Properties of Soliton-Transported Bio-energy in α-helix Protein Molecules with Three Channels

Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 369 376 c International Academic Publishers Vol. 48, No. 2, August 15, 2007 Properties of Soliton-Transported Bio-energy in α-helix Protein Molecules with Three Channels PANG Xiao-Feng 1,2 and LIU Mei-Jie 1 1 Institute of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu 610054, China 2 International Center for Material Physics, the Chinese Academy of Sciences, Shenyang 110015, China (Received August 15, 2006; Revised January 16, 2007) Abstract We study numerically the propagating properties of soliton-transported bio-energy excited in the α-helix protein molecules with three channels in the cases of the short-time and long-time motions and its features of collision at temperature T = 0 and biological temperature T = 300 K by the dynamic equations in the improved Davydov theory and fourth-order Runge Kutta method, respectively. From these simulation experiments we see that the new solitons in the improved model can move without dispersion at a constant speed retaining its shape and energy in the cases of motion of both short-time or T = 0 and long time or T = 300 K and can go through each other without scattering in their collisions. In these cases its lifetime is, at least, 120 ps at 300 K, in which the soliton can travel over about 700 amino acid residues. This result is consistent with analytic result obtained by quantum perturbed theory in this model. In the meanwhile, the influences of structure disorder of α-helix protein molecules, including the inhomogeneous distribution of amino acids with different masses and fluctuations of spring constant, dipole-dipole interaction, exciton-phonon coupling constant and diagonal disorder, on the solitons are also studied by the fourth-order Runge Kutta method. The results show that the soliton still is very robust against the structure disorders and thermal perturbation of proteins at biological temperature 300 K. Therefore we can conclude that the new soliton in the α-helix protein molecules with three channels is a possible carrier of bio-energy transport and the improved model is possibly a candidate for the mechanism of this transport. PACS numbers: 87.15.He, 31.50.+w, 36.20.-r,65.60.+w Key words: bio-energy transport, soliton, protein molecule, Runge Kutta method, thermal stabilization 1 Introduction What is life or life activity? In the light of biophysicist s view, the so-called life or life activity is just a processes of mutual changes and coordination and unity of bio-material, bio-energy and bio- information in the living systems. Their synthetic movements and cooperative changes are just total life activity. Therefore we can say that the bio-material is its foundation, the bio-energy is its center, the bio-information is the key of life activity, but the transformation and transfer of bio-information are always accompanied by the transport of bio-energy in living systems. Thus, the bio-energy transport is an elementary and important process in life activity. The bio-energies needed are mainly provided by that released in adenosine triphosphate (ATP) hydrolysis in the living systems, are mainly used in these processes of muscle contraction, DNA reduplication and neuroelectric pulse transfer on the neurolemma and work of calcium pump and sodium pump, etc. Therefore, there is always a process of bio-energy transport from the producing place to required organisms in the living systems. However, understanding of mechanism of the bio-energy transport in the living systems is a long-standing problem which retains interesting up to now. As an alternative to electronic mechanisms, one can assume that the energy is stored as vibrational energy in the C = O stretching model (amide-i) of polypeptide chains in the protein molecules. Following Davydov s idea, [1] one can take into account the coupling between the amide-i vibration (intramolecular excitation or exciton) and deformation of amino acid residures (or, acoustic phonon) in the α-helix proteins. Through the coupling, the vibrational quantum is self-trapped as a soliton, which can moves over a macroscopic distances along the molecular chains keeping its shape and energy and momentum and other quasi-particle properties. This is just Davydov theory of bioenergy transport in α-helical protein molecules, which was proposed by Davydov in 1970s. [1] Obviously, the above idea about the soliton mechanism of the bio-energy transport in protein molecules is surely among the most interesting one, namely it yields a competing picture for the mechanism of bio-energy transport in the protein molecules and consequently has been the subject of a large body of work. [2 15] A lot of problems related to the Davydov model have been extensively studied [2 15]. However, a considerable controversy has been arisen in recent years over concerning whether the Davydov soliton is sufficiently stable in the region of biological temperature at 300 K to provide a viable explanation for bio-energy transport, and, whether her lifetime at The project supported by National Natural Science Foundation of China under Grant No. 19974034

370 PANG Xiao-Feng and LIU Mei-Jie Vol. 48 300 K is enough long for it to be biological useful. For the thermal equilibrium properties of the Davydov soliton the result obtained by quantum Monte Carlo simulation [12] show that the correlations characteristic of the Davydov solitonlike quasi-particles only occur at low temperature of T < 10 K for the widely accepted parameter values, which is consistent at a qualitative level with the Cottingham et al. s result. [13] The latter is straightforwardly quantum-mechanical perturbation calculation. The lifetime of the Davydov soliton obtained by using this methods is too small (about 10 12 10 13 s) to be useful in the biological processes. This shows clearly that the Davydov solution is not a true wave function of the systems. However, a thorough 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. [3,9 15] The results do not completely rule out the Davydov theory. Indeed, it is possible that using another wave function and a more sophisticated Hamiltonian we can find a soliton with good thermal stability and a suitably long lifetime. As a matter of fact, Takeno s [16] and Pang s [15] studies showed that considering different couplings between the relevant modes in the vibronic Hamiltonian can enhance the binding energy and stability of the soliton. On the other hand, some scientists thought that the soliton with multiquanta state (n 2), e.g., Brown et al. s coherent state, [5] Kerr et al. s [11] and Schweitzer et al. s multiquanta state, Cruzeiro Hansson s, [9] and Forner s [10] two-quanta state, would be thermally stable in the region of biological temperature, and could provide a realistic mechanism for the bio-energy transport in the protein molecules. However, the standard coherent state is unsuitable for the biological protein molecules because the number of particles in this state is innumerable, which violates the conservation of number of particles of the systems. The assumption of a multiquanta state (n > 2) is also at odds with the fact that the energy released in the ATP hydrolysis (about 0.43 ev) can only excite two quanta of amide-i vibration. The numerical study of a two-quanta model by Forner [10] reveals remarkable differences from one-quantum dynamics, i.e., the soliton with two-quanta is more stable than that with one-quantum, but Cruzeiro Hansson had thought that Forner s two-quanta state in the semiclassical case was not exact. Thus he reconstructed a so-called two-quanta state of the semiclassical Davydov system. However we proved [17] that the Hansson s ansatz contain exactly four quanta, instead of two quanta, which is impossible because the energy released in ATP hydrolysis can only excite two quanta. Therefore Hansson s improvement is still not successful and the exact wave function of the model remains unknown. As a result, we improved and extended [17] the Davydov model including its Hamiltonian and wave function, namely, a new coupling interaction between the acoustic and amide-i vibrational modes has been added in originally Davydov s Hamiltonian, the exciton state with onequantum in the Davydov s wave function is replaced by the quasi-coherent two-quantum state in our model, [17] respectively. Thus, the new Hamiltonain and wave function in the improved model are different from that of the Davydov model. [1 14] We get new results which is shown in Table 1. [17] From the results we see that the new soliton is different from the Davydov s soliton, it is thermally stable at biological temperature 300 K. Therefore it is a possible carrier of bio-energy transport in the protein molecules. Table 1 Conparison of properties of the solitons in our model and Davydov model. Features of solitons Critical Number of Nonlinear Binding Lifetime Thermal tempera- amino acid Model interaction Amplitude Width energy at 300 K (s) stability ture (K) traveled by G (10 21 J) (10 10 m) (10 21 J) at 300 K soliton in lifetime Our model 3.8 1.72 4.95 7.8 10 9 10 10 Stable 320 Several hundreds Davydov model 1.18 0.974 14.88 0.188 10 12 10 13 Unstable < 200 Less than 10 However, these analytic results are only related to the periodic protein molecules with one-channel, some approximate ways containing long-wave approximation, continuum approximation and long-time approximation, and so on, are used in the calculation. As a matter of fact, the biological protein molecules consist of 20 different amino acid residues with molecular weights between 75 m p (glycine) and 204 m p (tryptophane), which correspond to the variation of mass between 0.67 M < M < 1.8 M, here M = 114 m p is an average mass of amino acid residue, m p is proton mass. In the meanwhile, the proteins are all many channels and nonuniform, instead of periodic, they have certain structure disorders. These nonuniformities result necessarily in the fluctuations of spring constant, dipole-dipole interaction, exciton-phonon coupling constant and diagonal disorder in the proteins. Thus the states of new solitons will

No. 2 Properties of Soliton-Transported Bio-energy in α-helix Protein Molecules with Three Channels 371 be changed. In such a case, it is very necessary to study the properties of the new soliton in the bio-energy transport process in α-helix protein molecules with three channels with structure disorder. In this paper, we will numerically simulate the behaviors of the new soliton in the α-helix protein molecules by fourth-order Runge Kutta method. [18] 2 Numerical Simulation Method The Hamiltonian and the wave function of the α-helix protein molecules with three channels in the improved model are represented by [17] H = H ex + H ph + H inx = n [ε 0 B nαb n+1α J(B nαb n+1α + B nα B + n+1α )] + n [ P 2 nα 2M + 1 2 W (q nα q n 1α ) 2] + n [χ 1 (q n+1α q n 1α )B nαb nα + χ 2 (q n+1α q n 1α )(B n+1α B nα + B nαb n+1α )], (1) Φ(t) = Φ(t) β(t) = 1 λ [1 + nα a nα (t)b nα + 1 2! ( nα { a nα (t)b nα) 2 ] 0 ex exp i n } [q nα (t)p nα π nα (t)u nα ] 0 ph, (2) respectively, where B n and B n are annihilation and creation operators of the exciton at the n-th site associated with the amide-i oscillator having energy ε 0 = 0.205 ev, J is a nearest neighboring dipole-dipole interaction of amide-i vibrational quanta. H ex here describes Boson-type Frenkel excitons excited by the energy released in ATP hydrolysis in the protein molecular systems, H ph describes a harmonic lattice in terms of the coordinate operator u n and momentum operator P n of animo acid residue, H int represents the interaction between the two modes of motion, M is mass of the amino acid molecule, W is force constant of the protein molecular chains, χ 1 and χ 2 are two coupled constants related to the interactions between the intramolecular excitations and displacement of animo acid residues, they represent the modulations of the one-site energy and resonant (or dipole-dipole) interaction energy of excitons caused by the molecular displacement, respectively. subscript α = 1, 2, 3 denotes the number of three channels in the protein molecules, 0 ph and 0 ex are vacuum states of phonon and exciton, respectively, λ is a normalization constant. The state in Eq. (2) contains two excitons due to a(t) ˆN a(t) = a(t) ˆB ˆB n n a(t) = 2, where ˆN = ˆB ˆB n n is the n n number operator of exciton. Therefore the wave function in Eq. (2) is a quasi-coherent state with two quanta. The a n (t) and q n (t) = Φ(t) u n Φ(t), π n (t) = Φ(t) P n Φ(t) are three undetermined functions, which can determine from Eqs. (1) and (2). In fact, from time-dependent Schrödinger equation with the Hamiltonian (1) and time-dependent wave function (2) and the equations: we can find out H Φ = i Φ, (3) t i t Φ(t) u n Φ(t) = Φ(t) [u n, H] Φ(t), (4) i t Φ(t) P n Φ(t) = Φ(t) [P n, H] Φ(t), (5) i ȧ nα (t) = ε 0 a nα (t) J[a n+1α (t) + a n 1α (t)] + χ 1 [q n+1α (t) q n 1α (t)]a nα (t) + χ 2 [q n+1α (t) q n 1α (t)][a n+1α (t) + a n 1α (t)] + 5 { w(t) 1 } [q mα (t)π mα (t) π mα (t) q mα (t)] a nα (t) + L[a nα+1 (t) + a nα 1 (t)], (6) 2 2 m M q nα = W [q n+1α (t) 2q nα (t) + q n 1α (t)] + 2χ 1 [ a n+1α (t) 2 a n 1α (t) 2 ] + 2χ 2 { a nα (t)[a n+1α (t) a n 1α (t)] + a nα (t)[a n+1α(t) a n 1α(t)] }, (7) where L is the coefficient of the neighboring interaction among the three channels. Using transformation: ( iε0 t ) a na (t) a na (t) exp, (8)

372 PANG Xiao-Feng and LIU Mei-Jie Vol. 48 we can eliminate the ε 0 a nα (t) in Eq. (6), where a nα (t) in Eqs. (6) and (7) is a complex function of time t. For the numerically integration of Eqs. (6) and (7), the a nα (t) should make transformation Thus equations (6) and (7) are changed as a nα (t) = ar nα (t) + iai nα (t) with a nα 2 = ar nα 2 + ai nα 2. (9) ȧr nα = J(ai n+1α + ai n 1α ) + χ 1 (q n+1α q n 1α )ai nα + χ 2 (q n+1 q n 1α )(ai n+1α + ai n 1α ) + L[ai nα+1 (t) + ai nα 1 (t)], (10) ȧi nα = J(ar n+1α + ar n 1α ) + χ 1 (q n+1α q n 1α )ar nα q nα = y nα M, + χ 2 (q n+1α q n 1α )(ar n+1α + ar n 1α ) + L[ai nα+1 (t) + ai nα 1 (t)], (11) ẏ nα = W [q n+1α 2q nα + q n 1α ] + 2χ 1 [ar 2 n+1α + ai 2 n+1α ai 2 n 1α ai 2 n 1α] + 4χ 2 [ar nα (ar n+1α ar n 1α ) + ai nα (ai n+1α ai n 1α )], (13) where ar na and ai na are real and imaginary part of a nα (t). We now calculate numerically the solutions of Eqs. (10) (13). Thus the dynamic properties of an amino acid molecule are now described by the above four equations. [17] Then, the protein molecules consisting of N amino acid molecules should associatively solve 4N-equations. From these equations we can find out their solutions, ar na and ai na, by numerical simulation of fourth-order Runge Kutta method [18] and using the following initial condition at a point n 0 : a nα (t) = A sech [(n n 0 )(χ 1 +χ 2 ) 2 /4JW ], where A is normalization factor. Thus we can determine the solutions of Eqs. (7) and (8). However, in the simulation calculation the following boundary conditions must be satisfied: (i) The energy of the soliton must retain constant up to 0.0012%, i.e., the energy must be conservative at any position and time; (ii) In motion of soliton, the probability of the soliton must be normalized at any time, or speaking, numbers of particle for the system must be conservative; (iii) The energy of the soliton is real, its imaginary part must approach zero up to an accuracy of 0.001 fev. In accordance with the three criterions and utilizing the above equations and initial conditions we can calculate the evolution of time and space for the probability from Eqs. (10) (13) by Matlab language and data-parallel programming, where the time step size is chosen as 0.01 ps. 3 Influences of Structure Disorder of Proteins on Solitons When the above initial condition is imported from the end of the molecular chain, we obtain the numerical solutions of Eqs. (10) (13) by the fourth order Runge Kutta method [17] and the values of parameters for the α-helix protein molecules with three channels in the improved model. For the uniform protein molecules the result is shown in Fig. 1. In this calculation the average values of the physical parameters for the α-helix protein molecules with three channels are used. They are M = 5.73 10 25 kg, W = 39 N m 1, χ 1 is 6.2 10 11 N, χ 2 = (10 18) 10 12 N, and L = 1.5 mev, ε 0 = 0.2035 ev, and J = 9.68 10 4 ev, respectively. [1 17] In Fig. 1(a) we show the behaviors of motion of the solution, when the above initial conditions are simultaneously linked on the first ends of the three channels. From this figure we see that this solution can retain the clock shape to move over a long distance in the range of spacings of 400 amino acid residues and the time of 40 ps without dispersion along the molecular chains, i.e., this solution is a soliton. Therefore equations (6) and (7) have exactly soliton solution with a clock shape. This is the same with the analytic results obtained for the protein with single channel in continuum approximation in the improved model, in which the dynamical equation is a standardly nonlinear Schrödinger equation. [17] In Fig. 1(b) we plot the feature of motion of the solution, when above initial condition is only linked with the first ends of one channel, but not linked with other two channels. We see from this figure that the new two waves with small amplitudes are generated, except for one soliton occurred in the channel linked by the above initial condition. Obviously, the new two waves are still excited by the above initial condition through the interactions among the three channels. Although the two excitations are small, they can move over long distances along the two chains keeping their amplitudes. Therefore, they are still some solitons with a small amplitude and clock shape. In order to confirm further the soliton feature of the solutions of Eqs. (6) and (7), we study further the collision property of the solitons with a clock shape, set up from opposite ends of the channels for the α-helix protein molecules with three channels, and the result is shown in Fig. 2, when the above initial conditions are simultaneously linked on the opposite ends of the three channels. From this figure we see clearly that initial two solitons with clock shapes separating 100 amino acid spacings in each channel collide with each other at about 17 ps. After this collision, two (12)

No. 2 Properties of Soliton-Transported Bio-energy in α-helix Protein Molecules with Three Channels 373 solitons in each channel go through each other without scattering and retain still their shapes of clock to propagate toward and separately along the three chains. Thus we judge from the above results that the solution of Eqs. (6) and (7) in α-helix protein is an exact soliton. Fig. 1 Features of soliton solution of Eqs. (10) (13) for α-helix protein with three channels. Fig. 2 The collision features of these solitons. Fig. 3 Behaviour of long-time motion for the solitons. Fig. 4 The state of soliton under influences of structure disorders of 0.67 M < M k < 2 M, (χ 1 + χ 2) = ±4%(χ 1 + χ 2), J = ±2% J, W = ±8% W, ε 0 = ε β n. ε = 0.11 mev, β n 1. In the above simulation we study only the behavior of the soliton in the cases of short time of 40 ps, which exhibits clearly transport feature of this soliton. However, what does its behavior be in the cases of the longer-time and larger spacings? Thus we study further the behaviors of long-time for the solution of Eqs. (10) (13) in the α-helix proteins with three channels. In Fig. 3 we show the result of soliton solutions of Eqs. (10) (13) obtained at 120 ps times and 300 amino acid spacings, when the above initial conditions are simultaneously linked on the first ends of the three channels. We can see clearly from Fig. 3 that the soliton retains still its amplitude and shape of clock to move in such a case. This result shows that the lifetime of the soliton is, at least, 120 ps. What means the lifetime of 120 ps? We know that the characteristic unit of time for the model is τ 0 = r 0 /v 0 = (M/W ) 1/2 0.98 10 13 s, which is the time to move over one amino acid propagating at the sound speed, v 0, in the molecular chain. 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 is the typical length of the protein chain and τ is lifetime of the soliton. Hence for L/r 0 = 100, τ/τ 0 > 500 is a reasonable criterion for the soliton to be a possible mechanism for energy transfer in the proteins. Therefore the lifetime of the soliton, τ = 120 ps, corresponds to τ/τ 0 > 700 > 500. This means that the soliton in the improved model is a possible carrier of bio-energy transport in the proteins. This conclusion agrees also with analytic results in Table 1. [17] This shows that our analytic results and the improved model are correct. [17] However, the 20 different amino acid residues with molecular weights between 75m p (glycine) and 204m p (tryptophane), which correspond to the variation of mass between 0.67 M < M < 1.8 M, are nonuniformly distributed in the protein molecules. This will result in changes or fluctuations of spring constant, dipole-dipole interaction, exciton-phonon coupling constant and diagonal disorder for the proteins, Thus the states of new solitons will change in such a case. Then in the nonuniform protein molecules we should introduce the random

374 PANG Xiao-Feng and LIU Mei-Jie Vol. 48 number generators, α k and β n, to designate the random features of the mass sequences and ground state energy, at the same time, represent the fluctuations of spring constant, dipole-dipole interaction, exciton-phonon coupling constant and diagonal disorder by W = W W, J = J J, (χ 1 + χ 2 ) = (χ 1 + χ 2 ) (χ 1 + χ 2 ), ε 0 = ε ε 0 = ε β n in the nonuniform proteins, [10,17] respectively. When the disorder of mass sequence is in the region of 0.67 M < M k < 2 M, or 0.67 < α k < 2, where M k = α k M, and fluctuations of (χ1 + χ 2 ), J, W, and ground state energy ε 0 are about (χ 1 + χ 2 ) = ±4%(χ 1 + χ 2 ), J = ±2% J, W = ±8% W, ε 0 = ε β n, ε = 0.11 mev, β n 1, respectively, the states of the new soliton obtained by the above equations and fourth-order Runge Kutta method [18] at T = 300 K are shown in Fig. 4. From these figures we see clearly that the new soliton is still stable at 300 K, when the structure nonuniformity occurs in the proteins. Therefore we can conclude that the new soliton is robust against the thermal perturbation and structure nonuniformity of protein molecules. Thus the new soliton in the improved model is a real carrier of the bio-energy transport in the protein molecules. 4 The States of Solitons in Proteins at Biological Temperature However, the α-helix protein molecules in the living systems work always at biological temperature of 300 K, therefore, we must study the transported behaviour of the soliton at 300 K and should add the effect of temperature on the soliton into the above equations. How do we consider this effect? As a matter of fact, this effect was studied in many models in the protein molecules. [1,3,4,9 15] We here adopt Lomdahl and Kerr s method [11] in the calculation because the Lomdahl and Kerr s numerical result exhibits just the thermal instability of the Davydov soliton. In the Lamdahl and Kerr s method [11] the decay term MΓ q n and random noise term, F n (t), resulting from the temperature, were added in displacement equation of the amino acid molecules Eq. (7). Thus The latter can now be represented by M q n (t) = W [q nt1 (t) 2q n (t) + q n 1 (t)] + 2χ 1 [ a n+1 2 a n 1 2 ] + 2χ 2 {a n(t)[a n+1 (t) a n 1 (t)] + a n (t)[a n+1(t) a n 1(t)] MΓ q n + F n (t), (14) where Γ is dissipation coefficient of vibration of amino acids. The correlation function of the random noise force is determined by F (x, t)f (0, 0) = 2MK B Jδ(x)δ(t) 1 r 0, where r 0 is a lattice constant. Assuming again that random noise force obeys the normal distribution with criterion deviation σ and zero expected value. Thus this distribution can be represented by N(F n ) = 1 2πσ exp ( F 2 n 2σ r=1 σ = 2MK BT Γ, τ where τ is a time constant, and the quantity Γ is an inverse number of the time constant of the heat bath, and vice versa. In practical calculation, the random noise force F n (t) is computed by the random number, which can be presented by F n (t) = σ L [X nr (t) 1/2]. We have assumed L = 12, the random number X nr (t) is in the region of (0 X nr 1) at each time step. Therefore the error where deviation difference of [X n (t) 1/2] is about 1/12, then criterion deviation of F n (t) is σ, its expected value is zero. Then, the size of random noise force is in the range of F n (t) 6 σ. Hence, F n (t) is Gaussian distribution at L. Thus we now can find out the soliton solution ), of the equations of motion, equations (6) and (14) with decay effect and random noise force by the above method and fourth-order Runge Kutta method. [18] This result at 300 K is shown in Fig. 5 for the α-helix protein molecules with three channels, when the above initial conditions are simultaneously linked on the first ends of the three channels. From this figure we see that the new soliton in the improved model can move along the three channels at the constant speed and amplitude without dispersion in such a case. So, the soliton is still thermally stable at the biological temperature 300 K. In Fig. 6 we report also the result of motion of the soliton in the case of long time of 120 ps and large spacings of 1000 sites at 300 K for the α-helix protein molecules, when the above initial conditions are simultaneously linked on the first ends of the three channels. We see from this figure that the solitons are undisturbed in such a case, and move really over a long time and large spacing along the protein molecular chains to retain its amplitude and velocity at the bio-temperatures. In Fig. 7 we plot the collision behaviors of the solitons with clock shape, set up from opposite ends of the channels in the α-helix protein molecules, when the above initial conditions are simultaneously linked on the opposite ends of the three channels. From this figure we see clearly that initial two solitons with clock shapes separating 100 amino acid spacings in each channel collide with each other at about 16 ps. After the collision, two solitons in each channel

No. 2 Properties of Soliton-Transported Bio-energy in α-helix Protein Molecules with Three Channels 375 go through each other to retain still their shapes of clock and to propagate toward and separately along the three chains. These results show clearly that although there is the large lattice fluctuations in the protein molecules due to the influence of temperature, the nonlinear coupling interaction between the amino acids and excitons is still able to stabilize the soliton, therefore this soliton is very robust against the thermal perturbation of environment. In this case the lifetime of the new soliton is also, at least, 120 ps. This means that the new soliton could play an important role in the biological processes. the new soliton in the improved model is a carrier of the bio-energy transport in the protein molecules. Fig. 7 300 K. The properties of collision of the solitons at Fig. 5 The behaviors of the new soliton at biological temperature 300 K. Fig. 6 300 K. The state of the soliton in long-time motion at However, the structure nonumiformity of the protein molecules are not considered in the above calculation. Thus we should study further the influences of structure nonuniformity of proteins on the new soliton at the biological temperatures of 300 K by the fourth-order Runge Kutta method [18] and Eqs. (6) and (14). In Fig. 8, we plot the states of the new soliton in the nonuniform α- helix proteins at T = 300 K, when the disorder of the mass sequence is in the region of 0.67 M < M k < 2 M, or 0.67 < α k < 2, and fluctuations of (χ 1 + χ 2 ), J, W, and ε 0 are about (χ 1 + χ 2 ) = ±4%(χ 1 + χ 2 ), J = ±2% J, W = ±4% W, ε 0 = ε β n, ε = 0.5 mev, β n 0.5. From these figures we see clearly that the new soliton is undisturbed and still thermally stable at 300 K, when the structure nonuniformity occurs in the proteins. Therefore we can conclude that the new soliton is robust against the thermal perturbation and structure nonuniformity of the protein molecules at the biological temperatures. Thus Fig. 8 The state of new soliton under influences of structure disorders at 0.67 M < M k < 2 M, (χ 1 + χ 2) = ±4%(χ 1 + χ 2), J = ±2% J, W = ±4% W, ε 0 = ε β n, ε = 0.5 mev, β n 0.5, in the nonuniform α- helix proteins at T = 300 K. 5 Conclusions In one word, we study numerically the properties of soliton solutions of equations of motion in the cases of short-time and long-time motion and its features of collision in the α-helix protein molecules with three channels at the biological temperature 300 K in the improved model by the fourth-order Runge Kutta method. We see clearly from these results that this soliton in the improved model is very stable whether in the cases of long- and short-time motions and mutual collision at 300 K, it can move along the protein molecular chains without dispersion at a constant speed retaining its shape and energy in the cases of motion of both short-time and T = 0 and long time and T = 300 K and can go through each other without scattering in the collision. In these case its lifetime is, at least, 120 ps at 300 K, in which the soliton can travel over about 700 amino acid residues. This result is consistent with analytic result obtained by quantum perturbed theory in this model. In the meanwhile, the influences of structure disorder of protein molecules, including the inhomogeneous distribution of amino acids with different masses and fluctuations of spring constant, dipole-dipole interaction, exciton-phonon coupling constant and diagonal disorder, on the solitons are also studied. The results show that the soliton is easily undisturbed and very robust against the structure disorders and thermal perturbation. Therefore the new soliton in the improved model is a possible carrier of bio-energy transport and the model is possibly a candidate for the mechanism of this transport.

376 PANG Xiao-Feng and LIU Mei-Jie Vol. 48 References [1] A.S. Davydov, J. Theor. Biol. 38 (1973) 559; Phys. Scr. 20 (1979) 387; Physica D 3 (1981) 1; Biology and Quantum Mechanics, Pergaman, New York (1982); Solitons in Molecular Systems, 2nd ed., Reidel, Dordrecht (1991). [2] M.J. Skrinjar, D.W. Kapor, and S.D. Stojanovic, Phys. Rev. A 38 (1988) 6402; Phys. Rev. A 40 (1989) 198; Phys. Lett. A 38 (1988) 489; Phys. Scr. 39 (1988) 658. [3] P.L. Christiansen and A.C. Scott, Davydov s Soliton Revisited, Plenum Press, New York (1990). [4] A.C. Scott. Phys. Rev. A 26 (1982) 578; Phys. Rev. A 27 (1983) 2767; Phys. Scr. 25 (1982) 651; Phys. Rep. 217 (1992) 1; Physica D 51 (1990) 333. [5] D.W. Brown, Phys. Rev. A 37 (1988) 5010; D.W. Brown and Z. Ivic, Phys. Rev. B 40 (1989) 9876; Z. Ivic and D.W. Brown, Phys. Rev. Lett. 63 (1989) 426; D.W. Brown, B.J. West, and K. Lindenberg, Phys. Rev. A 33 (1986) 4104 and 4110; D.W. Brown, K. Lindenberg, and B.J. West, Phys. Rev. B 35 (1987) 6169; Phys. Rev. B 37 (1988) 2946; Phys. Rev. Lett. 57 (1986) 234. [6] Pang Xiao-Feng, Chin. J. Biochem. Biophys. 18 (1986) 1; Chin. J. Atom. Mol. Phys. 6 (1986) 1986; Chin. Appl. Math. 10 (1986) 276. [7] L. Cruzeiro, J. Halding, P.L. Christiansen, et al., Phys. Rev. A 37 (1985) 703. [8] B. Mechtly and P.B. Shaw, Phys. Rev. B 33 (1988) 3075. [9] L. Cruzeio-Hansson, Phys. Rev. A 45 (1992) 4111; Phys. Rev. Lett. 73 (1994) 2927; Physica D 68 (1993) 65; L. Cruzeiro-Hansson, V.M. Kenker, and A.C. Scott, Phys. Lett. A 190 (1994) 59. [10] W.Förner, Phys. Rev. A 44 (1991) 2694; J. Phys. Condensed Matter 3 (1991) 1915, 3235; J. Phys. Condensed Matter 4 (1992) 4333; J. Phys. Condensed Matter 5 (1993) 823, 883, 3883, 3897; H. Motschman, W. Forner, and J. Ladik, J. Phys. Condensed Matter 1 (1989) 5083; Physica D 68 (1993) 68; J. Comput. Chem. 13 (1992) 275; W. Forner and L.J. Ladik, Reference [3] p. 267. [11] P.S. Lomdahl and W.C. Kerr, Phys. Rev. Lett. 55 (1985) 1235; Referece [3] p. 259; W.C. Kerr and P.S. Lomdahl, Phys. Rev. B 35 (1989) 3629; W.C. Kerr and P.S. Lomdahl, Phys. Rev. B 35 (1989) 3629; Reference [3] p. 23. [12] X. Wang, D.W. Brown, and K. Lindenberg, Phys. Rev. A 37 (1998) 3357; Phys. Rev. Lett. 62 (1989) 1796. [13] J.P. Cottingham and J.W. Schweitzer, Phys. Rev. Lett. 62 (1989) 1792; J.W. Schweitzer, Phys. Rev. A 45 (1992) 8914; Refenrence [3] p. 285. [14] H. Bolterauer and M. Opper, Z. Phys. B 82 (1991) 95; Bechtly and P.B. Shaw, Phys. Rev. B 35 (1989) 3629. [15] Pang Xiao-Feng, J. Phys. Condensed Matter 2 (1990) 9541; Phys. Rev. E 49 (1994) 4747; European Phys. J. B 10 (1999) 415; Physica D 154 (2001) 138; Chin. Phys. Lett. 10 (1993) 381, 437, 573; Chin. Phys. 9 (2000) 108; Commun. Theor. Phys. (Beijing, China) 35 (2001) 763; The Theory of Nonlinear Quantum Mechanics, Chongqing Press, Chongqing (1994); Chin. Sci. Bulletin 38 (1993) 1572, 1665; Acta Phys. Slovaca 48 (1998) 99; Chin. J. Infrared Mill. Waves 12 (1993) 377, 16 (1997) 64, 232; Acta Phys. Sinica 42 (1993) 1841; Acta Phys. Sinica 46 (1997) 625; Chin. J. Atom. Mol. Phys. 5 (1987) 383; Chin. J. Atom. Mol. Phys. 12 (1995) 411; Chin. J. Atom. Mol. Phys. 13 (1996) 70; Chin. J. Atom. Mol. Phys. 14 (1997) 232; J. Int. Inf. Mill. Waves 22 (2001) 277; 23 (2002) 343; X.F. Pang and H.J.W. Muller- Kersten, J. Phys. Condensed Matter 12 (2000) 885. [16] S. Takeno, Prog. Theor. Phys. 71 (1984) 395; 73 (1985) 853; J. Phys. Soc. JPN 59 (1991) 3127. [17] X.F. Pang, Phys. Rev. E 62 (2000) 6989; European Phys. J. B 19 (2001) 297; J. Phys. Condensed Matter 18 (2006) 613; Phys. Lett. A 335 (2005) 408; Commun. Theor. Phys. (Beijing, China) 35 (2001) 323; Commun. Theor. Phys. (Beijing, China) 37 (2002) 187; Commun. Theor. Phys. (Beijing, China) 41 (2004) 470; Commun. Theor. Phys. (Beijing, China) 43 (2005) 367; J. Int. Inf. Mill. Waves 22 (2001) 291; J. Phys. Chem. Solids 62 (2001) 793; Chin. J. Biomed. Engineering 8 (1999) 39; Chin. J. Biomed. Engineering 10 (2001) 613; Phys. Stat. Sol (b) 226 (2002) 317; Int. J. Mod. Phys. B 19 (2005) 4677; Int. J. Mod. Phys. B 20 (2006) 3027. [18] J. Stiefel, Einfuhrung in die Numerische Mathematik, Teubner Verlag, Stuttgart (1965); K.E. Atkinson, An Introdution to Numerical Analysis, Wiley, New York (1987).