the exciton-phonon interaction. The wave function of the systems proposed by Davydov is in the form of

$| D_{2} (t)$ ＞= $| ϕ_{D} (t) > 1 β (t) >$ = $\sum_{n} ϕ_{n} (t) B_{n}^{+} \exp (- \frac{i}{ℏ} \sum_{n} [β_{n} (t) P_{n} - π_{n} (t) u_{n}]) 10$ ＞ (2）

(Or)

$| D_{1} (t) > = \sum_{n} {φ_{n} (t) B_{n}^{+} \exp (\sum_{q} [α_{n q} (t) a_{q}^{+} - α_{n q}^{*} (t) a_{n}])} 10 > (3)$

where I0 ＞= I0 ＞ex I0＞ph, I0 ＞ ex and I0＞ph are the ground states of the exciton and phonon, respectively, $a_{q} (a_{q}^{+})$ is annihilation (creation) operator of the phonon with wave vector q, $ϕ_{n} (t) and β_{n} (t) = < Φ | u_{n} | Φ >$ and $π_{n} (t) = < | Φ | P_{n} | Φ >$ and $α_{n q} (t) = < D_{1} (t) | a_{q} | D_{1} (t) >$ are some undetermined functions of time. Obviously, $ϕ_{D} (t) > = \sum_{n} ϕ_{n} (t) B_{n}^{+} 10$ | ＞ex in Eq.(2) is an eigenstate of the number operator, $\hat{N} = \sum_{n} B_{n}^{+} B_{n}$ , corresponding to a single excitation, i.e., $\hat{N} | ϕ_{D} (t)$ ＞= $| ϕ_{D} (t)$ >. The Davydov soliton obtained from Eqs. (1) - (2) in the semiclassical limit and using the continuum approximation has the from

$ϕ_{D} (x, t) = {(\frac{μ_{D}}{2})}^{1 / 2} sech [\frac{μ_{D}}{r_{0}} (x - x_{0} - v t)] \exp {i [\frac{ℏ v}{2 J r_{0}^{2}} (x - x_{0}) - E_{ν} t / ℏ)]} (4)$

Corresponding to an excitation localized over a scale r0/ $μ_{D}$ , where

$μ_{D} = \frac{χ_{1}^{2}}{(1 - s^{2}) w J}, G_{D} = 4 J μ_{D}, s^{2} = \frac{v^{2}}{v_{0}^{2}}, v_{0} = r_{0} {(w/M)}^{1/2}$

Is the sound speed in the protein molecular chains, v is the velocity of the soliton, r0 is the lattice constant. Evidently, the soliton contains only one exciton, i.e., N=＜ $ϕ_{D} (t) | \hat{N} | ϕ_{D} (t) > = 1$ . This shows that the Davydov soliton is formed through self-trapping of one exciton with binding energy EBD, $E_{B D} = \frac{- χ_{1}^{4}}{3 J w^{2}}$ .

Davydov’s idea yields a compelling picture for the mechanism of bioenergy transport in protein molecules and consequently has been the subject of a large number of works [3-28]. A lot of issues related to the Davydov model, including the foundation and accuracy of the theory, the quantum and classical properties and the thermal stability and lifetimes of the Davydov soliton have extensively been studied by many scientists [7-26]. However, considerable controversy has arisen concerning whether the Davydov soliton is sufficiently stable in the region of biological temperature to provide a viable explanation for bio-energy transport. It is out of question that the quantum fluctuations and thermal perturbations are expected to cause the Davydov soliton to decay into a delocalized state. Some numerical simulations indicated that the Davydov soliton is not stable at the biological temperature 300K [7-11,24-26]. Other simulations showed that the Davydov soliton is stable at 300K [10-24], but they were based on classical equations of motion which are likely to yield unreliable estimates for the stability of the soliton [3]. The simulations based on the ID２＞ state in Eq. (2) generally show that the stability of the soliton decreases with increasing temperatures and that the soliton is not sufficiently stable in the region of biological temperature. Since the dynamical equations used in the simulations are not equivalent to the equation, the stability of the soliton obtained by these numerical simulations is unavailable or unreliable. The simulation [9] based on the ID１＞state in Eq. (3) with the thermal treatment of Davydov [3], where the equations of motion are derived from a thermally averaged Hamiltonian, yields the confusing result that the stability of the soliton is enhanced with increasing temperature, predicting that ID1＞- type soliton is stable in the region of biological temperature. Evidently, the conclusion is doubtful because the Davydov procedure in which an equation of motion for an average dynamical state from an average Hamiltonian, corresponding to the Hamiltonian averaged over a thermal distribution of phonons, is inconsistent with standard concepts of quantum-statistical mechanics in which a density matrix must be used to describe the system. Therefore, any exact fully quantum- mechanical treatment for the numerical simulation of the Davydov soliton does not exist. However, for the thermal equilibrium properties of the Davydov soliton, there is a quantum Monte Carlo simulation [13]. In the simulation, correlation characteristic of solitonlike quasiparticles occur only at low temperatures, about T<10k, for widely accepted parameter values. This is consistent at a qualitative level with the result of Cottingham et al [15,21]. The latter is a straightforward quantum-mechanical perturbation calculation. The lifetime of the Davydov soliton obtained by using this method is too small (about $10^{- 12} - 10^{- 13} \sec$ ) to be useful in biological processes. This indicates clearly that the Davydov solution is not a true wave function of the systems. A through study in terms of parameter values, different types of disorder, different thermalization schemes, different wave functions, and different associated dynamics leads to a very complicated picture for the Davydov model [10-12]. These results do not completely rule out the Davydov theory; however they do not eliminate the possibility of another wave function and a more sophisticated Hamiltonian of the system having a soliton with longer lifetimes and good thermal stability. Indeed, the question of the lifetime of the soliton in protein molecules is twofold. In Langevin dynamics, the problem consists of uncontrolled effects arising from the semiclassical approximation. In quantum treatments, the problem has been the lack of an exact wave function for the soliton. The exact wave function of the fully quantum Davydov model has not been known up to now. Different wave functions have been used to describe the states of the fully quantum-mechanical systems [4,5]. Although some of these wave functions lead to exact quantum states and exact quantum dynamics in the J = 0 state, they also share a problem with the original Davydov wave function, namely that the degree of approximation included when $J \neq 0$ is not known. Therefore, it is necessary to reform Davydov’s wave function. Scientists had though that the soliton with a multiquantum（ $n \geq 2$ ）, for example, the coherent state of Brown et al. [4], the multiquantum state of Kerr et al.[12] and Schweitzer et.al [15,21], the two-quantum state of Cruzeiro-Hansson [18] and Forner [22], and so on, would be thermally stable in the region of biological temperature and could provide a realistic mechanism for bioenergy transport in protein molecules. However, the assumption of the standard coherent state is unsuitable or impossible for biological protein molecules because there are innumerable particles in this state and one could not retain conservation of the number of particles of the system. The assumption of a multiquantum state (n > 2) along with a coherent state is also inconsistent with the fact that the bioenergy released in ATP hydrolysis can excite only two quanta of amide-I vibration. On the other hand, the numerical result shows that the soliton of two-quantum state is more stable than that with a one-quantum state. Cruzeiro - Hansson [18] had thought that Forner’s two-quantum state in the semiclassical case was not exact. Therefore, he constructed again a so-called exactly two-quantum state for the semiclassical Davydov system as follows [18]:

$I ϕ (t) > = \sum_{n, m = 1}^{N} φ_{nm} ({u_{1}}, {P_{1}}, t) B_{n}^{+} B_{m}^{+} | 0 >_{ex}, (5)$

Where, $B_{n} (B_{n}^{+})$ is the annihilation (creation) operator for an amide - I vibration quantum (exciton), is the displacement of the lattice molecules, $P_{1}$ is its conjugate momentum, and ${| 0 〉}_{e x}$ is the ground state of the exciton. He calculate the average probability distribution of the exciton per site, and average displacement difference per site, and the thermodynamics average of the variable, $P = B_{1}^{+} B_{1} - B_{2}^{+} B_{2}$ as a measure of localization of the exciton, versus quantity $ν = J W / χ_{1}^{2}$ and $l n β (β = 1 / K_{B} T)$ in the so-called two-quantum state. Eq. (5), where $χ_{1}$ is a nonlinear coupling parameter related to the interaction of the exciton-phonon in the Davydov model. Their energies and stability are compared with those of the one-quantum state. From the results of above thermal averages, he drew the conclusion that the wave function with a two-quantum state can lead to more stable soliton solutions than that with a one-quantum state, and that the usual Langevin dynamics ,whereby the thermal lifetime of the Davydov soliton is estimated, must be viewed as underestimating the soliton lifetime. However, by checking carefully Eq. (5), we can find that the Cruzeiro-Hansson wave function [18,24-26] does not represent exactly the two-quantum state. To find out how many quanta the state Eq.(1), indeed contains, the expectation value of the exciton number operator has to be computed. $N = {\sum_{n} B}_{n}^{+} B_{n}$ , in this state Eq.(5), and sum over the sites, i.e., the exciton numbers N are

$\begin{array}{l} N = < ϕ | \sum_{n} B_{n}^{+} B_{n} | ϕ > = \sum_{i j l m n} φ_{i m}^{*} φ_{j l} e x < 0 | B_{i} B_{m} B_{n}^{+} B_{n} B_{j}^{+} B_{l}^{+} | 0 >_{e x} (6) \\ = \sum_{n j} (φ_{n j}^{*} φ_{j n} + φ_{j n}^{*} φ_{j n}) + \sum_{n l} (φ_{n l}^{*} φ_{n l} + φ_{\ln}^{*} φ_{n l}) = 4 \end{array}$

Where, we use the relations

$[B_{n} . B_{j}^{+}] = σ_{n j}, \sum_{n l} {| φ_{n l} |}^{2} = 1 (7)$

$_{e x} 〈 0 | B_{n}^{+} {| 0 〉}_{e x} =_{e x} 〈 0 | B_{n}^{+} B_{n} {| 0 〉}_{e x} =_{e x} 〈 0 | B_{n}^{+} B_{n} B_{l} {| 0 〉}_{e x} = .... = 0 (8)$

Therefore, the state Eq. (5), as it is put forward in Ref.[10],deals with four excitons (quanta), instead of two excitons, in contradiction to the author’s statements. Obviously, it is impossible to create the four excitons by the energy released in the ATP hydrolysis (about 0.43 eV). Thus the author’s wave function is still not relevant to protein molecules, and his discussion and conclusion are all unreliable and implausible in that paper [10]. It is believed that the physical significance of the wave function, Eq. (5), is also unclear, or at least is very difficult to understand. As far as the physical meaning of Eq. (5) is concerned, it represents only a combinational state of single-particle excitation with two quanta created at sites n and m; $ϕ_{n m} ({u_{1}}, {P_{1}}, t)$ [18,26] is the probability amplitude of particles occurring at the sites n and m simultaneously. In general, n = ≠m and $ϕ_{n m} \neq ϕ_{n} ϕ_{m}$ in accordance with the author’s idea. In such a case it is very difficult to imagine the form of the soliton by the mechanism of self- trapping of the two quanta under the action of the nonlinear exciton-phonon interaction, especially when the difference between n and m is very large. Hansson has also not explained the physical and biological reasons and the meaning for the proposed trial state. Therefore, we think that the Cruzeiro-Hansson representation is still not an exact wave function suitable for protein molecules. Thus, the wave function of the systems is still an open problem today. On the basis of the work of Cruzeio-Hansson[18], [12,13,22], Schweitzer [21] and Pang [24-26] proposed a new model of the bioenergy transport in the protein molecules, in which both the Hamiltonian and the wave function of the Dovydov model [24] have been improved. A new coupling interaction between the acoustic and amide - I vibrational modes was added to the original Davydov's Hamiltonian which takes into account relative displacement of the neighbouring amino acids resulting from dipole - dipole interaction of the neighbouring amide -1 vibrational quanta. Davydov's wave function has been also replaced with a quasi-coherent two-quanta state to exhibit the coherent behaviors of collective excitations of the excitons and phonons [25,26] which are a feature of the energy released in ATP hydrolysis in the systems. The equation of motion and the properties of the new soliton in the new model are different from those in the Davydov model and as a result the soliton lifetime and stability are greatly enhanced. It is suggested that this model can resolve the controversy on the thermal stability and lifetime of the soliton excited in the protein molecules. The quantum properties of the new soliton will be studied here, but here attention is paid also to the problem of its lifetime and thermal stability at biological temperature 300K and the lifetime of the new soliton at 300K is calculated in detail by using the generally accepted values of the parameters appropriate to $α$ -helical protein molecules in terms of the quantum perturbation theory developed by Cottingham et al [15], which can take simultaneously into account the quantum and thermal effects. It can be seen that the lifetime of the new soliton at 300K is long enough to provide a viable explanation of the bio-energy transport in the proteins. The plan of this paper is as follows. In Section 2, the new model, including the extended Hamiltonian and the wave function, is presented. The equations of motion and the new soliton solution in this model are given in Section 3. In Section 4, the properties and thermal stability of the new soliton are discussed, and the possibility of the soliton being a suitable candidate for the mechanism of bioenergy transport in protein molecules is predicted on the basis of results obtained in this paper. In Section 5, the properties of the new soliton are described and its lifetime is calculated by using quantum-mechanical perturbution method. The detailed discussion of the properties and changes of the lifetimes of the soliton for a large range of parameter values is presented. The conclusions of this investigation are also given in this section.

Establishment of New theory of Bio - energy Transport in the Protein Molecules

Results obtained by many scientists over the years indicate that the Davydov model, whether it is the wave function or the Hamiltonian, is indeed too simple, i.e., it does not denote the elementary properties of the collective excitations occurring in protein molecules, and many improvements of it have been unsuccessful, as mentioned above. What is the source of this problem? It is well known that the Davydov theory on bioenergy transport was introduced into protein molecules from an exciton-soliton model in generally one-dimensional molecular chains [24]. Although the molecular structure of the alpha - helix protein is analogous to some molecular crystals, for example Acetanilide (ACN) (in fact, both are polypeptides; the alpha-helix protein molecule is the structure of three peptide channels, ACN is the structure of two peptide channels. If comparing the structure of alpha helix protein with ACN, we find that the hydrogen-boned peptide channels with the atomic structure along the longitudinal direction are the same except for the side group), a lot of properties and functions of the protein molecules are completely different from that of the latter. The protein molecules are both a kinds of soft condensed matter and bio-self-organization with action functions, for instance, self-assembling and self-renovating. The physical concepts of coherence, order, collective effects, and mutual correlation are very important in bio – self - organization, including the protein molecules, when compared with generally molecular systems [25,26]. Therefore, it is worth studying how we can physically describe these properties. It is noted that Davydov operation is not strictly correct. Therefore, it is believed that a basic reason for the failure of the Davydov model is just that it ignores completely the above important properties of the protein molecules. Let us consider the Davydov model with the present viewpoint. First, as far as the Davydov wave functions, both $| D_{1} > a n d | D_{2} >$ , are concerned [3], they are not true solutions of the protein molecules. On the one hand, there is obviously asymmetry in the Davydov wave function since the phononic parts is a coherent state, while the excitonic part is only an excitation state of a single particle. It is not reasonable that the same nonlinear interaction generated by the coupling between the excitons and phonons produces different states for the phonon and exciton. Thus, Davydov’s wave function should be modified [24-26], i.e., the excitonic part in it should also be coherent or quasicoherent to represent the coherent feature of collective excitation in protein molecules. However, the standard coherent [4] and large - n excitation states [12,22] are not appropriate for the protein molecules due to the reasons mentioned above. Similarly, Forner’s and Cruzeiro-Hansson’s two-quantum states do not fulfill the above request. In view of the above discussion, we proposed the following wave function of the protein molecular systems:

$\begin{array}{l} Φ (t) = {| φ}_{P} (t) > | β (t) > = \frac{1}{λ} [I + \sum_{n} φ_{n} (t) B_{n}^{+} + \frac{1}{2!} {(\sum_{n} φ_{n} (t) B_{n}^{+})}^{2}] | 0 >_{e x} \times (9) \\ \exp (- \frac{i}{ℏ} \sum_{n} [β_{n} (t) P_{n} - π_{n} (t) u_{n}]) {| 0 〉}_{p h} \end{array}$

Where, $B_{n}^{+}$ and $B_{n}$ are boson creation and annihilation operators for the exciton, $| 0 >_{e x} a n d | 0 >_{p h}$ are the ground states of the exciton and phonon, respectively $u_{n}$ and $P_{n}$ are the displacement and momentum operators of the amino acid residue at site n respectively. The $ϕ_{n} (t) . β_{n} (t) = < Φ (t) | u_{n} | Φ (t) > a n d π_{n} (t) = < Φ (t) | P_{n} | Φ (T) >$ are there sets of unknown functions, $λ$ is a normalization constant. It is assumed hereafter that $λ = 1$ for convenience of calculation, except when explicitly mentioned. A second problem arises for the Davydov Hamiltonian [24-26,28]. The Davydov Hamiltonian takes into account the resonant or dipole-dipole interaction of the neighboring amide - I vibrational quanta in neighboring amino acid residues with an electrical moment of about 3.5D, but why do we not consider the changes of relative displacement of the neighboring amino acid residues arising from this interaction ? It is reasonable to add the new interaction term $χ_{2} (u_{n + 1} - u_{n}) (B_{n + 1}^{+} B_{n} + B_{m}^{+} B_{n + 1})$ into the Davydov’s Hamiltonian to represent correlations of the collective excitations and collective motions in the protein molecules, as mentioned above [24-26]. Although the dipole- dipole interaction is small as compared with the energy of the amide-I vibrational quantum, the change of relative displacement of neighboring peptide groups resulting from this interaction cannot be ignored due to the sensitive dependence of dipole-dipole interaction on the distance between amino acids in the protein molecules, which is a kind of soft condensed matter and bio-self-organization. Thus, the Davydov Hamiltonian is replaced by

$\begin{array}{l} H = H_{e x} + H_{p h} + H_{int} = \sum_{n} [ε_{0} B_{n}^{+} B_{n} - J (B_{n}^{+} B_{n + 1} + B_{n}^{} B_{n + 1}^{+})] + \sum_{n} [\frac{P_{n}^{2}}{2 M} + \frac{1}{2} w (^{u_{n} - u_{n - 1}) 2}] \\ + \sum_{n} [χ_{1} (u_{n + 1} - u_{n - 1})] B_{n}^{+} B_{n} + χ_{2} (u_{n + 1} - u_{n}) \times (B_{n + 1}^{+} B_{n} + B_{n}^{+} B_{n + 1}) (10) \end{array}$

Where $ε_{0}$ = 0.205ev is the energy of the exciton (the C = 0 strechiong mode). The present nonlinear coupling constants are $χ_{1} a n d χ_{2}$ . They represent the modulations of the on-site energy and resonant (or dipole-dipole) interaction energy of excitons caused by the molecules displacements, respectively .M is the mass of a amino acid molcule and w is the elasticity constant of the protein molecular chains. J is the dipole-dipole interaction energy between neighboring sites. The physical meaning of the other quantities in $E q . (6)$ are the same as those in the above explanations. The Hamiltonian and wave function shown in Eqs.(9)-(10) are different from Davydov’s. We add a new interaction term, $\sum_{n} χ_{2} (u_{n + 1} - u_{n}) (B_{n + 1}^{+} B_{n} + B_{n}^{+} B_{n + 1}),$ into the original Davydov Hamiltonian. Thus the Hamiltonian now has better correspondence between the interactions and can also represent the features of mutual correlations of the collective excitations and of collective motions in the protein molecules. We should point out here that the different coupling between the relevant modes was also considered by Pang [24-26] and others [27-28] in the Hamiltonian of the vibron-soliton model for one-dimensional oscillator-lattice and protein systems, respectively, but the wave functions of the systems they used are different from Eqs. (9,10). Evidently , the present wave function of the exciton in Eq. (9) is not an excitation state of a single particle, but rather a coherent state, more precisely, a quasicoherent state, because it retain only fore three terms of the expansion of a standard coherent state, which can be viewed as an effective truncation of a standard coherent state. When $φ_{n} (t)$ is small, i.e., $| φ_{n} (t) | < < 1$ , Pang represented the wave function of the excitons, $| φ_{P} (t) >$ , in Eq.(9) as

$\begin{array}{l} | φ_{P} (t) > = \frac{1}{λ} [1 + \sum_{n} φ_{n} (t) B_{n}^{+} + \frac{1}{2!} {(\sum_{n} φ_{n} (t) B_{n}^{+})}^{2}] | 0 >_{e x} ~ \frac{1}{λ} \exp [- \frac{1}{2} \sum_{n} {| φ_{n} (t) |}^{2}] \times \\ \exp {\sum_{n} φ_{n} (t) B_{n}^{+}} | 0 >_{e x} = \frac{1}{λ} \exp {\sum_{n} [φ_{n} (t) B_{n}^{+} - φ_{n}^{*} (t) B_{n}]} | 0 >_{e x} (11) \end{array}$

The last representation in Eq. (11) is a standard coherent state. Therefore, the state of exciton denoted by the new wave function $| φ (t) >$ has a coherent feature. Thus the wave function in Eq. (11) is normalized at $λ = 1$ . Since ${\sum_{n} | φ_{n} (t) |}^{2} = 1$ is required in the calculation, then this condition of $| φ_{n} (t) | < < 1$ is naturally satisfied for the proteins consisting of several hundreds of amino acids. Just so, the wave function denoted in Eq.(9) represents exactly the coherent features of collective excitations of both the excitons and phonons caused by the nonlinear exciton - phonon interaction, this indicates that the wave function in Eq.(9) is justified for the proteins. However, it is not an eigenstate of the number operator, $\hat{N} = \sum_{n} B_{n}^{+} B_{n}$ , because of

$\hat{N} | φ_{P} > = \sum_{n} B_{n}^{+} B_{n} | φ_{P} 〉 = {\sum_{n} φ_{n} (t) B_{n}^{+} + {(\sum_{n} φ_{n} (t) B_{n}^{+})}^{2}} | 0 >_{e x} = 2 | φ_{P} 〉 - (2 + \sum_{n} φ_{n} (t) B_{n}^{+}) | 0 >_{e x} (12)$

Therefore, the $| φ_{P} 〉$ represents a coherent superposition of the excitonic state with two quanta and the ground state of the exciton, but it has a determinate numbers of quanta. From the expectation value of number operator $\hat{N}$ we find that this state contains the number of exciton is

$\begin{array}{l} N = < φ_{P} | \hat{N} | φ_{P} > = \sum_{n} 〈 φ_{P} | B_{n}^{+} B_{n} | φ_{P} 〉 = {\sum_{n} {| φ_{n} (t) |}^{2} + (\sum_{n} {| φ_{n} (t) |}^{2}) (\sum_{m} {| φ_{m} (t) |}^{2})} \\ = (\sum_{n} {| φ_{n} (t) |}^{2}) (1 + \sum_{m} {| φ_{m} (t) |}^{2}) = 2 (13) \end{array}$

Namely, it contains only two quanta. Where we utilize Eq. (8) and the following relation [24] is:

$\sum_{n} {| ϕ_{n} (t) |}^{2} = 1, \sum_{m} {| ϕ_{m} (t) |}^{2} = 1, [B_{n} . B_{m}^{+}] = δ_{n m} (14)$

$\begin{array}{l} _{e x} < 0 | B_{n}^{+} | 0 >_{e x} =_{e x} < 0 | B_{n}^{+} B_{n} | 0 >_{e x} =_{e x} < 0 | B_{n}^{+} B_{m} | 0 >_{e x} \\ =_{e x} < 0 | B_{n}^{+} B_{m} B_{l} | 0 >_{e x} =_{e x} < 0 | B_{n}^{+} B_{m} B_{l}^{+} B_{n} | 0 >_{e x} \\ =_{e x} < 0 | B_{n}^{+} B_{m} B_{i}^{+} B_{l} B_{j} | 0 >_{e x} =_{e x} < 0 | B_{n}^{+} B_{m} B_{l}^{+} B_{i} B_{j} B_{n} | 0 >_{e x} .... = 0 \end{array}$

Therefore, the new wave function is completely different from Davydov’s. The latter is an excitation state of a single particle with one quantum and an eigenstate of the number operator, but the former is not. The new state is a quasicoherent state. It contains only two excitons, which come from the second and third terms in Eq. (9), in which each term contributes only an exciton, but it is not an excitation state of two single parties. Hence, as far as the form of new wave function in Eq. (9) is concerned, it is either two - quanta states proposed by Forner [21] and Cruzeiro-Hansson [10,18] or a standard coherent state proposed by Brown et al. [4,2] and Kerr et al’s [13] and Schweitzer et al’s [15,21] multiquanta states. Therefore, the wave function, Eq. (9), is new for the protein molecular systems. It not only exhibitsthe coherent feature of the collective excitation of excitons and phonons caused by the nonlinear interaction generated by the exciton-phonon interaction, which , thus, also makes the wave function of the states of the system symmetrical, but it also agrees with the fact that the energy released in the ATP hydrolysis (about 0.43 eV) may only create two amide-I vibrational quanta which, thus, can also make the numbers of excitons maintain conservation in the Hamiltonian, Eq. (10). Meanwhile, the new wave function has another advantage, i.e., the equation of motion of the soliton can also be obtained from the Heisenberg equations of the creation and annihilation operators in quantum mechanics by using Eqs. (9) and (10), but the wave function of the states of the system in other models could not, including the one-quanta state [3] and the two-quanta state [12,22]. Therefore, the above Hamitonian and wave function, Eqs. (9) and (10), are reasonable and appropriate to the protein molecules.

The Dynamic Equation of Bio - energy Transport

We now derive the equations of motion from Pang’s model. First of all, we give the interpretation of $β_{n} (t)$ and $π_{n} (t)$ in Eq. (9). We know that the phonon part of the new wave function in Eq. (9) depending on the displacement and momentum operators is a coherent state of the normal model of creation and annihilation operators. A coherent state for the mode with wave vector $q$ is [3,12,24-26]

$| α (t) 〉 = \exp (\sum_{q} [α_{q} (t) a_{q}^{+} - α_{q}^{*} (t) a_{q}]) {| 0 〉}_{p h} (15)$

Utilizing the standard transformations

$u_{n} = \sum_{q} {[\frac{ℏ}{2 N M ω_{q}}]}^{1 / 2} e^{i q n r_{0}} (a_{- q}^{+} + a_{q}) P_{n} = i \sum_{q} {[\frac{M ℏ ω_{q}}{2 N}]}^{1 / 2} e^{i q n r_{0}} (a_{- q}^{+} - a_{q}) (16)$

We can get [12,23] $| α (t) 〉 = | β (t) 〉$ , where $| β (t) 〉$ is in Eq. (9), and $ω_{q} = 2 {(w / M)}^{1 / 2} \sin (r_{0} q / 2)$ , $r_{0}$ is the distance between neighboring amino acid molecules, and $a_{q} (a_{q}^{+})$ is the annihilation (creation) operator of the phonon with wave vector $q$ , where

$\begin{array}{l} 〈 α (t) | a_{q} | α (t) 〉 = a_{q} (t) = {(\frac{M ω_{q}}{2h})}^{1 / 2} β_{q} (t) + i {(\frac{1}{2M ℏ ω_{q}})}^{1 / 2} π_{q} (t) \\ β_{q} (t) = \frac{1}{\sqrt{N}} \sum_{n} e^{- {iqnr}_{0}} β_{n} (t) π_{q} (t) = \frac{1}{\sqrt{N}} \sum_{n} e^{- {iqnr}_{0}} π_{n} (t) 〈 Φ (t) | P_{n} | Φ (t) 〉 = π_{n} (t) (17) \end{array}$

Utilizing again the above results and the formulas of the expectation values of the Heisenberg equations of operators, $u_{n}$ and $P_{n}$ , in the state $| Φ (t) 〉$ .

$i ℏ \frac{\partial}{\partial t} 〈 Φ (t) | u_{n} | Φ (t) 〉 = 〈 Φ (t) | [u_{n}, H] | Φ (t) 〉 i ℏ \frac{\partial}{\partial t} 〈 Φ (t) | P_{n} | Φ (t) 〉 = 〈 Φ (t) | [u_{n}, H] | Φ (t) 〉 (18)$

We can obtain the equation of motion for the $β_{n} (t)$ as

$\begin{array}{l} M {\ddot{β}}_{n} (t) = w [β_{n + 1} (t) - 2 β_{n} (t) + β_{n - 1} (t)] + 2 χ_{1} [{| ϕ_{n + 1} (t) |}^{2} - {| ϕ_{n - 1} (t) |}^{2}] \\ 2 χ_{2} {ϕ_{n}^{*} (t) [ϕ_{n + 1} (t) - ϕ_{n - 1} (t)] + ϕ_{n} (t) [ϕ_{n + 1}^{*} (t) - ϕ_{n - 1}^{*} (t)]} (19) \end{array}$

From Eq. (19) we see that the presence of two quanta for the oscillators increases the driving force on the phonon field by that factor, when compared with the Davydov theory.

We now derive the equation of motion for the $φ_{n}$ . A basic assumption in the derivation is that $| Φ (t) 〉$ in Eq. (9) is a solution of the time-dependent $S h r \ddot{o} d i n g e r$ equation [24-26]:

$i ℏ \frac{\partial}{\partial t} | Φ (t) 〉 = H | Φ (t) 〉 (20)$

The left-hand side of Eq. (16) has [12,23]

$\begin{array}{l} i ℏ \frac{\partial}{\partial t} | Φ (t) 〉 = {i ℏ (\sum_{n} {\dot{ϕ}}_{n} (t) B_{n}^{+} + \sum_{n} {\dot{ϕ}}_{n} (t) ϕ_{n} (t) B_{n}^{+} B_{n}^{+} {| 0 〉}_{e x})} | β (t) 〉 \\ + | φ_{P} (t) 〉 {\sum_{n} {β_{n} (t) P_{n} - π_{n} (t) u_{n} + \frac{1}{2} [β_{n} (t) {\dot{π}}_{n} (t) - {\dot{β}}_{n} (t) π_{n} (t)]} | β (t) 〉} (21) \end{array}$

Now left-multiplying the both sides of Eq. (21) by $〈 Φ (t) |$ , the left-hand side of Eq. (21) can be

$i ℏ 〈 Φ (t) | u_{n} | Φ (t) 〉 = i ℏ \sum_{n} ϕ_{n}^{*} (t) ϕ_{n} (t) (\sum_{m} ϕ_{m}^{*} (t) ϕ_{m} (t) + 1) = \frac{5}{4} \sum_{n} [{\dot{β}}_{n} (t) π_{n} (t) - {\dot{π}}_{n} (t) β_{n} (t)] \sum_{n} {| ϕ_{n} (t) |}^{2} (22)$

Similarly, for the right-hand side of Eq. (20) we can have [12,23]

$\begin{array}{l} 〈 Φ (t) | (H_{ex} + H_{ph} + H_{int}) | Φ (t) 〉 = {\sum_{n} {ε_{0} | ϕ_{n} {(t)}^{2} | - J ϕ_{n}^{*} (t) [ϕ_{n + 1} (t) - ϕ_{n - 1} (t)]} \\ \times (1 + \sum_{m} {| ϕ_{m} (t) |}^{2}) + + {\sum_{n} {χ_{1} [β_{n + 1} (t) - β_{n - 1} (t)] {| ϕ_{n} (t) |}^{2} + χ_{2} [β_{n + 1} (t) - β_{n - 1} (t)] \\ \times φ_{n}^{*} (t) [φ_{n + 1} (t) - φ_{n - 1} (t)]}} (1 + \sum_{m} {| φ_{m} (t) |}^{2}) + \frac{5}{2} W (t) \sum_{n} {| φ_{n} (t) |}^{2} (23) \end{array}$

Where,

$W (t) = 〈 β (t) | H_{p h} | β (t) 〉 = \sum_{n} (\frac{1}{2 M} π_{n}^{2} (t) + \frac{1}{2} w {[β_{n} (t) - β_{n - 1} (t)]}^{2}) + \sum_{q} \frac{1}{2} ℏ ω_{q} (24)$

And utilizing Eqs. (8) and (12) - (14) and the relationships can be obtained:

$\begin{array}{l} \sum_{n} [β_{m + 1} (t) - 2 β_{m} (t) + β_{m - 1} (t)] β_{m} (t) = - \sum_{n} {[β_{m + 1} (t) - β_{m - 1} (t)]}^{2} \\ 〈 Φ (t) | \sum_{n} (B_{n}^{+} B_{n - 1} + B_{n} B_{n - 1}^{+}) | Φ (t) 〉 = \sum_{n} [ϕ_{n}^{*} (t) ϕ_{n + 1} (t) + ϕ_{n - 1}^{*} (t) ϕ (t)] (1 + \sum_{m} {| ϕ_{m} (t) |}^{2}) \\ 〈 Φ (t) | \sum_{n} (u_{n + 1} - u_{n - 1}) (B_{n}^{+} B_{n}) | Φ (t) 〉 = \sum_{n} {[β_{m + 1} (t) - β_{m - 1} (t)] {| ϕ_{n} (t) |}^{2}} (1 + \sum_{m} {| ϕ_{m} (t) |}^{2}) \\ 〈 Φ (t) | \sum_{n} (u_{n - 1} - u_{n}) (B_{n}^{+} B_{n - 1} + B_{n} B_{n - 1}^{+}) | Φ (t) 〉 = \sum_{n} {[β_{m + 1} (t) - β_{m - 1} (t)] [ϕ_{n}^{*} (t) ϕ_{n + 1} (t) + ϕ_{n - 1}^{*} (t) ϕ (t)]} \\ \times (1 + \sum_{m} {| ϕ_{m} (t) |}^{2}) (25) \end{array}$

From Eqs. (20) - (23) we can obtain

$\begin{array}{l} i ℏ \frac{\partial}{\partial t} ϕ_{n} (t) = ε_{0} ϕ_{n} (t) - J [ϕ_{n + 1} (t) + ϕ_{n - 1} (t)] + χ_{1} [β_{n + 1} (t) + β_{n - 1} (t)] ϕ_{n} (t) \\ - χ_{2} [β_{n + 1} (t) + β_{n} (t)] \times [ϕ_{n + 1} (t) + ϕ_{n - 1} (t)] \\ + \frac{5}{2} (W (t) - \frac{1}{2} \sum_{m} [{\dot{β}}_{m} (t) π_{m} (t) - {\dot{π}}_{m} (t) β (t)]) ϕ_{n} (t) (26) \end{array}$

In the continuum approximation we get from Eqs. (19) and (26)

$i ℏ \frac{\partial}{\partial t} ϕ (x, t) = R (t) ϕ (x, t) - J r_{0}^{2} \frac{\partial^{2}}{\partial x^{2}} ϕ (x, t) - G_{p} {| ϕ (x, t) |}^{2} ϕ (x, t) (27)$

And

$\frac{\partial β (x, t)}{\partial ξ} = \frac{\partial β (x, t)}{\partial x} = - \frac{4 (χ_{1} + χ_{1})}{w (1 - s^{2}) r_{0}} {| ϕ (x, t) |}^{2} (28)$

Here,

$ξ = x - v_{t} R (t) = ε_{0} - 2 J + \frac{5}{2} {W (t) - \frac{1}{2} \sum_{m} [{\dot{β}}_{m} (t) π_{m} (t) - {\dot{π}}_{m} (t) β (t)]}$ and $s = v / v {}_{0}$ . The soliton solution of Eq. (27) is thus

$ϕ (x, t) = {(\frac{μ_{p}}{2})}^{1 / 2} \sec h [(μ_{p} / r_{0}) (x - x_{0} - V_{t})] \times \exp {i [\frac{ℏ v}{2 J r_{0}^{2}} (x - x_{0}) - E_{v} \frac{t}{ℏ}]} (29)$

With $G_{P} = \frac{8 {(χ_{1} + χ_{2})}^{2}}{w (1 - s^{2})}, G_{P} = \frac{8 {(χ_{1} + χ_{2})}^{2}}{w (1 - s^{2})} (30)$

Although forms of the above equations of motion and the corresponding solution, Eqs. (27)- (30), are quite similar to that of the Davydov soliton, the properties of new soliton have very large differences from the latter because the parameter values in the equation of motion and the solution Eqs. (27) and (29), including $R (t), G_{P},$ and $μ_{P}$ , have obvious distinctions from that in the Davydov model. A straightforward result of Pang’s model is to increase the nonlinear interaction energy $G_{P} (G_{P} = 2 G_{D} [1 + 2 (χ_{2} / χ_{1}) + {(χ_{2} / χ_{1})}^{2}])$ and the amplitude of the new soliton and decrease its width due to an increase of [24-26] when compared with Davydov soliton [3], where $μ_{D} = x_{1}^{2} / w (1 - s^{2}) J$ , and $G_{D} = 4 x_{1}^{2} / w (1 - s^{2})$ are the corresponding values in the Davydov mode[3-8]l. Thus the localized feature of the new soliton is enhanced. Therefore its stability against the quantum fluctuation and thermal perturbations increased considerably as compared with the Davydov soliton.

The Properties of Carrier of Bio - energy Transport

The energy of soliton in Pang’s model becomes [24-26]

$\begin{array}{l} E = < Φ (t) | H | Φ (t) = \frac{1}{r_{0}} \int_{- \infty}^{\infty} 2 [J r_{0}^{2} {(\frac{\partial ϕ}{\partial x})}^{2} + R {| ϕ (x, t) |}^{2} - G_{p} {| ϕ (x, t) |}^{4} d x + \\ \frac{1}{r_{0}} \int_{- \infty}^{\infty} \frac{1}{2} [M {(\frac{\partial β (x, t)}{\partial t})}^{2} + w r_{0}^{} {(\frac{\partial β (x, t)}{\partial x})}^{2}] d x = E_{0} + \frac{1}{2} M_{s o l} v^{2} (31) \end{array}$

The rest energy of the new soliton is

$E_{0} = 2 (ε_{0} - 2 J) - \frac{8 {(x_{1} + x_{2})}^{4}}{3 w^{2} J} = E_{s}^{0} - W (32)$

Where, $W = [2 {(x_{1} + x_{2})}^{4}] / 3 w^{2} J$ is the energy of deformation of the lattice. The effective mass of the new soliton is

$M_{s o l} = 2 m_{e x} + \frac{8 {(x_{1} + x_{2})}^{4} (9 s^{2} + 2 - 3 s^{4})}{3 w^{2} J {(1 - s^{2})}^{3} v_{0}^{2}} (33)$

We utilize Eqs. (8) and (12)－(14) in the above calculations. In such a case, the binding energy of the new soliton is

$E_{B P} = \frac{- 8 {(x_{1} + x_{2})}^{4}}{3 J w^{2}} (34)$

$E_{B P}$ Is larger than that of the Davydov soliton.The latter is $E_{B D} = - x_{1}^{4} / 3 J w^{2}$ . They have the following relation:

$E_{B P} = 8 E_{B D} [1 + 4 (\frac{x_{2}}{x_{1}}) + 6 {(\frac{x_{2}}{x_{1}})}^{2} + 4 {(\frac{x_{2}}{x_{1}})}^{3} + {(\frac{x_{2}}{x_{1}})}^{4}] (35)$

We can estimate that the binding energy of the new soliton is about several decades larger than that of the Davydov soliton .This is a very interesting result. It is helpful to enhance thermal stability of the new soliton. Obviously, the increase of the binding energy of the new soliton comes from its two - quanta nature and the added interaction. $\sum_{i} χ_{2} (u_{n + 1} - u_{n}) (B_{n + 1}^{+} B_{n} + B_{n}^{+} B_{n + 1}^{})$ , in the Hamiltonian of the systems, Eq. (10). However, we see from Eq. (35) that the former plays the main role in the increase of the binding energy and the enhancement of thermal stability for the new soliton relative to the latter due to $χ_{2} < χ_{1}$ . The increase of the binding energy results in significant changes of properties of the new soliton,which are discussed as follows.

In comparing various correlations to this model, it is helpful to consider them as a function of a composite coupling parameter like that of Pouthier and Spatchek et al[32-35] and Scott[6]again, it is convenient to define another composite parameter[3,24-26]that can be written as

$4 π α_{P} = {(χ_{1} + χ_{2})}^{2} / 2 w ℏ ω_{D} (36)$

Where, $ω_{D} = {(w / M)}^{1 / 2}$ is the band edge for acoustic phonons (Debye frequency). If, $4 π α_{P} > 1$ it is said to be weak. Using widely accepted values for the physical parameters for the alpha helix protein molecule [2-23],

$\begin{array}{l} J = 1.55 \times 10^{- 22} J . w = (13 - 19.5) N / m . M = (1.17 - 1.91) \times 10^{- 25} k g \\ χ_{1} = 62 \times 10^{- 12} N . χ_{2} = (10 - 18) \times 10^{- 12} N . r_{0} = 4.5 \times 10^{- 10} m . (37) \end{array}$

We can estimate that the coupled constant lies in the region of $4 π α_{P} = 0.11 - 0.273$ , which is not a weakly coupled theory, the coupling strength is enhanced as compared with the Davydov model, the latter is $4 π α_{D}$ $= 0.036 - 0.045$ . Using the notation of Bullough et al. [29.30], Teki et al. [31,32], and Pouthier, et al.[33-35].

$γ = J / 2 ℏ w_{D} (38)$

In terms of the two composite parameters, and , the soliton binding energy for Pang’s model can be written by

$E_{B P} / J = 8 {(4 π α_{P} / γ)}^{2} / 3 M_{s o l} = 2 m_{e x} [1 + 32 {(4 π α_{P})}^{2} / 3] (39)$

From the above parameter values, we find $γ = 0.08$ . Utilizing this value, the $E_{B P} / J$ versus $4 π α$ relations in Eq. (39) are plotted in figure 1.

However, $E_{B P} / J = {(4 π α_{P} / γ)}^{2} / 3$ for the Davydov model (here $M_{s o l}^{'} = m_{e x} [1 + 2 {(4 π α_{P} / γ)}^{2} / 3],$ , $4 π α_{D} = χ_{1}^{2} / 2 w ℏ ω_{D}$ ), then the $E_{B D} / J$ versus $4 π α_{D}$ relation is also plotted in figure 2. From this figure we see that the difference of soliton binding energies between two models becomes larger with increasing $4 π α$ [24-26].

Also, we see clearly from Eqs. (28)－(32) and (35) that the localized feature of the new soliton is enhanced due to increases of the nonlinear interaction and of the binding energy of the new soliton resulting from the increases of exciton-phonon interaction in Pang’s model. Thus, the stability of the soliton against quantum and thermal fluctuations is also enhanced considerately [24-26]. As a matter of fact, the nonlinear interaction energy forming the new soliton in Pang’s model is $G_{P} = 8 {(χ_{1} + χ_{2})}^{2} / (1 - s^{2}) w = 3.8 \times 10^{- 32} J$ , and it is larger than the linear dispersion energy, $J = 1.55 \times 10^{- 32} J$ , i.e., the nonlinear interaction in Pang’s model is so large that it can actually cancel or suppress the linear dispersion effect in the equation of motion ,thus the new soliton is stable in such a case according the soliton theory [2,33-35]. On the other hand, the nonlinear interaction energy in the Davydov model is only $G_{D} = 4 χ_{1}^{2} / (1 - S^{2}) w = 1.8 \times 10^{- 21} J$ , and it is about three to four times smaller than $G_{P}$ . Therefore, the stability of the Davydov soliton is weaker as compared with the new soliton. Moreover, the binding energy of the new soliton in Pang’s model is $E_{B P} = (4.16 - 4.3) \times 10^{- 21} J$ in Eq. (31), which is somewhat larger than the thermal perturbation energy, $k_{B} T = 4.13 \times 10^{- 21} J$ , at $300 K$ and about four times larger than the Debye energy, $k Θ = ℏ ω_{D} = 1.2 \times 10^{- 21} J$ (there $ω_{D}$ is the Debye frequency). This shows that transition of the new soliton to a delocalized state can be suppressed by the large energy difference between the initial (solitonic) state and final (delocalized) state, which is very difficult to compensate with the energy of the absorbed phonon. Thus, the new soliton is robust against quantum fluctuations and thermal perturbation, therefore it has a large lifetime and good thermal stability in the region of biological temperature .In practice, according to Schweitzer et al.s studies (i.e the lifetime of the soliton increases as $μ_{p}$ and $T_{0} = ℏ V_{0} μ_{p} / K_{B} π$ increase at a given temperature)[15] and the above obtained results, an inference could roughly be drawn that the lifetime of the new soliton will increase considerably as compared with that of the Davydov soliton due to the increase of $μ_{p}$ and $T_{0}$ because the latter are about three times larger than that of the Davydov model. On the other hand, the binding energy of the Davydov soliton $E_{BD} = χ_{_{1}}^{4} / {3w}^{2} J = 0.188 \times 10^{- 21} J$ , and it is about 23 times smaller than that of the new soliton, about 22 times smaller than $K_{B} T$ , and about 6 times smaller than $K_{B} Θ$ , respectively. Therefore, the Davydov soliton is easily destructed by the thermal perturbation energy and quantum transition effects. Thus it indicates that the Davdov soliton has a very small lifetime, and it is unstable at the biological temperature $300 K$ . This conclusion is consistent at a qualitative level with the result s of Wang et al.[13,14] and Cottingham et al.[15,21]. One can sum up the differences between Pang’s model and Davydov’s model, Eqs. (1) - (4), as follows. First, the parameter μp is increased (μp =2μD $[1 + 2 (\frac{χ_{2}}{χ_{1}}) + {(\frac{χ_{2}}{χ_{1}})}^{2}]$ ,). Secondly the non-linear coupling energy becomes Gp = ${\frac{8 (χ_{1} + χ_{2})}{w (1 - s^{2})}}^{2}$ (Gp=2GD $[1 + 2 (\frac{χ_{2}}{χ_{1}}) + {(\frac{χ_{2}}{χ_{1}})}^{2}]$ , where $G_{D} = \frac{4 χ_{1}^{2}}{w (1 - s^{2})}$ is the nonlinear interaction in the Davydov model, resulting from the two - quanta nature and the enhancement of the coupling the coefficient ( 1+ 2). For -helical protein molecules, and using the parameter values listed in Eq. (37) the values of the main parameters in Pang’s model can be calculated. These values and the corresponding values in the Davydov model are simultaneously listed in table 1. From table 1 we can see clearly that the new model produces considerable changes in the properties of the new soliton, such as large increase of the nonlinear interaction, binding energy and amplitude of the soliton, and decrease of its width as compared to that of the Davydov soliton. This indicates that the soliton in Pang’s model is more localized and more robust against quantum and the thermal stability has been enhanced [2,27,28] which implies an increase in lifetime for the new soliton. From Eq. (19) it can also be found that the effect of the two - quanta nature is larger than that of the added interaction. We can therefore refer to the new soliton as quasi-coherent.

In the above studies, the influences of quantum and thermal effects on soliton state, which are expected to cause the soliton to decay into delocalized states, we postulate that the model Hamiltonian and the wavefunction in Pang’s model together give a complete and realistic picture of the interaction properties and allowed states of the protein molecules. The additional interaction term in the Hamiltonian gives better symmetry of interactions. The new wavefunction is a reasonable choice for the protein molecules because it not only can exhibit the coherent features of collective excitations arising from the nonlinear interaction between the excitons and phonons, but also retain the conservation of number of particles and fulfil the fact that the energy released by the ATP hydrolysis can only excite two quanta. In such a case, using a standard calculating method [2,26] and widely accepted parameters we can calculate the region encompassed of the excitation or the linear extent of the new soliton, $Δ X = 2 π r_{0} / μ_{p}$ , to be greater than the lattice constant r0, i.e., $Δ X > r_{0}$ as shown in table 1. Conversely, we can explicitly calculate the amplitude squared of the new soliton using Eq. (29) in its rest frame as $| φ (x) |^{2} = \frac{μ_{p}}{2} \sec h^{2} (\frac{μ_{p} x}{r_{0}})$ . Thus the probability to find the new soliton outside a range of width r0 is about 0.10. This number can be compatible with the continuous approximation since the quasi-coherent soliton can spread over more than one lattice spacing in the system in such a case. This proves that assuming the continuous approximation used in the calculation is valid. Therefore we should believe that the above calculated results obtained from Pang’s model is correct.

The lifetime of the carrier of Bio-Energy transport at biological temperature

Partially Diagonalized Form of the Model Hamiltonian: The lifetime of the soliton in the protein molecules is an centre problem in the process of bioenergy transport because the soliton possess certain biological meanings and can play an important role in the biological process, only if it has enough long lifetimes. Therefore, to calculate the lifetime of the new soliton in Pang’s model has important significance. For convenlence of calculation, we here represent the wave function of the system in Eq. (9) by [24-26]

$| Φ (t) ＞ = ϕ (t) ＞ | β (t) ＞ = U_{1} 10 ＞_{ex} U_{2} 10 ＞_{ph}, (40)$

Where,

$U_{1} = \frac{1}{λ} [1 + \sum_{n} ϕ_{n} (t) B_{n}^{+} + \frac{1}{2!} (\sum_{n} ϕ_{n} (t) B_{n}^{+})^{2}] (40a)$

$U_{2} = \exp {- \frac{i}{ℏ} \sum_{n} [β_{n} (t) P_{n} - π_{n} (t) u_{n}]} (40b)$

$= \exp {\frac{1}{\sqrt{N}} \sum_{q} α_{q} (t) a_{q}^{+} - α_{q}^{*} (t) a_{q}} (40c)$

Where, we assume ${\sum_{i} | ϕ_{i} |}^{2} = n$ , where n is an integer, denotes the number of particle. The wave function, Eq.(40), does not only exhibit coherent properties, but also agrees with the fact that the energy released in the ATP hydrolysis (about 0.43eV) excites only two amide-I vibrational quanta, instead of multiquanta (n > 2) [24-26]. Therefore, the Hamitonian and wave function of the systems, Eqs. (9) - (10), or (40) are reasonable and appropriate to the protein molecules. Using the standard transformation in Eq. (16), where, $ω_{q} = 2 {(w / M)}^{1 / 2} \sin (\frac{r_{0} q}{2})$ , Eq.(10) becomes

$\begin{array}{l} H = \sum_{n} [ε_{0} B_{n}^{+} B_{n}^{} - J (B_{n}^{+} B_{n + 1}^{} + B_{n + 1}^{+} B_{n}^{})] + \sum_{q} ℏ ω_{q} (a_{q}^{+} a_{q} + \frac{1}{2}) \\ + \frac{1}{\sqrt{N}} \sum_{q . n} [g_{1} (q) B_{n}^{+} B_{n}^{} + g_{2} (q) (B_{n}^{+} B_{n + 1}^{} + B_{n}^{+} B_{n + 1}^{})] (a_{q} + a_{- q}^{+}) e^{i n r_{0}^{} q} (41) \end{array}$

Where,

$g_{1} (q) = 2 χ_{1} i {[\frac{ℏ}{2 M ω_{q}}]}^{1 / 2} \sin r_{0} q; g_{2} (q) = χ_{2} {[\frac{ℏ}{2 M ω_{q}}]}^{1 / 2} (e^{i r_{0} q} - 1) (42)$

In a semiclassical and continuum approximations, from Eq. (41) we can obtain the envelope soliton solution Eq. (29) in Pang’s model, we now represent Eq. (29) by the following form [24-26]

$ϕ (x, t) = {(\frac{μ_{p}}{2})}^{1 / 2} S e c h [\frac{μ_{p}}{r_{0}} (x - v t)] \exp [\frac{i}{ℏ} (\frac{ℏ^{2} v x}{2 J r_{0}^{2}} - E_{s o l} t)] (43)$

where

$μ_{p} = \frac{2 {(χ_{1} + χ_{2})}^{2}}{w (1 - s^{2}) J} (44)$

The energy of the new soliton is

$E_{S 01} = 2 [(ε_{0} - 2 J) + \frac{ℏ^{2} v^{2}}{4 J r_{0}^{2}} - \frac{2 μ_{p}^{2}}{3} J] (45)$

Thus we can also find out that

$α_{q} (t) = \frac{i π (χ_{1} + χ_{2})}{w μ_{p} (1 - v^{2} / v_{0}^{2})} {[\frac{M}{2 ℏ ω_{q}}]}^{\frac{1}{2}} (ω_{q} + q v) \csc h (π q r_{0} / 2 μ_{p}) e^{i q v t} = α_{q} e^{i q v t} (46)$

This treatment yields a localized coherent structure of the excitons with size of order 2 $π$ r0/μp that propagates with velocity v and can transfer energy ES01＜ $2 ε_{0}$ . Unlike bare excitons that are scattered by the interactions with the phonons, this soliton state describes a quasi-particle consisting of the two excitons plus a lattice deformation and hence a priori includes interaction with the acoustic phonons. So the soliton is not scattered and spread by this interaction of the vibration of amino acids (lattices), and can maintain its form, energy, momentum and other quasiparticle properties moving over a macroscopic distance. The bell-shaped form of the soliton Eq. (43) does not depend on the excitation method. It is self-consistent. Since the soliton always move with velocity less than that of longitudinal sound in the chain they do not emit phonons, i.e., their kinetic energy is not transformed into thermal energy. This is one important reason for the high stability of the new soliton. In addition the energy of the soliton state is below the bottom of the bare exciton bands, the energy gap being $4 μ_{p}^{2} J / 3$ for small velocity of propagation. Hence there is an energy penalty associated with the destruction with transformation from the soliton state to a bare exciton state, i.e, the destruction of the soliton state requires simultaneous removal of the lattice distortion. We know in general that the transition probability to a lattice state without distortion is very small, in general, being negligible for a long chain. Considering this it is reasonable to assume that such a soliton is stable enough to propagate through the length of a typical protein structure. However, the thermal stability of the soliton state must be calculated quantitatively. The following calculation addresses this point explicitly [24-26]. We now diagonalize partially the model Hamiltonian in order to calculate the lifetime of the soliton, Eq. (43), using the quantum perturbation method [14]. Since one is interested in investigating the case where there is initially a soliton moving with a velocity v on the chains, it is convenlent to do the analysis in a frame of reference where the soliton is at rest. We should then consider the Hamiltonian in this rest frame of the soliton, $\tilde{H}$ -vP, where P is the total momentum, and P= $\sum_{q} ℏ q (a_{q}^{+} a_{q} - B_{q}^{+} B_{q})$ , Where, $B_{q}^{+} = \frac{1}{\sqrt{N}} \sum_{n} e^{i q n r_{0}} B_{n}^{+}$ . Also, in order to have simple analytical expressions we make the usual continuum approximation. This gives

$\begin{matrix} \tilde{H} = \int_{0}^{L} d x 2 [\begin{array}{l} (ε_{0} - 2 J) ϕ^{+} (x) ϕ (x) + J r_{0}^{2} \frac{\partial ϕ^{+}}{\partial x} \frac{\partial ϕ}{\partial x} - \frac{i ℏ v}{2} \\ (\frac{\partial ϕ^{+}}{\partial x} ϕ (x) - ϕ^{+} (x) \frac{\partial ϕ}{\partial x}) \end{array}] (47) \\ + \sum_{q} \begin{array}{l} ℏ (ω_{q} - q v) a_{q}^{+} a_{q} + \frac{1}{\sqrt{N}} \sum_{q} 2 [g_{1} (q) + 2 g_{2} (q)] \\ (a_{- q}^{+} + a_{q}) \int_{0}^{L} d x e^{i k x} ϕ^{+} (x) ϕ^{} (x) \end{array} \end{matrix}$

Where, $ϕ (x)$ represents now the field operator corresponding to Bn in the continuum limit (whereas before it only indicated a numerical value). Here L = Nr0, －π＜kr0＜π, and ≈ (w/M)1/2 r0•|q｜, x = nr0. Since the soliton excitation is connected with the deformation of intermolecular spacing, it is necessary to pass in Eq. (47) to new phonons taking this deformation into account. Such a transformation can be realized by means of the following transformation of phonon operators [29].

$b_{q} = a_{p} - \frac{1}{\sqrt{N}} α_{q}, b_{q}^{+} = a_{q}^{+} - \frac{1}{\sqrt{N}} α_{q}^{*}, (48)$

Which describe phonons relative to a chain with a particular deformation, where bq (b $_{q}^{+}$ ) is the annihilation (creation) operator of new phonon. The vacuum state for the new phonons is

$| \tilde{0} 〉_{p h} = \exp [\frac{1}{\sqrt{N}} \sum_{q} (α_{q} (t) a_{q}^{+} - α_{q}^{*} (t)) a_{q}] | 0 >_{p h} (49)$

Which is a coherent phonon state [30], i.e., bq| $\tilde{0} 〉_{p h} = 0$ . The Hamiltonian $\tilde{H}$ can now be rewritten [24-26] as

$\begin{matrix} \tilde{H} = \int_{0}^{L} 2 d x ϕ (x) [ε_{0} - 2 J + V (x) - J r_{0}^{2} \frac{\partial^{2}}{\partial x^{2}} + i ℏ \frac{\partial}{\partial x}] ϕ (x) + \\ \sum_{q} ℏ (ω_{q} - q v) [b_{q}^{+} b_{q} + \frac{1}{\sqrt{N}} (α_{q} b_{q}^{+} + α_{q}^{*} b_{q}^{+})] + W' + \\ \frac{1}{\sqrt{N}} \sum 2 [g_{1} (q) + 2 g_{2} (q)] (b_{- q}^{+} + b_{q}) \int_{0}^{L} d x e^{i q x} ϕ_{}^{+} (x) ϕ (x) (50) \end{matrix}$

Where,

$W' = \frac{1}{N} \sum_{q} ℏ (ω_{q} - q v) | α_{q}^{} |^{2} V (x) = \frac{1}{N} \sum_{q} [g_{1} (q) + 2 g_{2} (q)] (α_{- q}^{*} + α_{- q}^{}) e^{i q x} (51)$

To describe the deformation corresponding to a soliton in the subspace where there is

$\int_{0}^{L} d x ϕ_{}^{+} (x) ϕ_{}^{} (x) = 1$

From Eq (45) in such a case. From the above formulae we can obtain

$V (x) = - 2 J μ_{p}^{2} \sec h^{2} (μ_{p} x / r_{0}) (52)$

In order to partially diagonalize the Hamiltonian Eq.(50) we introduce the following canonical transformation[14,23]

$ϕ (x) = \sum_{j} A_{j} C_{j} (x), ϕ^{+} (x) = \sum_{j} C_{j}^{*} (x) A_{j}^{+} (53)$

Where,

$\int C_{1}^{*} (x) C_{j} (x) d x = δ_{l j}, \sum_{j} \begin{array}{l} C_{j}^{*} (x^{'}) C_{j} (x) \\ = δ (x - x^{'}), \int d x | C_{j} (x) |^{2} = 1 \end{array} (54)$

The operators $A_{s}^{+}$ and $A_{k}^{+}$ are the creation operators for the bound states Cs(x) and delocalized state Ck(x), respectively. The detailed calculation of the partial diagonalization and of corresponding Cs(x) and Ck(x) are described in Appenix A. The partially diagonalized Hamiltonian obtained is as follows

$\begin{array}{l} \tilde{H} = W' + E_{s} A_{s}^{+} A_{s} + \sum_{k} E_{k} A_{k}^{+} A_{k} + \sum_{q} ℏ (ω_{q} - q v) b_{q}^{+} b_{q} + \\ \frac{1}{\sqrt{N}} \sum_{q} ℏ (ω_{q} - q v) (b_{q}^{+} α_{q} + α_{q}^{*} b_{q}) (1 - A_{s}^{+} A_{s}) + \frac{1}{\sqrt{N}} \\ \sum_{k k^{'} q} F (k, k^{'}, q) (b_{- q}^{+} + b_{q}) A_{k^{'}}^{+} A_{k} - \frac{1}{\sqrt{N}} \\ \sum_{k q} \tilde{F} (k, q) (b_{- q}^{+} + b_{q}) (A_{s}^{+} A_{- k} - A_{k}^{+} A_{s}) \end{array}$

and

$C_{s} (x) = {(\frac{μ_{p}}{2 r_{0}})}^{1 / 2} \sec h (μ_{p} x / r_{0}) \exp [i ℏ x v / 2 J r_{0}^{2}], with E_{s} = 2 [ε_{0} - 2 J - \frac{ℏ^{2} v^{2}}{2 J r_{0}^{2}} - μ_{p}^{2} J] (56a)$

$C_{k} (x) = \frac{μ_{p} \tanh (μ_{p} x / r_{0}) - i k r_{0}}{\sqrt{N r_{0}} [μ_{p} - i k r_{0}]} \exp [i k x + \frac{i ℏ v x}{2 J r_{0}^{2}}], (56b)$

with

$E_{k} = 2 [ε_{0} - 2 J - \frac{ℏ^{2} v^{2}}{2 J r_{0}^{2}} - J {(k r_{0})}^{2}]$

Where,

$F (k, k^{'}, q) = 2 [g_{1} (q) + 2 g_{2} (q)] \int_{0}^{L} d x e^{i q x} C_{k^{'}}^{*} (x) C_{k} (x)$

$\begin{array}{l} \approx 2 [g_{1} (q) + 2 g_{2} (q)] {1 - \frac{i μ_{p} q r_{0}}{[μ_{p} + i (k + q) r_{0}] [μ_{p} - i k r_{0}]}} (57) \\ \approx F [k, (k + q), q] δ_{k^{'} k + q} \end{array}$

$\tilde{F} (k, q) = 2 [g_{1} (q) + 2 g_{2} (q)] \int_{0}^{L} d x e^{i q x} C_{k^{'}}^{*} (x) C_{s} (x)$

$= \frac{2 π}{{\sqrt{2 μ}}_{p}} [g_{1} (q) + 2 g_{2} (q)] {\frac{i q r_{0}}{[μ_{p} + i k r_{0}]}} \sec h [π (k - q) r_{0} / 2 μ_{p}] (58)$

Where, $α_{q}$ is determined by V(x) and the condition, ( $ω_{q}$ －vq) $α_{q}$ =( $ω_{q}$ +qv) $α_{q}^{*}$ , which is required to get the factor, (1－ $A_{s}^{+} A_{s}$ ),in the $\tilde{H}$ in Eq.(55). Thus we find

$α_{q} = \frac{i π (χ_{1} + χ_{2})}{w μ_{p} (1 - s^{2})} {[\frac{M}{2 ℏ ω_{q}}]}^{1 / 2} (ω_{q} + q v) \csc h (π q r_{0} / 2 μ_{p}) (59)$

and

$W' = \frac{2}{3} μ_{p}^{2} J$

For this $α_{q}$ the | $\tilde{0}$ ＞ph in Eq.(49) is just the coherent phonon state introduced by Davydov. However, the bound state Cs(x) in Eq.(56a), unlike the unbounded state Ck(x) in Eq.(56b), is self-consistent with the deformation. Such a self-consistent state of the intramolecular excitation and deformation forms a soliton which in the intrinsic reference frame is stationary. For the new soliton described by the state vector $| ψ > = \frac{1}{\sqrt{2!}} {(A_{s}^{+})}^{2} | 0 >_{e x} | \tilde{0} >_{p h}$ the average energy of $\tilde{H}$ in Eq.(55) is

$< ψ | \tilde{H} | ψ > = 2 (ε_{0} - 2 J - \frac{ℏ^{2} v^{2}}{4 J r_{0}^{2}}) - \frac{4}{3} J μ_{p}^{2} (60)$

A Evidently, the average energy of $\tilde{H}$ in the soliton state | $ψ$ ＞, Eq.(60), is just equal to the above soliton energy Esol, or the sum of the energy of the bound state in Eq.(56a), Es, and the deformation energy of the lattice, W’, i.e. $< ψ | \tilde{H} | ψ > =$ , Esol=Es+W’. This is an interesting result, which shows clearly that the quasi-coherent soliton formed by this mechanism is just a self-trapping state of the two excitons plus the corresponding deformation of the amino acid lattice. However, it should be noted that $| ψ >$ is not an exact eigenstate of $\tilde{H}$ owing to the presence of the terms in $\tilde{H}$ with As and A -k.BA

Transition Probability and Decay Rate of the New Soliton: We now calculate the transition probability and decay rate of the quasi-coherent soliton arising from the perturbed potential by using the first-order quantum perturbation theory developed by Cottingham, et al. [14], in which the influences of the thermal and quantum effects on the properties of the soliton can be taken into account simultaneously. For the discussion of the decay rate and lifetime of the new soliton state it is very convenlent to divide $\tilde{H}$ in Eq. (55) into H0+V1+V2, where

$\begin{array}{l} H_{0} = W' + E_{s} A_{s}^{+} A_{s} + \sum_{k} E_{k} A_{k}^{+} A_{K} + \sum_{q} ℏ (ω_{q} - v q) b_{q}^{+} b_{q} + \\ \frac{1}{\sqrt{N}} \sum_{q} ℏ (ω_{q} - v q) (α_{q} b_{q}^{+} + α_{q}^{*} b_{q}) (1 - A_{s}^{+} A_{s}) (61) \end{array}$

$V_{1} = \frac{1}{\sqrt{N}} \sum_{k k^{'} q} F (k, k + q, q) (b_{- q}^{+} + b_{q}) A_{k^{'}}^{+} A_{k} (62)$

$V_{2} = \frac{1}{N} \sum_{k q} \tilde{F} (k, q) (b_{- q}^{+} + b_{q}) (A_{s}^{+} A_{k} - A_{s}^{+} A_{- k}), V = V_{1} + V_{2} (63)$

Where, H0 describes the relevant quasi-particle excitations in the protein. This is a soliton together with phonons relative to the distorted amino acid lattice. The resulting delocalized excitations belongs to an exciton-like band with phonons relative to a uniform lattice. The bottom of the band of the latter is at the energy $4 J μ_{p}^{2} / 3$ relative to the soliton, in which the topological stability associated with removing the lattice distortion is included. We now calculate the decay rate of the new soliton along the following lines by using Eq. (61) and V2 in Eq. (63) and quantum perturbation theory. Firstly, we compute a more general formula for the decay rate of the soliton containing n quanta in the system in which the three terms contained in Eq. (40a) is replaced by (n+1) terms of the expression of a coherent state, $\exp {\sum_{n} [φ_{n} (t) B_{n}^{+} - φ_{n}^{*} (t) B_{n}]} | 0 >_{e x} .$ . Finally we find out the decay rate of the new soliton with two-quanta. In such a case H0 is chosen such the ground state, |n> has energy $W' + n E_{s}^{'}$ in the subspace of excitation number equal to n, i.e., $< n | \sum_{i} B_{i}^{+} B_{i} | n > = < n l (A_{s}^{+} A_{s} + \sum_{k} A_{k}^{+} A_{k}) l n > = n$ . In this subspace the eigenstates have the simple form

|n-m,k1k2…km, {nq｝>

$\frac{1}{\sqrt{(n - m)!}} {(A_{S}^{+})}^{n - m} A_{k_{1}}^{+} A_{k_{2}}^{+} \dots A_{k_{m}}^{+} | 0 >_{e x} \underset{q}{Π} \frac{{(d_{q}^{+})}^{n_{q}}}{\sqrt{n_{q}!}} 1 \tilde{0} >_{p h}^{n - m} (64)$

Where, $d_{q} = b_{q} + \frac{m}{n} \frac{1}{\sqrt{N}} α_{q} = a_{q} - \frac{n - m}{n} \frac{1}{\sqrt{N}} α_{q}$ (m≤n, n and m are all intgers) (65)

With, dq| $\tilde{0} >_{p h}^{n - m}$ =0. The corresponding energy of the systems is

$\begin{array}{l} E_{n - m; k_{1} ... k_{m_{1}}; {n_{q}}}^{(0)} = (1 - (^{m / n) 2}) W' + \\ (n - m) {E^{'}}_{s} + \sum_{j = 1}^{m} {E^{'}}_{k_{1}}^{} + \sum_{q} ℏ (ω_{q} - v q) n_{q} (66) \end{array}$

${E^{'}}_{s}$ is the energy of a bound state with one exciton, ${E^{'}}_{k}$ is the energy of the unbound(delocalized) state with one exciton. When m=0 the excitation state is a n-type soliton plus phonons relative to the chain with the deformation corresponding to the n-type soliton. For m=n the excited states are delocalized and the phonons are relative to a chain without any deformation. Furthermore except for small k, the delocalized states approximate ordinary excitons. Thus the decay of the soliton is just a transition from the initial state with the n-type soliton plus the new phonons:

A $| n > = \frac{1}{\sqrt{n!}} \underset{q}{Π} \frac{{(b_{q}^{+})}^{n_{q}}}{{(n_{q}!)}^{1 / 2}} {(A_{s}^{+})}^{n} | 0 >_{e x} | \tilde{0} >_{p h} (67)$

With corresponding energy Es {nq｝=W+n ${E^{'}}_{s} + \sum_{q} ℏ (ω_{q} - v q)$ nq to the final state with delocalized excitons and the original phonons:

 $| α k > = \underset{q}{Π} \frac{{(a_{q}^{+})}^{n_{q}}}{\sqrt{n_{q}^{}!}} | 0 >_{p h} {(A_{k}^{+})}^{n} | 0 >_{e x} (68)$

With corresponding energy Ek{nq}=n ${E^{'}}_{k}^{} + \sum_{q} ℏ (ω_{q} - v q)$ nq caused by the part, V2, in the perturbation interaction V. In this case, the initial phonon distribution will be taken to be at thermal equilibrium. The probability of the above transitions in lowest order perturbation theory is given by

$\begin{array}{l} \bar{W} = \frac{1}{ℏ^{2}} \int_{0}^{t} \begin{array}{l} d t^{'} \int_{0}^{t} d t^{″} \\ {\sum_{α k^{'}} \sum_{l} P_{l}^{(p h)} < n | \exp (\frac{i H_{0} t^{″}}{ℏ}) V_{2} \exp (\frac{- i H_{0} t^{″}}{ℏ}) | α k^{'} > \end{array} \cdot (69) \\ < α k^{'} | e x p (\frac{i H_{0} t^{'}}{ℏ}) V_{2} \exp (\frac{- i H_{0} t^{'}}{h}) | n >} \end{array}$

We should calculate the transition probability of the soliton resulting from the perturbed potential, (V1 + V2), at first-order in perturbation theory. Following Cottingham and Schweitzer [14] we estimate only the transition from the soliton state to delocalized exciton states caused by the potential V2, which can satisfactorily be treated by means of perturbation theory since the coefficient $\tilde{F}$ (k,q) defined by Eq. (58) is proportional to an integral over the product of the localized state and a delocalized state, and therefore is of order 1/ $\sqrt{N}$ . The V1 term in the Hamiltonian is an interaction between the delocalized excitons and the phonons. The main effect of V1 is to modify the spectrum of the delocalized excitatons in the weak coupling limit (Jμp/ KBT0<<1, the definition of T0 is given below). As a result the delocalized excitons and phonons will have their energies shifted and also have finite lifetimes. These effects are ignored in our calculation since they are only of second order in V1. The sum over l in Eq. (69) indicates a sum over an initial set of occupation numbers for phonons relative to the distorted amino acid lattice with probability distribution $P_{l}^{p h}$ , which is taken to be the thermal equilibrium distribution for a given temperature T. Since

$\begin{array}{l} e^{- i H_{0} t} | n, {n_{q}} > = e x p {- i (W' + n {E^{'}}_{q}) \\ t / ℏ - i \sum_{q} (ω_{q} - q v) b_{q}^{+} b_{q} t} | n, {n_{q}} > \end{array}$

and

$\begin{array}{l} e^{i H_{0} t} | n - 1, {{n^{'}}_{q}^{}} > = e x p \\ {- i [(1 - \frac{1}{n^{2}}) W' + (n - 1) {E^{'}}_{s} + {E^{'}}_{k}] \\ t / ℏ - i \sum_{q} (ω_{q} - q v) d_{q}^{+} d_{q}^{} t} | n - 1, {{n^{'}}_{q}} > \end{array}$

Where, dq=bq+ $\frac{1}{n} \frac{1}{\sqrt{N}} α_{q}$ ,

Using the explicit form for V2 and the fact that the sum over states｜ $k^{'} α, {{n^{'}}_{q}} >$ contains a complete set of phonons for each values of k′, one can rewrite $\bar{W}$ as

$\begin{array}{l} \bar{W} = \frac{1}{ℏ^{2}} \frac{π^{2}}{2 n μ_{1} N^{2}} \sum_{k} \sum_{k^{'}} \sum_{k^{″}} [g_{1}^{*} (k) + 2 g_{2}^{*} (k)] [g_{1} (k^{″}) + (70) \\ 2 g_{2} (k^{″})] \frac{(k r_{0}) (k^{″} r_{0})}{{(n μ_{1}^{})}^{2} + {(k^{'} r_{0})}^{2}} S e c h [\frac{π r_{0}}{2 n μ_{1}} (k - k^{'})] \cdot \\ \sec h [\frac{π r_{0}}{2 n μ_{1}} (k^{″} - k^{'})] \int_{0}^{t} d t^{'} \int_{0}^{t} d t^{″} {\exp [\frac{- i}{ℏ} (\begin{array}{l} n (n^{2} - \frac{2}{3} n) \\ μ_{1}^{2} J + n J (^{k^{'} r_{0}) 2} \end{array}) (t^{'} - t^{″})] \cdot \\ < < \exp [i \sum_{q} \begin{array}{l} (ω_{q} - q v) \\ b_{q}^{+} b_{q} (t^{'} - t^{″})] (b_{k}^{+} + b_{- k} \end{array}) \exp [i \sum_{q} \begin{array}{l} (ω_{q} - q v) \\ a_{q}^{+} a_{q} (t^{'} - t^{″}) (b_{- k^{″}}^{+} + b_{k^{″}} \end{array}) > >} \end{array}$

Where,

$\begin{array}{l} g_{1} (k) + 2 g_{2} (k) = 2 χ_{1} (^{\frac{ℏ}{2 M ω_{k}}) 1 / 2} [A (\cos (r_{0} k) - 1) \\ + i (A + 1) \sin (r_{0} k)] \approx 2 i (A + 1) (r_{0} k) χ_{1} (^{\frac{ℏ}{2 M ω_{k}}) 1 / 2}, \\ μ_{1} = \frac{χ_{1}^{2} (1 + A^{2})}{ω (1 - s^{2}) J}, A = χ_{2} / χ (71) \end{array}$

Here, A is a new parameter introduced to describe the rate between the new nonlinear interaction term and the one in the Davydov’s model. To estimate the lifetime of the soliton we are interested in the long-time behavior of $\frac{d \bar{w}}{d t}$ . By straightforward calculation, the average transition probability or decay rate of the soliton is given by

$\begin{matrix} Γ_{n} = \lim_{t \to \infty} \frac{d \bar{W}}{d t} = \frac{4}{ℏ} [\frac{π^{2}}{2 n μ_{1} N^{2}}] \sum_{k k^{'} k''} [[g_{1}^{*} (k) + 2 g_{_{2}}^{*} (k)] [g_{1} (k^{″}) + 2 g_{2} (k^{″})] \\ \frac{(k r_{0}) (k^{″} r_{0})}{{(n μ_{1})}_{}^{2} + {(k^{'} r_{0})}^{2}} \cdot \sec h [\frac{π r_{0}}{2 n μ_{1}} (k - k^{'})] \cdot \sec h [\frac{π r_{0}}{2 n μ_{1}} (k^{″} - k^{'})] \\ Re {\int_{0}^{\infty} d t \exp [- \frac{i}{ℏ} (n (n^{2} - \frac{2}{3} n) μ_{1}^{2} J + n J {(k^{'} r_{0})}^{2}) t] \cdot \\ < < \exp [i \sum_{q} (ω_{q} - q v) b_{q}^{+} b_{q}^{} t] (b_{k}^{+} + b_{- k}) \\ \exp [- i \sum_{q} (ω_{q} - q v) a_{q}^{+} a_{q} t] (b_{- k^{″}}^{+} + b_{k^{″}}) > >}} \\ = \frac{4}{ℏ^{2}} \frac{π^{2}}{2 n μ_{1} N^{2}} \sum_{k k^{'} k''} {[g_{1}^{*} (k) + 2 g_{_{2}}^{*} (k)] [g_{1} (k^{″}) + 2 g_{2} (k^{″})] \\ \frac{(k r_{0}) (k^{″} r_{0})}{{(n μ_{1})}_{}^{2} + {(k^{'} r_{0})}^{2}} \sec h [\frac{π r_{0} (k - k^{'})}{2 n μ_{1}}] \cdot \end{matrix}$ $\sec h [\frac{π r_{0}}{2 n μ_{1}} (k^{″} - k^{'})] Re \int_{0}^{\infty} \begin{array}{l} d t U (k, k^{″} t) \exp \\ [- \frac{i}{ℏ} (n (n^{2} - \frac{2}{3} n) μ_{1}^{2} J + n J {(k^{'} r_{0})}^{2}) t] \end{array}} (72)$

Where, the thermal average is

$\begin{array}{l} U (k, k^{″}, t) = < < \exp [i \sum_{q} (ω_{q} - q v) b_{q}^{+} b_{q}^{} t] (b_{k}^{+} + b_{- k}) \\ \exp [- i \sum_{q} (ω_{q} - q v) a_{q}^{+} a_{q} t] (b_{- k^{″}}^{+} + b_{k^{″}}) > > \end{array}$

With

$\begin{array}{l} < < A > > = T r {A \exp [- β \underset{q}{Σ} ℏ (ω_{q} - q v) b_{q}^{+} b_{q}]} / T r {\exp [- β Σ ℏ (ω_{q} - q v) b_{q}^{+} b_{q}]} \\ = T r {A \exp [- β \sum_{q} ℏ (ω_{q} - q v) b_{q}^{+} b_{q}]} / Z_{p h} (73) \end{array}$

and Zph= $\prod_{q} {(1 - \exp [- β ℏ (ω_{q} - q v)]}}^{- 1}, (β = \frac{1}{K_{B} T})$

This rather unusual expression of Γn occurs because the phonons in the final state are related to a different deformation. However, the analytical evaluation of $U (k, k^{″}, t)$ is a critical step in the calculation of the decay rate Γn. It is well known that the trace contained in $U (k, k^{″}, t)$ can be approximately calculated by using the occupation number states of single-particles and coherent state. However the former is both a very tedious calculation, including the summation of infinite series, and also not rigorous because the state of the excited quasiparticles is coherent in Pang’s model. Here we use the coherent state to calculate the $U (k, k^{″}, t)$ as it is described in Appendix B. The decay rate obtained finally is,

$\begin{matrix} Γ_{n} = \lim_{t \to \infty} \frac{d \bar{W}}{d t} = \frac{2}{n μ_{1} ℏ^{2}} \frac{π^{2}}{N^{2}} \sum_{k k^{'}} \begin{array}{l} [| g_{1} (k) + 2 g_{2} (k) |^{2} \\ \frac{{(r_{0} k)}^{2} \sec h^{2} [π (k - k^{'}) r_{0} / 2 n μ_{1}]}{{(n μ_{1})}_{}^{2} + {(k^{'} r_{0})}^{2}} \end{array} Re \int_{0}^{\infty} d t \cdot \\ {\begin{cases} \exp [- i (n J (^{k^{'} r_{0}) 2} + n (n^{2} - \frac{2}{3} n) μ_{1}^{2} J t / ℏ + R_{n} (t) + \\ ξ_{n} (t)] \frac{\exp [i (ω_{k} - k v) t]}{\exp [β ℏ (ω_{k} - k v)] - 1} \end{cases}}] (74) \end{matrix}$

Where,

$\begin{array}{l} R_{n} (t) = - \frac{1}{n^{2} N} \sum_{k} | α_{k} |^{2} {i - \exp [- i (ω_{k} - k v) t]}, \\ ξ_{n} (t) = - \frac{4}{n^{2} N} \sum_{k} \frac{| α_{k} |^{2} \sin^{2} [\frac{1}{2} (ω_{k} - k v) t]}{\exp [β ℏ (ω_{k} - k v)] - 1} (75) \end{array}$

This is just a generally analytical expression for the decay rate of the soliton containing n quanta at any temperature within lowest order perturbation theory. Note that in the case where a phonon with wavevector k in Eq. (75) is absorbed, the delocalized excitation produced does not need to have wavevector equal to k. The wavevector here is only approximately conserved by the sech 2[π (k-k’) $r_{0} / 2 n μ_{1}$ ] term. This is, of course, a consequence of the breaking of the translation symmetry by the deformation. Consequently, we do not find the usual energy conservation. The terms, Rn (t) and $ξ_{n} (t)$ , occur because the phonons in the initial and final states are defined relative to different deformations [24-26]. We should point out that the approximations made in the above calculation are physically justified because the transition and decay of the soliton is mainly determined by the energy of the thermal phonons absorbed. Thus the phonons with large wavevectors, which fulfil wavevector conservation, make a major contribution to the transition matrix element, while the contributions of the phonons with small wavevector, which do not fulfil wavevector conservation, are very small, and can be neglected.

From Eqs. (74) and (75) we see that the $Γ_{n}$ and Rn (t) and $ξ_{n} (t)$ and $μ = n μ_{1}$ mentioned above are all changed by increasing the number of quanta, n. Therefore, the approximation methods used to calculate $Γ_{n}$ and related quantities (especially the integral contained in $Γ_{n}$ ) should be different for different n. We now calculate the explicit formula of the decay rate of the new soliton with two-quanta (n = 2) by using Eqs. (74)-(75) in Pang’s model. In such a case we can compute explicitly the expressions of this integral and R2 (t) and $ξ_{2} (t)$ contained in Eqs.(74)-(75) by means of approximation. As a matter of fact, in Eq.(75) at n=2 the functions R2 (t) and $ξ_{2} (t)$ can be exactly evaluated in terms of the digamma function and its derivative. In the case when the soliton velocity approaches zero and the phonon frequency $ω_{q}$ is approximated by $\sqrt{w / M}$ |q|r0, as it is shown in Appendix C. For t $\to \infty$ (because we are interested in the long-time steady behaviour) the asymptotic forms of R2(t) and $ξ_{2} (t)$ are

$R_{2} (t) = - R_{0} [\ln (\frac{1}{2} ω_{α} t) + 1.578 + \frac{1}{2} i π] (76)$

$ξ_{2} (t) \approx - π R_{0} k_{B} T t / ℏ (w h e r e c o t h \frac{1}{2} ω_{α} t ~ 1) (77)$

$\underset{t \to \infty}{L i m} ξ_{2} (t) = - η t, η = π R_{0} / β ℏ = π R_{0} k_{B} T / ℏ (78)$

Where, $\begin{array}{l} R_{0} = \frac{4 {(χ_{1} + χ_{2})}^{2}}{π ℏ w} (^{M / w) 1 / 2} = \frac{2 J μ_{p} r_{0}}{π ℏ v_{0}}, (79) \\ ω_{α} = \frac{2 μ_{p}}{π} (^{\frac{w}{M}) 1 / 2}, T_{0} = ℏ ω_{α} / K_{B} \end{array}$

At R0＜1 and T0＜T and R0 T/T0＜1 for the protein molecules, one can evaluate the integral including in Eq.(74) by using the approximation which is shown in Appendix C. The result is

$\begin{matrix} \frac{1}{π ℏ} Re \int_{0}^{\infty} d t \exp {\begin{cases} - i [2 J (^{k' r_{0}) 2} + \frac{4}{3} J μ_{p}^{2} - \\ ℏ ω_{k}] t / ℏ + R_{2} (t) + ξ_{2} (t) \end{cases}} \\ \approx \frac{1}{π ℏ} (^{2.43 ω_{α}) - R_{0}} Γ (1 - R_{0}) [^{η^{2} + (^{δ (k, k^{'}) / ℏ) 2}] - (1 - R_{0}) / 2} (80) \\ [1 - \frac{1}{2} {[\frac{π R_{0}}{2} + (1 - R_{0}) (\frac{δ (k, k^{'})}{η ℏ})]}^{2}] \end{matrix}$

Where,

$δ (k, k^{'}) = 2 J {(k^{'} r_{0})}^{2} + \frac{4}{3} μ_{p}^{2} J - ℏ ω_{k}, Φ_{1} = \frac{R_{0} π}{2}, Φ_{2} = [(1 - R_{0}) \tan^{- 1} (\frac{δ (k, k^{'})}{η ℏ})] (81)$

The decay rate of the soliton, in such an approximation, can be represented, from Eqs. (74) and (80), by

$\begin{array}{l} Γ_{2} = \lim_{t \to \infty} \frac{d \bar{W}}{d t} = \frac{2}{μ_{p}} {(\frac{π}{N})}^{2} \\ \sum_{k k^{'}} [\frac{\begin{array}{l} (^{k r_{0}) 2} | g_{1} (k) + 2 g_{2} (k) |^{2} \\ \sec h^{2} [(π r_{0} / 2 μ_{p}) (k - k^{'})] \end{array}}{[μ_{p}^{2} + {(k^{'} r_{0})}^{2}] [\exp (β ℏ ω_{k}) - 1]} {(2.43 ω_{α})}^{- R_{0}} \\ {\frac{{(η^{2} + \frac{1}{ℏ^{2}} {[\frac{4}{3} μ_{p}^{2} J + 2 {(k^{'} r_{0})}^{2} J - ℏ ω_{k}]}^{2})}^{(1 + R_{0}) / 2}}{ℏ^{2} η^{2} + {[\frac{4}{3} μ_{p}^{2} J + 2 {(k^{'} r_{0})}^{2} J - ℏ ω_{k}]}^{2}}} \\ {1 - \frac{1}{2} {[\frac{R_{0} π}{2} + (1 - R_{0}) [\frac{\frac{4}{3} μ_{p}^{2} J + 2 {(k^{'} r_{0})}^{2} J - ℏ ω_{k}}{ℏ η}]]}^{2}} (82) \end{array}$

This is the final analytical expression for the decay rate of the quasi-coherent solition with two-quanta. Evidently, it is different from that in the Davydov model [15,21]. To emphasis the difference of the decay rate between the two models we rewrite down the corresponding quantity for the Davydov soliton [15,21].

$\begin{array}{l} Γ_{D} = \frac{1}{ℏ^{2}} \frac{χ_{1}^{2}}{μ_{D}} {(\frac{2 π}{N})}^{2} \sum_{k k^{'}} (\frac{ℏ}{2 M ω_{k}}) \\ \frac{{(k r_{0})}^{2} s i n^{2} (k r_{0}) \sec h^{2} [(π r_{0} / 2 μ_{D}) (k - k^{'})]}{[μ_{D}^{2} + {(k^{'} r_{0})}^{2}] [\exp (β ℏ ω_{k}) - 1]} {(\frac{ω_{α}^{D}}{η_{D}})}^{- R_{0}^{D}} \cdot \\ \frac{ℏ^{2} η_{D}}{ℏ^{2} η_{D}^{2} + [J μ_{D}^{2} / 3 + J {(k^{'} r_{0})}^{2} - ℏ ω_{k}]} (83) \end{array}$

Where,

$η_{D} = π R_{0}^{D} K_{B} T / ℏ, R_{0}^{D} = \frac{2 χ_{1}^{2}}{π ℏ w} {(\frac{M}{w})}^{1 / 2}, ω_{α}^{D} = \frac{2 μ_{D}}{π} {(\frac{M}{w})}^{1 / 2} (84)$

Equation (83) can also be found out from Eq. (74) at n = 1 by using the Cottingham et al’s approximation.The two formulae above, Eqs. (82) and (83), are completely different, not only for the parameter’s values, but also the factors contained in them. In Eq.(82) the factor, ${1 - \frac{1}{2} {[\frac{R_{0} π}{2} + (1 - R_{0}) [(\frac{4}{3} μ_{p}^{2} J + 2 {(k^{'} r_{0})}^{2} J - ℏ ω_{k}) / ℏ η]]}^{2}}$ is added, while in Eq.(83) the factor, ${(\frac{ω_{α}}{η_{D}})}^{- R_{0}^{D}} η_{D}$ replaces the term ${(2.43 ω_{d})}^{- R_{0}} \cdot (η^{2} + \frac{1}{ℏ^{2}} [\frac{4}{3} μ_{p}^{2} J +$ $2 {(k^{'} r_{0})}^{2} J - ℏ ω_{k}]^{2})^{(\frac{1 + R_{0}}{2})}$ ) in Eq. (82) due to the two-quanta nature of the new wavefunction and the additional interaction term in the new Hamiltonian. In Eq. (82) the η, R0 and T0 are not small, unlike in the Davydov model. Using Eq. (72) and table 1 we find out the values of $η$ , R0 and To at T = 300K in both models, which are listed in table 2. From this table we see that the $η$ , Ro and To for Pang’s model are about 3 times larger than the corresponding values in the Davydov model due to the increases of μp and of the non-linear interaction coefficient Gp. Thus the approximations used in the Davydov model by Cottingham, et.al [14] can not be applied in our calculation of lifetime of the new soliton, although we utilized the same quantum-perturbation scheme. Hence we can audaciously suppose that the lifetimes of the quasi-coherent soliton will greatly change.

Discussion for the Lifetime of the New Soliton and Results

The above expression, Eq. (82), allows us to compute numerically the decay rate, $Γ_{2}$ , and the lifetimes of the new soliton, τ = 1/ $Γ_{2}$ , for values of the physical parameters appropriate to -helical protein molecules. Using the parameter values given in Eq. (37), tables 1 and 2 , v = 0.2 v0 and assuming the wavevectors are in the Brillouin zone, the values of $Γ_{2}$ between 1.54×1010S-1 - 1.89×1010S-1 can be obtained. This corresponds to the soliton lifetimesτ, of between 0.53×10-10S 0.65×10-10S at T=300K, or /0=510-630, where 0=r0/v0 is the time for travelling one lattice spacing at the speed of sound, equal to (M/w)1/2=0.96×10-13S. In this amount of time, the new soliton, travelling at two tenths of the speed of sound in the chain, would travel several hundreds of amino acid lattice spacings, that is several hundred times more than the Davydov soliton for which /0<10 at 300K [15,21]（i.e., the Davydov soliton traveling at a half of the sound speed can cover less than 10 lattice spacing in its lifetime）. The lifetime is sufficiently long for the new soliton excitation to be a carrier of bio-energy. Therefore the quasi-coherent soliton is a viable mechanism for the bio-energy transport at biological temperature in the above range of parameters. Attention is being paid to the relationship between the lifetime of the quasi-coherent soliton and temperature. Fig.3 shows the relative lifetimes /0 of the new soliton versus temperature T for a set of widely accepted parameter values as shown in Eq. (37). Since one assumes that v＜v0, the soliton will not travel the length of the chain unless /0 is large compared with L/r0, where L=Nr0 is the typical length of the protein molecular chains. Hence for L/r0≈100, /0＞500 is a reasonable criterion for the soliton to be a possible mechanism of the bio-energy transport in protein molecules. The lifetime of the quasi-coherent soliton shown in Fig.4 decreases rapidly as temperature increases, but below T=310K it is still large enough to fulfill the criterion. Thus the new soliton can play an important roles in biological processes. For comparison, log $(τ / τ_{0})$ versus the temperature relationships was plotted simultaneously for the Davydov soliton and the new soliton with a quasi-coherent two-quanta state in Fig.4. The temperature-dependence of log ( $(τ / τ_{0})$ ) of the Davydov soliton is obtained from Eq. (83). We find that the differences of values of $(τ / τ_{0})$ between the two models are very large. The value of $(τ / τ_{0})$ of the Davydov soliton really is too small, and it can only travel fewer than ten lattice spacings in half the speed of sound in the protein chain. Hence it is true that the Davydov soliton is ineffective for biological processes [3-23].

The dependency of the soliton lifetime on the other parameters can also be studied by using Eq. (82). Parameter values near the above accepted values shown in Eq. (37) are chosen. In Pang’s model we know from Eq. (82) that the lifetime of the soliton depends mainly on the following parameters: coupling constants (1+2), M, w, J, phonon energy  k, as well as on the composite parameters μ (μ = μp), R0 and T/T0. At a given temperature, /0 increases as μand T0 increase. The dependences of the lifetime /0, at 300K on (1+2) and μ are shown in Figs.5 and 6, respectively. Since μ is inversely proportional to the size of the soliton, and determines the binding energy in the new model, it is an important quantity. It is regarded as an independent variable. In such a case the other parameters in Eq. (82) adopt the values in Eq. (37). It is clear from Figs.5 and 6 that the lifetime of the soliton, /0, increases rapidly with increasing μand (1+2). Furthermore, when μ≥5.8 and (1+2) ≥ 7.5×10-11N, which are values appropriate to the new model, we find /0 ＞500. For comparison, the corresponding result obtained using Eq. (83) is shown for the original Davydov model as a dashed line in figure 6. Here we see that the increase in lifetime of the Davydov soliton with increasing μ is quite slow and the difference between the two models increases rapidly with increasing μ. The same holds for the dependency on the parameter (1+2), but the result for the Davydov soliton is not drawn in figure 5. These results show again that the quasi-coherent soliton in Pang’s model is a likely candidate for the mechanism of bio-energy transport in the protein molecules. In addition it shows that a basic mechanism for increasing the lifetime of the soliton in the biomacromolecules is to enhance the strength of the exciton-phonon interaction.

In figure 7 $τ / τ_{0}$ 0 versus $η$ is plotted. Since – $η$ designates the influence of the thermal phonons on the soliton, it is also an important quantity. Thus, it is regarded here as an independent variable. The other parameters in Eq. (82) take the values in Eq. (37). From this figure we see that /0 increases with increasing $η$ . Therefore, to enhance $η$ can also increase the value of $τ / τ_{0}$ . In order to understand the behavior of the quasi-coherent soliton lifetime in very wide ranges, it is necessary to study $τ / τ_{0}$ in the limit $ω_{a} t$ →0 in Eq. (75) or Eqs. (C1) and (C3) (i.e., this is in the initial case) in which we can evaluate analytically the values of R2 (t) and 2 (t). In fact, for $ω_{a} t$ ＜1 both R2 (t) and 2 (t) have power-series expansions. To the lowest order as $ω_{a} t$ →0, it can be found from Eq. (75)

R2 (t) ≈—R0[iπ2ωαt/6+3ζ(3)(ωαt)2] (85)

$ξ_{2} (t) \approx - \frac{R_{0} K_{B}^{2} T T_{0} π^{2}}{3 ℏ^{2}} t^{2}, (86)$

$using coth (π ω_{α} t) \approx [{(π ω_{α} t)}^{- 1} + \frac{π}{3} ω_{α} t]$

Thus, $\begin{array}{l} \frac{1}{π ℏ} Re \int_{0}^{\infty} d t \exp {\begin{cases} - i [2 J {(k' r_{0})}^{2} + \frac{4 J μ_{p}^{2}}{3} - ℏ ω_{k}] \\ \frac{t}{ℏ} + R_{2} (t) + ξ_{2} (t) \end{cases}} \approx \\ [4 π (3 ζ (3) R_{0} K_{B}^{2} T_{0}^{2} \\ {+ R_{0} π^{2} K_{B}^{2} T T_{0} / 3)]}^{- \frac{1}{2}} \exp {- \frac{\begin{array}{l} [2 J (^{k' r_{0}) 2} + \frac{4}{3} μ_{p}^{2} J - ℏ ω_{k} + \\ ℏ (R_{0} π^{2} K_{B} T)]^{2} \end{array}}{4 [3 ζ (3) R_{0} K_{B}^{2} T_{0}^{2} + R_{0} π^{2} K_{B}^{2} T T_{0} / 3]}} \end{array}$

When, T/T0＞1 and π4R0T/2μT0＞1. The above integral is a generalization of the usual - function for energy conservation in zero-temperature perturbation theory. Thus we can obtain from Eqs.(74) and (87) at n=2 the decay rate of the soliton as

$\begin{array}{l} Γ_{2} = \frac{2 π^{3}}{μ_{p} ℏ N^{2} K_{B}} {(\frac{π}{R_{0} T_{0} [3 ζ (3) T_{0} + π^{2} T / 3]})}^{- \frac{1}{2}} \\ \sum_{k k^{'}} \frac{{(k r_{0})}^{2} | g_{1} (k) + 2 g_{2} (k) |^{2}}{μ_{p}^{2} + {(k^{'} r_{0})}^{2}} \sec h^{2} [(\frac{π r_{0}}{2 μ_{p}}) (k - k^{'})] \\ {\exp [\frac{{[2 J {(k^{'} r_{0})}^{2} + \frac{4}{3} μ_{p}^{2} J - ℏ ω_{k} + \frac{1}{6} R_{0} π^{2} K_{B} T_{0}]}^{2}}{4 [3 ζ (3) R_{0} K_{B}^{2} T_{0}^{2} + R_{0} K_{B}^{2} T T_{0} π^{2} / 3]}] [\exp (β ℏ ω_{k}) - 1]}^{- 1} \end{array}$

The expression of the decay rate of the quasi-coherent soliton in this limit is different from Eq. (84). Therefore, studying properties of the lifetime of the new soliton in such a case helps in understanding the behavior of the soliton. A summary of the results obtained from Eq. (88) are given in figures 8-11. The dependency of lifetime on temperature T is shown in figure 8, which has been obtained from the numerical evaluation of Eq. (88).

In figures 9 and 10 we plot /0 versus (1+2) and versus  at T = 300K, respectively. From figures 8-10 it can be seen that /0 increases as T decreases and as  and (1+2) increase. Furthermore, it is clear from this Gaussian expression in Eq.(88) that the lifetime of the new soliton will be large if μand (1+2)are larger, but the Gaussian expression is very small for k and k’ between -/r0 and +/r0, i.e., in the Brillouin zero. Obviously, the temperature dependence of the lifetime of the new soliton is mainly due to the temperature dependence of the width of the Gaussian, which decreases with decreasing temperature. The dashed line in figure 10 is the result for the Davydov soliton under the same conditions. It is clear that the lifetime of the Davydov soliton is lower than that of the new soliton, especially at large $μ$ , although at low the difference between them is small. Taking figure 4 also into account we find that the lifetime of the Davydov soliton is indeed generally low. However this is not the case for the new soliton. In figure 11, /0 is plotted as a function of T0 at T = 300K. T0 is related to the Debye temperature of the systems, therefore it is also an important quantity. It is regarded here as an independent variable and evaluate other parameters as in Eq. (37). From this figure it can seen that the lifetime of the new soliton is large if T0 is either large or small, because the Gaussian expression in Eq. (88) is very small for k and k’ between -/r0 and +/r0. As a point of reference, note that these parameters have the values T/T0≈1.03 —1.06, JT/KBT = 4.10 at 300K and μ = 5.81-5.96 depending on whether the widely accepted or the“three- channel”parameter values for the protein are assumed. From these results, it is clear that using widely accepted and realistic parameter values, the new model can satisfy the relation /0≥500 at 300K and large μand large T0. Hence the proposed new soliton model provides a viable candidate for the biological processes.

Here a new theory of bio-energy transport is proposed to study the properties of the nonlinear excitation and motion of the soliton along protein molecules. In this theory, Davydov’s Hamiltonian and wave function of the systems are simultaneously improved and extended, a new interaction is added into the original Hamiltonian, and the original wave function of the excitation state of single particles is replaced by a new wave function of a two-quanta quasi-coherent state. From this model, a lot of interesting and new results are obtained. The soliton has sufficiently long lifetime and can pay an important role in biological processes. Therefore, it is an exact carrier of bio-energy in living systems. Present problem is why the quasi-coherent soliton has such long lifetime? From Eqs. (35) and (45) and tables 1 and 2 it can be seen that the binding energy and localization of the new soliton increase due to the increase of the nonlinear interactions of exciton-phonon interaction, i.e., the new wave function with two-quanta state and the new Hamiltonian with the added interaction produce considerable changes to the properties of the soliton. In fact, the nonlinear interaction energy in the new model is Gp=8(1+2)2 /(1-s2)w=3.8×10-21J, and it is larger than the linear dispersion energy, J=1.55×10-22J, i.e., the nonlinear interaction is so large that it can really cancel or suppress the linear dispersion effects in the equation of motion of this model. From this point the soliton is stable according to the conditions of formation and stability of the soliton in the soliton theory [27,28]. By comparison, the non-linear interaction energy in the Davydov model is GD＝4 / (1-s2) w≈1.18×10-21J and it is 3-4 times smaller than Gp. Thus the stability of the Davydov soliton is weak compared to that of the new soliton. Moreover, the binding energy of the quaasi-coherent soliton in Pang’s model is EBP = 4 $μ_{p}^{2}$ J/3 = 7.8×10-21J in Eq. (19), which is about 2 times larger than the thermal energy, KBT = 4.14×10-21J, at 300K, and about 6 times larger than the Debye energy, $K_{B} Θ$ = ωD = 1.2×10-21J (here D is Debye frequency), and it is approximately equal to = 8.2×10-21J, i.e., it has same order of magnitude of the energy of the amide-I vibrational quantum, $ε_{0} / 4$ . This shows that the quasi-coherent soliton is robust against the quantum fluctuation and thermal perturbation of the systems due to the large energy gap between the soliton state and the delocalized state. In contrast, the binding energy of the Davydov soliton is only EBD = $\frac{χ_{1}^{4}}{3 w^{2} J}$ = 0.188×10-21J , which is about 41 times smaller than that of the new soliton, about 23 times smaller than KBT and about six times smaller than $K_{B} Θ$ =ωD=1.2×10-21J, respectively. Therefore, it is easily destroyed by thermal and quantum effects. Hence the Davydov soliton has very small lifetime (about 10-12～10-13s), and it is unstable at 300K [15-18,24-26]. Therefore, the quasi-coherent soliton can provide a realistic mechanism for the bio-energy transport in protein molecules. The two-quanta nature of the quasi-coherent soliton plays a more important role in the increase of lifetime relative to that of the added interaction because of the following facts. (1) The change of the nonlinear interaction energy GP=2G $_{D} [1 + 2 (\frac{χ_{2}}{χ_{1}}) + {(\frac{χ_{2}}{χ_{1}})}^{2}]$ by p produced the added interaction in the new model are G = GP (20)－GP (2=0) = 1.08GD < GP(2 = 0) = 2GD and  = P(20)－P(2 = 0) = 1.08D<P(2 = 0) = 2D, respectively, i.e., the roles of the added interaction on Gp and p are smaller than that of the two-quanta nature. The two parameters GP and P are responsible for the lifetime of the soliton. Thus the effect of the former on the lifetimes is smaller than the latter. (2) The contribution of the added interaction to the binding energy of the soliton is about, ${E^{'}}_{B P} = E_{B D} {[1 + (\frac{χ_{2}}{χ_{1}})]}^{4} = 2.6 E_{B D}$ , which is smaller than that of the two-quanta nature which is E = 16EBD. Putting them together in Eq. (35) we see that EBP 41EBD. (3)From the (1+2)-dependence of /0 in figure 5, /0100 has already been found directly at 2 = 0 which is about 20 times larger than that of the Davydov soliton under the same conditions. This shows clearly that the major effect in the increase of the lifetime is due to the modified wave function. Therefore, it is very reasonable to refer to the new soliton as the quasi-coherent soliton [30-35]. The above calculation helps to resolve the controversies on the lifetime of the Davydov soliton, which is too small in the region of biological temperature. However, by modifying the wave function and the Hamiltonian of the model, a stable soliton at biological temperatures could be produced. This result was obtained considering a new coupled interaction between the acoustic and amide-I vibration modes and a wave function with quasi-coherent two-quanta state. In such a way, the quasi-coherent soliton is a viable mechanism for the bio-energy transport in living systems. Therefore, it can be seen that Pang’s model is completely different from the Davydov’s model. Thus, the equation of motion and properties of the soliton occurring in Pang’s model are also different from that in the Davydov’s model. The distinction of features of the solitons between the two models is shown in table 3 [15]. From the table 3 we know that our new model repulse and refuse the shortcomings of the Davydov model [3], the new soliton in Pang’s model is thermal stable at biological temperature 300K, and has so enough long lifetime, thus it can plays important role in biological processes.

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Conclusion

So, a plausible, logically grounded hypothesis partially confirmed in the experiment was stated that the main reason for chronic cerebrospinal venous insufficiency, venous congestion of the brain, venous reflux, pathological expansion of the jugular veins, cerebral circulatory disorders, VVD, disorders of certain functions of the central nervous system may be one and the same mechanism, the same underlying cause: the presence of relatively “large” periodically open Arterio Venous Anastomoses (AVA) that are in the extract the venous network, i.e. in the zone outside the brain. The work of large AVA is hidden from the eyes of doctors, basically it is also hidden from the eyes and sensations of patients. Only the most sensitive people can sometimes notice unusual pulsations in the abdomen, in the region of the liver, after which, due to overflow of hollow (cava) veins, naturally, cardiac extra systoles are possible. I think that in the near future there will be a rethinking of the causes of many cardiovascular diseases and brain diseases, a serious study of the functioning of anastomoses, the study of measures to counter the pathological effect of abnormal AVA, the development of “smart” artificial arteriovenous shunts with a controlled lumen. All this is necessary to preserve human health and increase the duration of active life. An old proverb is confirmed: an old man is as old as his vessels. And the vessels are aging, as we see from the presented hypothesis, because of the abnormally working AVA anastomoses, which themselves are vessels that connect arteries and veins directly. For its part, the growth and discovery of AVA, apparently, are due to stress, hypodynamia, because of alcohol abuse, because of unhealthy lifestyles, because of poor heredity, etc. These are open AVA that lead to many diseases, including some diseases of the brain, cardiovascular and central nervous systems.

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