Keywords: Form; Living system; Bioenergy; protein; Biological energy; Soliton; ATP hydrolysis; Amide; Exciton; Life time; Amino acid; Quasicoherent state.
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 oxidized 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 preceded 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 oxidized 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 oxidized forms of these molecules serve for primary acceptors of electrons and hydrogen atoms through attaching two hydrogen atoms [3], which is expressed by $$NAD{P}^{+}+2{H}^{+}+2{e}^{}\to NADP\cdot H+{H}^{+}$$ Where NADP molecule becomes the reduced molecule NADPH. 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]: $$NA{D}^{+}+2{H}^{+}+2{e}^{}\to NAD{P}^{+}+{H}^{+}$$ 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 pyridinedependent hydrogenases, and with the participation NAD^{+} and NADP^{+} molecules two hydrogen atoms, including two protons and two electrons, are removed from this group of atoms. One proton and two electrons combine with the NAD^{+} and 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 oxidize 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] $${H}^{+}+NAD\cdot H+3{H}_{3}P{O}_{4}+3ADP+1/2{O}_{2}\to NA{D}^{+}+4{H}_{2}O+3ATP$$ Where ADP is called the adenosine diphosphate. The abbreviated form of this reaction can be written as $$ADP+{P}_{i}\to ATP+{H}_{2}0$$ 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 $$AT{P}^{4}+{H}_{2}O\to AD{P}^{3}+HP{O}_{4}^{2}+{H}^{+}+0.43eV$$ Its abbreviated form is of $$ATP+{H}_{2}O\to ADP+{P}_{i}$$ In this process ATP molecules are transformed as ADP molecules and the bioenergy of about 0.43eV is also released. Then it is referred to as dephosphorylation reaction of ATP molecules.
We know from the above representations that an increase in free energy $\Delta \text{G}$ in reaction and its decrease in reaction depend on their temperatures, concentrations of the ions Mg^{2+} and Ca^{2+} and on the pH value of the medium. Under the standard conditions $\Delta {G}_{0}$ =0.32 eV ( ~7.3 kcal/mole). If the appropriate corrections are made taking into consideration the physiological pH values and the concentration of Mg^{2+} and Ca^{2*} 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 ~0.54eV ( ~12.5 kcal/mole) for the free energy in the hydrolysis of ATP molecule. 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, Mg^{2+}, Ca^{2+}.
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 F and F, which are joined to each other by the small proteins F5 and F6 . These proteins form the F  F complex or the enzyme ATPase, in which F 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 F  F is positioned so that the enzyme F is on the inside of the membrane [13].
The enzyme F 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 F  F 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 F and ATP. The largest two proteins in F, which is composed of five protein molecules, take part in the synthesis and dissociation of ATP molecules, the other three are apparently inhibitors controlling these reactions.
After removing enzyme F molecules from mitochondria the remaining F enzymes increase greatly the permeability of protons in the coupling membranes, which confirms that the enzyme F has really a channel for the passage of protons which is constructed by the enzyme F. However, the complete mechanism for the synthesis of ATP molecules by the enzyme ATPase is still not clearly known up to now.
Davydov model of bioenergy transport work at α − helical proteins as shown in Figure.1.
The Davydov soliton obtained from Eqs. (1)(2) in the semi classical limit and using the continuum approximation has the from $${\varphi}_{D}(x,t)={\left(\frac{{\mu}_{D}}{2}\right)}^{1/2}\text{sech}\left[\frac{{\mu}_{D}}{{r}_{0}}(x{x}_{0}vt)\right]\mathrm{exp}\left\{i\left[\frac{\hslash v}{2J{r}_{0}^{2}}(x{x}_{0}){E}_{\nu}t/\hslash )\right]\right\}\text{(4)}$$ Corresponding to an excitation localized over a scale r_{/} D_{μ} , where ${\mu}_{D}=\frac{{\chi}_{1}^{2}}{(1{s}^{2})wJ},{\text{G}}_{\text{D}}=4J{\mu}_{D},{\text{s}}^{\text{2}}=\frac{{v}^{2}}{{v}_{0}^{2}},{\text{v}}_{\text{0}}={r}_{\text{0}}{\text{(w/M)}}^{\text{1/2}}$ 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., ${\varphi}_{D}(t)\left\widehat{N}\right{\varphi}_{D}(t)>=1$ This shows that the Davydov soliton is formed through selftrapping of one exciton with binding energy E_{BD}, ${E}_{BD}=\frac{{\chi}_{1}^{4}}{3J{w}^{2}}$
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 [328]. 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 [726]. 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 bioenergy 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 [711, 2426]. Other simulations showed that the Davydov soliton is stable at 300 K [1024], 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 ID_{2} 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 quantumstatistical 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 quantummechanical 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 [1012]. 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 quantummechanical 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 $J\ne 0$ is not known. Therefore, it is necessary to reform Davydov’s wave function. Scientists had though that the soliton with a multiquantum $\text{n}\ge \text{2}$ , 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 twoquantum state of CruzeiroHansson [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 amideI vibration. On the other hand, the numerical result shows that the soliton of twoquantum state is more stable than that with a onequantum state.
CruzeiroHansson [18] had thought that Forner’s twoquantum state in the semiclassical case was not exact. Therefore, he constructed again a socalled exactly twoquantum state for the semiclassical Davydov system as follows [18]: $$\text{I}\varphi \left(\text{t}\right)>={\displaystyle \sum _{\text{n},\text{m}=\text{1}}^{\text{N}}{\phi}_{\text{nm}}}\left(\left\{{\text{u}}_{\text{1}}\right\},\left\{{\text{P}}_{\text{1}}\right\},\text{t}\right){\text{B}}_{\text{n}}^{+}{\text{B}}_{\text{m}}^{\text{+}}\text{0}{>}_{\text{ex}},\text{(5)}$$ where ${B}_{n}({B}_{n}^{+})$ is the annihilation (creation) operator for an amideI vibration quantum (exciton), ${u}_{1}$ is the displacement of the lattice molecules, ${P}_{1}$ is its conjugate momentum, and ${0\rangle}_{ex}$ 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 $\nu =JW/{\chi}_{1}^{2}$ and $ln\beta (\beta =1/{K}_{B}T)$ in the socalled twoquantum state. Eq.(5), where ${\chi}_{1}$ is a nonlinear coupling parameter related to the interaction of the excitonphonon in the Davydov model. Their energies and stability are compared with those of the onequantum state. From the results of above thermal averages, he drew the conclusion that the wave function with a twoquantum state can lead to more stable soliton solutions than that with a onequantum 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 CruzeiroHansson wave function[18,2426] does not represent exactly the twoquantum 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. $N={{\displaystyle {\sum}_{n}B}}_{n}^{+}{B}_{n}$ , in this state Eq.(5), and sum over the sites, i.e., the exciton numbers N are $$\begin{array}{l}N=<\varphi {\displaystyle \sum _{n}{B}_{n}^{+}}{B}_{n}\varphi >={\displaystyle \sum _{ijlmn}{\phi}_{im}^{\ast}}{\phi}_{jl}ex<0{B}_{i}{B}_{m}{B}_{n}^{+}{B}_{n}{B}_{j}^{+}{B}_{l}^{+}0{>}_{ex}\text{}(6)\\ \text{}={\displaystyle \sum _{nj}\left({\phi}_{nj}^{\ast}{\phi}_{jn}+{\phi}_{jn}^{\ast}{\phi}_{jn}\right)}+{\displaystyle \sum _{nl}\left({\phi}_{nl}^{\ast}{\phi}_{nl}+{\phi}_{\mathrm{ln}}^{\ast}{\phi}_{nl}\right)}=4\end{array}$$ where we use the relations $$[{B}_{n}.{B}_{j}^{+}]={\sigma}_{nj},{\displaystyle {\sum}_{nl}{\left{\phi}_{nl}\right}^{2}}=1\text{(7)}$$ $${}_{ex}\langle 0{B}_{n}^{+}{0\rangle}_{ex}{=}_{ex}\langle 0{B}_{n}^{+}{B}_{n}{0\rangle}_{ex}{=}_{ex}\langle 0{B}_{n}^{+}{B}_{n}{B}_{l}{0\rangle}_{ex}=\mathrm{....}=0\text{}(8)$$ 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 singleparticle excitation with two quanta created at sites n and m;[18,26] is the probability amplitude of particles occurring at the sites n and m simultaneously. In general, n=≠m and 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 excitonphonon 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 CruzeiroHansson 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 CruzeioHansson [12,13,18,22], Schweitzer [21] and Pang [2426] 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 amideI vibrational modes was added to the original Davydov’s Hamiltonian which takes into account relative displacement of the neighbouring amino acids resulting from dipoledipole interaction of the neighbouring amide1 vibrational quanta. Davydov’s wave function has been also replaced with a quasicoherent twoquanta state to exhibit the coherent behaviors of collective excitations of the excitons and phonons [2526] 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 300 K and the lifetime of the new soliton at 300K is calculated in detail by using the generally accepted values of the parameters appropriate to 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 300 K is long enough to provide a viable explanation of the bioenergy 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 quantummechanical 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.
Let us consider the Davydov model with the present viewpoint. First, as far as the Davydov wave functions, both ${D}_{1}>and{D}_{2}>$ 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 [2426], 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 largen excitation states [12,22] are not appropriate for the protein molecules due to the reasons mentioned above. Similarly, Forner’s and CruzeiroHansson’s twoquantum states do not fulfill the above request. In view of the above discussion, we proposed the following wave function of the protein molecular systems: $$\begin{array}{l}\Phi (t)={\phi}_{P}\left(t\right)>\beta \left(t\right)>=\frac{1}{\lambda}\left[I+{\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}+\frac{1}{2!}}{\left({\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}}\right)}^{2}\right]0{>}_{ex}\times \text{(9)}\\ \mathrm{exp}\left(\frac{i}{\hslash}{\displaystyle \sum _{n}[{\beta}_{n}(t){P}_{n}{\pi}_{n}(t){u}_{n}]}\right){0\rangle}_{ph}\end{array}$$ where ${B}_{n}^{+}$ and ${B}_{n}$ are boson creation and annihilation operators for the exciton, $0{>}_{ex}and0{>}_{ph}$ 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 ${\varphi}_{n}\left(t\right).{\beta}_{n}\left(t\right)=<\Phi \left(t\right){u}_{n}\Phi \left(t\right)>and{\pi}_{n}\left(t\right)=<\Phi \left(t\right)\left{P}_{n}\right\Phi \left(T\right)>$ are there sets of unknown functions, $\lambda $ is a normalization constant. It is assumed hereafter that $\lambda =1$ for convenience of calculation, except when explicitly mentioned.
A second problem arises for the Davydov Hamiltonian [2426,28]. The Davydov Hamiltonian takes into account the resonant or dipoledipole interaction of the neighboring amideI 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 ${\chi}_{2}\left({u}_{n+1}{u}_{n}\right)\left({B}_{n+1}^{+}{B}_{n}+{B}_{m}^{+}{B}_{n+1}\right)$ into the Davydov’s Hamiltonian to represent correlations of the collective excitations and collective motions in the protein molecules, as mentioned above [2426]. Although the dipole dipole interaction is small as compared with the energy of the amideI vibrational quantum, the change of relative displacement of neighboring peptide groups resulting from this interaction cannot be ignored due to the sensitive dependence of dipoledipole interaction on the distance between amino acids in the protein molecules, which is a kind of soft condensed matter and bioselforganization. Thus, the Davydov Hamiltonian is replaced by $$\begin{array}{l}H={H}_{ex}+{H}_{ph}+{H}_{\mathrm{int}}={\displaystyle \sum _{n}\left[{\epsilon}_{0}{B}_{n}^{+}{B}_{n}J\left({B}_{n}^{+}{B}_{n+1}+{B}_{n}^{}{B}_{n+1}^{+}\right)\right]}+{\displaystyle {\sum}_{n}[\frac{{P}_{n}^{2}}{2M}+\text{\hspace{0.05em}}\frac{1}{2}}w{(}^{{u}_{n}}]\\ +{\displaystyle \sum _{n}\left[{\chi}_{1}\left({u}_{n+1}{u}_{n1}\right)\right]}{B}_{n}^{+}{B}_{n}+{\chi}_{2}\left({u}_{n+1}{u}_{n}\right)\times \left({B}_{n+1}^{+}{B}_{n}+{B}_{n}^{+}{B}_{n+1}\right)\text{(10)}\end{array}$$ Where ${\epsilon}_{0}=0.205\text{ev}$ is the energy of the exciton (the C=0 strechiong mode). The present nonlinear coupling constants are ${\chi}_{1}and{\chi}_{2}$ They represent the modulations of the onsite energy and resonant (or dipoledipole) 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 dipoledipole interaction energy between neighboring sites. The physical meaning of the other quantities in Eq.(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 ${\sum}_{n}{\chi}_{2}}\left({u}_{n+1}{u}_{n}\right)\left({B}_{n+1}^{+}{B}_{n}+{B}_{n}^{+}{B}_{n+1}\right),$ 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 [2426] and others [2728] in the Hamiltonian of the vibronsoliton model for onedimensional oscillatorlattice 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 for three terms of the expansion of a standard coherent state, which can be viewed as an effective truncation of a standard coherent state. When ${\phi}_{n}\left(t\right)$ is small, i.e., $\left{\phi}_{n}\left(t\right)\right<<1$ Pang represented the wave function of the excitons, ${\phi}_{P}\left(t\right)>$ , in Eq.(9) as $$\begin{array}{l}{\phi}_{P}\left(t\right)>=\frac{1}{\lambda}\left[1+{\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}+\frac{1}{2!}{\left({\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}}\right)}^{2}}\right]0{>}_{ex}~\frac{1}{\lambda}\mathrm{exp}[\frac{1}{2}{\displaystyle \sum _{n}{\left{\phi}_{n}(t)\right}^{2}}]\times \\ \mathrm{exp}\left\{{\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}}\right\}0{>}_{ex}=\frac{1}{\lambda}\mathrm{exp}\left\{{\displaystyle \sum _{n}\left[{\phi}_{n}\left(t\right){B}_{n}^{+}{\phi}_{n}^{\ast}\left(t\right){B}_{n}\right]}\right\}0{>}_{ex}\text{(11)}\end{array}$$ The last representation in Eq.(11) is a standard coherent state. Therefore, the state of exciton denoted by the new wave function $\phi \left(t\right)>$ has a coherent feature, thus the wave function inEq.11) is normalized at $\lambda =1$ Since ${{\displaystyle {\sum}_{n}\left{\phi}_{n}\left(t\right)\right}}^{2}=1$ required in thecalculation, then this condition of $\left{\phi}_{n}\left(t\right)\right<<1$ naturally satisfied for the roteins 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, $\widehat{N}={\displaystyle {\sum}_{n}{B}_{n}^{+}{B}_{n}}$ because of $$\widehat{N}{\phi}_{P}>={\displaystyle \sum _{n}{B}_{n}^{+}}{B}_{n}{\phi}_{P}\rangle =\left\{{\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}}+{\left({\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}}\right)}^{2}\right\}0{>}_{ex}=2{\phi}_{P}\rangle \left(2+{\displaystyle \sum _{n}{\phi}_{n}\left(t\right){B}_{n}^{+}}\right)0{>}_{ex}(12)$$ Therefore, the ${\phi}_{P}\rangle $ 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 $\widehat{N}$ we find that this state contains the number of exciton is $$\begin{array}{l}N=<{\phi}_{P}\left\widehat{N}\right{\phi}_{P}>={\displaystyle \sum _{n}\langle {\phi}_{P}{B}_{n}^{+}{B}_{n}{\phi}_{P}\rangle}=\left\{{\displaystyle \sum _{n}{\left{\phi}_{n}\left(t\right)\right}^{2}+\left({\displaystyle \sum _{n}{\left{\phi}_{n}\left(t\right)\right}^{2}}\right)\left({\displaystyle \sum _{m}{\left{\phi}_{m}\left(t\right)\right}^{2}}\right)}\right\}\\ =\left({\displaystyle \sum _{n}{\left{\phi}_{n}\left(t\right)\right}^{2}}\right)\left(1+{\displaystyle \sum _{m}{\left{\phi}_{m}\left(t\right)\right}^{2}}\right)=2\text{(13)}\end{array}$$ namely, it contains only two quanta. Where we utilize Eq.(8) and the following relation [24] is: $$\sum _{n}{\left{\varphi}_{n}\left(t\right)\right}^{2}}=1,{\displaystyle \sum _{m}{\left{\varphi}_{m}\left(t\right)\right}^{2}}=1,[{B}_{n}.{B}_{m}^{+}]={\delta}_{nm}\text{(14)$$ $$\begin{array}{l}{}_{ex}<0\left{B}_{n}^{+}\right0{>}_{ex}{=}_{ex}<0\left{B}_{n}^{+}{B}_{n}\right0{>}_{ex}{=}_{ex}<0\left{B}_{n}^{+}{B}_{m}\right0{>}_{ex}\\ \text{}{=}_{ex}0\left{B}_{n}^{+}{B}_{m}{B}_{l}\right0{}_{ex}{=}_{ex}0\left{B}_{n}^{+}{B}_{m}{B}_{l}^{+}{B}_{n}\right0{}_{ex}\\ \text{}{=}_{ex}0\left{B}_{n}^{+}{B}_{m}{B}_{i}^{+}{B}_{l}{B}_{j}\right0{}_{ex}{=}_{ex}0\left{B}_{n}^{+}{B}_{m}{B}_{l}^{+}{B}_{i}{B}_{j}{B}_{n}\right0{}_{ex}\mathrm{....}=0\end{array}$$ 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 twoquanta 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 exhibits the coherent feature of the collective excitation of excitons and phonons caused by the nonlinear interaction generated by the excitonphonon 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 amideI 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 onequanta state [3] and the twoquanta state [12,22]. Therefore, the above Hamitonian and wave function, Eqs. (9) and (10), are reasonable and appropriate to the protein molecules.
We now derive the equation of motion for the${\phi}_{n}$ . A basic assumption in the derivation is that $\Phi (t)\rangle $ in Eq. (9) is a solution of the timedependent equation [2426]: $$\text{i}\hslash \frac{\partial}{\partial \text{t}}\Phi (\text{t})\rangle =\text{H}\Phi (\text{t})\rangle \text{(20)}$$ The lefthand side of Eq. (16) has [12,23] $$i\hslash \frac{\partial}{\partial t}\Phi (t)\rangle =\left\{i\hslash \left({\displaystyle \sum _{n}{\dot{\varphi}}_{n}(t){B}_{n}^{+}+{\displaystyle \sum _{n}{\dot{\varphi}}_{n}(t){\varphi}_{n}(t){B}_{n}^{+}{B}_{n}^{+}{0\rangle}_{ex}}}\right)\right\}\beta (t)\rangle $$ $$+{\phi}_{P}(t)\rangle \left\{{\displaystyle \sum _{n}\left\{{\beta}_{n}(t){P}_{n}{\pi}_{n}(t){u}_{n}+\frac{1}{2}\left[{\beta}_{n}(t){\dot{\pi}}_{n}(t){\dot{\beta}}_{n}(t){\pi}_{n}(t)\right]\right\}\beta (t)\rangle}\right\}\text{(21)}$$ Now leftmultiplying the both sides of Eq.(21) by $\langle \Phi (\text{t})$ ,the lefthand side of Eq.(21) can be $$\text{i}\hslash \langle \Phi (\text{t})\left{\text{u}}_{\text{n}}\right\Phi (\text{t})\rangle \text{i}\hslash {\displaystyle \sum _{\text{n}}{\varphi}_{\text{n}}^{*}(\text{t}){\varphi}_{\text{n}}(\text{t})}\left({\displaystyle \sum _{\text{m}}{\varphi}_{\text{m}}^{*}(\text{t}){\varphi}_{\text{m}}(\text{t})+\text{1}}\right)$$ $$=\frac{\text{5}}{\text{4}}{\displaystyle \sum _{\text{n}}\left[{\dot{\beta}}_{n}(t){\pi}_{n}(t){\dot{\pi}}_{n}(t){\beta}_{n}(t)\right]}{\displaystyle \sum _{\text{n}}{\left{\varphi}_{\text{n}}(\text{t})\right}^{\text{2}}}\text{(22)}$$ Similarly, for the righthand side of Eq. (20) we can have [12,23] $$\langle \Phi (\text{t})\left\left({\text{H}}_{\text{ex}}+{\text{H}}_{\text{ph}}+{\text{H}}_{\text{int}}\right)\right\Phi (\text{t})\rangle =\{{\displaystyle \sum _{\text{n}}\left\{{\epsilon}_{\text{0}}\left{\varphi}_{\text{n}}{(\text{t})}^{\text{2}}\right\text{J}{\varphi}_{\text{n}}^{*}(\text{t})\left[{\varphi}_{\text{n}+\text{1}}(\text{t}){\varphi}_{\text{n}\text{1}}(\text{t})\right]\right\}}\times \left(1+{\displaystyle \sum _{m}{\left{\varphi}_{m}(t)\right}^{2}}\right)$$ $$+\{{\displaystyle \sum _{n}\{{\chi}_{1}\left[{\beta}_{n+1}(t){\beta}_{n1}(t)\right]{\left{\varphi}_{n}(t)\right}^{2}+{\chi}_{2}\left[{\beta}_{n+1}(t){\beta}_{n1}(t)\right]}$$ $$\times {\phi}_{n}^{*}(t)\left[{\phi}_{n+1}(t){\phi}_{n1}(t)\right]\}\}\left(1+{\displaystyle \sum _{m}{\left{\phi}_{m}(t)\right}^{2}}\right)+\frac{5}{2}W(t){\displaystyle \sum _{n}{\left{\phi}_{n}(t)\right}^{2}}\text{(23)}$$ where $$W(t)=\langle \beta (\text{t})\left{H}_{ph}\right\beta (\text{t})\rangle ={\displaystyle \sum _{n}\left(\frac{1}{2M}{\pi}_{n}^{2}(t)+\frac{1}{2}w{\left[{\beta}_{n}(t){\beta}_{n1}(t)\right]}^{2}\right)}+{\displaystyle \sum _{q}\frac{1}{2}\hslash {\omega}_{q}}\text{(24)}$$ and utilizing Eqs.(8) and (12)(14) and the relationships can be obtained: $$\sum _{\text{n}}\left[{\beta}_{\text{m}+1}(t)2{\beta}_{\text{m}}(t)+{\beta}_{\text{m}1}(t)\right]{\beta}_{m}(t)}={\displaystyle \sum _{\text{n}}{\left[{\beta}_{\text{m}+1}(t){\beta}_{\text{m}1}(t)\right]}^{2}$$ $$\langle \Phi (\text{t})\left{\displaystyle \sum _{n}\left({B}_{n}^{+}{B}_{n1}+{B}_{n}{B}_{n1}^{+}\right)}\right\Phi (\text{t})\rangle ={\displaystyle \sum _{n}\left[{\varphi}_{n}^{*}(t){\varphi}_{n+1}(t)+{\varphi}_{n1}^{*}(t)\varphi (t)\right]\left(1+{\displaystyle \sum _{m}{\left{\varphi}_{m}(t)\right}^{2}}\right)}$$ $$\langle \Phi (\text{t})\left{\displaystyle \sum _{n}\left({u}_{n+1}{u}_{n1}\right)\left({B}_{n}^{+}{B}_{n}\right)}\right\Phi (\text{t})\rangle ={\displaystyle \sum _{n}\left\{\left[{\beta}_{\text{m}+1}(t){\beta}_{\text{m}1}(t)\right]{\left{\varphi}_{n}(t)\right}^{2}\right\}\left(1+{\displaystyle \sum _{m}{\left{\varphi}_{m}(t)\right}^{2}}\right)}$$ $$\begin{array}{l}\langle \Phi (\text{t})\left{\displaystyle \sum _{n}\left({u}_{n1}{u}_{n}\right)\left({B}_{n}^{+}{B}_{n1}+{B}_{n}{B}_{n1}^{+}\right)}\right\Phi (\text{t})\rangle =\\ {\displaystyle \sum _{n}\left\{\left[{\beta}_{\text{m}+1}(t){\beta}_{\text{m}1}(t)\right]\left[{\varphi}_{n}^{*}(t){\varphi}_{n+1}(t)+{\varphi}_{n1}^{*}(t)\varphi (t)\right]\right\}}\times \left(1+{\displaystyle \sum _{m}{\left{\varphi}_{m}(t)\right}^{2}}\right)\text{(25)}\end{array}$$ From Eqs.(20)(23) we can obtain $$i\hslash \frac{\partial}{\partial t}{\varphi}_{n}(t)={\epsilon}_{0}{\varphi}_{n}(t)J\left[{\varphi}_{n+1}(t)+{\varphi}_{n1}(t)\right]+{\chi}_{1}\left[{\beta}_{n+1}(t)+{\beta}_{n1}(t)\right]{\varphi}_{n}(t)$$ $${\chi}_{2}\left[{\beta}_{n+1}(t)+{\beta}_{n}(t)\right]\times \left[{\varphi}_{n+1}(t)+{\varphi}_{n1}(t)\right]$$ $$+\frac{5}{2}\left(W(t)\frac{1}{2}{\displaystyle \sum _{m}\left[{\dot{\beta}}_{m}(t){\pi}_{m}(t){\dot{\pi}}_{m}(t)\beta (t)\right]}\right){\varphi}_{n}(t)\text{(26)}$$ In the continuum approximation we get from Eqs.(19) and (26) $$i\hslash \frac{\partial}{\partial t}\varphi (x,t)=R(t)\varphi (x,t)J{r}_{0}^{2}\frac{{\partial}^{2}}{\partial {x}^{2}}\varphi (x,t){G}_{p}{\left\varphi (x,t)\right}^{2}\varphi (x,t)\text{(27)}$$ and $$\frac{\partial \beta (x,t)}{\partial \xi}=\frac{\partial \beta (x,t)}{\partial x}=\frac{4\left({\chi}_{1}+{\chi}_{1}\right)}{w\left(1{s}^{2}\right){r}_{0}}{\left\varphi (x,t)\right}^{2}\text{(28)}$$ here $\xi =x{v}_{t}$ $\text{}R(t)={\epsilon}_{0}2J+\frac{5}{2}\left\{W(t)\frac{1}{2}{\displaystyle \sum _{m}\left[{\dot{\beta}}_{m}(t){\pi}_{m}(t){\dot{\pi}}_{m}(t)\beta (t)\right]}\right\}$ and $s=v/v{}_{0}$ . The soliton solution of Eq.(27) is thus $$\varphi (x,t)={\left(\frac{{\mu}_{p}}{2}\right)}^{1/2}\mathrm{sec}h\left[\left({\mu}_{p}/{r}_{0}\right)\left(x{x}_{0}{V}_{t}\right)\right]\times \mathrm{exp}\left\{i\left[\frac{\hslash v}{2J{r}_{0}^{2}}\left(x{x}_{0}\right){E}_{v}\frac{t}{\hslash}\right]\right\}\text{(29)}$$ with $${\mu}_{P}=\frac{2{\left({\chi}_{1}+{\chi}_{2}\right)}^{2}}{w\left(1{s}^{2}\right)J},{G}_{P}=\frac{8{\left({\chi}_{1}+{\chi}_{2}\right)}^{2}}{w\left(1{s}^{2}\right)}\text{(30)}$$ 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 ${\mu}_{P}$ have obvious distinctions from that in the Davydov model. A straightforward result of Pang’s model is to increase the nonlinear interaction energy ${G}_{P}\left({G}_{P}=2{G}_{D}\left[1+2\left({\chi}_{2}/{\chi}_{1}\right)+{\left({\chi}_{2}/{\chi}_{1}\right)}^{2}\right]\right)$ and the amplitude of the new soliton and decrease its width due to an increase of [2426] when compared with Davydov soliton [3], where ${\mu}_{D}={x}_{1}^{2}/w(1{s}^{2})J$ , and ${G}_{D}=4{x}_{1}^{2}/w(1{s}^{2})$ are the corresponding values in the Davydov mode [38]. 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.
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. [3235] and Scott [6] again, it is convenient to define another composite parameter [3,2426] that can be written as $$4\pi {\alpha}_{P}={\left({\chi}_{1}+{\chi}_{2}\right)}^{2}/2w\hslash {\omega}_{D}\text{(36)}$$ where ${\omega}_{D}={\left(w/M\right)}^{1/2}$ is the band edge for acoustic phonons (Debye frequency). If, $4\pi {\alpha}_{P}>1$ it is said to be weak. Using widely accepted values for the physical parameters for the alpha helix protein molecule [223],$$J=1.55\times {10}^{22}J.w=(1319.5)N/m.M=(1.171.91)\times {10}^{25}kg$$ $${\chi}_{1}=62\times {10}^{12}N.{\chi}_{2}=(1018)\times {10}^{12}N.{r}_{0}=4.5\times {10}^{10}m.\text{(37)}$$ We can estimate that the coupled constant lies in the region of $4\pi {\alpha}_{P}=0.110.273$ , which is not a weakly coupled theory, the coupling strength is enhanced as compared with the Davydov model, the latter is $4\pi {\alpha}_{D}=0.0360.045$ Using the notation of Bullough et al. [29,30], Teki et al. [31,32], and Pouthier et al. [33 35] $$\gamma =J/2\hslash {w}_{D}\text{(38)}$$ In terms of the two composite parameters, $4\pi {\alpha}_{P}$ and $\gamma $ ,the soliton binding energy for Pang’s model can be written by $${E}_{BP}/J=8{\left(4\pi {\alpha}_{P}/\gamma \right)}^{2}/3,{M}_{sol}=2{m}_{ex}\left[1+32{\left(4\pi {\alpha}_{P}\right)}^{2}/3\right]\text{(39)}$$ From the above parameter values, we find $\gamma =0.08$ Utilizing this value, the ${E}_{BP}/J$ versus $4\pi \alpha $ relations in Eq.(39) are plotted in Fig.1. However, ${E}_{BP}/J={\left(4\pi {\alpha}_{P}/\gamma \right)}^{2}/3$ for the Davydov model (here ${M}_{sol}^{\text{'}}={m}_{ex}\left[1+2{\left(4\pi {\alpha}_{P}/\gamma \right)}^{2}/3\right],4\pi {\alpha}_{D}={\chi}_{1}^{2}/2w\hslash {\omega}_{D}),$ then the ${E}_{BD}/J$ versus $4\pi {\alpha}_{D}$ 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\pi \alpha $ [2426].
As a matter of fact, the nonlinear interaction energy forming the new soliton in Pang’s model is ${\text{G}}_{\text{P}}=\text{8}{\left({\chi}_{\text{1}}+{\chi}_{\text{2}}\right)}^{\text{2}}/\left(\text{1}{\text{s}}^{\text{2}}\right)w=3.8\times {10}^{32}J$ ,and it is larger than the linear dispersion energy $\text{J}=\text{1}.\text{55}\times {\text{10}}^{\text{32}}\text{J}$ ,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 [2,3335]. On the other hand, the nonlinear interaction energy in the Davydov model is only ${\text{G}}_{\text{D}}=\text{4}{\chi}_{\text{1}}{}^{\text{2}}/\left(\text{1}{\text{S}}^{\text{2}}\right)w=1.8\times {10}^{21}J$ and it is about three to four times smaller than ${\text{G}}_{\text{P}}$ .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 ${E}_{BP}=\left(4.164.3\right)\times {10}^{21}J$ in Eq.(31), which is somewhat larger than the thermal perturbation energy, ${k}_{B}T=4.13\times {10}^{21}J$ at $300K$ and about four times larger than the Debye energy $k\Theta =\hslash {\omega}_{D}=1.2\times {10}^{21}J$ (there ${\omega}_{D}$ 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 ${\mu}_{p}$ and ${T}_{0}=\hslash {V}_{0}{\mu}_{p}/{K}_{B}\pi $ 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 ${\mu}_{p}$ 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 ${\text{E}}_{\text{BD}}={\chi}_{{}_{\text{1}}}^{\text{4}}/{\text{3w}}^{\text{2}}\text{J}=0.188\times {10}^{21}J$ 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}\Theta $ , 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 $300K$ .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 $\left[1+2(\frac{{\chi}_{2}}{{\chi}_{1}})+{(\frac{{\chi}_{2}}{{\chi}_{1}})}^{2}\right],).$ Secondly the nonlinear coupling energy becomes Gp = ${\frac{8({\chi}_{1}+{\chi}_{2})}{w(1{s}^{2})}}^{2}$ (G_{p}=2G_{D} $\left[1+2(\frac{{\chi}_{2}}{{\chi}_{1}})+{(\frac{{\chi}_{2}}{{\chi}_{1}})}^{2}\right]$ , where ${G}_{D}=\frac{4{\chi}_{1}^{2}}{w(1{s}^{2})}$ is the nonlinear interaction in the Davydov model, resulting from the twoquanta nature and the enhancement of the coupling the coefficient $({\chi}_{1}+{\chi}_{2})$ . For $\alpha $ 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.
Parameters 
μ 
G 
Amplitude of soliton 
Width of soliton Δ X(×1010m) 
Binding energy of soliton EB 
Our 
5.94 
3.8 
1.72 
4.95 
7.8 
Davydov 
1.90 
1.18 
0.974 
14.88 
0.188 
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 wave function 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 wave function 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 fulfill 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, $\Delta X=2\pi {r}_{0}/{\mu}_{p}$ to be greater than the lattice constant r0, i.e., $\Delta X>{r}_{0}$ 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 $\phi (x){}^{2}=\frac{{\mu}_{p}}{2}\mathrm{sec}{h}^{2}(\frac{{\mu}_{p}x}{{r}_{0}})$ 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 quasicoherent 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.
For convenlence of calculation, we here represent the wave function of the system in Eq. (9) by [2426] $$\Phi (t)>=\varphi (t)>\beta (t)>={\text{U}}_{\text{1}}\text{10}{\uff1e}_{\text{ex}}{\text{U}}_{\text{2}}\text{10}{\uff1e}_{\text{ph}},\text{(40)}$$ where $${\text{U}}_{\text{1}}=\frac{1}{\lambda}[1+{\displaystyle \sum _{n}{\varphi}_{n}(t){B}_{n}^{+}+}\frac{1}{2!}({\displaystyle \sum _{n}{\varphi}_{n}(t){B}_{n}^{+}{)}^{2}}]\text{(40a)}$$ $${\text{U}}_{\text{2}}=\mathrm{exp}\left\{\frac{i}{\hslash}{\displaystyle \sum _{n}[{\beta}_{n}(t){P}_{n}{\pi}_{n}(t){u}_{n}}]\right\}\text{(40b)}$$ $$=\mathrm{exp}\left\{\frac{1}{\sqrt{N}}{\displaystyle \sum _{q}{\alpha}_{q}(t){a}_{q}^{+}{\alpha}_{q}^{*}(t){a}_{q}}\right\}\text{(40c)}$$ where we assume ${{\displaystyle \sum _{i}\left{\varphi}_{i}\right}}^{2}=n$ ,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 amideI 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 ${\omega}_{q}=2{(w/M)}^{1/2}\mathrm{sin}(\frac{{r}_{0}q}{2})$ ,Eq.(10) becomes $$\begin{array}{l}H={\displaystyle \sum _{n}\left[{\epsilon}_{0}{B}_{n}^{+}{B}_{n}^{}J({B}_{n}^{+}{B}_{n+1}^{}+{B}_{n+1}^{+}{B}_{n}^{})\right]}+{\displaystyle \sum _{q}\hslash {\omega}_{q}({a}_{q}^{+}{a}_{q}+\frac{1}{2})}\\ +\frac{1}{\sqrt{N}}{\displaystyle \sum _{q.n}[{g}_{1}(q){B}_{n}^{+}{B}_{n}^{}+{g}_{2}(q)({B}_{n}^{+}{B}_{n+1}^{}+{B}_{n}^{+}{B}_{n+1}^{})]({a}_{q}+{a}_{q}^{+}){e}^{in{r}_{0}^{}q}}\text{(41)}\end{array}$$ where $${g}_{1}(q)=2{\chi}_{1}i{\left[\frac{\hslash}{2M{\omega}_{q}}\right]}^{1/2}\mathrm{sin}{r}_{0}q;{g}_{2}(q)={\chi}_{2}{\left[\frac{\hslash}{2M{\omega}_{q}}\right]}^{1/2}({e}^{i{r}_{0}q}1)\text{(42)}$$ In a semi classical 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[2426] $$\varphi (x,t)={\left(\frac{{\mu}_{p}}{2}\right)}^{1/2}Sech[\frac{{\mu}_{p}}{{r}_{0}}(xvt)]\mathrm{exp}\left[\frac{i}{\hslash}\left(\frac{{\hslash}^{2}vx}{2J{r}_{0}^{2}}{E}_{sol}t\right)\right]\text{(43)}$$ where $${\mu}_{p}=\frac{2{({\chi}_{1}+{\chi}_{2})}^{2}}{w(1{s}^{2})J}\text{(44)}$$ The energy of the new soliton is $${\alpha}_{q}(t)=\frac{i\pi ({\chi}_{1}+{\chi}_{2})}{w{\mu}_{p}(1{v}^{2}/{v}_{0}^{2})}{\left[\frac{M}{2\hslash {\omega}_{q}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}({\omega}_{q}+qv)\mathrm{csc}h(\pi q{r}_{0}/2{\mu}_{p}){e}^{iqvt}={\alpha}_{q}{e}^{iqvt}\text{(46)}$$ This treatment yields a localized coherent structure of the excitons with size of order 2 r0/μp that propagates with velocity v and can transfer energy ES01< . Unlike bare excitons that are scattered by the interactions with the phonons, this soliton state describes a quasiparticle 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 bellshaped form of the soliton Eq. (43) does not depend on the excitation method. It is selfconsistent. 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 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, $\tilde{H}\text{vP},$ where P is the total momentum, and $\text{P}={\displaystyle \sum _{q}\hslash q({a}_{q}^{+}{a}_{q}{B}_{q}^{+}{B}_{q})}$ ,where ${B}_{q}^{+}=\frac{1}{\sqrt{N}}{\displaystyle \sum _{n}{e}^{iqn{r}_{0}}}{B}_{n}^{+}$ . Also, in order to have simple analytical expressions we make the usual continuum approximation. This gives $$\begin{array}{c}\tilde{H}={\displaystyle {\int}_{0}^{L}dx2\left[({\epsilon}_{0}2J){\varphi}^{+}(x)\varphi (x)+J{r}_{0}^{2}\frac{\partial {\varphi}^{+}}{\partial x}\frac{\partial \varphi}{\partial x}\frac{i\hslash v}{2}\left(\frac{\partial {\varphi}^{+}}{\partial x}\varphi (x){\varphi}^{+}(x)\frac{\partial \varphi}{\partial x}\right)\right]}\\ +{\displaystyle \sum _{q}\hslash ({\omega}_{q}qv){a}_{q}^{+}{a}_{q}+\frac{1}{\sqrt{N}}{\displaystyle \sum _{q}}2[{g}_{1}(q)+2{g}_{2}(q)]({a}_{q}^{+}+{a}_{q}){\displaystyle {\int}_{0}^{L}dx{e}^{ikx}{\varphi}^{+}(x){\varphi}^{}(x)}}\text{(47)}\end{array}$$ where $\text{}\varphi (x)$ 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 ${\omega}_{q}$ ≈(w/M)^{1/2} r0•q ,x=nr_{0}. 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] $${\text{b}}_{\text{q}}={a}_{p}\frac{1}{\sqrt{N}}{\alpha}_{q},{b}_{q}^{+}={a}_{q}^{+}\frac{1}{\sqrt{N}}{\alpha}_{q}^{*},\text{(48)}$$ which describe phonons relative to a chain with a particular deformation, where b_{q} (b_{q}^{+}) is the annihilation (creation) operator of new phonon. The vacuum state for the new phonons is $$\stackrel{~}{0}{\rangle}_{ph}=\mathrm{exp}\left[\frac{1}{\sqrt{N}}{\displaystyle \sum _{q}({\alpha}_{q}(t){a}_{q}^{+}{\alpha}_{q}^{*}(t)){a}_{q}}\right]0>{}_{ph}\text{(49)}$$ which is a coherent phonon state[30], i.e., $\tilde{0}{\rangle}_{ph}=0$ . The Hamiltonian $\stackrel{~}{H}$ can now be rewritten[ 2426] as $$\begin{array}{c}\stackrel{~}{H}={\displaystyle {\int}_{0}^{L}2dx\varphi (x)[{\epsilon}_{0}2J+V(x)J{r}_{0}^{2}\frac{{\partial}^{2}}{\partial {x}^{2}}+i\hslash \frac{\partial}{\partial x}]}\varphi (x)+\\ {\displaystyle \sum _{q}\hslash ({\omega}_{q}qv)[{b}_{q}^{+}{b}_{q}+\frac{1}{\sqrt{N}}({\alpha}_{q}{b}_{q}^{+}+{\alpha}_{q}^{*}{b}_{q}^{+})]}+W\text{'}+\\ \frac{1}{\sqrt{N}}{\displaystyle \sum 2[{g}_{1}(q)+2{g}_{2}(q)]({b}_{q}^{+}+{b}_{q}){\displaystyle {\int}_{0}^{L}dx{e}^{iqx}{\varphi}_{}^{+}(x)}}\varphi (x)\text{}(50)\end{array}$$ where $$W\text{'}=\frac{1}{N}{\displaystyle \sum _{q}\hslash}({\omega}_{q}qv){\alpha}_{q}^{}{}^{2},V(x)=\frac{1}{N}{\displaystyle \sum _{q}[{g}_{1}(q)+2{g}_{2}(q)]({\alpha}_{q}^{*}+{\alpha}_{q}^{}){e}^{iqx}}\text{(51)}$$ To describe the deformation corresponding to a soliton in the subspace where there is $${\int}_{0}^{L}dx{\varphi}_{}^{+}(x){\varphi}_{}^{}(x)}=1$$ from Eq(45) in such a case. From the above formulae we can obtain $$\text{V}\left(\text{x}\right)=2J{\mu}_{p}{}^{2}\mathrm{sec}{h}^{2}({\mu}_{p}x/{r}_{0})\text{(52)}$$ In order to partially diagonalize the Hamiltonian Eq.(50) we introduce the following canonical transformation[14,23] $$\varphi (x)={\displaystyle \sum _{j}{A}_{j}}{C}_{j}(x),\text{}{\varphi}^{+}(x)={\displaystyle \sum _{j}{C}_{j}^{*}(x){A}_{j}^{+}}\text{(53)}$$ where $$\int {C}_{1}^{*}(x){C}_{j}(x)dx={\delta}_{lj},{\displaystyle \sum _{j}{C}_{j}^{*}({x}^{\prime}){C}_{j}(x)=\delta (x{x}^{\prime}),{\displaystyle \int dx{C}_{j}(x){}^{2}=1}}}\text{(54)$$ The operators A_{s}^{+} and A_{k}^{+} the creation operators for the bound states C_{s}(x) and delocalized state C_{k}(x), respectively. The detailed calculation of the partial diagonalization and of corresponding C_{s}(x) and C_{k}(x) are described in Appenix A. The partially diagonalized Hamiltonian obtained is as follows $$\begin{array}{c}\tilde{H}=W\text{'}+{E}_{s}{A}_{s}^{+}{A}_{s}+{\displaystyle \sum _{k}{E}_{k}{A}_{k}^{+}{A}_{k}+}{\displaystyle \sum _{q}\hslash ({\omega}_{q}qv){b}_{q}^{+}{b}_{q}}+\\ \text{}\frac{1}{\sqrt{N}}{\displaystyle \sum _{q}\hslash ({\omega}_{q}qv)({b}_{q}^{+}{\alpha}_{q}+{\alpha}_{q}^{\ast}{b}_{q}})(1{A}_{s}^{+}{A}_{s})+\frac{1}{\sqrt{N}}{\displaystyle \sum _{k{k}^{\prime}q}F(k,{k}^{\prime},q)({b}_{q}^{+}+{b}_{q})}{A}_{{k}^{\prime}}^{+}{A}_{k}\\ \text{}\frac{1}{\sqrt{N}}{\displaystyle \sum _{kq}\tilde{F}(k,q)({b}_{q}^{+}+{b}_{q})}({A}_{s}^{+}{A}_{k}{A}_{k}^{+}{A}_{s})\text{(55)}\end{array}$$ and $${C}_{s}(x)={(\frac{{\mu}_{p}}{2{r}_{0}})}^{1/2}\mathrm{sec}h({\mu}_{p}x/{r}_{0})\mathrm{exp}[i\hslash xv/2J{r}_{0}^{2}],\text{}$$ $$\text{with}{E}_{s}=2\left[{\epsilon}_{0}2J\frac{{\hslash}^{2}{v}^{2}}{2J{r}_{0}^{2}}{\mu}_{p}^{2}J\right]\text{(56a)}$$ $${C}_{k}(x)=\frac{{\mu}_{p}\mathrm{tanh}({\mu}_{p}x/{r}_{0})ik{r}_{0}}{\sqrt{N{r}_{0}}[{\mu}_{p}ik{r}_{0}]}\mathrm{exp}[ikx+\frac{i\hslash vx}{2J{r}_{0}^{2}}],\text{(56b)}$$ with $${E}_{k}=2\left[{\epsilon}_{0}2J\frac{{\hslash}^{2}{v}^{2}}{2J{r}_{0}^{2}}J{(k{r}_{0})}^{2}\right]$$ where $$F(k,{k}^{\prime},q)=2[{g}_{1}(q)+2{g}_{2}(q)]{\displaystyle {\int}_{0}^{L}dx{e}^{iqx}{C}_{{k}^{\prime}}^{\ast}}(x){C}_{k}(x)$$ $$\approx 2[{g}_{1}(q)+2{g}_{2}(q)]\left\{1\frac{i{\mu}_{p}q{r}_{0}}{[{\mu}_{p}+i(k+q){r}_{0}][{\mu}_{p}ik{r}_{0}]}\right\}\approx F[k,(k+q),q]{\delta}_{{k}^{\prime}k+q}\text{(57)}$$ $$\tilde{F}(k,q)=2[{g}_{1}(q)+2{g}_{2}(q)]{\displaystyle {\int}_{0}^{L}dx{e}^{iqx}{C}_{{k}^{\prime}}^{\ast}}(x){C}_{s}(x)$$ $$=\frac{2\pi}{{\sqrt{2\mu}}_{p}}[{g}_{1}(q)+2{g}_{2}(q)]\left\{\frac{iq{r}_{0}}{[{\mu}_{p}+ik{r}_{0}]}\right\}\mathrm{sec}h[\pi (kq){r}_{0}/2{\mu}_{p}]\text{(58)}$$ where ${\alpha}_{q}$ is determined by V(x) and the condition $({\omega}_{q}\text{vq)}{\alpha}_{q}=({\omega}_{q}+\text{qv)}{\alpha}_{q}^{\ast}$ which is required to get the factor, $(1{A}_{s}^{+}{A}_{s})$ in the $\tilde{H}$ in Eq.(55). Thus we find $${\alpha}_{q}=\frac{i\pi ({\chi}_{1}+{\chi}_{2})}{w{\mu}_{p}(1{s}^{2})}{\left[\frac{M}{2\hslash {\omega}_{q}}\right]}^{1/2}({\omega}_{q}+qv)\mathrm{csc}h(\pi q{r}_{0}/2{\mu}_{p})\text{(59)}$$ and $W\text{'}=\frac{2}{3}{\mu}_{p}^{2}J$ For this ${\alpha}_{q}$ the $\stackrel{~}{0}{>}_{\text{ph}}$ in Eq.(49) is just the coherent phonon state introduced by Davydov. However, the bound state C_{s}(x) in Eq.(56a), unlike the unbounded state C_{k}(x) in Eq.(56b), is selfconsistent with the deformation. Such a selfconsistent 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 $\psi >=\frac{1}{\sqrt{2!}}{({A}_{s}^{+})}^{2}0{>}_{ex}\tilde{0}{>}_{ph}$ he average energy of $\tilde{H}$ in Eq.(55) is $$<\psi \left\tilde{H}\right\psi >=2({\epsilon}_{0}2J\frac{{\hslash}^{2}{v}^{2}}{4J{r}_{0}^{2}})\frac{4}{3}J{\mu}_{p}^{2}\text{(60)}$$ Evidently, the average energy of $\tilde{H}$ in the soliton state $\psi >$ Eq.(60), is just equal to the above soliton energy E_{sol}, or the sum of the energy of the bound state in Eq.(56a), Es, and the deformation energy of the lattice,W’, i.e., $<\psi \left\stackrel{~}{H}\right\psi >={E}_{sol}={E}_{s}+W\u2019$ This is an interesting result, which shows clearly that the quasicoherent soliton formed by this mechanism is just a selftrapping state of the two excitons plus the corresponding deformation of the amino acid lattice. However, it should be noted that $\psi >$ is not an exact eigenstate of $\tilde{H}$ owing to the presence of the terms in $\tilde{H}$ with ${A}_{k}^{+}{A}_{s}$ and ${A}_{s}^{+}{A}_{k}$
For the discussion of the decay rate and lifetime of the new soliton state it is very convenlent to divide $\stackrel{~}{H}$ in Eq.(55) into H_{0}+V_{1}+V_{2} , where $${H}_{0}=W\text{'}+{E}_{s}{A}_{s}^{+}{A}_{s}+{\displaystyle \sum _{k}{E}_{k}{A}_{k}^{+}{A}_{K}}+{\displaystyle \sum _{q}\hslash ({\omega}_{q}vq){b}_{q}^{+}{b}_{q}+}$$ $$\frac{1}{\sqrt{N}}{\displaystyle \sum _{q}\hslash ({\omega}_{q}vq)({\alpha}_{q}{b}_{q}^{+}+{\alpha}_{q}^{*}{b}_{q})(1{A}_{s}^{+}{A}_{s})}\text{(61)}$$ $${V}_{1}=\frac{1}{\sqrt{N}}{\displaystyle \sum _{k{k}^{\prime}q}F(k,k+q,q)({b}_{q}^{+}+{b}_{q})}{A}_{{k}^{\prime}}^{+}{A}_{k}\text{(62)}$$ $${V}_{2}=\frac{1}{N}{\displaystyle \sum _{kq}\stackrel{~}{F}(k,q)({b}_{q}^{+}+{b}_{q})}({A}_{s}^{+}{A}_{k}{A}_{s}^{+}{A}_{k}),V={V}_{1}+{V}_{2}\text{(63)}$$ where H_{0} describes the relevant quasiparticle 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 excitonlike band with phonons relative to a uniform lattice. The bottom of the band of the latter is at the energy $4J{\mu}_{p}^{2}/3$ 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 $\mathrm{exp}\left\{{\displaystyle \sum _{n}\left[{\phi}_{n}\left(t\right){B}_{n}^{+}{\phi}_{n}^{\ast}\left(t\right){B}_{n}\right]}\right\}0{>}_{ex}$ Finally we find out the decay rate of the new soliton with twoquanta. In such a case H0 is chosen such the ground state, n> has energy $W\text{'}+n{E}_{s}^{\text{'}}$ in the subspace of excitation number equal to n, i.e., $<n{\displaystyle \sum _{i}{B}_{i}^{+}{B}_{i}n>=<nl({A}_{s}^{+}{A}_{s}+{\displaystyle \sum _{k}{A}_{k}^{+}{A}_{k}})ln>=n}$ In this subspace the eigenstates have the simple form nm,k1k2… km, {nq}> $$\frac{1}{\sqrt{(nm)!}}{({A}_{S}^{+})}^{nm}{A}_{{k}_{1}}^{+}{A}_{{k}_{2}}^{+}\cdots {A}_{{k}_{m}}^{+}0{>}_{ex}\underset{q}{\Pi}\frac{{({d}_{q}^{+})}^{{n}_{q}}}{\sqrt{{n}_{q}!}}1\tilde{0}{>}_{ph}^{nm}\text{(64)}$$ where ${d}_{q}={b}_{q}+\frac{m}{n}\frac{1}{\sqrt{N}}{\alpha}_{q}={a}_{q}\frac{nm}{n}\frac{1}{\sqrt{N}}{\alpha}_{q}\text{(m\u2264n,nandmareallintgers)(65)}$ with ${d}_{q}\tilde{0}{>}_{ph}^{nm}=0$ The corresponding energy of the systems is $${E}_{nm;{k}_{1}\mathrm{...}{k}_{{m}_{1}};\left\{{n}_{q}\right\}}^{(0)}=(1{(m/n)}^{2})W\text{'}+(nm){{E}^{\prime}}_{s}+{\displaystyle \sum _{j=1}^{m}{{E}^{\prime}}_{{k}_{1}}^{}+{\displaystyle \sum _{q}\hslash ({\omega}_{q}vq){n}_{q}}}\text{(66)}$$ ${{E}^{\prime}}_{s}$ is the energy of a bound state with one exciton, ${{E}^{\prime}}_{k}$ is the energy of the unbound(delocalized) state with one exciton. When m=0 the excitation state is a ntype soliton plus phonons relative to the chain with the deformation corresponding to the ntype 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 ntype soliton plus the new phonons: $$n>=\frac{1}{\sqrt{n!}}\underset{q}{\Pi}\frac{{({b}_{q}^{+})}^{{n}_{q}}}{{({n}_{q}!)}^{1/2}}{({A}_{s}^{+})}^{n}0{>}_{ex}\stackrel{~}{0}{>}_{ph}\text{(67)}$$ With corresponding energy Es{nq} =w+n ${{E}^{\prime}}_{s}+{\displaystyle \sum _{q}\hslash ({\omega}_{q}vq)}$ n_{q} to the final state with delocalized excitons and the original phonons: $$\alpha k>=\underset{q}{\Pi}\frac{{({a}_{q}^{+})}^{{n}_{q}}}{\sqrt{{n}_{q}^{}!}}0{>}_{ph}{({A}_{k}^{+})}^{n}0{>}_{ex}\text{(68)}$$ with corresponding energy E_{k}{n_{q}}=n ${{E}^{\prime}}_{k}^{}+{\displaystyle \sum _{q}\hslash ({\omega}_{q}vq)}$ n_{q} caused by the part, V_{2}, 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 $$\begin{array}{l}\overline{W}=\frac{1}{{\hslash}^{2}}{\displaystyle {\int}_{0}^{t}d{t}^{\prime}{\displaystyle {\int}_{0}^{t}d{t}^{\u2033}}\{{\displaystyle \sum _{\alpha {k}^{\prime}}{\displaystyle \sum _{l}{P}_{l}^{(ph)}<n\mathrm{exp}\left(\frac{i{H}_{0}{t}^{\u2033}}{\hslash}\right){V}_{2}\mathrm{exp}\left(\frac{i{H}_{0}{t}^{\u2033}}{\hslash}\right)}}\alpha {k}^{\prime}>}\cdot \text{(69)}\\ \alpha {k}^{\prime}exp\left(\frac{i{H}_{0}{t}^{\prime}}{\hslash}\right){V}_{2}\mathrm{exp}\left(\frac{i{H}_{0}{t}^{\prime}}{h}\right)n\}\end{array}$$ We should calculate the transition probability of the soliton resulting from the perturbed potential, (V_{1}+V_{2}), at firstorder 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 V_{1} term in the Hamiltonian is an interaction between the delocalized excitons and the phonons. The main effect of V_{1} is to modify the spectrum of the delocalized excitatons in the weak coupling limit (J_{μp}/ K_{B}T_{0} << 1 , the definition of T_{0} 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 V_{1}.
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 , which is taken to be the thermal equilibrium distribution for a given temperature T. Since ${P}_{l}^{ph}$ ,which is taken to be the thermal equilibrium distribution for a given temperature T. Since $${e}^{i{H}_{0}t}n,\{{n}_{q}\}>=exp\{i(W\text{'}+n{{E}^{\prime}}_{q})t/\hslash i{\displaystyle \sum _{q}({\omega}_{q}qv)}{b}_{q}^{+}{b}_{q}t\left\}\rightn,\left\{{n}_{q}\right\}>$$ and $${e}^{i{H}_{0}t}n1,\{{{n}^{\prime}}_{q}^{}\}>=exp\{i[(1\frac{1}{{n}^{2}})W\text{'}+(n1){{E}^{\prime}}_{s}+{{E}^{\prime}}_{k}]t/\hslash i{\displaystyle \sum _{q}({\omega}_{q}qv){d}_{q}^{+}}{d}_{q}^{}t\}n1,\left\{{{n}^{\prime}}_{q}\right\}>$$ $${\text{whered}}_{\text{q}}{\text{=b}}_{\text{q}}+\frac{1}{n}\frac{1}{\sqrt{N}}{\alpha}_{q},$$ using the explicit form for V_{2} and the fact that the sum over states ${k}^{\prime}\alpha ,\text{}\left\{{{n}^{\prime}}_{q}\right\}$ contains a complete set of phonons for each values of k′, one can rewrite $\overline{W}$ as $$\begin{array}{l}\text{}\overline{W}=\frac{1}{{\hslash}^{2}}\frac{{\pi}^{2}}{2n{\mu}_{1}{N}^{2}}{\displaystyle \sum _{k}{\displaystyle \sum _{{k}^{\prime}}{\displaystyle \sum _{{k}^{\u2033}}[{g}_{1}^{\ast}(k)+2{g}_{2}^{\ast}(k)]}}}[{g}_{1}({k}^{\u2033})+2{g}_{2}({k}^{\u2033})]\frac{(k{r}_{0})({k}^{\u2033}{r}_{0})}{{(n{\mu}_{1}^{})}^{2}+{({k}^{\prime}{r}_{0})}^{2}}Sech\left[\frac{\pi {r}_{0}}{2n{\mu}_{1}}(k{k}^{\prime})\right]\cdot \\ \mathrm{sec}h\left[\frac{\pi {r}_{0}}{2n{\mu}_{1}}({k}^{\u2033}{k}^{\prime})\right]{\displaystyle {\int}_{0}^{t}d{t}^{\prime}}{\displaystyle {\int}_{0}^{t}d{t}^{\u2033}\{\mathrm{exp}\left[\frac{i}{\hslash}\left(n({n}^{2}\frac{2}{3}n){\mu}_{1}^{2}J+nJ{({k}^{\prime}{r}_{0})}^{2}\right)({t}^{\prime}{t}^{\u2033})\right]\cdot}\text{(70)}\\ \text{}\mathrm{exp}[i{\displaystyle \sum _{q}({\omega}_{q}qv){b}_{q}^{+}{b}_{q}({t}^{\prime}{t}^{\u2033})]({b}_{k}^{+}+{b}_{k}})\mathrm{exp}[i{\displaystyle \sum _{q}({\omega}_{q}qv){a}_{q}^{+}{a}_{q}({t}^{\prime}{t}^{\u2033})({b}_{{k}^{\u2033}}^{+}+{b}_{{k}^{\u2033}}})\}\end{array}$$ where $${g}_{1}(k)+2{g}_{2}(k)=2{\chi}_{1}{(\frac{\hslash}{2M{\omega}_{k}})}^{1/2}[A(\mathrm{cos}({r}_{0}k)1)+i(A+1)\mathrm{sin}({r}_{0}k)]\approx 2i(A+1)({r}_{0}k){\chi}_{1}{(\frac{\hslash}{2M{\omega}_{k}})}^{1/2},$$ $${\mu}_{1}=\frac{{\chi}_{1}^{2}(1+{A}^{2})}{\omega (1{s}^{2})J},A={\chi}_{2}/{\chi}_{1}\text{(71)}$$ 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 longtime behavior of $\frac{d\overline{w}}{dt}$ . By straightforward calculation, the average transition probability or decay rate of the soliton is given by $$\begin{array}{c}{\Gamma}_{n}=\underset{t\to \infty}{\mathrm{lim}}\frac{d\overline{W}}{dt}=\frac{4}{\hslash}\left[\frac{{\pi}^{2}}{2n{\mu}_{1}{N}^{2}}\right]{\displaystyle \sum _{k{k}^{\prime}k\text{'}\text{'}}[[{g}_{1}^{*}(k)+2{g}_{{}_{2}}^{*}(k)]}[{g}_{1}({k}^{\u2033})+2{g}_{2}({k}^{\u2033})]\frac{(k{r}_{0})({k}^{\u2033}{r}_{0})}{{(n{\mu}_{1})}_{}^{2}+{({k}^{\prime}{r}_{0})}^{2}}\cdot \mathrm{sec}h\left[\frac{\pi {r}_{0}}{2n{\mu}_{1}}(k{k}^{\prime})\right]\cdot \\ \mathrm{sec}h\left[\frac{\pi {r}_{0}}{2n{\mu}_{1}}({k}^{\u2033}{k}^{\prime})\right]\mathrm{Re}\{{\displaystyle {\int}_{0}^{\infty}}dt\mathrm{exp}\left[\frac{i}{\hslash}\left(n({n}^{2}\frac{2}{3}n){\mu}_{1}^{2}J+nJ{({k}^{\prime}{r}_{0})}^{2}\right)t\right]\cdot \\ <<\mathrm{exp}[i{\displaystyle \sum _{q}({\omega}_{q}qv){b}_{q}^{+}{b}_{q}^{}t]({b}_{k}^{+}+{b}_{k}})\mathrm{exp}[i{\displaystyle \sum _{q}({\omega}_{q}qv){a}_{q}^{+}{a}_{q}t]({b}_{{k}^{\u2033}}^{+}+{b}_{{k}^{\u2033}}})>>\}\}\\ =\frac{4}{{\hslash}^{2}}\frac{{\pi}^{2}}{2n{\mu}_{1}{N}^{2}}{\displaystyle \sum _{k{k}^{\prime}k\text{'}\text{'}}\{[{g}_{1}^{*}(k)+2{g}_{{}_{2}}^{*}(k)]}[{g}_{1}({k}^{\u2033})+2{g}_{2}({k}^{\u2033})]\frac{(k{r}_{0})({k}^{\u2033}{r}_{0})}{{(n{\mu}_{1})}_{}^{2}+{({k}^{\prime}{r}_{0})}^{2}}\mathrm{sec}h\left[\frac{\pi {r}_{0}(k{k}^{\prime})}{2n{\mu}_{1}}\right]\cdot \end{array}$$ $$\mathrm{sec}h\left[\frac{\pi {r}_{0}}{2n{\mu}_{1}}({k}^{\u2033}{k}^{\prime})\right]\mathrm{Re}{\displaystyle {\int}_{0}^{\infty}dtU(k,{k}^{\u2033}t)\mathrm{exp}[\frac{i}{\hslash}\left(n({n}^{2}\frac{2}{3}n){\mu}_{1}^{2}J+nJ{({k}^{\prime}{r}_{0})}^{2}\right)t]}\}\text{(72)}$$ where the thermal average is $$U(k,{k}^{\u2033},t)=<<\mathrm{exp}[i{\displaystyle \sum _{q}({\omega}_{q}qv){b}_{q}^{+}{b}_{q}^{}t]({b}_{k}^{+}+{b}_{k}})\mathrm{exp}[i{\displaystyle \sum _{q}({\omega}_{q}qv){a}_{q}^{+}{a}_{q}t]({b}_{{k}^{\u2033}}^{+}+{b}_{{k}^{\u2033}}})>>$$ with $$<<A>>=Tr\{A\mathrm{exp}[\beta {\displaystyle \sum _{q}\hslash ({\omega}_{q}qv){b}_{q}^{+}{b}_{q}}]\}/Tr\{\mathrm{exp}[\beta {\displaystyle \sum _{q}\hslash ({\omega}_{q}qv){b}_{q}^{+}{b}_{q}}]\}$$ $$Tr\{A\mathrm{exp}[\beta {\displaystyle \sum _{q}\hslash ({\omega}_{q}qv){b}_{q}^{+}{b}_{q}}]\}/{Z}_{ph}\text{(73)}$$ $$\text{and}{Z}_{ph}=\prod _{q}{(1\mathrm{exp}[\beta \hslash ({\omega}_{q}qv)]\}}^{1},\text{(}\beta =\frac{\text{1}}{{\text{K}}_{\text{B}}T}\text{)}$$ 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 $U(k,{k}^{\u2033},t)$ is a critical step in the calculation of the decay rate Γn. It is well known that the trace contained in $U(k,{k}^{\u2033},t)$ can be approximately calculated by using the occupation number states of singleparticles 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 $U(k,{k}^{\u2033},t)$ as it is described in Appendix B. The decay rate obtained finally is $$\begin{array}{c}{\Gamma}_{n}=\underset{t\to \infty}{\mathrm{lim}}\frac{d\overline{W}}{dt}=\frac{2}{n{\mu}_{1}{\hslash}^{2}}\frac{{\pi}^{2}}{{N}^{2}}{\displaystyle \sum _{k{k}^{\prime}}[{g}_{1}(k)+2{g}_{2}(k){}^{2}\frac{{({r}_{0}k)}^{2}\mathrm{sec}{h}^{2}[\pi (k{k}^{\prime}){r}_{0}/2n{\mu}_{1}]}{{(n{\mu}_{1})}_{}^{2}+{({k}^{\prime}{r}_{0})}^{2}}}\mathrm{Re}{\displaystyle {\int}_{0}^{\infty}dt}\cdot \\ \left\{\mathrm{exp}[i(nJ{({k}^{\prime}{r}_{0})}^{2}+n({n}^{2}\frac{2}{3}n){\mu}_{1}^{2}Jt/\hslash +{R}_{n}(t)+{\xi}_{n}(t)]\frac{\mathrm{exp}[i({\omega}_{k}kv)t]}{\mathrm{exp}[\beta \hslash ({\omega}_{k}kv)]1}\right\}]\text{(74)}\end{array}$$ Where $${R}_{n}(t)=\frac{1}{{n}^{2}N}{\displaystyle \sum _{k}\left{\alpha}_{k}{}^{2}\right\{i\mathrm{exp}[i({\omega}_{k}kv)t]\}},\text{}{\xi}_{n}(t)=\frac{4}{{n}^{2}N}{\displaystyle \sum _{k}\frac{{\alpha}_{k}{}^{2}{\mathrm{sin}}^{2}[\frac{1}{2}({\omega}_{k}kv)t]}{\mathrm{exp}[\beta \hslash ({\omega}_{k}kv)]1}}\text{(75)}$$ 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 wave vector k in Eq.(75) is absorbed, the delocalized excitation produced does not need to have wave vector equal to k. The wave vector here is only approximately conserved by the sech^{2} $[\pi \left(kk\u2019\right){r}_{0}/2n{\mu}_{1}]$ 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, R_{n}(t) and ${\xi}_{n}(t)$ , occur because the phonons in the initial and final states are defined relative to different deformations[2426].
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 wave vectors, which fulfill wave vector conservation, make a major contribution to the transition matrix element, while the contributions of the phonons with small wave vector, which do not fulfill wave vector conservation, are very small, and can be neglected.
From Eqs.(74) and (75) we see that the Γn and R_{n}(t) and ${\xi}_{n}(t)$ and $\mu =n{\mu}_{1}$ mentioned above are all changed by increasing the number of quanta, n. Therefore, the approximation methods used to calculate Γn and related quantities (especially the integral contained in Γn) should be different for different n. We now calculate the explicit formula of the decay rate of the new soliton with twoquanta (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 R_{2} (t) and ${\xi}_{2}(t)$ contained in Eqs.(74)(75) by means of approximation. As a matter of fact, in Eq. 75) at n=2 the functions R_{2} (t) and ${\xi}_{2}(t)$ 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 q ω is approximated by $\sqrt{w/M}$ qr0, as it is shown in Appendix C. For $t\to \infty $ (because we are interested in the longtime steady behavior) the asymptotic forms of R_{2}(t) and ${\xi}_{2}(t)$ are $${R}_{2}(t)={R}_{0}[\mathrm{ln}(\frac{1}{2}{\omega}_{\alpha}t)+1.578+\frac{1}{2}i\pi ]\text{(76)}$$ $${\xi}_{2}(t)\approx \pi {R}_{0}{k}_{B}Tt/\hslash (wherecoth\frac{1}{2}{\omega}_{\alpha}t~1)\text{(77)}$$ $$\text{i}\text{.e}\text{.,}\underset{t\to \infty}{Lim}{\xi}_{2}(t)=\eta t,\eta =\pi {R}_{0}/\beta \hslash =\pi {R}_{0}{k}_{B}T/\hslash (78)$$ where ${R}_{0}=\frac{4{({\chi}_{1}+{\chi}_{2})}^{2}}{\pi \hslash w}{(M/w)}^{1/2}=\frac{2J{\mu}_{p}{r}_{0}}{\pi \hslash {v}_{0}},\text{}{\omega}_{\alpha}=\frac{2{\mu}_{p}}{\pi}{(\frac{w}{M})}^{1/2},{\text{T}}_{\text{0}}=\hslash {\omega}_{\alpha}/{K}_{B}\text{(79)}$ At R_{0}< 1 and T_{0}< T and R_{0}T/T_{0}< 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 $$\begin{array}{c}\frac{1}{\pi \hslash}\mathrm{Re}{\displaystyle {\int}_{0}^{\infty}dt}\mathrm{exp}\left\{i[2J{(k\text{'}{r}_{0})}^{2}+\frac{4}{3}J{\mu}_{p}^{2}\hslash {\omega}_{k}]t/\hslash +{R}_{2}(t)+{\xi}_{2}(t)\right\}\\ \approx \frac{1}{\pi \hslash}{(}^{2}\Gamma (1{R}_{0}){[}^{{\eta}^{2}}\left[1\frac{1}{2}{\left[\frac{\pi {R}_{0}}{2}+(1{R}_{0})\left(\frac{\delta (k,{k}^{\prime})}{\eta \hslash}\right)\right]}^{2}\right]\text{}(80)\end{array}$$ where $$\delta (k,{k}^{\prime})=2J{({k}^{\prime}{r}_{0})}^{2}+\frac{4}{3}{\mu}_{p}^{2}J\hslash {\omega}_{k},{\Phi}_{1}=\frac{{R}_{0}\pi}{2},{\Phi}_{2}=[(1{R}_{0}){\mathrm{tan}}^{1}\left(\frac{\delta (k,{k}^{\prime})}{\eta \hslash}\right)]\text{(81)}$$ The decay rate of the soliton, in such an approximation, can be represented, from Eqs.(74) and (80), by $$\begin{array}{l}{\Gamma}_{2}=\underset{t\to \infty}{\mathrm{lim}}\frac{d\overline{W}}{dt}=\frac{2}{{\mu}_{p}}{\left(\frac{\pi}{N}\right)}^{2}{\displaystyle \sum _{k{k}^{\prime}}[\frac{{(k{r}_{0})}^{2}{g}_{1}(k)+2{g}_{2}(k){}^{2}\mathrm{sec}{h}^{2}[(\pi {r}_{0}/2{\mu}_{p})(k{k}^{\prime})]}{[{\mu}_{p}^{2}+{({k}^{\prime}{r}_{0})}^{2}][\mathrm{exp}(\beta \hslash {\omega}_{k})1]}{(2.43{\omega}_{\alpha})}^{{R}_{0}}}\\ \left\{\frac{{\left({\eta}^{2}+\frac{1}{{\hslash}^{2}}{[\frac{4}{3}{\mu}_{p}^{2}J+2{({k}^{\prime}{r}_{0})}^{2}J\hslash {\omega}_{k}]}^{2}\right)}^{(1+{R}_{0})/2}}{{\hslash}^{2}{\eta}^{2}+{[\frac{4}{3}{\mu}_{p}^{2}J+2{({k}^{\prime}{r}_{0})}^{2}J\hslash {\omega}_{k}]}^{2}}\right\}\left\{1\frac{1}{2}{\left[\frac{{R}_{0}\pi}{2}+(1{R}_{0})\left[\frac{\frac{4}{3}{\mu}_{p}^{2}J+2{({k}^{\prime}{r}_{0})}^{2}J\hslash {\omega}_{k}}{\hslash \eta}\right]\right]}^{2}\right\}\text{(82)}\end{array}$$ This is the final analytical expression for the decay rate of the quasicoherent solition with twoquanta. 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] $$\begin{array}{l}{\Gamma}_{D}=\frac{1}{{\hslash}^{2}}\frac{{\chi}_{1}^{2}}{{\mu}_{D}}{\left(\frac{2\pi}{N}\right)}^{2}{\displaystyle \sum _{k{k}^{\prime}}\left(\frac{\hslash}{2M{\omega}_{k}}\right)}\frac{{(k{r}_{0})}^{2}si{n}^{2}(k{r}_{0})\mathrm{sec}{h}^{2}[(\pi {r}_{0}/2{\mu}_{D})(k{k}^{\prime})]}{[{\mu}_{D}^{2}+{({k}^{\prime}{r}_{0})}^{2}][\mathrm{exp}(\beta \hslash {\omega}_{k})1]}{\left(\frac{{\omega}_{\alpha}^{D}}{{\eta}_{D}}\right)}^{{R}_{0}^{D}}\cdot \text{(83)}\\ \text{}\frac{{\hslash}^{2}{\eta}_{D}}{{\hslash}^{2}{\eta}_{D}^{2}+[J{\mu}_{D}^{2}/3+J{({k}^{\text{'}}{r}_{0})}^{2}\hslash {\omega}_{k}]}\text{}\end{array}$$ where $${\eta}_{D}=\pi {R}_{0}^{D}{K}_{B}T/\hslash ,{R}_{0}^{D}=\frac{2{\chi}_{1}^{2}}{\pi \hslash w}{(\frac{M}{w})}^{1/2},{\omega}_{\alpha}^{D}=\frac{2{\mu}_{D}}{\pi}{(\frac{M}{w})}^{1/2}\text{(84)}$$ 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, $$\{1\frac{1}{2}{\left[\frac{{R}_{0}\pi}{2}+(1{R}_{0})\left[(\frac{4}{3}{\mu}_{p}^{2}J+2{({k}^{\text{'}}{r}_{0})}^{2}J\hslash {\omega}_{k})/\hslash \eta \right]\right]}^{2}\}$$ is added, while in Eq.(83) the factor, ${(\frac{{\omega}_{\alpha}}{{\eta}_{D}})}^{{R}_{0}^{D}}{\eta}_{D}$ replaces the term $${(2.43{\omega}_{d})}^{{R}_{0}}\cdot {({\eta}^{2}+\frac{1}{{\hslash}^{2}}{[\frac{4}{3}{\mu}_{p}^{2}J+2{({k}^{\text{'}}{r}_{0})}^{2}J\hslash {\omega}_{k}]}^{2})}^{(\frac{1+{R}_{0}}{2})})$$ in Eq.(82) due to the twoquanta nature of the new wave function 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 $\eta $ ,R0 and To at T=300K in both models, which are listed in Table 2. From this table we see that the $\eta $ , 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 nonlinear interaction coefficient Gp. Thus the approximations used in the Davydov model by Cottingham, et al. [14] cannot be applied in our calculation of lifetime of the new soliton, although we utilized the same quantumperturbation scheme. Hence we can audaciously suppose that the lifetimes of the quasicoherent soliton will greatly change.

R_{o} 
T_{o} (K) 
$$\eta (\times {10}^{13}/s)$$ 
New model 
0.529 
294 
6.527 
Davydov model 
0.16 
95 
2.096 
Attention is being paid to the relationship between the lifetime of the quasicoherent soliton and temperature. Fig.3 shows the relative lifetimes τ/τ_{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< v_{0}, the soliton will not travel the length of the chain unless τ/τ_{0} is large compared with L/r_{0}, where L=Nr0 is the typical length of the protein molecular chains. Hence for L/r0≈100,τ/τ_{0}>500 is a reasonable criterion for the soliton to be a possible mechanism of the bioenergy transport in protein molecules. The lifetime of the quasicoherent soliton shown in Figure.4 decreases rapidly as temperature increases, but below T=310K it is still large enough to fulfill the criterion.
Model 
nonlinear interaction G(10^{21}J) 
Amplitude 
Width 10^{10}m 
Binding energy 
Lifetime at 300K 
Critical temperature 
Number of amino acid traveled by soliton in lifetime 
Our model 
3.8 
1.72 
4.95 
7.8 
10^{9}10^{10} 
320 
Several handreds 
Davydov model 
1.18 
0.974 
14.88 
0.188 
10^{12}10^{13} 
<200 
Less than 10 
For comparison, log versus the temperature relationships was plotted simultaneously for the Davydov soliton and the new soliton with a quasicoherent twoquanta state in Figure 4. The temperaturedependence of log ( τ/τ0) of the Davydov soliton is obtained from Eq. (83). We find that the differences of values of ( τ/τ0) between the two models are very large. The value of ( τ/τ0) 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 [323].
For comparison, the corresponding result obtained using Eq. (83) is shown for the original Davydov model as a dashed line in Fig.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 (χ_{1}+χ_{2}), but the result for the Davydov soliton is not drawn in Figure5. These results show again that the quasicoherent soliton in Pang’s model is a likely candidate for the mechanism of bioenergy 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 excitonphonon interaction.
In order to understand the behavior of the quasicoherent soliton lifetime in very wide ranges, it is necessary to study τ/τ_{0} in the limit ${R}_{2}\left(t\right)\approx \u2014{R}_{0}\left[i{\pi}^{2}{\omega}_{\alpha}t/6+3\zeta \left(3\right){\left({\omega}_{\alpha}t\right)}^{2}\right]\to 0$ 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 R_{2}(t) and ξ_{2}(t). In fact, for ${\omega}_{a}t<1$ both R_{2}(t) and ξ_{2}(t) have powerseries expansions. To the lowest order as ${\omega}_{a}t\to 0$ , it can be found from Eq.(75) $${R}_{2}\left(t\right)\approx \u2014{R}_{0}\left[i{\pi}^{2}{\omega}_{\alpha}t/6+3\zeta \left(3\right){\left({\omega}_{\alpha}t\right)}^{2}\right]\text{(85)}$$ $${\xi}_{2}(t)\approx \frac{{R}_{0}{K}_{B}^{2}T{T}_{0}{\pi}^{2}}{3{\hslash}^{2}}{t}^{2}\text{,(86)}$$ $$\text{usingcoth(}\pi {\omega}_{\alpha}t)\approx [{(\pi {\omega}_{\alpha}t)}^{1}+\frac{\pi}{3}{\omega}_{\alpha}t]\text{(87)}$$ Thus $$\begin{array}{l}\frac{1}{\pi \hslash}\mathrm{Re}{\displaystyle {\int}_{0}^{\infty}}dt\mathrm{exp}\left\{i\left[2J{(k\text{'}{r}_{0})}^{2}+\frac{4J{\mu}_{p}^{2}}{3}\hslash {\omega}_{k}\right]\raisebox{1ex}{$t$}\!\left/ \!\raisebox{1ex}{$\hslash $}\right.+{R}_{2}(t)+{\xi}_{2}(t)\right\}\approx [4\pi (3\zeta (3){R}_{0}{K}_{B}^{2}{T}_{0}^{2}\\ \\ \text{}{+{R}_{0}{\pi}^{2}{K}_{B}^{2}T{T}_{0}/3)]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}\mathrm{exp}\left\{\frac{{[2J{(k\text{'}{r}_{0})}^{2}+\frac{4}{3}{\mu}_{p}^{2}J\hslash {\omega}_{k}+\hslash ({R}_{0}{\pi}^{2}{K}_{B}T)]}^{2}}{4[3\zeta (3){R}_{0}{K}_{B}^{2}{T}_{0}^{2}+{R}_{0}{\pi}^{2}{K}_{B}^{2}T{T}_{0}/3]}\right\}\text{(88)}\end{array}$$ when T/T_{0}>1 and π^{40T/2μT0>1. The above integral is a generalization of the usual δ  function for energy conservation in zerotemperature perturbation theory. Thus we can obtain from Eqs.(74) and (87) at n=2 the decay rate of the soliton as $$\begin{array}{l}{\Gamma}_{2}=\frac{2{\pi}^{3}}{{\mu}_{p}\hslash {N}^{2}{K}_{B}}{\left(\frac{\pi}{{R}_{0}{T}_{0}[3\zeta (3){T}_{0}+{\pi}^{2}T/3]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}{\displaystyle \sum _{k{k}^{\prime}}\frac{{(k{r}_{0})}^{2}{g}_{1}(k)+2{g}_{2}(k){}^{2}}{{\mu}_{p}^{2}+{({k}^{\prime}{r}_{0})}^{2}}\mathrm{sec}{h}^{2}\left[\left(\frac{\pi {r}_{0}}{2{\mu}_{p}}\right)(k{k}^{\prime})\right]}\\ \text{}{\left\{\mathrm{exp}\left[\frac{{\text{[}2J{\text{(}{k}^{\prime}{r}_{0}\text{)}}^{2}+\frac{4}{3}{\mu}_{p}^{2}J\hslash {\omega}_{k}+\frac{\text{1}}{\text{6}}{\text{R}}_{\text{0}}{\pi}^{\text{2}}{\text{K}}_{\text{B}}{\text{T}}_{\text{0}}\text{]}}^{\text{2}}}{4[3\zeta (3){R}_{0}{K}_{B}^{2}{T}_{0}^{2}+{R}_{0}{K}_{B}^{2}T{T}_{0}{\pi}^{2}/3]}\right]\left[\mathrm{exp}(\beta \hslash {\omega}_{k})1\right]\right\}}^{1}\text{}\end{array}$$ }
The twoquanta nature of the quasicoherent 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 energyGP=2G_{D} ${}_{}\left[1+2\left(\frac{{\chi}_{2}}{{\chi}_{1}}\right)+{\left(\frac{{\chi}_{2}}{{\chi}_{1}}\right)}^{2}\right]$ by μ_{p} produced the added interaction in the new model are ΔG =G_{P} (χ2≠0) (χ2=0)=1.08GD < GP(χ2=0)=2GD and Δμ=μ_{P}(χ2≠0)μ_{P}(χ2=0)=1.08μD< μ_{P}(χ2=0) =2μD,respectively, i.e., the roles of the added interaction on Gp and μ_{p} are smaller than that of the twoquanta nature. The two parameters GP and μ_{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 ${{E}^{\prime}}_{BP}={E}_{BD}{\left[1+\left(\frac{{\chi}_{2}}{{\chi}_{1}}\right)\right]}^{4}=2.6{E}_{BD}$ which is smaller than that of the twoquanta nature which is E =16E_{BD}. Putting them together in Eq.(35) we see that EBP≈ 41EBD. (3)From the (χ_{1}+χ_{2}) dependence of τ/τ_{0} in Fig.5, τ/τ_{0}≈100 has already been found directly at χ_{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 quasicoherent soliton [3035].
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 amideI vibration modes and a wave function with quasicoherent twoquanta state. In such a way, the quasicoherent soliton is a viable mechanism for the bioenergy 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.
In getting Eq. (A1) the variable x was assumed to be continuous and the chain length to tend to infinity L=Nr0→∞. Thus this wave vector k has a continuous value between ∞ and ∞. In subsequent calculation we mainly use a discrete description. The continuous description is transformed into a discrete one according to the rules $${\int}_{\infty}^{\infty}dx/{r}_{0}}\to {\displaystyle \sum _{n},}{\displaystyle {\int}_{\infty}^{\infty}dx\to \frac{2\pi}{N{r}_{0}}{\displaystyle \sum _{k},}}\delta (k{r}_{0}{k}^{\prime}{r}_{0})\to \frac{N}{2\pi}{\delta}_{k{k}^{\prime}},{C}_{s}(x)\to {C}_{s}(n),{C}_{k}(x)\to {\left(\frac{N}{2\pi}\right)}^{1/2}{C}_{k}(n)$$ Utilizing Eqs. (50)(51), (53) and (54), then the partially diagonalized Hamiltonian in the new representation is just Eq.(55).
${\xi}_{2}(t)$ can be easily elvaluated when v≈0 and R_{0}< 1 at sufficiently high temperature T>T_{0}(T_{0}= ℏ ωα/KB). In this case it is $${\xi}_{2}(t)=\frac{{R}_{0}}{{\omega}_{\alpha}}\left[\frac{T}{{T}_{0}}\right]{\displaystyle {\int}_{0}^{\infty}d{\omega}_{k}}\frac{{\mathrm{sin}}^{2}[\frac{1}{2}{\omega}_{k}t]}{s{h}^{2}({\omega}_{k}/{\omega}_{\alpha})}=\frac{{R}_{0}T}{{T}_{0}}[1\pi {\omega}_{\alpha}t\mathrm{coth}(\pi {\omega}_{\alpha}t)]\text{(C3)}$$ where we use the relation . $\mathrm{exp}(\beta \hslash {\omega}_{k})\approx 1+\beta \hslash {\omega}_{k}$ As t →∞ (because we are interested in the longtime steady behaviour) the leading terms in the above asymptotic formulae of R_{2}(t) and ξ_{2} (t) can be represented by $${R}_{2}(t)={R}_{0}[\mathrm{ln}(\frac{1}{2}{\omega}_{\alpha}t)+1.578+\frac{1}{2}i\pi ]\text{(C4)}$$ $${\xi}_{2}(t)\approx \pi {R}_{0}{k}_{B}Tt/\hslash \text{(C5)}$$ (where we approximated coth $\frac{1}{2}{\omega}_{\alpha}t~1$ ) , i.e., $$\underset{t\to \infty}{Lim}{\xi}_{2}(t)=\eta t,\eta =\pi {R}_{0}/\beta \hslash =\pi {R}_{0}{k}_{B}T/\hslash \text{(C6)}$$ Except at low temperature, the ${x}^{\prime}(={\omega}_{\alpha}t)$ dependent term in the real part of R_{2}(t) is small with respect to ${\xi}_{2}(T)$ for parameter values of interest and can be neglected. Furthermore, since R_{0}< 1 ( but it is not very small, about R_{0}≈0.529) and T_{0}< T (but it is not too small, about T_{0}≈294K) and R_{0} T/T_{0}< 1 for the protein molecules, then one can evaluate the integral in Eq.(72) by using the following approximation and utilizing the above results of Eqs.(C4C6). $$\begin{array}{c}\frac{1}{\pi \hslash}\mathrm{Re}{\displaystyle {\int}_{0}^{\infty}dt}\mathrm{exp}\left\{i[2J{(k\text{'}{r}_{0})}^{2}+\frac{4}{3}J{\mu}_{p}^{2}\hslash {\omega}_{k}]t/\hslash +R(t)+\xi (t)\right\}\\ \approx \frac{1}{\pi \hslash}{(}^{2}\Gamma (1{R}_{0}){[}^{{\eta}^{2}}\{\mathrm{cos}(\frac{\pi {R}_{0}}{2})\cdot \\ \mathrm{cos}\left[(1{R}_{0}){\mathrm{tan}}^{1}\left(\frac{\delta (k,{k}^{\prime})}{\eta \hslash}\right)\right]\mathrm{sin}(\frac{\pi {R}_{0}}{2})sin\left[(1{R}_{0}){\mathrm{tan}}^{1}\left(\frac{\delta (k,{k}^{\prime})}{\eta \hslash}\right)\right]\}\text{(C7)}\\ =\frac{1}{\pi \hslash}{(}^{2}\Gamma (1{R}_{0}){[}^{{\eta}^{2}}\mathrm{cos}({\Phi}_{1}+{\Phi}_{2})\\ \approx \frac{1}{\pi \hslash}{(}^{2}\Gamma (1{R}_{0}){[}^{{\eta}^{2}}\left[1\frac{1}{2}{\left[\frac{\pi {R}_{0}}{2}+(1{R}_{0})\left(\frac{\delta (k,{k}^{\prime})}{\eta \hslash}\right)\right]}^{2}\right]\end{array}$$ where $$\delta (k,{k}^{\prime})=2J{({k}^{\prime}{r}_{0})}^{2}+\frac{4}{3}{\mu}_{p}^{2}J\hslash {\omega}_{k},{\Phi}_{1}=\frac{{R}_{0}\pi}{2},{\Phi}_{2}=[(1{R}_{0}){\mathrm{tan}}^{1}\left(\frac{\delta (k,{k}^{\prime})}{\eta \hslash}\right)]\text{(C8)}$$
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