The properties of nonlinear excitations and verification of validity of theory of energy transport in the protein molecules

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1 Ope Access Aals of Proteomics ad Bioiformatics Review Article The properties of oliear excitatios ad verificatio of validity of theory of eergy trasport i the protei molecules Pag Xiao-Feg* Istitute of Life Sciece ad Techology, Uiversity of Electroic Sciece ad Techology of Chia. Chegdu , Chia *Address for Correspodece: Pag Xiao- Feg, Istitute of Life Sciece ad Techology, Uiversity of Electroic Sciece ad Techology of Chia. Chegdu , Chia. pagxf006@yahoo.com.c Submitted: 7 March 018 Approved: 06 April 018 Published: 09 April 018 Copyright: 018 Xiao-Feg P. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Keywords: Noliear excitatio; Protei; Eergy trasport; Solito; Stability verificatio; Validity; Numerical simulatio; Bidig eergy; Dyamic equatio Abstract Based o differet properties of structure of helical protei molecules some theories of bio-eergy trasport alog the molecular chais have bee proposed ad established, where the eergy is released by hydrolysis of adeosie triphosphate (ATP). A brief survey of past researches o differet models ad theories of bio-eergy, icludig Davydov s, Brow et al s, Schweitzer s, Cruzeiro-Hasso s, Forer s ad Pag s models were first stated i this paper. Subsequetly we studied ad reviewed maily ad systematically the properties ad stability of the carriers (solitos) trasportig the bio-eergy at physiological temperature 300K i Pag s ad Davydov s theories. However, these theoretical models icludig Davydov s ad Pag s model were all established based o a periodic ad uiform proteis, which are differet from practically biological proteis molecules. Therefore, it is very ecessary to ispect ad verify the validity of the theory of bio-eergy trasport i really biological protei molecules. These problems were extesively studied by a lot of researchers ad usig differet methods i past thirty years, a cosiderable umber of research results were obtaied. I here reviewed the situatios ad progresses of study o this problem, i which we reviewed the correctess of the theory of bio-eergy trasport icludig Davydov s ad Pag s model ad its ivestigated progresses uder iflueces of structure ouiformity ad disorder, side groups ad imported impurities of protei chais as well as the thermal perturbatio ad dampig of medium arisig from the biological temperature of the systems. The structure ouiformity arises from the disorder distributio of sequece of masses of amio acid residues ad side groups ad imported impurities, which results i the chages ad fluctuatios of the sprig costat, dipole-dipole iteractio, excito-phoo couplig costat, diagoal disorder or groud state eergy ad chai-chai iteractio amog the molecular chaels i the dyamic equatios i differet models. The iflueces of structure ouiformity, side groups ad imported impurities as well as the thermal perturbatio ad dampig of medium o the bio-eergy trasport i the proteis with sigle chai ad three chais were studied by differetly umerical simulatio techique ad methods cotaiig the average Hamiltoia way of thermal perturbatio, fourth-order Ruge-Kutta method, Mote Carlo method, quatum perturbed way ad thermodyamic ad statistical method, ad so o. I this review the umerical simulatio results of bio-eergy trasport i uiform protei molecules, the ifluece of structure ouiformity o the bio-eergy trasport, the effects of temperature of systems o the bio-eergy trasport ad the simultaeous effects of structure ouiformity, dampig ad thermal perturbatio of proteis o the bio-eergy trasport i a sigle chais ad helical molecules were icluded ad studied, respectively. The results obtaied from these studies ad reviews represet that Davydov s solito is really ustable, but Pag s solito is stable at physiologic temperature 300K ad uderiflueces of structure ouiformity or disorder, side groups, imported impurities ad dampig of medium, which is cosistet with aalytic results. Thus we ca still coclude that the solito i Pag s model is exactly a carrier of the bio-eergy trasport, Pag s theory is appropriate to helical protei molecules. Itroductio As it is well kow, the so-called life is just processes of mutual chages ad coordiatio of the bio-material, bio-eergy ad bio-iformatio, their sythetic movemets ad cooperative chages are total life activity i the live systems i the light of biophysicist s view, where the bio-material is the foudatio if life, the bio-eergy is its ceter, the bio-iformatio is the key of life activity, but the trasformatio ad How to cite this article: Xiao-Feg P. The properties of oliear excitatios ad verificatio of validity of theory of eergy trasport i the protei molecules. A Proteom Bioiform. 018; :

2 trasfer of bio-iformatio are always accompaied by the trasport of bio-eergy i livig systems [1]. Thus, the bio-eergy ad its trasport are a fudametal ad importat process i life activity. As a matter of fact, may biological processes, such as muscle cotractio, DNA reduplicatio, euroelectric pulse trasfer o the eurolemma ad work of calcium pump ad sodium pump, ad so o, are associated with bio-eergy trasport i the life bodies, where the eergy is released by the hydrolysis of adeosie triphosphate (ATP). Namely, a ATP molecule reacts with water, which results i the eergy release of 0.43eV uder ormal physiological coditios. The reactio ca be represeted by 4 3 ATP H O ADP HPO H ev where ADP is the adeosie diphosphate. Just so, there are always a biological process of eergy trasport from productio place to absorptio place i the livig systems. Therefore, ivestigatio of bio-eergy trasport alog protei molecules ad determiatio its rules have a importat sigiicace i life sciece. I geeral, the bio-eergy is trasported alog the protei molecules. However, uderstadig the mechaism of the trasport i livig systems has bee a log-stadig problem that remais of great iterest up to ow. Noliear excitatio ad theory of eergy trasport i protei molecules Davydov theory of solito excitatio ad its improvemet Geerally speakig, the eergy ca be coverted to a particular vibratioal excitatio withi a protei molecule. A likely recipiet exchage is the amide-i vibratio. Their vibratio is primarily a stretch ad cotractio of the C= O bod of the peptide groups. The amide-i vibratio is also a promiet feature i ifrared ad Rama spectra of protei molecules. Experimetal measuremet shows that oe of the fudametal frequecies of the amide-i vibratio is about 0.05eV. This eergy is about half the eergy released durig the ATP hydrolysis. Moreover, it remais early costat from protei to protei, idicatig that it is rather weakly coupled to other degrees of freedom. All these factors ca lead to the assumptio that the eergy released by ATP hydrolysis might stay localized ad stored i the amide-i vibratio excitatio. As a alterative to electroic mechaisms [-4], oe ca assume that the eergy is stored as vibratioal eergy of the C=0 stretchig mode (amide-i) i a protei polypeptide chai. Followig Davydov s idea [5-1],oes take ito accout the couplig betwee the amide-i vibrarioal quatum (excito ) ad the acoustic phoo (molecular displacemets) i the amio acid residues; Through the couplig, oliear iteractio appears i the motio of the vibrartioal quata, which could lead to a self-trapped state of the vibratioal quatum. The latter plus the deformatioal amio acid lattice together ca travel over macroscopic distaces alog the molecular chais, retaiig the wave shape, eergy, mometum ad other properties of the quasiparticle. I igure 1, structure of α helical protei this way, the bio-eergy ca be trasported as a localized wave packet or solito. This is just the Davydov s model of bio-eergy trasport i proteis, which was proposed i the 1970s [5-7]. Davydov model of bio-eergy trasport works at helical proteis as show i igure 1. Followig Davydov idea [5-14, the Hamiltoia describig such system is of the form H D P 1 0B B J(B B1 B B1 ) w(u u 1 ) M (u u )B B H H H (1) ex ph it Published: April 09,

3 Figure 1: I Davydov s model, where 0 =0.05ev is the amide-i quatum eergy, -J is the dipole-dipole iteractio eergy betwee eighbourig amides, B (B ) is the creatio (aihilatio) operator for a amide-i quatum (excito) i the site, u is the displacemet operator of lattice oscillator at site, P is its cojugate mometum operator, M is the mass of a amio acid residue, w is the elasticity costat of the protei molecular chais, ad χ 1 is a oliear couplig parameter ad represets the couplig size of the excito- phoo iteractio i the protei molecules. The wave fuctio of states of the systems i Davydov model is of the form [5-14]: D (t) = i D (t) 1(t ) = () tb exp [ () tp () tu]10. () or * D1 () t () tb exp q() taq q() ta 10 q (3) where I0>=I0> ex Io> ph, I0> ex ad I0> ph are the groud states of the excito ad phoo, respectively, a q (a q ) is aihilatio (creatio) operator of the phoo with ware vector q, ( t ) ad () t D u D ad () t D P D ad q ( t) D1(t) a q D1(t) are some udetermied fuctios of time. The Davydov solito obtaied from Eqs.(1)-() i the semiclassical limit ad usig the cotiuum approximatio has the from of 1/ D D v D( x, t) sech (x x0 vt ) expi (x x0) E t / ) r0 Jr 0 (4) which correspods to a excitatio localized over a scale r0/ D, where /(1 ), 1/ D 1 s wj GD 4 JD, s v / v0, v 0 r0(w/m) is the soud speed i the protei molecular chais, v is the velocity of the solito, r 0 is the lattice costat. Evidetly, the solito cotais oly oe excito because N=< () t Nˆ () t 1. This shows that the Davydov solito is formed through self-trappig of oe excito, its D D Published: April 09,

4 4 bidig eergy is E /3 BD 1 Jw. Davydov s idea yields a compellig picture for the mechaism of bioeergy trasport i protei molecules ad cosequetly has bee the subject of a large umber of works [15-103]. A lot of issues related to the Davydov model, icludig the foudatio ad accuracy of the theory, the quatum ad classical properties ad the thermal stability ad lifetimes of the Davydov solito have extesively bee studied by may scietists. However, cosiderable cotroversy has arise i recet years cocerig whether the Davydov solito is suficietly stable i the regio of biological temperature to provide a viable explaatio for bio-eergy trasport. It is out of questio that the quatum luctuatios ad thermal perturbatios are expected to cause the Davydov solito to decay ito a delocalized state. Some umerical simulatios idicated that the Davydov solito is ot stable at the biological temperature 300K [36-49]. Other simulatios showed that the Davydov solito is stable at 300K [36-49], but they were based o classical equatios of motio which are likely to yield ureliable estimates for the stability of the solito [-4]. The simulatios based o the ID > state i Eq.() geerally show that the stability of the solito decreases with icreasig temperatures ad that the solito is ot suficietly stable i the regio of biological temperature. Sice the dyamical equatios used i the simulatios are ot equivalet to the oliear Schro diger equatio, the stability of the solito obtaied by these umerical simulatios is uavailable or ureliable. The simulatio[9] based o the ID 1 > state i Eq. (3) with the thermal treatmet of Davydov [15-0], where the equatios of motio are derived from a thermally averaged Hamiltoia, yields the cofusig result that the stability of the solito is ehaced with icreasig temperature, predictig that ID 1 >- type solito is stable i the regio of biological temperature. Evidetly, the coclusio is doubtful because the Davydov procedure i which a equatio of motio for a average dyamical state obtaied from a average Hamiltoia, correspodig to the Hamiltoia averaged over a thermal distributio of phoos, is icosistet with stadard cocepts of quatum-statistical mechaics i which a desity matrix must be used to describe the system. Therefore, ay fully exact quatum- mechaical treatmet for the umerical simulatio of the Davydov solito does ot exist. However, for the thermal equilibrium properties of the Davydov solito, there is a quatum Mote Carlo simulatio [64,65]. I this study the correlatio characteristic of solitolike quasiparticles occur oly at low temperatures, about T<10k for widely accepted parameter values. This is cosistet at a qualitative level with the result of Cottigham et al. [66,67]. The latter is a straightforward quatum- mechaical perturbatio calculatio, the lifetime of the Davydov solito obtaied by usig this method is too small (about sec) to be useful i biological processes. This idicates clearly that the Davydov solutio is ot a true wave fuctio of the systems. A through study i terms of parameter values, differet types of ouiformity, differet thermalizatio schemes, differet wave fuctios, ad differet associated dyamics leads to a very complicated picture for the Davydov model [44-63]. These results do ot completely rule out the Davydov theory, however they do ot elimiate the possibility of aother wave fuctio ad a more sophisticated Hamiltoia of the system havig a solito with loger lifetimes ad good thermal stability. Ideed, the questio of the lifetime of the solito i protei molecules is twofold. I Lagevi dyamics, the problem cosists of ucotrolled effects arisig from the semiclassical approximatio. I quatum treatmets, the problem has bee the lack of a exact wave fuctio for the solito. The exact wave fuctio of the fully quatum Davydov model has ot bee kow up to ow. Differet wave fuctios have bee used to describe the states of the fully quatum-mechaical systems [1-3]. Although some of these wave fuctios lead to exact quatum states ad exact quatum dyamics i the J=0 state, they also share a problem with the origial Davydov wave fuctio, amely that the degree of approximatio icluded whe J 0 is ot kow. Published: April 09,

5 Therefore, it is ecessary to reform Davydov s wave fuctio. Scietists had though that the solito with a multiquatum, for example, the coheret state of Brow et al. [1-8], the multiquatum state of Kerr et al. [6,63] ad Schweitzer et al. [66,67], the two- quatum state of Cruzeiro-Hasso [44-49] ad Forer [74], ad so o, would be thermally stable i the regio of biological temperature ad could provide a realistic mechaism for bio-eergy trasport i protei molecules. However, the assumptio of the stadard coheret state is usuitable or impossible for biological protei molecules because there are iumerable particles i this state ad oe could ot retai coservatio of the umber of particles of the system. The assumptio of a multiquatum state (>) alog with a coheret state is also icosistet with the fact that the bioeergy released i ATP hydrolysis ca excite oly two quata of amide-i vibratio. O the other had, the umerical result shows that the solito of two-quatum state is more stable tha that with a oe-quatum state [74]. Cruzeiro-Hasso [44-49] had thought that Forer s two-quatum state i the semiclassical case was ot exact. Therefore, he costructed agai a so-called twoquatum state for the semiclassical Davydov system as follows [44-49]: + + ( t) {u 1},{P 1}, t B, 1 m Bm 0 m ex (5) where B is the aihilatio (creatio) operator for a amide-i vibratio B quatum (excito), u 1 is the displacemet of the lattice molecules, P 1 is its cojugate mometum, ad 0 ex is the groud state of the excito. He calculated the average probability distributio of the excito per site, ad average displacemet differece per site, ad the thermodyamics average of the variable, P B1 B1 B B, as a measure of localizatio of the excito, versus quatity JW / 1 ad I 1/ k BT i the socalled two-quatum state, Eq.(5), where χ 1 is a oliear couplig parameter related to the iteractio of the excito-phoo i the Davydov model. Their eergies ad stability are compared with those of the oe-quatum state. From the results of above thermal averages, he drew the coclusio that the wave fuctio with a two-quatum state ca lead to more stable solito solutios tha that with a oe-quatum state, ad that the usual Lagevi dyamics, whereby the thermal lifetime of the Davydov solito is estimated, must be viewed as uderestimatig the solito lifetime. However, by checkig carefully Eq.(5), Pag foud that the Cruzeiro-Hasso wave fuctio does ot represet exactly the two-quatum state. To id out how may quata the state Eq.(5) ideed cotais, the expectatio value of the excito umber operator Nˆ B B has to be computed i this state Eq.(5), the the excito umbers N cotaied are ijlm j j j j l l l l j im jl ex i m j l ex N BB 0BB BBBB 0 4 Published: April 09, l Therefore, the state, Eq.(5), as it is put forward i Ref.[44-49],deals with four excitos (quata), istead of two excitos i cotradictio to the author s statemets. Obviously, it is impossible to create the four excitos by the eergy released i the ATP hydrolysis (about 0.43 ev). Thus the author s wave fuctio i Eq.(5) is still ot relevat to protei molecules, ad his discussio ad coclusio are ureliable ad implausible i that paper [44-49]. It is believed that the physical sigiicace of the wave fuctio, Eq.(5), is also uclear, or at least is very dificult to uderstad. As far as the physical meaig of Eq.(5) is cocered, it represets oly a combiatioal state of sigle-particle excitatio with two quata created at sites ad m; m { u1}, {P 1}, t is the probability amplitude of particles occurrig at the sites ad m simultaeously. I geeral, = m

6 ad m m i accordace with the author s idea. I such a case it is very dificult to imagie the form of the solito by the mechaism of self- trappig of the two quata uder the actio of the oliear excito-phoo iteractio, especially whe the differece betwee ad m is very large.meawhile, Hasso has also ot explaied the physical ad biological reasos ad the meaig for the proposed trial state. Therefore, we thik that the Cruzeiro-Hasso represetatio is still ot a exact wave fuctio suitable for protei molecules. Thus, the wave fuctio of the systems is still a ope problem today, a correct theory of bio-eergy trasport eed still to costruct further. Pag s model of solito excitatio ad properties of eergy trasport From serious study of Davydov model we ca id that it is ideed too simple, i.e., it does ot deote the elemetary properties of the collective excitatios occurrig i protei molecules, ad may improvemets of it have also bee usuccessful. I fact, Davydov operatio is ot strictly correct. A basic reaso for the failure of the Davydov model is just that it igores completely the above importat properties of the protei molecules. Let us cosider the Davydov model with the preset viewpoit. First, as far as the Davydov wave fuctios, both D 1 ad D, are cocered, they are ot true solutios of the protei molecules. O the oe had, there is obviously asymmetry i the Davydov wave fuctio sice the phooic parts is a coheret state, while the excitoic part is oly a excitatio state of a sigle particle. It is ot reasoable that the same oliear iteractio geerated by the couplig betwee the excitos ad phoos produces differet states for the phoo ad excito. Thus, Davydov s wave fuctio should be modiied [7-103], i.e., the excitoic part i it should also be coheret or quasicoheret to represet the coheret feature of collective excitatio i protei molecules. However, the stadard coheret [1-8] ad large- excitatio states [6,63] are ot appropriate to the protei molecules due to the reasos metioed above. Similarly, Forer s ad Cruzeiro-Hasso s two-quatum states do ot fulill the above request. O the basis of the work of Cruzeio-Hasso, Forer, Schweitzer ad Takeo ad Pag, both the Hamiltoia ad the wave fuctio of the Dovydov model have bee improved ad developed by Pag [104-15], i which Davydov s wave fuctio has bee replaced with a quasi-coheret two-quata state for exhibitig the coheret behaviors of collective excitatios [16-19] which are a feature of the eergy released i ATP hydrolysis i the systems. The ew wave fuctio is represeted [104-15] by 1 1 t t t 1 t B t B 0 ex! (6) i exp t P t u 0 ph where B ad B are boso creatio ad aihilatio operators for the excito, 0 ex ad 0 ph are the groud states of the excito ad phoo, respectively u ad P are the displacemet ad mometum operators of the lattice oscillator at site, respectively. The t, t t u t ad t t P t are three sets of ukow fuctios, λ is a ormalizatio costat. It is assumed hereafter that λ=1 for coveiece of calculatio, except whe explicitly metioed. A secod problem arises from the Davydov Hamiltoia [5-14]. The Davydov Hamiltoia takes ito accout the resoat or dipole-dipole iteractio of the eighborig amide-i vibratioal quata i eighborig peptide groups with a electrical momet of about 3.5D, but why do we cosider ot the chages of relative displacemet of the eighborig peptide groups arisig from this iteractio? This u u B B B B meas that it is reasoable to add the ew iteractio term Published: April 09,

7 ito the Davydov s Hamiltoia for represetig the correlatios of the collective excitatios ad collective motios i the protei molecules, as metioed above [76-103]. Although the dipole- dipole iteractio is small as compared with the eergy of the amide-i vibratioal quatum, the chage of relative displacemet of eighborig peptide groups resultig from this iteractio caot be igored due to the sesitive depedece of the dipole-dipole iteractio o the distace betwee amio acids i the protei molecules, which is a kid of soft codesed matter ad bio-self-orgaizatio. Thus, the Davydov Hamiltoia is replaced by P 1 H H H H B B J B B B B [ w( u u ) ] ex ph it M u u BB u u B B BB (7) Where 0 =0.05eV is the eergy of the excito (the C=0 stretchig mode). The preset oliear \ couplig costats are χ 1 ad χ 1. They represet the modulatios of the o-site eergy ad resoat (or dipole-dipole) iteractio eergy of excitos caused by the molecules displacemets, respectively. M is the mass of a amio acid molecule ad w is the elasticity costat of the protei molecular chais. J is the dipoledipole iteractio eergy betwee eighborig sites. The physical meaig of the other quatities i Eq.(7) are the same as those i the above explaatios. The Hamiltoia ad wave fuctio show i Eqs.(6)-(7) are differet from Davydov s. We add a ew iteractio term, u 1 1 1, u B B BB ito the origial Davydov Hamiltoia. Thus the Hamiltoia ow has better oe-by oe correspodece of the iteractios ad ca represet the features of mutual correlatios of the collective excitatios ad of collective motios i the protei molecules. However, we here should poit out that the differet couplig betwee the relevat modes was also cosidered by Takeo et al. [75-77, ] ad Pag [78-103] i the Hamiltoia of the vibro-solito model for oe-dimesioal oscillatorlattice ad protei systems, respectively, but the wave fuctios of the systems they used are differet from Eqs.(6)-(7). Obviously, the preset wave fuctio of the excito i Eq.(6) is ot a excitatio state of sigle particle, but rather a coheret state, more precisely, a quasicoheret state, because it retai oly fore three terms of the expasio of a stadard coheret state, which thus ca be viewed as a effective trucatio of a stadard coheret state. It is clear that whe small ϕ (t), i.e., ϕ (t) << 1, we ca represet the wave fuctio of the excitos, ϕ (t) >, i Eq.(6) [104-15] by 1 t 1 t 1 t B t B 0 ex ~ exp[ ( t) ]! (8) exp t B 0 ex exp t B t B 0 ex The last represetatio i Eq.(8) is a stadard coheret state. Therefore, the state of excito deoted by the wave fuctio ϕ (t) > has a coheret feature. I the meawhile, we ca verify that the ew wave fuctio i Eq.(6) is also ormalized at λ=1. Sice the coditio of t 1 is required i the calculatio, the the above coditio of ϕ (t) << 1 also is aturally satisied for the protei molecules cosistig of several hudreds or thousads of amio acid residures. Thus the above represetatio i Eq.(8) is justiied ad correct for the protei molecules. Sice the coheret state is certaily ormalized, the the wave fuctio ϕ (t) > i Eq.(6), which ca be represeted as a stadard coheret state, is exactly ormalized at λ=1. Clearly, the above demostratio Published: April 09,

8 for the ormalizatio is correct, reasoable ad credible because it is obtaied from a strict mathematical, physical ad biological theory. Thus we have ot ay reasos to doubt the ormalizatio of wave fuctio i Eq.(6). However, it is ot a eigestate of the umber operator because of ˆ N t B B () t t B t B 0 t t B 0 ex ex (9) However, i this state the umbers of quata are determiate istead of iumerable. Sice the third term i the excito part i Pag s wave fuctio cotais the coeficiet of 1/, which guaratees that the third term i the excito s wave fuctio cotribute oly oe quatum, the we id that the state cotais umber of excito by computig the expectatio value of the umber operator N ^ B B i this state ad sum over the states, i.e., ˆ m N t N t t B B t t t t m t m t m 1 Published: April 09, (10) Therefore, t > cotais oly two quata, istead of oe quatum or three quata, i.e., it represets exactly a coheret superpositio of the excitoic state with two quata ad the groud state of the excito. Thus the ew wave fuctio ot oly exhibits the coheret feature of the collective excitatio of excitos ad phoos caused by the oliear iteractio geerated by virtue of the excito-phoo iteractio, which makes the wave fuctio of the states of the system symmetrical, but also agrees with the fact that the eergy released i the ATP hydrolysis (about 0.43 ev) ca oly create two amide-i vibratioal quata which, thus, ca also make the umbers of excitos maitai coservatio i the Hamiltoia, Eq.(7).The it is correct ad available. We here refer to it as a two quata quasicoheret state. Obviously, it is completely differet from Davydov s, which is a excitatio state of a sigle particle with oe quatum ad a eigestate of the umber operator. At the same time, the ew wave fuctio i Eq.(6) is either two- quata states proposed by Forer [74] ad Cruzeiro-Hasso [44-49] or a stadard coheret state proposed by Brow et al. [1-8], ad Kerr et al. [6,63,] ad Schweitzer et al s [66-67] multiquata states. Therefore, the wave fuctio, Eq.(6), is ew for the protei molecular systems. I the meawhile, the ew wave fuctio has the followig advatages, i.e., the equatio of motio of the solito i the system ca also be obtaied from the Heiseberg equatios of the creatio ad aihilatio operators i quatum mechaics usig Eqs.(6) ad (7). However, it is improssible for the wave fuctio of state of the system i other models, icludig the oe-quata state [5-14] ad the two-quata state [44-49]. Therefore, the above Hamitoia ad wave fuctio, Eqs.(6) ad (7),are both ew ad appropriate to the protei molecules. We kow fro Eq.(6) that the phoo part depedig o the displacemet ad mometum operators i the ew wave fuctio i Eq.(6) is a coheret state of the ormal model creatio ad aihilatio operators. A coheret state for the mode with wave vector q is deoted by [5-14,6-63,104-14] * ( t) exp [ q( t) aq q( t) aq] 0 ph. Utilizig the stadard trasformatios: q 1 u iqr0 e ( aq aq q NM q ) M q iqr0 P i e ( aq aq ), q N we ca get ( t) ( t), where (t) is same i Eq.(6) ad ( ) 1 q w M si( r0 q ) [6,63], r 0 is the distace betwee eighborig amio acid molecules, ad a ( a q q ) is the aihilatio (Creatio) operator of the phoo with wave vector q. Utilizig the 1

9 above results ad the formulas of the expectatio values of the Heiseberg equatios of operators, u ad P, i the state (t). i ( t) u ( t) ( t) u, H ( t) t i ( t) P ( t) ( t) u, H ( t) t ad the time-depedet Shrodiger equatio [79-105], i ( t) t cotiuum approximatio we get we ca obtai: H ( t), i the i ( xt, ) Rt ( ) ( xt, ) Jr0 ( xt, ) G (, ) (, ) p xt xt t x (11) ad ( xt, ) ( xt, ) M wr 4 o 11r0 ( x, t) (1) t x x here 5 1 R ( t) J W ( t) 0 m ( t) m ( t) m ( t) ( t) m ad W ( t) ( t) H ph ( t), 1 1 ( t) w M ( t) 1 ( t) q 1 q. The solito solutio of Eq.(11)[130-13] is thus 1 p v t ( xt, ) sec hp r0xx0 vtexp i xx0ev Jr0 with 1 P w s J 1 s v v, 81 G P w 1 s, 0 (13) (14) The above treatmet yields a localized coheret structure with size of order π r 0 /μ p that propagates with velocity v ad ca trasfer eergy E S01 < 0. Ulike bare excitos that are scattered by the iteractios with the phoos, this solito state describes a quasi-particle cosistig of the two excitos plus a lattice deformatio ad hece a priori icludes iteractio with the acoustic phoos. So the solito is ot scattered ad ca spread through maitaiig its form, eergy, mometum ad other quasiparticle properties after movig over a macroscopic distace. The bell-shaped form of the solito i Eq.(13) also does ot deped o the excitatio method. It is selfcosistet. Sice the solito always move with velocity v less tha that of logitudial soud v 0 i the chais the they do ot emit phoos, i.e., its kietic eergy is ot trasformed ito thermal eergy. This is oe importat reaso for the high stability of the ew solito. I additio the eergy of the solito state is below the bottom of the bare excito bads, the eergy gap betwee the beig 4 p J / 3 for small velocity of propagatio. Hece there is a eergy pealty associated with the destructio with trasformatio from the solito state to a bare excito state, i.e, the destructio of the solito state requires simultaeous removal of the lattice distortio. We kow that the trasitio probability to a lattice state without distortio is very small, i geeral, beig egligible for a log chai. Cosiderig this it is reasoable to assume that such a solito is stable eough to propagate through the legth of a typical protei structure. However, the thermal stability of the solito state must be calculated quatitatively. The followig calculatio addresses this poit explicitly. Published: April 09,

10 Although forms of the above equatios of motio ad the correspodig solutio, Eqs.(11)- (13),are quite similar to those of the Davydov solito, the properties of the ew solito have very large differeces from the latter because the parameter values i the equatio of motio ad the solutio Eqs.(11) ad (13),icludig R(t), G p ad μ p, have obvious distictios from those i the Davydov model. A straightforward result of the ew model is to icrease the oliear iteractio eergy G p (G p = G D ) ) ad the amplitude of the ew solito, ad decrease its width due to a icrease of p( p D[1 ( x x1) ( x x1) ]) whe compared with Davydov solito, where D x1 w(1 s ) J ad GD 4 x1 w(1 s ) are the correspodig values i Davydov model. Thus the localized feature of the ew solito is ehaced. Therefore, its stability agaist the quatum luctuatio ad thermal perturbatios is icreased cosiderably as compared with the Davydov solito. The eergy of the ew solito i Eq.(13) i the improved model ca be represeted [130-13] by 1 4 E () t H () t Jr0 R( x,) t Gp ( x,) t dx r 0 x 1 1 ( xt, ) ( xt, ) 1 M wr0 dx E0 Msolv r (15) 0 t x 4 8( x1 x ) 0 The rest eergy of the ew solito is E0 ( 0 J ) Es W, 3w J 4 where W [ ( x1 x ) ] 3w J is the eergy of deformatio of the amoo acid 4 4 8x1 x 9s 3s. lattice. The effective mass of the ew solito is Msol mex 3 3wJ1s v I such a case, the bidig eergy of the ew solito is E BP x1 x x x x x 8E BD 3Jw x 1 x1 x1 x 1 0 (16) 4 E BP is larger tha that of the Davydov solito.the latter is EBD x1 3Jw. We ca estimate that the bidig eergy of the ew solito is about several decades larger tha that of the Davydov solito.this is a very iterestig result. It is helpful to ehace thermal stability of the ew solito. Obviously, the icrease of the bidig eergy of the ew solito comes from its two-quata ature ad the added iteractio, u 1 ub 1B BB 1 i, i the Hamiltoia of the systems, Eq.(7). However, we see from Eq.(16) that the former plays the mai role i the icrease of the bidig eergy ad the ehacemet of thermal stability for the ew solito relative to the latter due to χ < 1. The icrease of bidig eergy results i a sigiicat chage of property of the ew solito, which are discussed as follows. I comparig various correlatios to this model, it is helpful to cosider them as a fuctio of a composite couplig parameter like that of Youg et al. [133], ad Scott [15-0], that ca be writte as 4 P 1 w, where 1 D D wm is the bad edge for acoustic phoos (or Debye frequecy). If 4πα P << 1, it is said to be weak. Usig widely accepted values for the physical parameters i the alpha helix protei molecule [5-15]: J J. w ( ) N m N. 1 (10 18) 10 N. M ( ) kg r m (17) Published: April 09,

11 we ca estimate that the coupled costat lies i the regio of 4πα P = , but 4 α D = for the Davydov model. Hece, the ew model is ot a weakly coupled theory as compared with the Davydov model. Usig agai the otatio of Vezel ad Fischer [134], Nagy [135] ad Wager ad Kogeter [136], it is coveiet to deie aother composite parameter [15-0]: J wd. I terms of the two composite parameters, 4πα P ad γ, the solito bidig eergy i the ew model ca be writte by E J 84 3, sol ex P BP P M m (18) From the above parameter values i Eq.(18), we obtai γ = Utilizig these values, the E BD /J versus 4πα relatios i Eq.(18) are plotted i igure. However, E J 4 3 for the Davydov model, where BD D ' M 14 3 sol m ex P ad 4 D 1 w ), the the E D BD /J versus 4πα D relatio is also plotted i igure. From this igure we see that the differece of solito bidig eergies betwee two models becomes larger with icreasig 4πα. Meawhile, we see clearly from Eqs.(1)-(15) ad (16) that the localized feature of the ew solito is ehaced due to icreases of the oliear iteractio ad its bidig eergy resultig from the icreases of excito-phoo iteractio i the improves model. Thus, the stability of the ew solito agaist quatum ad thermal luctuatios is ehaced cosiderately. I fact, the oliear iteractio eergy formig ew silito i the ew model is G 8 1 s w J, ad it is larger tha the liear dispersio eergy, P 1 J = J, i.e., the oliear iteractio i this model is so large that it ca actually cacel or suppress the dispersio effect i the equatio of motio,thus the ew solito is stable i such a case accordig to the solito theory [5-14,137] (Figure ). O the other had,the oliear iteractio eergy i the Davydov model is G 4 1s w J, ad it is about three to four times smaller tha G p. D 1 Therefore, the stability of the Davydov solito is weaker as compared with the ew solito. Moreover, the bidig eergy of the ew solito i the improved model is E BP = ( ) 10-1 J i Eq.(15), which is somewhat larger tha the thermal perturbatio eergy, K B T = J, at 300K ad about four times larger tha the Debye eergy, 1 KB D 1.10 J (there ω D is the Debye frequecy).this shows that trasitio of the ew solito to a delocalized state by the heat eergy ca be suppressed by the large eergy differece betwee the iitial (solitoic) state ad ial (delocalized) state, which is very dificult to compesate for with the eergy of the absorbed phoo. Thus, the ew solito is robust agaist quatum ad thermal luctuatios, therefore it has a large lifetime ad good thermal stability i the regio of biological temperature.i Figure : The bidig eergy (E B ) of the solitos i uits of dipole-dipole iteractio eergy (J) vs the coupled costat, 4πα, relatio i Pag s model ad the Dvydovy s model. Published: April 09,

12 practice, accordig to Schweitzer et al. s studies (i.e the lifetime of the solito icreases as μ p ad T0 v0p KB icrease at a give temperature)[67-68] ad from the above results obtaied we ifer that the lifetime of the ew solito will icrease cosiderably relative to that of the Davydov solito due to the icreae of μ p ad T 0 because the latter are about three times larger tha that of the Davydov model. O the other had,the bidig eergy of the Davydov solito, 4 1 EBD 3w J J, is about 3 times smaller tha that of the ew solito, 1 about times smaller tha K B T, ad about 6 times smaller tha K B Θ, respectively. Therefore, the Davydov solito is easily destructed by the thermal perturbatio eergy ad quatum trasitio effects. Thus we ca aturally obtai that the Davdov solito has oly a very small lifetime, ad is ustable at the biological temperature 300K. This coclusio is cosistet at a qualitative level with the result s of Wag et al. [64,65] ad Cottigham et al. [66,67]. I the above ivestigatio of the ilueces of quatum ad thermal effects o solito state, which are expected to cause the solito to decay ito delocalized states, we postulate that the model Hamiltoia ad the wavefuctio i the ew improved model together give a complete ad realistic picture of the iteractio properties ad allowed states of the protei molecules. The additioal iteractio term i the Hamiltoia gives a oe-to-oe correspodece of iteractios i the ew model. The ew wavefuctio is a reasoable choice for the protei molecules because it ot oly ca exhibit the coheret features of collective excitatios arisig from the oliear iteractio betwee the excitos ad phoos, but also retai the coservatio of umber of particles ad fulil the fact that the eergy released by the ATP hydrolysis ca oly excite two quata. I such a case, usig a stadard calculatig method [,6] ad widely accepted parameters i Eq.(17) we ca calculate the regio ecompassed of the excitatio or the liear extet of the ew solito, X r0 /, which is greater tha the lattice costat p r 0 i.e., X r 0. Meawhile, we ca explicitly calculate the amplitude squared of the p p ew solito usig Eq.(13) i its rest frame, which is ( x) sec h ( x ). Thus the r0 probability to id the ew solito outside a rage of width r 0 is about This meas that the ew solito is very well localized i this coditio. Meawhile, this umber ca be compatible with the cotiuous approximatio sice the quasi-coheret solito ca spread over more tha oe lattice spacig i the system i such a case. Thus, this proves that assumig of the cotiuous approximatio used i the calculatio is valid because the solito widths is large tha the order of the lattice spacig, the the solito stability is improved. Fially we calculate the values of the mai parameters i the ew model by the above values. These values ad the correspodig values i the Davydov model are simultaeously listed i table 1. From table 1 we ca see clearly that the ew model produces cosiderable chages i the properties of the solito, for example, large icrease of the oliear iteractio, bidig eergy ad amplitude of the ew solito, ad decrease of its width as compared to those of the Davydov solito. This shows that the solito i the ew model is more localized ad more robust agaist quatum ad thermal luctuatios, thus its stability is ehaced [5-14, ], which implies a icrease i lifetime for the ew solito. From Eq.(16) we also id that the effect of the two-quata ature is larger tha that of the added iteractio. We thus ca refer to the ew solito as quasi-coheret state, which is ovel ad correct wave fuctio. Table 1: Compariso of parameters used i the Davydov model ad Pag s model. Parameters Models Pag s Model Davydov model μ G ( 10-1 J) Amplitude of solito A Width of solito x ( m) Bidig eergy of solito E B ( 10-1 J) Published: April 09,

13 The ecessity verifyig the validity of these theoretical models We exhibit the properties, successful ad problems of differet models i the above ivestigatio, which but leaves behid may questios that are worth to study cotiuously. Actually, the above results were obtaied aalytically, based o may hypotheses, i which the protei molecules, which were used by the researchers, was regarded as a periodically ad uiformly iiite-log chais composed of amio acid residues with same weight. Obviously, the proteis are ot completely coformable with the biological protei molecules i the livig systems. As it is kow, the biological proteis are a iite log structure, which are composed of several hudreds or thousads amio acid residues with differet molecular weights betwee 75 m p (glycie) ad 04 m p (tryptopha), which correspod to variatios i mass betwee 0.67M M 1.80M, where M =114mp is the average mass of a amio acid residue ad m p is the proto mass. Ad the biological proteis adhere all some side groups, which will affect the structure ad dyamic features of protei molecules. This meas that there are a structure ouiformity ad disorder i biological-protei molecules. These structural ouiformities result ecessarily i the luctuatios of the sprig costat, the dipole-dipole iteractio, the excito-phoo couplig costat ad the diagoal disorder i the equatios of motio. Thus, the states of the solitos obtaied from the theoretical models, which are established based o uiform proteis, will be chaged uder iluece of these structure disorders i the biological protei molecules. Otherwise, i the above ivestigatio all physical parameters of the protei molecules were represeted by their average values, ad some approximatio methods, such as log- wavelegth approximatio, cotiuum approximatio, or log-time approximatio, were also used i cocrete calculatios, which caot be evidetly used i the biological proteis. Careri et al. [19, ], demostrated that eve relatively small amouts of disorder i a amorphous ilm of acetailide (ACN), a protei-like crystal, is eough to destroy the spectral sigature of a solito. Therefore, we have the reasos to doubt the real existece of the solitos ad him correctess of the above theory of bio-eergy trasport i protei molecules. At the same time, the biological proteis work always at physiological temperature ad biological solutio cotaiig water molecules ad other ios. The thermal perturbatio ad dampig of medium arisig from the temperature will also chage the states ad lifetimes of the solitos, which are the carrier of the bio-eergy trasport. These water molecules ad other ios existed i the solutio serve as some imported impurities to iluece the dyamic features of the protei molecules. O the other had, for the α-helix protei molecules, which are costructed by three chaels, there are also chai-chai iteractio amog the three chaels, which carry also a dispersive effect ad further iluece the states of the solitos i differet models. I such a case, we have the reaso to doubt the validity of the above theories of bio-eergy trasport. This meas that we must verify whether these models are appropriate to biological proteis? Whether ca the models represet the real features of biological protei molecules? Namely, it is quite ecessary to verify the validity of these theories i the biological protei molecules ad to check the states ad properties of the bio-solitos uder the ilueces of these structure ouiformlity, chai-chai iteractio ad temperature ad dampig of the medium i the protei molecules. However, what are methods which ca be used to verify ad check the correctess of the theories of bio-eergy trasport? The umerical simulatio is just a best method studyig ad solvig these problems. I fact, the umerical simulatio ca work at the protei molecules with a iite legth, it abados the various approximatios used i aalytic calculatios, ad ca iclude the effects of structure ouiformity, imported impurities ad side groups of proteis ad ilueces of temperature ad dampig of medium o the bio-eergy trasport. Therefore, the umerical calculatio ca be used Published: April 09,

14 to study the properties of the solitos or bio-eergy trasport i the really biologicalprotei molecules. I particle, a lot of scietists ad researchers, cotaiig Davydov, Scott et al., Wag X et al, Forer ad Pag XF, used the differet umerical-simulatio methods to ivestigate extesively the properties of bio-eergy trasport i sigle chai protei ad t α-helix protei molecules with three chaels, respectively, where a lot of researched results, which are eough to assess ad check the correctess of theory of bio-eergy trasport i the biological proteis, ivolvig Davydov s ad Pag s models, were obtaied. These umerical- simulatio techiques cotai the average Hamiltoia way of thermal perturbatio, fourth-order Ruge-Kutta method, Mote Carlo method, quatum perturbed way ad thermodyamic ad statistical method, ad so o. We here review widely ad i-depth the progress of differet umerical simulatios ad give their results, which ca be used to assess ad verify the validity ad availability of the theories of bio-eergy trasport metioed above. For this purpose we here stated ad discussed followig four problems. Properties of eergy trasport i uiform protei molecules Determiatios of value of basic parameters i eergy trasport theory As it is kow, the properties of bio-eergy trasport are closely related to the structure features of α-helix protei molecules. If the parameters of structure of proteis are differet, the the property of the eergy trasport will also be chaged. Therefore, correct determiatio of the values of these parameters cotaied i these theories is very importat for verifyig the validity of the theory of bio-eergy trasport. As it is kow, there are ive basic parameters, the mass of amio acid residue, M, the sprig costat, w, dipole-dipole iteractio costat, J, excito-phoo couplig costat χ 1 ad groud state eergy ε 0 i Davydov model, but aother couplig costat χ to be icluded i Pag s model for the protei molecules, of sigle chai. For these parameters i the protei molecules, the widely accepted ad available values are give i Eq.(17), where m p = kg is proto masse, r 0 is lattice costat or distace betwee two amio acids. For α-helix proteis there is agai a chai-chai iteractio coeficiet L, which is L=1.5meV, amog the three chais. The above values are i essece their average values, such as M here is a average value of 0 differet amio acids. I these parameters the determiatio of excito-phoo couplig costats plays a key part i the correctess of the bio-eergy trasport theory, meawhile, it is also quite dificult to determie. As it is kow, χ 1 ad χ represet the chages of groud state eergy ad dipole-dipole iteractioal eergy whe the amio acid residue displaces oe uit distace, respectively. Experimetal value of χ 1 is i the rage of 35-6PN, its cocrete value was obtaied umerical calculatio. A primary motivatio for early umerical simulatio of Davydov theory [5-14] was just to determie the magitude of the excito-phoo couplig coeficiet χ 1. Oes thought that χ 1 should be eough large to support solito trasport of the bio-eergy i the α-helical protei. To proceed seriously, the followig assumptios are made. (1) Iitial amide-i eergy was localized i a sigle tur of the helix, A typical iitial coditio to lauch a symmetrical solito is /3 ad 0 for >1.This meas that two amide-i quata were put ito the irst tur at oe ed of the helix at time t=0. () All importat dipole-dipole iteractios were calculated. The other larger iteractios was cosidered ad eight loger rage iteractio terms were also ivolved, except for the eighborig iteractios J ad A umerical code that embodied these two assumptios is developed i the early 1980s [15,16, 68,144]. This code deoted a helix of 00 turs, which is about the legth of the alpha-helix i myosi, with free ed boudary coditios. With the above iitial coditios we ca obtai that a threshold value for a solito to be formed is the excito-phoo couplig coeficiet to equal to χ 1 50pN. Published: April 09,

15 If the two quata of the above iitial coditios were cosidered by the wave Q fuctio proposed by Kerr et al. [145], (1 / Q)( ( t) B ) 0, the the ex threshold value will be icreased. However, the excito-phoo couplig coeficiet for a really biological alpha-helix protei molecules should be large eough to esure the solito formatio, which but have bee challeged by Mechtly ad Shaw [70]. They costructed a wave fuctio for the Davydov Hamiltoia, which is more accurate tha Davydov s product wave fuctio because it is a eigestate of the traslatio operator over the rage of 0 4πα 1, where 4πα is a couplig parameter which was deied i Ref.[15-0]. The its dyamic equatios differ from the Davydov s. Sice umerical simulatio of the dyamic equatios with the above iitial coditios idicates the solito is formed oly for 4πα >1. Thus Machtly et al. [70], obtaied that the previous threshold value of χ 1 for solito formatio is uderestimated by a factor of at least four. I this case the effective mass of the solito rises quadratically, the Heiseberg s ucertaity relatio x v M becomes closer to that of a classical object. So, this result is doubted. / eff Rhodes ad Nichclls [146] studied also this problem, i which he itroduce a wave fuctio that is based o a time depedet uitary trasformatio. The umerical calculatios shows delocalizatio of the wave packet as is required by the above ucertaity relatio. From the above ivestigatios we ca coclude the followig results. (A) Numerical studies of Davydov model shows a threshold level of the excitophoo couplig coeficiet χ 1 above which amide-i eergy, that is iitially localized o a sigle tur of the alpha helix, is coverted ito a solito. The threshold level is χ 1 50pN, which agrees with idepedet experimetal measuremet of χ 1 > 35pN to 6pN (B). Numerical simulatio of improved Davydov s wave fuctios [70,146] that idicates the larger threshold values of χ 1 ware misleadig. The larger threshold values are a artifact of the wave packet feature of the improved wave fuctio. Therefore it ot credible. The results of differet methods of umerical simulatio for the solutio of dyamic equatios i Davydov model i uiform protei chais Hyma, McLaughli, ad Scott [68] studied irst the properties of the bio-eergy trasport i alpha-helix protei molecules with three chaels i Davydov model [5-14] usig umerical simulatio mother with the help of the computer at the Los Alamos Scietiic Laboratory i I this case the iteractio amog the chaels is icluded i the Hamiltoia of the protei molecules. From the average Hamiltoia they gave the discrete forms of dyamic equatio of the excito ad phoo i the biological protei with three chaels, which are deoted by i [ W ( u u )] J( ) L( ) (19) 0 1 1, 1, 1, 1,, 1, 1 Mu wu ( u u ) ( ), 1,,3. (0) 1,, 1, 1 1, 1, 1 u, where W [ M( ) w( u 1, u, ) ]. Equatios (19) ad (0) were t itegrated umerically by them for studyig the excited states of three chais of helical protei molecules. Each chai cotaied 00 peptide groups. The molecule was characterized by the followig quatities: ε 0 = 0.05eV; M= 70m P, ν 0 = 10 4 cm / s, J = J, L = J, Published: April 09,

16 w = 76N / m, χ 1 = (5-7) N. The iitial coditios at t=0 were take as ϕ a =1 for =1 or =0 for 1, u x = 0. The calculatios were performed for differet values of the couplig parameter χ 1 of itrapeptide Amide I excitatios with displacemets of their equilibrium positios. It was show that for the above iitial coditios distict solitos are formed ad propagate i the proteis with χ N. Solitos with χ1 close to the critical value propagate with velocity ~ m/s. Therefore, the distace aometers, correspodig to the legth of the alpha-helical myosie molecule i muscle ibers [5,6], could be traversed by solitos (where the frictio forces of medium is eglected) withi 130 PS. Hyma et al. [68], foud that the coclusio obtaied from the umerical studies of Davydov s dyamic model i the alpha-helical proteis coirm his predictio of the formatio of solitos; these solitos are robust, localized, ad a dyamic etity that coupled amide-i vibratios to logitudial soud waves, thus they may provide a eficiet mechaism of eergy trasport i biological systems. Both the umerical studies ad aalytical computatios show a threshold level of oliearity below which the solitos will ot be formed. A rough estimate idicates that this oliearity has the required order of magitude. Hyma et al s result is iterestig i that the process of formatio of solitos from a deiite iitial state is ivestigated ad the role of the discreteess of the chai, which, apparetly reduces to the fact that solitos are formed i the chai oly with supercritical values of the couplig parameter χ1, determiig the oliearity of the system, is clariied. I 1979, Eilbeck made a computer ilm that demostrates the propagatio of a iteral vibratioal excitatio of a edge group alog a peptide groups (PG) chai. This ilm clearly shows that for a above threshold value of the couplig parameter χ1 betwee the vibratioal excitatio ad displacemets of PG alog the molecular chais, the excitatio propagates i the form of a solito, i.e., i the form of a local pulse, whose shape ad width remai costat durig the motio. Eilbeck s ilm is importat for two reasos: irst, it coirms the previous calculatios performed at the Los Alamos Laboratory ad, secod, it clearly demostrates the stability of solitos relative to iteractios with soud waves. The soud wave was excited together with the solito. Movig faster tha the solito, it relected several times from the eds of the chai passig through the solito ad ot causig ay chages i it. Eilbeck s ilm ad the umerical calculatios i Ref. 68 show that the solito forms at the very begiig of the peptide chai. Therefore, solitos ca arise withi comparatively short sectios of alpha-helical proteis. Scott [15-18,68] modiied further Davydov s dyamic-equatios for the alpha helix solito to iclude te additioal dipole-dipole iteractios ad represet helical symmetry, Thus the equatios (19)-(0) were replaced by i [ W ( u u )] J( ) L( ) 0 1 1, 1, 1, 1,, 1, 1 [( u u )] ( u u )] ] NF PF QF RF SF ' 1,, 1,, 1, 1 N P Q R S TF UF VF WF XF ZF T U V W X Z (1) Mu wu ( u u ) ( ) 1,, 1, 1 1, 1, [ ( ) ( ) ] ' * * * , 1, () where the J,L, N,F,Q,R,S,T, S, T,U,V,W, X, Z are the dipole-dipole iteractio eergies betwee pairs of amide-i bods,which are 0.143, 0,31, 0.073, 0.03, 0.019, 0.01, Published: April 09,

17 0.0089, , , , 0.00, , respectively. The term NF N is shorthad for the dipole-dipole iteractio betwee laterally adjacet amide-i bods, which ca be deoted as i N (, 1, 1)..., 1,,3..., 1,,3. Scott et al. [15-18,68] foud the solutios of the above equatios by umerical calculatio method, ad obtaied that there is a widow for solito formatio whe χ 1 lies i the rage of N <χ N, i.e., the excito-phoo couplig parameter i alpha-helix protei molecules is large eough to trasport the formatio of Davydov solitos uder ormal physiological coditio. Meawhile, they also gaied the followig coclusios from the studies: (1) It should be possible to create Davydov solitos by direct stimulatio with ifrared radiatio of 6.06 μm (correspodig to a amide-i absorptio at 1650 cm -1 ). This is clear because i the umerical study oly bod eergy was itroduced as a iitial coditio. () Davydov solitos should display a rather sharp iteral resoace (related to exchage of eergy betwee the three logitudial spies of the alpha helix) with a period of about psec or a free-space wavelegth of about 600 μm. (3) A piece of alpha-helix with uit cells (i.e., 3 peptide uits) may show a broad resoace ( Q = 1) with a period of about 0.54 psec. Sice the molecular weight (W 100 ) of a strad of helix is about 114, this resoace should appear at a free-space wavelegth of 1.4W cm. For this resoace to appear at 3 cm ( X-bad i RADAR jargo) the molecular wavelegth would be about W 100 =1000. For the resoace to appear at 30 cm (1 GHz) W 100 = (4) Mechaical effects iduced directly by the solito are limited to the kietic part of the total eergy. For the example studied here this is less tha 0.0% (0. 000) of the total eergy. Mechtly et al. [70], develop a theory of excito evolutio o a zero-temperature Davydov lattice which is free of certai deiciecies foud i the stadard Davydov theory. The approach makes use of a time-depedet uitary trasformatio o a Davydov Hamiltoia parametrized by a dimesioless lattice costat ad a dimesioless excito-phoo couplig costat α. The trasformatio geerator is expaded i a ormal-ordered series of multiphoo operators with expasio coeficiets chose to elimiate various terms i the trasformed Schrodiger equatio. At the oe-phoo level, they obtaied equatios of motio which differ from those of Davydov. I the small-polaro trasportless limit (iiite a) the equatios are exact. I the large-polaro cotiuum limit (vaishig a) the equatios become ield equatios whose statioary solutios are those of Gross s iterpolatio theory. To use umerical simulatio method for a oe-spie model of a α-helix (a =.7) they foud that the solito (a movig dispersioless excitatio) formatio durig evolutio from a localized iitial state above a threshold value of 4πα > 1, which is a order of magitude higher tha the threshold foud i stadard Davydov theory. This lower threshold for the Davydo theory is due to its iadequate treatmet of the excito- phoo iteractio i weak couplig. Sice the value of the couplig-costat threshold obtaied withi Mechtly et al s oe-phoo approximatio exceeds the rage of values for the α-helix suggested by MacNeil et al. [70], it is temptig to coclude that a α-helix ca ot support solitos at T=O. Thus it would be straightforward to exted the preset aalysis to test this coclusio o more realistic models of a α-helix. As a check o the validity of the oephoo approximatio itself, a extesio of the preset theory to the two-phoo level is doe further. Published: April 09,

18 Förer [50] studied the chages of dyamical properties of Davydov solito i a uiform protei molecular chai usig a fourth-order Ruge-Kutta method [147,148]. I this method the dyamic equatios of simulatio i Davydov model, which ca be obtaied from Eq.(1)-(), are as follows i [ ( u u )] ( J J ) (3) , 1( 1) 1 1( ) P w ( u u ) w ( u u ) ) (4) where P M u, ' exp[ i0t/ ]. I this method, he used a time step size of 0.01ps, the total eergy was coserved to 3eV (0.015%); a possible imagiary part of the eergy,which ca occur due to umerical iaccuracies was zero to a accuracy of 0.00feV, the orm is coserved up to 0.4pp (parts per millio). Otherwise, the chai eds are ixed ad a iitial excitatio with oe quatum is put at the site N-1, which N is the umber of uits chose to be N=00 i the calculatio. For the lattice u (0)=P (0)=0 were used. This simulatio result shows that there is a widow of the characteristic parameters of χ 1 ad w for the occurrece of Davydov solito, which are 60PN< χ 1 < 140PN ad 30N/m,< w < 90N/m. Meawhile, the motios of Davydov solito with chagig the time ad positio are show i igure 3, where we see clearly that the solito is dispersive at χ 1 =0 PN, is gradually formed with icreasig χ 1, but the solito is still somewhat dispersive at χ 1 =60PN ad is stable at χ 1 =100PN, which is larger tha the widely accepted value of 6PN. The above results of umerical simulatios i a uiform protei molecular chai exhibit clearly that the Davydov solito ca be form, but is ot stable for small χ 1, which is lower tha the widely accepted values of physical parameters show i Eq.(1). This coclusio is cosistet from Lomdahl et al s result of umerical simulatio [6], which idicates that the solitos are ucleated from radom iitial coditios, but could be dispersed i motio alog the chai as show i igure 3. a (b) (c) Figure 3: The chages of moved state of Davydov solito with D asatz i a uiform protei molecular chai obtaied by Förer [50]. Published: April 09,

19 The properties of umerical simulatio of eergy trasport i Pag s model The results i a sigal protei chai: We irst adopt the discrete dyamic equatios (6)-(7) i Pag s model to simulate umerically the dyamic properties of the solito i sigle chai proteis. We ow replace the ϕ (t) ad β (t) i Eqs. (6)-(7) by a (t) = a (t)r + ia(t)i ad q (t), respectively, where ar ad ai are real ad imagiary parts of a (t), ad use the trasformatio: a (t) α exp[ir(t)t / h to elimiate the term R(t)a (t) i Eq. (11). Thus equatios (11) ad (1) ca the rewritte as[104-15] ar J( ai ai ) ( q q ) ai ( q q )( ai ai ) (5) ar J( ai ai ) ( q q ) ai ( q q )( ai ai ) (6) y w( q q q ) ( ar ai ar ai ) 4 [ ar ( ar ar ) ai ( ai ai )] (7) where q y / M, a ar ai. Equatios (5)-(7) are dyamic equatios, which ca determie the states ad behaviors of the solito. We will obtai their solutios umerically usig the fourth-order Ruge-Kutta method [ ]. Obviously, there are four equatios for oe peptide group. Therefore, for protei molecules of sigle chai costructed by N amio acids we have to solve a system of 4N associated equatios. To use the fourth-order Ruge-Kutta method [ ] we eed irst to discretize the equatios, i which, the time be discretized ad deoted by j, the step size of the space variable is deoted by h. I the umerical simulatio, Pag et al., used ev for eergy, A 0 for legth, ad ps for time [104-15]. The followigs are details of the umerical simulatio: a time step size of ps, the total eergy E=<Φ(t) H Φ(t)> was coserved to 0.001% ( a possible imagiary part of the eergy ca be developed durig the simulatio due to umerical iaccuracies which was kept to below fev), the orm is coserved up to 0.3pp (parts per millio). A ixed chai with N uits ad a iitial excitatio i the forms of a (o)=asech[(- 0 ) (χ + χ 1 ) /4Jw] at the site were used i this calculatio, where A is the ormalizatio costat,. For the lattice, the iitial coditios of q (0)=π (0)=0 was applied. Pag et al., chose N=50 i this simulatios. Meawhile, the simulatio was performed usig a data parallel algorithms ad MALAB σοφτωαρε. Utilizig the above average values for the parameter, which are ow deoted by M, w, 0, J, 1 ad, we calculate umerically the solutio of Eqs.(5)-(7) by usig the fourth-order Ruge-Kutta method i uiform ad periodic proteis, where a is the umber desity of the solito occurrig at the th amio acid residue. Thus, we ca plot the state of motio of the solito i time-place. The result is show i igure 4. This igure shows that the amplitude of the solutio ca retai costacy. I igures 5,6 we show the propagatio behavior of the solutio for a log time period of 300Ps, ad the collisio property of two solitos, respectively. From the igures we see that the solutio is very stable while i motio for a log time period. Thus, Eqs. (5)- (7) have exactly a solito solutio i uiform ad periodic proteis, which coicide with the above aalytic results. Figure 4: Solito solutio of Eqs. (5) - (7) i uiform chais. Published: April 09,

20 Figure 5: State of Pag s solito i the cases of large time of 300ps ad log distaces of 400. Figure 6: The collisio behavior of two Pag s solitos for Eqs. (5) - (7). The results i α-helix protei molecules with three chaels I the meawhile, we umerically simulated also the properties of the solutios of the dyamic equatios for i the α -helix protei molecules with three chaels. I this case, the Hamiltoia ad the wave fuctio i Eqs. (6) ad (7) i Pag s model[104-15] are ow, respectively, replaced by P 1 H [ 0B B1 J( B B1 B B1)] [ w( q q 1) ] [ 1( q 1 q 1) M (8) B B ( q q )( B B B B ) L( B B B B )], 1,, () t a() t () t [1 a () t B ( a () t B )] 0 ex! i exp{ [ q ( t) P ( t) u ]} 0 ph (9) where L is the coeficiet of the chai-chai iteractios amog the three chaels. From the time-depedet Schrödiger equatio ad the Heiseberg equatios we ca get ia () t 0a () t J[ a 1() t a 1()] t 1[ q 1() t q 1()] t a () t [ q 1() t q 1()] t (30) 5 1 [ a 1() t a 1()] t { W() t [ qm () t m () t m () t q m()]} t a () t L[ a 1() t a 1()] t m Mq w[ q () t q () t q ()] t [ a () t a () ] t { a ( t)[ a ( t) a ( t)] a ( t)[ a ( t) a ( t)]} * * * (31) Applyig the method ad way ad fourth-order Ruge-Kutta method [ ] metioed above we ca simulate umerically the features of the solutios of Eqs. (8) - (31). I this case the iitial coditio of a () t Asec h[( 0)( 1 ) /4 Jw] was used i the calculatio, where α=1,,3, A is the ormalizatio factor. Meawhile we used M = kg= amu (atomic mass uits), w =39N/m, ε 0 =0.035eV, J = ev, 1 = N, =(10-18) 10-1 N ad L =1.5meV for the α-helix protei molecules i the simulatio. Published: April 09,

21 The umerical solutios of Eqs. (8)-(9) for the α-helix protei molecules with three chaels are show i igure 5. I igure 7a we show the motio behaviors of the solutio, where the iitial coditio of a () t Asec h[( 0)( 1) /4 Jw] is simultaeously motivated o the irst eds of the three chaels. From this igure we see that this solutio ca retai a clock shape while movig over a log distace i the rage of the spacig of 400 amio acid residues at a time of 40ps without dispersio alog the molecular chais, i.e., this solutio is a solito. This is similar to the above aalytic ad umerical results for the proteis with a sigle chai. I igure 7b ad 7c we plot the features of the solutios, with the iitial coditios: a a (t=0)=0, here α=1,, a3() t Asec h[( 0)( 1 ) /4 Jw] ad, a () t Asec h[( 0)( 1) /4 Jw] where α=1,, a 3 (t=0)=0 are used, respectively. These iitial coditios deote that the irst eds of oe chael ad two chaels are motivated, ad that the other two chaels ad the sigle chael are ot liked, respectively. Figure 7b idicates that two ew waves with small amplitudes are geerated due to the mutual iteractios amog the three chaels, except for oe solito occurrig o the chael liked by iitial coditios. Although the two excitatios are small, they ca move over log distaces alog the two chais while keepig their amplitudes. Therefore, they are still solitos with small amplitudes. However, there exists a strage pheomeo i igure 7c, i which the amplitudes of solitos geerated i the two motivated chais are small, but the solito i a umotivated chai is larger, obviously, it is due to the superpositio of waves iduced by the other two chais. From this study we kow that the solitos formed have higher eergy whe the iitial coditios are simultaeously motivated o the irst eds of the three chaels. We further study the collisio property of solitos, set up from opposite eds of the chaels for α-helix protei molecules; the result is show i igure 8a, where the above iitial coditios simultaeously motivate the opposite eds of the three chaels, where iitial two solitos separatig 100 amio acid spacigs i each chael collide with each other at about 17ps. After this collisio, the two solitos i each chael go through each other without scatterig to propagate toward ad separately alog the three chais, satisfyig the rules o collisio of macroscopic particles, which demostrates that the solitos have a corpuscle feature. I igure (a) (b) (c) Figure 7: Features of solito solutios of Eqs. (8)-(9) for α-helix proteis uder differet iitial coditios. Published: April 09,

22 8b ad 8c we plot the collisio features of the two solitos geerated from opposite eds of the chaels with the above iitial coditios motivatig the opposite eds of oe chael ad two chaels, ad where they are ot liked with the opposite eds of the other two chaels ad the sigle chael, respectively. I the two cases the solito feature of the solutios after collisio is substadard, especially where the iitial coditio motivates oly the opposite eds of the sigle chael, as show i igure 8b. Thus, the solutios of Eqs.(8)-(9) have a better solito feature, where the opposite eds of the three chaels are motivated simultaeously by the above iitial coditios i the α -helix protei molecules. Hece, the solito excited i the α -helix protei molecules has higher stability i the case of simultaeous motivatio of the iitial coditio o the three chaels. I igure 9 we show the states of solito solutios of Eqs. (30) - (31) at a time of 10ps ad at 300 amio acid spacigs, where the above iitial coditios are simultaeously liked o the irst eds of the three chaels. We see that the solito still retais its amplitude ad shape while movig. This result shows that the lifetime of the solito is at least 10Ps., thus it is very stable. The above results show that the states of Pag s solito are stable i a uiform protei molecular chai, thus the bio-eergy trasport ca be carried out i this system. O the other had, Förer [149] studied also the states of Davydov solito i α-helix proteis with three chaels usig wave fuctio, i which he used the dyamic equatios i Eqs.(1)-(13) at =0 ad 5 { W() t 1 [ qm () t m () t m () t q m()]} t a () t m =0, but χ 1 is replaced by χ i Eqs.(1)-(13), where the symmetric A model (a =a =a ), 1 3 the liear combiatio of the two degeerate E modes (a 1 =0, a =a 3 =-1/ ) ad a asymmetric local excitatio(l mode) of oe uit o a sigle chai are used. I this case the result of umerical simulatio for Davydov solito is show i igure 10, where L= 1.54meV, w=13n/m. M=114m P, J=0.967meV ad χ=6pn were utilized. Figure 10 idicates that i the case of the A mode the solito cosists of three idetical parallelmovig localized excitatios o all three chais, while, i the E mode the solito moves oly o two chais. I the case of the local excitatio the solito is foud maily o oe chai, with a small fractio of the excitatio trasferred to the others, but the (a) (b) (c) Figure 8: The collisio features of Pag;s solitos i differet coditios. Published: April 09,

23 Figure 9: The behavior of log-time motio for Pag s solito. Figure 10: The time evolutio of the Davydov solito i the case of three iteractig chais for three differet kid of iitial excitatios (A,E, L modes) ; first =1-100, secod chai, =101-00, third chai, = ([149]). pheomeo reported by Scott [15,18] that solitos o oe chai jump to aother after some time have ot observed. Maybe the built-up of a small fractio of the excitatio o the other chais i the case of the asymmetric iitial coditio might lead to a trasfer of the whole solito after loger times. Ifluece of structure ouiformity o the eergy trasport i protei molecules The results i Davydov model As metioed above, Förer [50] obtaied that Davydov solito with D is stable at χ 1 =100PN, which is larger tha the widely accepted value of 6PN. Thus he study further the ilueces of structure ouiformity o the states of the Davydov solito with D i a sigle chai protei molecule usig Eqs.(3)-(4) ad the fourth-order Ruge-Kutta method [ ] i this case. As metioed above, protei molecules cosist of 0 differet amio acid molecules with weights betwee 75 m p (glycie) ad 04 m p (tryptophae), which correspod to weight variatios betwee 0.67 M ad 1.80 M. Therefore it is a aperiodic system, ad there is a disordered distributio i mass sequeces of amio acids. To study structure ouiformity arisig from the mass disorder distributio of amio acid residues, which results ecessarily i chages of characteristic parameters i the dyamic equatios i Davydov s model, Förer itroduced a radom umber geerator to create radom sequeces of the mass alog the chai. Disorder i the mass sequece destroys the solito oly for a very large disorder stregth, with M values 0.01M M 50M. For 0.01M M 10M the solito velocity is reduced from 0.73 to 0.59 km/s, the soud velocity to.1km/s. I the case of the mass variatio of atural amio acids ( 0.67M M 1.80M ) o chage i solito dyamics is virtually foud, thus the average mass approximatio is justiied. For the chage of w, he foud Published: April 09,

24 o chage i solito dyamics up to radom variatio of ± 0% w. For ± 30% w the solito velocity is somewhat reduced to 0.68km/s. Fially the solito disperses slowly ad its propagatio is irregular for ±40% w. For the luctuatios of J aloe or together with the atural mass variatio the solito is stable up to ±5% J. Therefore, Davydov solito is far more sesitive to variatio i J tha i other parameters. If i additio W is aperiodic, the solito is stable up to ±10% w, while at ±0% w slowly dispersive behavior appears. Fially, if χ aloe is 1, aperiodic, or if χ 1, is aperiodic together with the atural mass variatio, χ 1, ca be varied up to ± 0% 1 without destructio of the solito. However, if disorder i w is also itroduced, χ ca be varied up to ±15% 1, 1 ad w up to ±40% w. Fially if all four parameters are radomly varied the maximal possible disorder that would still allow the existece of a solito is ±0% w, ±.5% J, ad ±10% 1. For this disorder stregth he has calculated 10 differet radomly chose sequeces to id out whether the solito properties deped oly o the magitude of disorder or also o the idividual sequeces. He foud that oly the solito velocity is affected; i this case it varies betwee 0.61 ad 0.80 km/s. Figure 11 shows the effects of disorder i the sequece of masses of amio acid residues o the Davydov solito by Förer[59], i which the mass at site 100 is icreased to 100 M ad 00 M, respectively, but all other masses have bee still kept equal to M. We id that o visible perturbatio occurs i the solito motio, a small fractio of the soud eergy is trapped at the impurity ad the major fractio is scatterig back This deoted that a impurity at oe site, which may be some other molecules boud to the protei at this site (like reactive cetres as e.g. heme groups) does ot disturb obviously the solito, uless it does ot iluece the couplig costat. I igure 1 he gave the ilueces of a radom series of masses for the whole chai, k M, o the Davydov solito, where α is a radom-umber geerator with equal probability withi a prescribed iterval ad is i the rages of 0.67 k 1.80 ad 0.67 k 00. Figure 10 shows that the aperiodicity due to the irst smaller itervals for k does ot sigiicatly affect the Davydov solito motio, but i the case of the larger itervals, the solito disperses. (a) (b) Figure 11: The evolutio of Davydov solito as fuctio of site ad time i a sigle chai protei with a impurity at site 100 for differet impurity massesof 100 D (a) ad 00 M (b), respectively. (a) (b) Figure 1: The evolutio of Davydov solito as fuctio of site ad time i a sigle chai protei with a radom mass sequece D for differet itervals of the M, (a) ad 0.67 k 00 (b). k Published: April 09,

25 Figure 13 shows Förer s six results for the moved states of Davydov solito uder differet luctuatios of w, J ad χ 1 (or x k ) i the case of 0.67M M 1.80M ad 1 6PN [50,150]. Up to a radom variatio of ± 0% w, we see o chages i the dyamics of the solito. For ±30% w ( Figure 13a) the solito velocity is somewhat reduced to o.68km/s. Fially, for ±50% w (Figure 13b) the excitatio decreases slowly ad the propagatio is irregular. Figure 13c shows the state of the solito i the case of the luctuatios of ±5% J. ad ±10% w i which the solito is stable, but it disperses slowly at ±0% w as show i igure 13d. If disorder also i w is itroduced, the solito is stable, whe χ 1 ca be varied up to ±15% 1 ad w up to ±40% w (Figure 13e). If ially, all parameters are radomly varied the maximal possible disorder which still allows the existece of a solito is ±0% w, ±.5%, J ad ±10 1. Figure 11f shows a dispersive example of the solito at ±15% w, ±5%, J ad ±15% 1. The case of diagoal disorder ε he foud that for a isolated impurity i the middle of the chai (,100 ) the solito ca pass the impurity oly if ε <0.5 mev[50,150]. I other cases it is rejected or destroyed. I the case of a radom sequece (, 1, β is radom) oly for ε <1me ca the solito pass the chai. For higher values of ε the excitatio disperses quickly. Therefore, we ca obtaied from Förer s study that the Davydov solito i wave fuctio D remais stable agaist strog disorder i the sequeces of masses, sprig costats, ad couplig costats. But weak diagoal disorder ad small dipole dipole iteractio costats ca all destroy the solitos. I the meawhile, Förer [50,150-15] studied also the ilueces of structure ouiformity o the states of the Davydov solito with wave fuctio D 1, i which the quatum ature of the lattice plays a greater role tha i the classical D state,. Thus he obtaied that the Davydov solito appears oly from oliearities roughly 3 Figure 13: The states of motio of Davydov solito as fuctio of site(k) ad time(t) for some typical examples of ouiformity at 0.67M M 1.80M ad 1 6PN (see Förer s work [150]. Published: April 09,

26 to 4 times larger tha those i D models, for example the Davydov solito with D 1 asatz is stable at 1 =140PN, which is more larger tha the widely accepted value of 6PN. The sesitivity of such solitos to these disorders is practically opposite that for the D state. Thus it is ot ecessary to study further the ilueces of structure ouiformity of protei molecules o the states ad properties o Davydov s solitos. The results i Pag s model The results i sigle chai protei I order to study the iluece of a radom series of masses o Pag s solito, Pag itroduced a parameter α k, represetig the size of the disorder distributio, which is a radom-umber geerator with equal probability withi a prescribed iterval ad ca deote the mass at each poit i the molecular chai, i.e. M k = k M [38-46]. Numerical simulatio shows that whe the α k is small, such as, a iterval 0.67 M M k 1.80 M, the state of Pag s solito is ot ilueced. Up to the α k itervals of, for example, 0.67 α k 300, the stability of Pag s solito may still remai, but i the case of large itervals such as 0.67 α 700, the vibratioal eergy is dispersed, as show k i igure 14. The iterval of 0.67 α k 300, over which the motio of Pag s solito is uperturbed, is evidetly larger tha the above atural iterval of masses of amio acids. Thus, Pag s solito is very robust agaist mass disorder i proteis. However, i the above calculatio we do ot itroduced the luctuatios of the sprig costat w, dipole-dipole iteractio costat J, couplig costat (χ 1 + χ ), ad the groud state eergyε 0 arisig from the structure ouiformity. Accordig to the above Förer s method [50,150-15] the chages i parameters are represeted by the luctuatios of the average icremets, w=w- w, J=J - J, (χ 1 + χ )=(χ 1 + χ )-( 1 ) ad ε 0 = ε 0 -ε 0, respectively, where w, J,((χ 1 + χ ) ad ε 0 are the values of the parameters i the protei molecules with the structural disorders. However, for the variatio i groud state eergy arisig from imported impurities or from the mass disorder, we use geerally the radom umber geerator,, to desigate its radom feature, i.e., the ε0 is deoted by effects of the luctuatios of the parameters o Pag s solito. 0. I the followig we study the collective I igure 15. we show the chages i stability of Pag s solito with icreasig luctuatios of the sprig costat w i the case of a mass iterval of 0.67α k,. We see that up to a radom variatio of ±40% w, the dyamics of the solito have ot chaged. For ± 50% w, the solito disperses somewhat, but its velocity is oly somewhat dimiished, whe compared to the earlier w case. Fially, for ±70% w, the solito disperses ad its propagatio is irregular,as show i igure 15. (a) (b) Figure 14: The state of Pag s solito at 0.67<α K <300(a) ad at 0.67< D K <700 (b). Published: April 09,

27 (a) (b) (c) (a) Figure 15: States of Pag s solito i the case of %w (b), 60%w(c), 70%w (d). k ad chages W= 0%w (a), The solito i Pag s model [113-15] is also more sesitive to the variatios i the dipole-dipole iteractio costat J caused by the structural disorder whe compared with the other parameters. The simulatio shows that for a variatio i J aloe, the solito is stable up to ±9% J, ad it disperses at J =±15% J. If we simultaeously cosider the collective effects of disorder of mass sequece ad luctuatio of J o Pag s solito, the its state is obviously chaged. I igure 16 we show the collective effect of the luctuatios J =±10% J, =±0% J, o the solito i the case of a mass iterval of From these igures we see clearly that Pag s solito is stable at J10% J, but it disperses sigiicatly at J =±0% J. The umerical calculatio shows that, arisig from the disorder of structure, if the couplig costat ( 1 + ) aloe is chaged, ( 1 + ) ca be varied to ±5% ( 1 ), ad i this case Pag s solito does ot disperse. However, for a luctuatio together with atural mass variatio, the stability of Pag s solito will be chaged. I igure 17 we illustrate the chages i the states of ew solitos with icreasig luctuatios of ( 1 ad ) at 0.67 M M M. We see from this igure that oly at ( 1 + )< 30%( 1 + ) Pag s solitos are stable, but they obviously disperse at ( 1 + )=35% ( ). 1 It is quite ecessary to collect the combied effect of radom variatios of the above physical parameters resultig from structural disorders o the properties of Pag s solito. The chages of states of solitos with icreasig iteractio costats i the case of 0.67 k ad J =±10% J is show i igure 18. We id o chage i the dyamics of Pag s solito at w=± 5% W ad J = ± 10% J, but it begis dispersig at w= 30%w, J 15% J, ad disperses cosiderably at w= 5%w, J 15% J. We are more iterested i the collective effect of simultaeous radom-variatios of the above ive parameters resultig from structural ouiformity o Pag s solito. I geeral, Pag s solito is very sesitive to the diagoal disorder, which is the chage i groud state eergy, 0, caused by differet side groups of amio acids ad local geometric distortios due to the impurities imported. We foud that for a isolated impurity i the middle of the chai, which causes the chage i the groud eergy to be 0 =, the solito ca pass the impurity oly if <1meV. I other cases it Published: April 09,

28 (a) (b) Figure 16: The states of Pag s solito i the case of the mass iterval 0.67 J 10% J (a) ad J 0% J (b). k as it chages at (a) (b) Figure 17: The states of Pag s solito i the case of the mass iterval % 1 (a) ad 35% 1 (b). k as it chages at (a) (a) (b) (b) (c) Figure 18: The stability of Pag s solito at 0.67 k ad ( 1 ) 10% 1 ad w 30%w, J 15% J (a) ad w 5%w, J 15% J (b) ad w 5%w, J 10% J (c). is relected or dispersed. I the case of a radom sequece, ε 0 =, 0.5, is a radom parameter, ad oly whe ε<1mev ca the solito pass the impurity to propagate alog the chais. For higher values of ε the solito is dispersed. Whe the diagoal disorder occurs together with luctuatios of the other four parameters, the state of Pag s solito is chaged obviously. The states of solitos with a icreasig sprig costat i the cases of 0.67 α, ( + )= 5% ( k 1 1 ), J=±5% J ad ε 0 =, 0.5, ε<1mev are give i igure 19. We see from this igure that whe the ive parameters are all radomly varied, a maximal disorder that would occur still with the solito i motio, is w=±10% w, J=±5% J, ( 1 + )= ±5%( 1 ), 0.67 M M M ad ε 0 =, 0.5, ε=1mev. I other cases the solito is dispersed or relected by the impurity. I summary, we have looked at the stability of Pag s solitos uder the iluece of various structural disorders i the proteis via the fourth-order Ruge-Kutta method [3-4] which ca help idetify ad check the stability of state of Pag s Published: April 09,

29 (a) (b) Figure 19: The states of Pag s solito at 0, 0.5, 1meV ad J 5% J, ( 1 ) 5% 1,0.67 k for w 10%w (a), w 0%w (b). solito trasportig bio-eergy. The above results obtaied show that the solito i Pag s model [113-15] is very robust agaist these structural disorders. However, the Davydov solito with oe quatum is ot so, ad smaller structural disorders i the proteis will destroy its stability as metioed above, which are obtaied by Forer er al. [50,149-15], what are the reasos why the solito i Pag s model is comparably more stable tha Davydov s? Clearly, this is due to the fact that the Pag model with a quasi-coheret two-quata state differs from the Davydov model with oe quatum state. Although Eqs. (5)- (7) ca become the dyamic equatios i the Davydov model whe =0 ad a are replaced by A (i such a case, the ormalizatio coditio of the Davydov wavefuctio the becomes A ( t) a ( t). This agai shows clearly that the ew wavefuctio i Pag model cotais exactly two quata), the stability of Pag s solito is greatly icreased due to cosiderable icreases of oliear couplig eergy, G p, ad a bidig eergy, E BP, which are larger by about three ad twety times tha Davydov s solito[113-15]. The results i α-helix protei molecules with three chaels i Pag s model. We ivestigated further the properties of Pag s solito i α-helix protei molecules with three chaels by fourth order Ruge-Kutta method [ ] ad.usig the above methods. (1) The idividual ilueces of differet luctuatios ad disorders o the motio of Pag s solito. () I accordace with the above method we study the ilueces of the radom series of masses, i which whether of the smaller or larger itervals for α k, such as, 0.67 α k 100 or 0.67 α k 300, o Pag s solito. This result maifests that this ouiformity does ot sigiicatly affect the stability of Pag s solito, its vibratioal eergy is ot dispersed as show i igure 0, where other parameters take mea values metioed above. The iterval over which Pag s solito moves uperturbed is evidetly larger tha the variatio i the masses of the atural amio acids (0.67 α k 1.80). Therefore, the solito i Pag s model is very robust agaist mass ouiformities i α-helix protei molecules [113-13]. Numerical simulatio results show that we do ot id ay chages i the dyamics of the solito up to a radom luctuatio of ±5% W. The results of our calculatios are show i igure 1, i the cases where luctuatios are ±15% W ad±5% W, respectively. For ± 30% W, Pag s solito velocity is oly somewhat dimiished whe compared with the case of W. Fially, for ±40% W, Pag s solito disperses completely ad propagatio is irregular. The solito is very sesitive to variatios i the dipole-dipole iteractio J i the α-helix protei molecules. The solito is stable up to 5% as show i igure, where we deote the features of the solitos uder chages of ±3% J ad J=±5% J, but the solito disperses at J=±7% J. Published: April 09,

30 (a) (b) Figure 0: The states of Pag s solitos i the ouiform masses of 0.67<α K <100 (a) ad 0.67<α K <300(b). (a) (b) Figure 1: The states of Pag s solitos at the fluctuatios of W=±15% W (a) ad W=±5% W (b). (a) (b) Figure : The features of Pag s solitos uder the chages J=±3% J (a) ad J=±5% J (b). I igure 3 we exhibit the chages of feature of Pag s solito uder the iluece of luctuatios of Δ( 1 + )=±6%( 1 ) ad ±9%( 1 ).These igures show that Pag s solitos are stable whe ( 1 + ) varies up to ±9%( 1 ). I the meawhile, Pag s solitos disperse ad split gradually ito some small waves whe ( 1 + ) are larger tha this value. With β 0.5 for ε==0.57mev ad 1.6meV, we show the behaviors of Pag s solito i igure 4. These igures show that i the case of a radom sequece, oly if ε<1.3mev ad β 0.5 ca Pag s solito pass through the chai ad is stable. For higher values of ε, Pag s solito is relected or dispersed. I igure 5 we show the variatios i features of Pag s solitos due to the luctuatios of ΔL=±5% L ad ±6% L. These igures show that the solito is stable oly if ΔL<±7% L. From the simulatio we kow that the smaller the luctuatio of the chai-chai iteractio, the higher the stability of Pag s solito. Whe ΔL=0 ad L=0, the stability of Pag s solito is the highest. (3) Associated effects of the luctuatios of six structural parameters o the Pag s solitos. We studied the associated ilueces of the chage of mass of amio acids ad the luctuatios i dipole-dipole iteractio, the excito-phoo couplig costat, the sprig costat, the diagoal disorder, ad the chai-chai iteractio arisig from the structure disorder of the α-helix protei molecules, o the behaviors of Pag s solito. Published: April 09,

31 (a) (b) Figure 3: The chages of phase of Pag s solito as affected by fluctuatios of ( 1 + )=±6%( 1 ) ad ( 1 + )=±9%( ), respectively. 1 ( a) (b) Figure 4: The behavior of Pag s solitos due to the chages ε 0 =ε β with β <0.5 ad ε=0.57mev(a) ad 1.6meV(b), respectively. (a) (b) Figure 5: The variatios of feature of Pag s solitos i fluctuatio of L= ±5% L (a) ad ±6% L (b). I igure 6 we show the chages of property of Pag s solitos uder the iluece of differet luctuatios of structure parameters, i which igure 6a deotes the state ad behaviour of Pag s solito with luctuatios of 0.67<α K <, ΔW=±5% W, ΔJ=±1% J Δ( 1 + )=±1%( 1 ) ad ΔL= ±1% L ; igure 6b results whe 0.67<α K <, ΔW=±10% W, ΔJ=±% J Δ( 1 + )=±3%( 1 ) ad ΔL= ±% L ; igure 6c results whe 0.67< α K <, ΔW=±10% W, ΔJ=±% J Δ( 1 + )=±5%( 1 ) ad ΔL= ±% L ; igure 4d is the result whe 0.67<α K <, ΔW=±10% W, ΔJ=±% J Δ( 1 + )= ±6%( 1 ) ad ΔL= ±% L. From these igures we see clearly that Pag s solitos show i igure 6a, b ad c are very stable uder the actios of differet structure disorders, but Pag s solito show i igure 6d is already ustable due to the iluece of the structure disorder, its amplitude chages, thus it is already dispersed. This maifests that the solitos excited i -helix protei molecules are ot stable i cases of large structure disorders. I comparig these results i igure 6 with those show i igure 1-5, we id that the collective effects of these structure disorders ad quatum luctuatios chage the states ad features of Pag s solitos, makig their amplitudes, eergies ad velocities decrease, although such ilueces caot destroy the solitos, which ca still trasport steadily alog the molecular chais while retaiig eergy ad mometum whe the quatum luctuatios are small, i.e., Pag s the solitos are quite robust agaist these disorder effects. However, Pag s the solitos may be dispersed or disrupted i cases of very large structure disorders. (4) The collective effects of various ouiformities o the motio of Pag s solito. We iwwally study the collective effects of the chage i mass of amio acids ad the luctuatios of dipole-dipole iteractio, the excito-phoo couplig costat, the Published: April 09,

32 (a) (b) ( c) (d) Figure 6: The chages i properties of the solitos uder the ifluece of differet fluctuatios, here (a) is the result whe 0.67<αK <, W=±5%, J=±1% J ( 1 + )=±1%( 1 ) ad L= ±1% L ; (b) is the result whe 0.67<α K <, W=±10% W, ( 1 + )= ±3%( 1 ), J=±% J ad L= ±% L ; (c) is the result whe 0.67< α K <, W=±10%, J=±% J ( + )=±5%( 1 1 ) ad L= ±% L ; ad (d) is the result whe 0.67<α K <, W=±10% W, J=±% J, ( 1 + )=±6%( 1 ) ad L= ±% L. ε=0.35mev, β <0.5. sprig costat, the diagoal disorder ad the chai-chai iteractios arisig from the structure disorder of α-helix protei molecules, o the properties of Pag s solito. The results obtaied are deoted i igure 7, i which igure 7a represets the features of Pag s solito formed uder luctuatio coditios of 0.67<α K <, ΔW=±3% W, J=±1% J Δ( 1 + )=±1%( 1 ), ΔL= ±1% L ad Δε 0 =ε β, ε=0.35mev, β <0.5; igure 7b results whe 0.67<α K <, ΔW=±6% W, J=±1% J, Δ( 1 )= ±%( 1 ), ΔL= ±1% L ad Δε 0 =ε β, ε=0.1mev, β <0.5; igure 7c results whe 0.67< α K <,ΔW=±8% W, J=±1% J Δ( 1 )=±3%( 1 ) ad ΔL= ±1% L ad Δε 0 =ε β, ε=0.1mev, β <0.5. We see from this igure that i these coditios the itrisic ature of Pag s solito durig bio-eergy trasport i the -helix protei molecules ca still be maitaied, but the solito begis to disperse whe these structure disorders are larger tha the values 0.67< α K <, ΔW=±8% W, ΔJ=±1% J Δ( 1 )=±3%( 1 ) ad ΔL= ±1% L ad Δε 0 =ε β, ε=0.1mev, β <0.5. Therefore, we ca coclude that the solito i Pag s model is exactly robust agaist various structure disorders of the α-helix protei molecules. Hwowever, the actual degree of structure ouiformity or disorder i protei molecules ukow up to ow. The structure ouiformity i masses of amio acid residues should smaller tha the mass iterval of the atural amio acid residues sice the proteis are a biological self-orgaizatio with higher order, i which the amio acid residues are ot free particles but are covaletly boud i the mai polypeptide chai. The structure ouiformity i the other parameters should be small due to small ilueces of the side groups o the geometry of the mai chai ad disorder distributio of the mass. Thus we ca coclude that the aturally occurrig the structure ouiformity i the parameters should be smaller tha the maximal structure ouiformity i which the Pag s solito, is stable. Natural structure ouiformity may iterfere with the solitos oly whe J ad ε are obviously varied sice the stability iterval i these cases is smaller relative to other parameters. However, the luctuatios of J ad ε for Pag s solito are also larger as metioed above. Thus it is ot ecessary to doubt that the structure ouiformity or disorder i protei molecule ca destroy the states of solito i Pag s model, thus the solito trasport of bio-eergy ca still maitai although the ilueces of atural structure ouiformity or disorder o it occur i biological protei molecules. The iflueces of temperature of systems o the eergy trasport i protei molecules Published: April 09,

33 (a) (b) (c) Figure 7: The features of the solito uder the ifluece of differet structure disorders, where (a) results i the case of 0.67<α K <, W=±3% W, J=±1% J ( 1 )=±1%( 1 ), L= ±1% L ad ε 0 =ε β, ε=0.35mev, β <0.5; (b) is the result i the case of 0.67<α K <, W=±6% W, J=±1% J ( 1 )=±%( 1 ), L= ±1% L ad ε 0 =ε β, ε=0.1mev, β <0.5; ad (c) is the result whe 0.67< α K <, W=±8% W, J=±1% J ( 1 )=±3%( 1 ) ad L= ±1% L ad ε 0 =ε β, ε=0.1mev, β <0.5. As it is kow, the thermal stability ad lifetime of the solitos, which trasport the bio-eergy i protei molecules, at the physiological temperature 300K are a key idicator to ispect ad check the validity of the theory of bio-eergy trasport ad true existece of solitos. Therefore, plety of researchers study this problem usig differet methods. The followig statemets are a simple survey o these ivestigatios. Ivestigatios o thermal stability of the Davydov solitos i rage of physiological temperature i uiform protei molecules First of all, Davydov ivestigated [153] the ilueces of thermal perturbatio at 300K o the solito through the thermodyamic average of Hamiltoia of the protei molecules, i which he thought that the thermal perturbatio results i the thermal excitatio of the phoos i this system. Thus the Davydov s wave fuctio, D, i Eq.() was replaced by 1 D1 ( t) ( t), where. If the Hamiltoia of the system 1/ q { q} q ( q!) ( aq) 0 q q ph i Eq.(1) i Ref.[1].is represeted by the creatio ad aihilatio operators of the excito ad phoo, Davydov deoted the thermodyamic average of the Hamiltoia by H' H, where ex it H ( H H ) U* H phu with ad () t U 0 ph * exp{ q () taq q () taq }0 q. Utilizig familiar method Davydov obtaied the ph dyamic equatio of the solito i this case. From this dyamic equatio Davydov obtaied that the effective mass ad oliear iteractio eergy deped o the temperature of the system, i.e., with icreasig temperature the effective mass of Davydov solito icreases ad the oliear iteractio eergy decreases, the solito s size icreases ad its properties come ever closer to these of a excito. Meawhile he obtaied also the depedece of the solito parameters o its velocity. Whe v< v o ad v approaches v 0 (v 0 is the soud speed of protei molecules), the oliear iteractio eergy icreases. At suficietly high temperatures, a loss of localizatio of Davydov solito occurs. I this case the solito eergy icreases ad its size decreases. Subsequetly, L. Cruzeiro et al. [154,] used also the thermodyamic average Hamiltoia i H HT metioed above, but H D1 ( H H H ) D1. From agai H H they obtaied the dyamic equatios of ad q, respectively, T ad q T * i t t * q * q q U ( aq aq ) U where, which are all the oliear Schrodiger equatios. They foud further out the solutios of the two equatios usig umerical ex it ph Published: April 09,

34 simulatio method. The results obtaied for the probability of a solito excitatio at physiological temperature 310K are show i igure 8, where M=114m P, J=7.8cm -1 ad w=13n/m are used, but 0 =1666cm -1 is removed from the umerical calculatios through a gauge trasformatio, the iitial coditio used are 4(0) 4(0) ad q (0) 0. Figure 8(a), (b) ad (c) are the results of N, N 10 ad N, respectively. Figure 8(a) shows a case i which the iitial excitatio is completely dispersed after 10 psec. Ideed, oly dispersive waves, which travel at about 1/3 the maximum speed of 3 soud i the chai [ v0=r0 w/m = m/sec ] is formed ad is ot accompaied by ay molecular displacemet. They are therefore excitos. Excitos are geerated which travel i opposite directios i the chai ad iterfere with each other 10 psec afterwards. This iterferece is a artiicial pheomeo due to the periodic boudary coditios. Figure 8(b) shows aother situatio i which part of the iitial excitatio is ot dispersed ad remais pied i the same bods where it was iitially located. Fially, i igure 8(c), where most of the iitial excitatio is ot dispersed but remais localized where it was put iitially. This result shows that the cotiuous trasitio from dispersio (igure 8(a) to a localized state (igure 8(c)) of the iitial excitatio is observed at biological temperatures 310K. I this case, the cosequece of icreasig temperature is similar to that of icreasig the oliearity, i.e., as temperature icreases, the threshold for localizatio of the iitial excitatio decreases. Icreasig the temperature thus produces a decrease i the effective dispersio, as was poited out by Davydov [153]. Thus they obtaied that the Davydov solito is stable at 310 K. Cruzeiro-Hasso et al[4,45,155]studied also the iluece of temperature of the system o the states of Davydov solito. Their method is to determie a appropriate feature of the system, such as localizatio, umber probability () t ad amio acid s displacemet ( un/ un/ 1), to select a appropriate observable Q capable of describig the feature, ad to compute from Gibbs statistical-mechaics prescriptio the equilibrium average value of Q at physiological temperature regio by Q dydtrq [ exp( H / KT B )] / dydtr[exp( H / KT B )], where K B is the Boltzma costat. The itegratios are over all the vibratioal coordiates ad mometa, ad the trace is over the quasiparticle states. Their results show that thermal destabilizatio is ot associated with trasitios to excito states ad that it ivolves istead disordered states with eergies itermediate betwee these of the miimum (a) (b) (c ) Figure 8: The evolutio of the probability of Davydov solito i the time ad space at 310K (see Cruzeiro et al, paper [45]). Published: April 09,

35 eergy solito state ad those of the excito states. A basic duality emerges regardig the effect of temperature o the stability of oliear structures: temperature is foud to help the oliearity i certai parameter ad temperature regimes by iducig disorder ad to destroy the oliearity i other regimes, e.g., always at large temperatures as a cosequece of Boltzma equalizatio. Thus they obtaied that the Davydov system exhibits uiversal behavior at physiological temperatures. I the meawhile, Bolterauer [7,156] argued that their classical thermalizatio scheme should lead to icorrect results whe applied to a quatum system. Bolterauer ad other workers foud the Davydov solitos to be stable at T =300 K. As it is kow, Davydov s model yields ideed a compellig picture for the mechaism of bio-eergy trasport i protei molecules, but there are cosiderable cotroversy cocerig whether the Davydov solito is thermal stable at 300K ad ca provide a viable explaatio for bio-eergy trasport. It is out of questio that the thermal perturbatios are expected to cause the Davydov solito to decay ito a delocalized state. The above simulatios showed that the Davydov solito is stable at 300K [ ], but they were based o classical equatios of motio which are likely to yield ureliable estimates for the stability of the Davydov s solito. Sice the dyamical equatios used i the simulatios are ot equivalet to the Schr o diger equatio, the stability of the solito obtaied by these umerical simulatios is uavailable or ureliable. Lawrece et al. [69,157], proposed a model icorporatig excito-phoo iteractio as a mechaism for localizig ad stabilizig eergy trasport i the logchai proteis. This is due to the fact that previous aalytical ad umerical studies [9-1,15,16,153,154] have ot adequately addressed the effects of thermal phoos, which may act to disperse excito eergy. Thus they iished a umerical calculatio, i which they idicated that the excitatios are strogly dispersed at physiological temperamet. They thik that the propagatio of the excito-phoo state at low temperatures makes a trasitio from a solitary wave mode to a statioary self-trapped mode as the couplig betwee excitos ad phoos is icreased. Thus they obtaied a ew result of calculatios of excito-ormal-mode couplig i the formamide dimer, which idicate that more sophisticated models are ecessary to yield the true couplig costat i proteis. Thus they calculated a parametric study of the Davydov model of eergy trasport i alpha-helical proteis. Previous ivestigatios have show that the Davydov model predicts that oliear iteractios betwee phoos ad amide-i excitatios ca stabilize the latter ad produce a log-lived Davydov solito, which propagates alog the alpha-helix proteis. The dyamics of this solitary wave are approximately those of solitos described usig the oliear Schr6diger equatio. However, the preset study based o the ew calculatio exteds these previous ivestigatios by aalyzig the effect of helix legth ad oliear couplig eficiecy o the phoo spectrum i short ad medium legth alpha-helical segmets. The phoo eergy accompayig amide-i excitatio shows periodic variatio i time with luctuatios that follow three differet time scales. The phoo spectrum is highly depedet upo chai legth but a majority of the eergy remais localized i ormal mode vibratios eve i the log chai alpha-helices. Variatio of the phoo-excito couplig coeficiet chages the amplitudes but ot the frequecies of the phoo spectrum. The computed spectra cotai frequecies ragig from 00 GHz to 6 THz, ad as the chai legth is icreased, the log period oscillatios icrease i amplitude. The most importat predictio of their study, however, is that the dyamics predicted by the umerical calculatios have more i commo with dyamics described by usig the Frohlich polaro model tha by usig the Davydov solito. Thus, they ially cocluded that the relevace of the Davydov solito mode, which was applied to eergy trasport i alpha-helical proteis, is questioable. Published: April 09,

36 Lomdahl et al. [158], ivestigated the temperature effect o the Davydov solito through addig the dampig force ad oise force, f mu F t, ito the discrete dyamic equatio (0)of displacemet of amio acids, u() t, ad let L 0 i Eq.(19). I accordace with statistical physics, the thermal oise term F (t) is related with the temperature of the systems, its correlatio fuctio ca be represeted by F( x, t) F(0,0) MK BT ( x) ( t) / ', where ' is the dampig costat, T is the temperature of the system. It is assumed that the radom deviatio of the oise obey the ormal distributio with criterio deviatio ad a zero expectatio value. That is, 1 NF ( ) exp[ F / ], where σ=mk B TΓ/τ,τ is the time costat, ad Γis the reciprocal time costat of L 1 the heat bath. It ca be show that F ( t) [ X r ( t) ], where X r1 r (t) is a radom umber 1 betwee 0 ad 1. If we choose L=1, the the deviatio of [ X ur ( t) ] is 1/1, ad the stadard deviatio of F (t) is. The domai of the radom oise force is just F ( t) 6. I the meawhile, they veriied umerically to high accuracy that over suficietly 1 1 log time itervals this gives for the mea kietic eergy mu t NKBT. They used these equatios to study the properties of solutio of the dyamic equatios by the solito detector. To check the cosistecy of the result, the calculatios were also doe i the covetioal microcaoical esemble. The system was prepared at T=300K, it was the allowed to evolve oly uder the iluece of the dyamic equatios. The result of these simulatios shows that the Davydov solito is istable at 300K ad disappear i a few picosecods as show i igure 9, which is similar with the results obtaied by Lawrece et al. [69,157]. Lomdahl et al. [158], give further the states ad features of Davydov solito cotaiig Q' D BB D ad 6 quata uder ilueces of dampig force ad thermal oise at T=300K usig the above method, which are show i igures 9,30, respectively. These results idicate clearly that the Davydov solito formed i these coditios caot still be thermally stable at biological temperature 300K l, whether it cotais oe quatum or two or six quata. O the other had, Förer s [150] used the above Lomhahl et al s method, formulae ad the fourth-order Ruge-Kutta method [ ] to calculate the chages of stability of the Davydov solito with varyig temperature of the system. They id that the Davydov solito is oly thermally stable at T<40K, but begis to disperse at 40K, ad is destroyed completely at 300K, as show i igures Therefore the Davydov solito is ot thermally stable at 300K; its critical temperature is oly 40K. These are cosistet with the aalytic results [ ]. Wag et al. [64], studied the effects of thermal perturbatio o properties Figure 9: The solito detector of Davydov model(q =) at T=300K([69]). Published: April 09,

37 Figure 30: The solito detector of Davydov model(q =6) at T=300K([69]). Figure 31: The state of the Davydov solito at 30K. Figure 3: The state of Davydov solito at 40K (See Ref.[150]) (See Ref.[150]). Figure 33: The state of Davydov solito at300k[150]. of Davydov solito by quatum Mote Carlo approach. I this case, the thermal equilibrium expectatio value of ay physical observable Q i the caoical esemble is give Q Tr[ Qexp( H)] / Tr[exp( H), which is formulated by Feyma s path itegral for the quatum propagator with imagiary time. The partitio Published: April 09,

38 fuctio Z Trexp( H)] ca be calculated by irst dividig the imagiary time ito L itervals / L. Betwee each pair of itervals they isert a complete set of basis states, represetig all L-polygoal arcs through the Hilbert space coectig the iitial ad ial imagiary times. The basis states are products of excito umber states ad the positio eigestates of the lattice. Usig the checkerboard decompositio techique, they separated the Davydov Hamiltoia i Eq.(1) ito two parts, H=H [P ]+H 1 [u ], which P M u, H( P ) cotais all lattice mometum operators P ad all excito hoppig terms from eve-umbered sites, ad Hu ( ) cotais all lattice positio operators together with all excito hoppig terms from odd-umbered sites. Thus they foud out. M u u w z du u u Tr J B B L i1 i i exp{ [ ( ) ( 1i i ) ]} exp[ ( 1 i i 1 eve 1 1 1i 1i i i odd BB )] exp[ { [( u u ) BB ( u u) B B )] JB ( B BB )}], Where i u i is the positio of the th lattice mass at the ith cut. The above formula is just the basis of quatum Mote Carlo simulatio. To utilize this formula Wag et al gaied the followig coclusios: (1) A coheret structure exists for temperatures below 7 K; () the basic uit of this coheret structure is highly localized ad bears a close resemblace to the Davydov solito if discreteess correctios to the latter are take ito accout; ad (3) above 7 K thermal perturbatio are effective i destroyig the iteral coherece of this basic uit, its destructio beig essetially complete above 11. K. These results are largely cosistet with the above dyamical simulatios of Lomdahl ad Kerr based o D asatz states ad Cottigham et al. s straightforward quatum-mechaical perturbatio calculatio [66], i which the lifetime of the Davydov solito obtaied by usig this method is too small (about ) to be useful i biological processes. A major differece is that their method gave quatum luctuatios, the equilibrium quatity is ot seriously affected by quatum luctuatios, but it is likely that dyamical properties would be affected by the presece of itrisic quatum oise. At the same time, Förer[150] studied further the behaviors of the Davydov solito usig the perturbatio method, i which he diagoalize the Hamiltoia partially by Cottigham et al., way [66]. Thus the Hamiltoia of the systems is deoted as H=H 0 +V, here H 0 is the diagoal part of H ad V the o-diagoal part. Thus he treated V as a perturbatio ad thought that D is a exact eigestate of H 0 ad calculated further the lifetime of Davydov solito by irst-order perturbatio theory i a cyclic chai of 01 uits usig the symmetric iteractio, where oly pied solito were foud umerically, i which 1 6 pn,w=13n/m, M=114m p, J=0.967meV,ad ( t 9) A sec h [( 100) 1 / wj ] as the iitial coditios are used. I Figure (34), exhibited the states ad chages of the Davydov solito ad wt () D / at 0.meV as the time chages at 300K, where D / exp( it/ ) Asec h[( 0) 1 / wj.these results idicate clearly that the Davydov solito is still ustable at 300K. Summarily, we caot demostrate still that the Davydove solito is thermally stable, although may differet techiques ad methods of thermal perturbatio of eviromet were used. Therefore we ca coclude from the above ivestigatios that the Davydov solito is ustable at physiological temperature 300K, thus it caot play a importat role i biological processes of bio-eergy trasport i the protei molecules. The iflueces of temperature of proteis o bio-eergy trasport i Pag s model The results i a sigle chai: Sice protei molecules work always at a biological temperature of 300K, we should irstly decide whether or ot the thermal motio of the Published: April 09,

39 Figure 34: The Davydov solito detector plot at 300K (a) ad Time evolutio of w(t) at J=0.meV(b) ([150]). lattice still permits solito motio i such a case. I accordace with thermodyamic theory, from a C=O vibratioal eergy of ε 0 = Nm ad a temperature of 300K, oe ids that the Boltzma factor is i this case. This meas that oly three out of 10,000 excitos are excited i the thermal equilibrium at 300K. The oe ca safely assume that the temperature of medium primarily affects solito motio via oly the lattice of the amio acid residue. Thus, prior to solito commecemet the system is i equilibrium with the temperature of eviromet ad the lattice is i thermal motio (which ca be described as a liear combiatio of its ormal modes), while the excito system is i its groud state. With solito commecemet a oequilibrium state is created, the state of the lattice chages, ad the behavior of the solito is ilueced by the thermal perturbatio via the lattice. This amouts to assume that the time the solito eeds to travel through the protei is small compared to the time the temperature of eviromet eeds to re-establish equilibrium with the system due to high velocity of the solito. However, how do we represet the effect of the heat bath o the lattice? Here we [113-15] adopt the above Lomdahl ad Kerr s [64,158] ad Lawrece ad co-worker s [69,157] method, which foud that the Davydov solito is destroyed at 300K by usig its dyamic equatios, to study the ilueces of the temperature of the systems o bio-eergy trasport i Pag s model.. Thus we added the decay term Mu ad radom oise term, F (t), resultig from the temperature ad dampig of medium, ito the displacemet equatio of the amio. acid molecules, i Eq. (7), where Mu ad F (t) are decided by the above formulae, Г is a dissipatio coeficiet for the vibratio of amio acids, which is about 10 8 s -1 for the proteis. Thus we ca study the ilueces of dissipatio force ad radom oise forces, arisig from the heat bath, o the states of Pag s solito by the fourth-order Ruge-Kutta method [ ]. Whe the solito starts at t=0, the iitial coditio of a (o)=asech[(- 0 ) ( 1 ) /4J w] is added i oe ed of the protei molecule, the lattice eergy luctuatios associated with the eviromet are larger by roughly three orders of magitude tha the local lattice eergies associated with the solito motio, but the solito ca still moves through the chai completely udisturbed at the biological temperature (300K) as show i igure 35, where the above average values of the physical parameters of proteis were used. Therefore, despite the large lattice-eergy luctuatios due to the heat bath, the oliear iteractio betwee the lattice ad the amide oscillators (excitos) is still able to stabilize the solito, or i other words, the thermal perturbatio at 300K caot destroy Pag s solito. The state of the solito i the case of a log time period (300ps) at 300K ad a higher temperature of 310 K as show i igures 36,37, respectively. These results [113-15] show clearly that Published: April 09,

40 the solito i Pag s model is thermally stable i the regio betwee 300K to 310K. The lifetime of Pag s solito is at least about 300ps at 300K. This meas that Pag s solito has a log eough lifetime eablig it to play a importat role i biological processes. These results agree with aalytic data i table I Eq.(19). However, at the high temperatures of 30K ad 35K, Pag s solito disperses gradually as show i igure 38,39, respectively. Thus the critical temperature of Pag s solito is about 35K. Therefore Pag s solito is thermally stable at 300K.These results are evidetly differet from those of Davydov solito i igures obtaied by FÖrer [150-15]. The results i -helix protei molecules: For the α -helix protei molecules the ilueces of biological temperature o the bio-eergy trasport ca be studied i accordace with the above method, i which we still add the decay term M q. ad radom oise term, F (t), resultig from the temperature ito the displacemet equatio of the amio acid molecules i Eq.(31) accordig to the above way ad Figure 35: The behavior of Pag s solito. Figure 36: The state of Pag s solito through at 300K the time period of 300ps at 310K. Figure 37: The state of Pag s solito at 310 K. Published: April 09,

41 Figure 38: The state of Pag s solito at 30K. Figure 39: The state of Pag s solito at 35K. Lomdahl ad Kerr s method[64,158]. We foud the solito solutio of the equatios of motios i Eqs.(30) ad (31) with decay effect ad radom oise force usig the above method ad the fourth-order Ruge-Kutta method[ ]. The results at 300K are show i igure 40 for the α -helix protei molecules with the above iitial coditios, which are simultaeously liked o the irst eds of the three chaels. From this igure we ca see that the solito i Pag s model ca still move alog the three chaels at a costat speed ad amplitude without dispersio. So Pag s solito is thermally stable at 300K. I igure 41 we also show the results of solito motio at a log time period of 10Ps ad large spacigs of 1000 sites (i.e., 333 amio acid residues are cotaied i each chael) at 300K for the α -helix protei molecules, whe the above iitial coditios are simultaeously liked o the irst eds of the three chaels. We see from this igure that the solitos are udisturbed i such coditios, ad really move over a log time period ad through large spacigs alog the protei molecular chais while retaiig their amplitudes ad velocities at bio-temperatures. I igure 4 we plot the collisio behaviors of the solitos with clock shapes, set up from opposite eds of the chaels i the α -helix protei molecules, whe the above iitial coditios are simultaeously liked o the opposite eds of the three chaels. From this igure we see clearly that the iitial two solitos with clock shapes, separatig 100 amio acid spacigs i each chael, collide with each other at about 16ps. After the collisio, the two solitos i each chael go through each other still retaiig their clock shapes ad propagatig toward ad separately alog the three chais. These results show clearly that although there are large lattice luctuatios i the protei molecules due to the iluece of temperature, the oliear iteractio betwee the amio acids ad excitos is still able to stabilize the solito, therefore Pag s solito is very robust agaist thermal perturbatio of the eviromet. The, the lifetime of Pag s solito is, at least, 10Ps. This meas that Pag s solito ca play a importat role i biological processes. Published: April 09,

42 Figure 40: The behaviors of Pag s solito. Figure 41: The state of the solito i log-time at biological temperature of 300K. motio at 300K. Figure 4: The properties of collisio of the solitos at 300K. What does this mea? As metioed above, the characteristic uit of time for the 1 13 theories is 0 r0 / v0 ( M/ W) s, ad L / r0 100, / is a reasoable criterio for the solito to be a possible mechaism for eergy trasport i proteis, where is the lifetime of the solito. 300 ps meas / Thus, Pag s solito is very thermally stable at 300K, i.e., Pag s olito has a log eough lifetime eablig it to play a importat role i biological processes. We studied also the chages of states of the solitos with icreasig temperature. I igure 43 we exhibit the trasport properties of Pag s solito i the α-helix proteis at differet temperatures of 95K, 305K, 310K, 315K, 30K ad 35K, respectively. These igures show that the amplitudes of the solito decrease with icreasig temperature, ad it begis to disperse at 30K. This meas that Pag s solito must exped a part of itself eergy to retai ad suppress the icreasig destructive effect of thermal Published: April 09,

43 perturbatio arisig from the rise i temperature of the system, thus its amplitude or eergy does decrease. Thus we estimated that the critical temperature of Pag s solito is about 30K i this case. At the same time, we foud that the trasported velocity of Pag s solito also decreases with the icrease i temperature of the system I table we give cocrete data of these velocities of Pag s solito at differet temperatures. Otherwise, the temperature-depedece of velocity of Pag s solito is show i igure 44. Here, the decrease i solito velocity with icreasig temperature is also obvious. Evidetly, this is due to the ehacemet of disordered thermal motio of the medium resultig from the rise i temperature, which icreases the resistace of motio of the solito. Thus, the velocity of Pag s solito ecessarily decreases. The above results of our ivestigatio maifest clearly that the oliear iteractio betwee the amio acids ad excitos is still able to stabilize the solito, although it udergoes destructive ilueces due to icreases of thermal perturbatio. Therefore, the solito is very robust agaist the iluece of thermal perturbatio of the eviromet. (a) (b) (c) (d) (e) (f) Figure 43: The chages of state of the solitos with icreasig temperatures of the α-helix protei molecules at temperatures of 95K(a), 305K(b), 310K(c) 315K(d), 30K(e) ad 35K(f), respectively. Figure 44: The temperature-depedece of velocity of the solito excited i the -helix protei molecules. Published: April 09,

44 Table : The values of velocity of the solitos i the α-helix protei molecules at differet temperatures. Temperature (K) velocity(m/s) The simultaeous effects of structure ouiformity, dampig ad temperature of protei molecules o the eergy trasport The results i sigle chai protei i Pag s model: However, the iluece of structure ouiformity or disorder i the protei molecules o the Pag s solito has ot bee cosidered i the above calculatios. I practice, the structure ouiformity arisig from the disorder i the mass sequece of amio acid molecules, side groups ad imported impurities is always existet i the proteis. Therefore, it is quite ecessary to further study the iluece of structure ouiformity o Pag s solito i the regio of K usig the above method [114-15]. The behavior of Pag s solito is show i igure 45, whe the disorder of the mass sequece is i the regio of 0.67 M M k M, where J 5% J, ( 1 ) 5%( 1 ), w 10%w, 0 ad ε=0.4mev, β 0.5, for T=300K, T=310K, T=315K ad T=30K, respectively. These igures show clearly that Pag s solito is still thermally stable at T<30K, but begis to disperse at T=30K. Thus, the critical temperature of Pag s solito is ot more tha 30K whe structure ouiformity exists. Therefore, we ca coclude that Pag s solito is very robust agaist thermal perturbatio ad structure ouiformity amog protei molecules at biological temperatures. The results i α-helix protei molecules with three chaels i Pag s model At the same time, we cotiuously studied the iluece of structure ouiformity of proteis o Pag s solito at the biological temperature of 300K i α-helix proteis usig the same method metioed above. I igure 46, we plot the states of Pag s solito at T=300K, while the disorder of the mass sequece is i the regio of 0.67 < k <, ad ( 1 ) 4%( 1 ), J % J, w 4%w, 0, ε=0.5mev, β 0.5 are existed. From these igures we see clearly that Pag s solito is udisturbed ad still thermally stable at 300K whe these structure ouiformities occur i the proteis. Therefore, we ca coclude that Pag s solito is robust agaist thermal perturbatio ad structure ouiformity of the α-helix protei molecules at biological temperatures. Thus, the solito i Pag s model [113-15] is a exact carrier for bio-eergy trasport i the α-helix protei molecules with three chaels. I igure 47 we exhibit the trasport properties of the solito i the α-helix proteis at differet temperatures of 300K, 310K, 315K ad 30K, respectively, whe the luctuatios of six parameters of structure are 0.67M Mk M, Δ( 1 )=±1%( ), J=±0.7% J, ΔW=±7% W, ΔL= ±0.8% L ad 0, ε=0.4mev, β 1 (a) (b) (c) (d) Figure 45: The state of Pag s solito uder the ifluece of disorder at 0.67M M5 M, J 5% J, ( ) 5%( ),ε=0.41mev, β <1, for T=300K(a), 1 1, w 10%w T=310K(b), T=315K(c), ad T=30K(d)., 0 Published: April 09,

45 Figure 46: The states of Pag s solito uder the iflueces of the structure ouiformities of 0.67 M M k M, J % J, ( 1) 4%( 1 ), w 4% w 0,, ε=0.5mev, β 0.5 at 300K i the -helix proteis. (a) (b) (c) (d) Figure 47: Whe the fluctuatios of structure parameters are 0.67M Mk M, J=±0.7% J, ( 1 )=±1%( 1 ), W=±7% W, L= ±0.8% L ad 0, ε=0.4mev, β 0.5 the peculiarities of the solito att=300k(a), 310K(b),315K(c) ad 30K,respectively This igure shows that the amplitudes of Pag s solito decrease with icreasig temperature, ad it begis to disperse at 315K. This maifests that the solito must exped itself a part of eergy to suppress the icrease of destructive effect resultig from the thermal perturbatio due to the lift of temperature of the systems, thus its amplitude or eergy does depress. We drew from this result that the critical temperature of Pag s solito is about 315K i this coditio. Obviously, decreases of the velocity of Pag s solito with icreasig temperature are due to the ehacemet of disorder thermal motio of medium, resultig from the lift of temperature, which icreases the resistace of motio of the solito. However, the Pabg s solito is still robust agaist the structure ouiformity of the protei molecules ad thermal perturbatio ad dampig of the medium, thus its critical temperature ca reach still 315K i this case. Thus we coclude that Pag s solito ca really play a importat role i the bioeergy trasport, Pag s theory of bio-eergy trasport is really appropriate to the α -helix proteis, which is cosistet with aalytic results [ ]. CONCLUSION I this review article we must look at whether the theories icludig Davydov s ad Pag s models are appropriate to the biological proteis i the livig systems. This is due to the fact that these theoretical models were built based o a periodic, uiform ad iiite proteis molecules, which are a ideal protei model ad differet from the biological proteis molecules i livig systems. I this ivestigatio the ilueces of structure ouiformity ad disorder, side groups ad imported impurities of protei chais as well as the thermal perturbatio ad dampig of medium arisig Published: April 09,