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Soliton excitations in alpha-helical protein structures
('l~aos, Solitons & Fractals Vol. 1. No. 6. pp.501-509, 1991 Printed in Great Britain
0960-0779/9255.00 + 00 © 1992 Pergamon Press Ltd
Soliton Excitations in Alpha-Helical Protein Structures S. I C H I N O S E
Faculty of Social Research, Nara University, 1500 Misasagi-cho, Nara 631, Japan
Abstract--We propose a general theory for soliton excitations in alpha-helical protein structures. In
this framework, the interaction between peptide groups is separated into the Amide I excitonphonon coupling effect and the effect of anharmonicity of phonon. In most of previous theories, only one of the two effects has been taken into account. Expressions for both effects are taken so that they coincide with the previous theoretical results in limiting cases. In the general case we show that both effects should be taken into account.
1. INTRODUCTION It had been recognized for some time that proteins may be considered as instruments of energy manipulation in biological systems, in which structure and molecular vibrations are crucial to the concept of molecular machines. Many theories [1-4] have suggested that the energy of the hydrolysis of A T P can be transferred without losses along protein molecules as special collectively excited states, i.e. solitons. However, thus far no completely satisfying theory exists for the most essential mechanism of energy transfer in the protein :molecule. The relative flexibility of large protein molecules is reflected in the large n u m b e r of possible conformations which are transformed into one another on changing the external conditions. One of the most important and interesting secondary structures is the alpha-helical structure which is formed as the result of intramolecular hydrogen bonds between the peptide groups of the protein molecule. The energy of formation of one hydrogen bond between peptide groups is of the order of 0.21 eV. The A m i d e I vibration in peptide groups has an energy equal to 0.21 eV and a large electrical dipole m o m e n t . When the adenosine triphosphate (ATP) molecule is hydrolysed, an energy about 0.49 eV is released which is insufficient to excite the electronic states of the molecule. It is only possible to excite the intramolecular A m i d e I vibration in a peptide group of protein. The vibrational dipole m o m e n t (0.3D) is also sufficiently large to provide a strong resonance interaction leading to collective vibrational excitons. In general, the interaction effect of peptide groups is separated into the A m i d e I e x c i t o n - p h o n o n coupling effect and the effect of anharmonicity of phonon. Expressions for both effects are taken so that they coincide with theoretical results in limiting cases. What is essential here is that the effect depends on the soliton velocity V. The first effect is important when V < c2 (longitudinal sound velocity), while the second one is when V > c2. In most of previous theories, only one of the two effects has been taken into account. Theories [1-3], where the effect of A m i d e I e x c i t o n - p h o n o n interaction is considered, are valid only when V < c2. On the other hand, theories [4, 5] where only the anharmonicity of interaction between molecules is taken into account are appreciable when V > c2. Such difficulties should be r e m o v e d by the inclusion of both effects. 501
502
S. |CHINOSE
Quite recently, the present author [5] made an investigation of the trapped states of the Amide I exciton by an acoustic soliton in the chain of peptide groups, where it is assumed that Amide I exciton interacts with the phonon in the chain with an exponential interaction potential. He showed there that the non-linearity of the hydrogen bond generated both the self-trapping mechanism for Amide I vibrations and the acoustic soliton formation for lattice vibrations by neglecting the coupling among three chains for alpha-helical protein. But, to be precise, we cannot neglect such coupling in the real alpha-helical protein. Actually, in a more detailed theoretical description of excited states of the molecule it is necessary to take into account the resonance interaction between the peptide groups of different chains. From these observations we try to calculate the energy and momentum of the soliton moving with the velocity V. After presenting our model for alpha-helical protein and giving a brief review on basic equations of a model protein system (Section 2), we calculate the energy and momentum of the excitation based on the continuum approximation (Section 3). In Section 4 we give a summary and comments on some other works. 2. MODEL ALPHA-HELICAL PROTEIN SYSTEM We consider an exponential lattice coupled with the Amide I intramolecular excitation which is described by the following Hamiltonian H = Hlatt q- H A + Hint,
(2.1)
with H,att = ~ [gMu~. .2
+ K b - t { - (u.-1.~, - u.,,~)
n,ol
+ b-l[exp(b(u._l,~-
u . , ~ ) ) - 1]}1,
(2.2)
2 "~ 1 HA = 1 E (l'-tgl2a + I'-re qT,o~) -- ~L1 Z (q..q.+l,~ + tt,ol
n,o~
1 + q.,~q.-l.) - -~L2 Z (q.~,q..~+a + q..q.,~,-1),
(2.3)
nff
1
Hint ~-~~ ~ -~-(Un+lor -- u._,~,)q2..
(2.4)
Here Hlatt is the lattice (intermolecular) vibration Hamiltonian, in which u,~, M, K, and b are the displacement of the peptide group (n, er) from its equilibrium position, the molecular mass, a coefficient of longitudinal elasticity of the harmonic lattice, and a parameter of anharmonicity, respectively, where l is the length of a single unit cell along one chain. The Amide I exciton subsystem is described by the Hamiltonian HA, in which qn~, ~, and e are normal coordinate, effective mass, and frequency, respectively, of the relevent molecular vibration of the peptide group (n, o0. L~ represents a force constant describing the Amide I exciton transfer between a particular Amicle I bond and its next neighbors in the longitudinal direction. L2 represents the corresponding coupling to lateral neighbors. The quantity Hint is the Amide I exciton-lattice interaction Hamiltonian, in which )~ is an interaction constant. Equation (2.4) may be understood by considering an example in which the frequency of the relevant Amide I exciton depends on the instantaneous position x , , = nl + u,,~ of the molecules. The interaction constant X is readily obtained by expanding (x,~) to first order with respect to the u's. In general, the coupling between two subsystems gives rise to two effects, i.e. the polaron interaction at each site
Soliton excitations
503
and the modification of the transfer integral. For simplicity, we only take into account the polaron effect through the Z term which lowers the site energy of the Amide I exciton (e). Equations of motion for Wn, and q~,~ are readily obtained as follows; I 'd2wnc~ - dt 2
Kb-1 ( 2 sinh~ 1 92 )~ n M
exp(blw,~)
-
t~;~ 2 ~ (qn-La
-
2
q.+L~)
=
O,
(2.5)
L1 L2 d2qn---£a2dt + eZqn~ - --it (qn+l,~ + q~-L.) - ~ (q~,~+l + q~,.-1) X + -i (u"+l"" - u.-1,~)q.. = 0, where lwn~ = Un-l,ec
-
-
(2.6)
Una. We then put
(2.7)
qn,(t) = ( p n , ( t ) e x p [ i m * n V - i~ot] + c.c.
in equations (2.5) and (2.6) where m* is the effective mass of the vibrational Amide I exciton and to is the energy of the bound state of vibrational exciton in the field of the local deformation moving with the velocity V along the chain. We employ a rotating-wave approximation to neglect q~,,exp[+ 2im*nV-T-2o)t] and d26p.Jdt 2 in comparison with cod(pn./dt, thus arriving at the set of equations Kb-I M
l -d2w
dt 2
2sinh~-~n
exp(blwn~) -
(Iq,._,d
-
= o,
d0~ L1 (cp._l~ - dpn+1.)sin(m*lV), dt 2to/~ (-
2
+
-
L1
_ _
(2,8)
(2.9)
(~On+l,. + (pn_l,a)cos(m*lV) -_L2 _ (q~.,0,+l +qb.,.-1) + Z~ (u.+L. - u.-x,.)q~.,¢ = 0,
(2.10)
where m* = I~to/Lll 2. When the displacement fields q~.~, w.. change slowly with n we can employ the so-called continuum approximation under the assumption that the velocity of soliton V satisfies the inequality m ' I V << 1.
(2.11)
Keeping fourth-order derivatives with respect to the coordinate x, we can transform equation (2.8) to the form ~2Wo~ 3t 2
232Wa (1C2)2~4W0~ c23X----;
12
3x 4
]/K-lc2 32(W2) 4/tZ 9Z]q~,]2 _ 0, - - 3X - - T - + M ~ 3x
(2.12)
where c2 = ~/(K/M). In the absence of coupling (2' = 0), (2.12) is called the Boussinesq equation. The solution of such an equation was studied by Toda and Wadati [6]. Then equations (2.9) and (2.10) are transformed to equations involving partial derivatives,
Lll 2 32~ 3X 2
3q~ _ _V3_.a~ ~t 5x ' 3u~ L2 ]./ (qh~+l + qh,-l) + 23X-~x q~¢ = [092 -- m0(V)2JqG,
where too(V) 2 = 0)2 + (LI/21~)m*V 2 with o)~ = e 2 - L,/p.
(2.13)
(2.14)
504
S. ICHINOSE
To study stationary solutions, belonging to the class of rapidly decreasing functions, we introduce the following dimensionless variable ~ = (x - Vt)/l and assume that w,(x, t) = w~(~), q~a(X, t) = ~,(~). Then equation (2.13) becomes an identity, and equations (2.12) and (2.14) are transformed as follows 1 d2w, 12 d~ 2 + (1 - s2)wa + ~/w 2 -
2Z ~ Mc22 di)2.
d2~ 1 - - - ~ 1 - 2 "[" 2)~w,~, + 1--~2(~o:+1 + (~o<-1) "q- ~"~(~a<-----0,
(2.15) (2.16)
where if2 = w 2 - co0(V) 2, s = VIc2, L1 = till'g, L2 = L21ll, (~a = qba(2~) 1/2" We are now in a position to solve these equations for the most simple case which corresponds to that with w , = w,+l + W~_l = w,,
~ , = (~,+1 = ~ - 1 = ~s
(2.17)
In this case all of the three chains of peptide groups take part in the formation of these excitations. The soliton excitations of this type are symmetric. From the conditions for the w's and q~'s we obtain the system of equations 1 d 2 Ws 12 ~d~ + (1 - s2)Ws + ,~w~ = ~.-c2A'f2%2"r"~2
L1 - - ~ + 2Xw, dp, + f2q)s = 0,
(2.18) (2.19)
where ~ = f~ + 2L2. It is convenient to replace the set of equations (2.18) and (2.19) by an integral equation. Integrating them and taking account of the boundary conditions (the functions q% and w s and their derivatives vanish at ~--~ ~) and using the replacement ~--0~,(~) in the integrands, we obtain the following equations 1 /dwl2 2~ 3 4X ~2 8Z ~0~' --12 ~ d ~ ] = (s2 - 1)w~ - ~yw, + --Mc~ (p,w, - --Mc2 w(v)rdr, dq)
]
+ 4% f0
w(r)rdr + 0~
= 0.
(2.20) (2.21)
In the case of bell-like solitons with centers at ~ = 0 the equality
(ad~,t1~=0 = 0
(2.22)
holds. Using this equality, we find the magnitude of the spectral parameter from equation (2.21) 4Z (3,0 - w(v)~d~, ~so = $~(0). (2.23) =
J0
Substituting this value into equation (2.21), we find the measure ( L1/1/2 [(~)s12 =
[\$s0l ff,0 w (Ovdr _
ff, Ws(r)rdT;]-l/2d~bs"
(2.24)
The desired amplitude ~,0 is determined from the normalization condition for the function ~ ( ~ ) . Under the displacement ~ by q)~(~) this condition becomes as follows
2,0
= ½c,
Soliton excitations
505
where C is the constant. Substituting (2.24) into this equation, we find the following one
fo ~;s°rz[( f-~---]zJo w,(y)ydy - fr .Io w~(y)ydy
]-1/2dr ~ [ ~ ]1/2~__~. C,_1j
(2.25)
which allows us to evaluate the maximum value ~s0 of the envelope ~(~). Finally, integrating equation (2.20), we obtain the required integral equation Ws(~)s) = 12
Ju
[
(2.26) We note that this integral equation is exactly equivalent to the set of differential equations, (2.20) and (2.21). If the norm of the corresponding integral operator is less than unity, the solution of (2.26) can be sought by the method of successive approximations.
3. SOLITONS OF ALPHA-HELICAL PROTEIN MOLECULE
We now proceed with the analysis. As mentioned in the previous section, the solution of the integral equation, (2.26), can be obtained using the method of successive approximations. Integrating equation (2.24), we find the envelope ~,(~) = ~ , ( - ~ ) in the implicit form
=
.JgSs,~,,(\~sO]fo
Ws(y)ydy - f]w~(y)ydy] -1/2 dr.
(3.1)
As the zeroth approximation we choose the following function = ¢, ,
(3.2)
where o~ is a positive number which is to be determined. Equation (2.25), taking into account (3.2), defines the amplitude ~,0 of the function
= ~PsO= (\ - - ~ ] 1 .
(3.3)
If cr is, however, a positive root of the following cubic equation ?~z 3 +(1-s2)z 6L 1
2X _ 0 ,
Mc~
(3.4)
the integral is simplified and becomes easily evaluated. Then the first-order approximation function takes the following form 2
w~l)(~)
sech2(~e),
--
X
sech(~),
~(1)
__= - - E 1 ~ 2 .
(3.5) (3.6)
The dimensionless parameter 0cgx /~ = 2L1
(3.7)
depending on the velocity, characterizes the inverse width of the soliton. Here g is given by g = [Llf//6vy] 1/2 with v = 1/12. It is easily proved that if the relation g = 1, or ~-,1~' = 6vx,
(3.8)
506
S. ICrUNOSE
holds, the first-order approximation coincides with the zeroth one. Therefore, if the relation (3.8) is satisfied, the solution ws(~) -
sech2(goff),
(3.9)
Z ~s(~) =
sech(go~),
~ = - Llgg,
(3.10)
become the exact one, if go is defined by the expression go-
2E~'
(3.11)
where o~ is a positive root of equation (3.4). Using equation (3.11), we can transform equation (3.4) to the following one
+ (1 - s2)go
X2
F_,IMC
--
0.
(3.12)
It involves directly go, which enters into the exact solutions. At a fixed value of s, equation (3.12) has the single positive root go(s) = go is s 2 < 1 and s 2 > 1. This is shown in Fig. 1. It follows from Fig. 1 that the sound velocity c2(s = 1) is not in any way special. The solutions have meaning for subsonic (s 2 < 1) and for supersonic (s 2 > 1) velocities. When ~s(~) is known, we can determine the displacement field
ws(~) = ws[~,(~)]
(3.13)
and with its help we can obtain the energy and momentum of the soliton moving with the velocity V E,(V) = {m
V
+ wo(V) 2 - r , ~ g o ( V ) 2 + -~Mcz[~--ff}
1 + s2 +
go(V) 2
go(V) 3, (3.14)
+16
3 Lt 2
Such a soliton excitation is stable if its energy is less than that of the bottom of the energetic band of free Amide I exciton, i.e. if E ( V ) < ~o0(V)2.
0
2
~-
Fig. 1. The dependence of /a0 on the velocity. Letters attached to each curve denote: (A) the subsonic case, (B) the supersonic case, (C) the general case, respectively.
Solitonexcitations (i)
507
Subsonic soliton
If the dispersion and the anharmonicity are absent and the parameters v and y approach zero simultaneously in such a way that the relation (3.8) remains valid, equation (3.12) has a positive solution only at the subsonic velocities /~sub) = L, I M ( c ~ - V2) '
V 2 < c~.
(3.16)
Then the functions (3.9) and (3.10) with /~0 =/~sub) coincide with those studied in both theories of Davydov and Takeno [1-3]. Substituting (3.16) into (3.14) and (3.15) we obtain the energy and momentum carried by the soliton of this kind E~SUb)(v) = ½m*V 2 + wo(V) 2 - ~q[/t~sub)]2 +
8. 2//~1~2[ ] gJvlc2(-~y ) 1 + s 2 + g1 [~sub)]Z [~sub)]3 p!sub)(v)
[m* + 16 sub 3 LI 2 -~-M[/~ )] (~-Z) ]V.
(3.17) (3.18)
k
As shown in Figs 2 and 3, if the soliton velocity approaches cz, both the soliton energy and is momentum grow infinitely. We note that such difficulty should be removed by the inclusion of anharmonicity into the interaction between molecules. (ii)
Supersonic soliton
If equation (3.12) holds as the ratio X2/ffq approaches zero then in the chain only supersonic acoustic solitons are excitated which are described for v = 1/12 by the function [4, 6]
~/~super)_ [3(s 2 _ 1)]1/2, V ----SC2, S2 > 1.
(3.19)
In this case there occurs a mutual compensation of the effects of dispersion and anharmonicity (y) of displacements. Then the energy and momentum of the acoustic soliton are given by the analytical expression E~upe~)(V) = ~/3.Z3(s 2 - 1)3/2(1 + 9s2)/30y 2,
(3.20)
P~Super)(v) = ~/3.tcZ(s 2 - 1)3/2y-2MV.
(3.21)
With a decrease in the velocity V, the soliton energy and momentum are lowered. Hence, in the limit V---> c2 they vanish as shown in Figs 2 and 3. This means that the longitudinal sound velocity c2 is in some way special. Such difficulty should be also removed by the inclusion of the coupling between the Amide I excitation mode and the intermolecular lattice phonon. (iii)
General case
As the velocity increases, ~ ( V ) increases also and the sound velocity c2(s = 1) is not in any way special as shown in Fig. 1. In the course of g~o), the above mentioned difficulties have been removed. A restriction on the velocity is determined by the condition of validity of the continuum approach /z~a)
<<
1
(3.22)
and also the requirement that the solitary excitation energy should be below the bottom of
508
S. ICHINOSE
,v f II
I
I/
0
I
$
L
Fig. 2. The dependence of the soliton energy on the velocity. Letters attached to each curve denote: (A) the subsonic case, (B) the supersonic case, (C) the general case, respectively.
I
I
Q.
JVI 0
I
2
Fig. 3. The dependence of the momentum of soliton on the velocity. Letters attached to each curve denote: (A) the subsonic case, (B) the supersonic case, (C) the general case, respectively.
the energy band of an intramolecular Amide I exciton. The stabilization of those kinds of excitations studied is provided by the influence of the four following effects; (1) the resonance interaction L1; (2) the Amide I exciton-phonon interaction X; (3) the dispersion of phonons v; (4) the anharmonocity (7) of displacements of molecules in the chain. In this general framework, the effect of interaction is separated into the Amide I exciton-phonon coupling effect and the effect of anharmonicity of phonon. Expressions for both effects are taken so that they coincide with theoretical results in limiting cases. CONCLUDING
REMARKS
The results obtained in this paper may be summarized as follows; In the present framework, the interaction between peptide groups is separated into the Amide I exciton-phonon coupling effect and the effect of anharmonicity of phonon. Expressions for both effects are taken so that they coincide with theoretical results in limiting cases. The first effect is important when V << c2, while the second one is when V >> c2. In most of previous theories, only one of the two interaction effects has been taken into account. Theories [1-3] where the effect of Amide I exciton-phonon interaction is considered are valid only when V << c2. On the other hand, theories [4, 5] where only the
Soliton excitations
509
anharmonicity of interaction between molecules is taken into account are applicable when V << c2. In the intermediate region V ~ c2, we have no such theories. Separation of the interaction effect into two categories has no clear meaning in this region. In general cases we showed both effects should be taken account. Throughout this paper we have neglected the effect of discreteness of the lattice. In actual situations this approximation is not always allowed. This point will be investigated in future publications. REFERENCES 1. A. S. Davydov, The theory of contraction of proteins under their excitation, J, Theor. Biol. 38, 559 (1973); Studia Biophys. 47,221 (1974); Physica Scripta 20,387 (1979). 2. A. C. Scott, Dynamics of Davydov solitons, Phys. Rev. A26,578 (1982); The vibrational structures of Davydov solitons, Physica Scripta 25,651 (1982). 3. S. Takeno, Vibron solitons in one-dimensional molecular crystals, Progr. Theor. Phys. 71,395 (1984). 4. S. Yomosa, Toda-lattice solitons ~r-helical proteins, J. Phys. Soc. Japan 34, 3692 (1984). 5. S. Ichinose, Migration of energy in the hydrogen-bonded chain of peptide groups, Unpublished. 6. M. Toda and M. Wadati, A soliton and two solitons in an exponential lattice and related equations, J. Phys. Soc. Japan 34, 18 (1973).
0960-0779/9255.00 + 00 © 1992 Pergamon Press Ltd
Soliton Excitations in Alpha-Helical Protein Structures S. I C H I N O S E
Faculty of Social Research, Nara University, 1500 Misasagi-cho, Nara 631, Japan
Abstract--We propose a general theory for soliton excitations in alpha-helical protein structures. In
this framework, the interaction between peptide groups is separated into the Amide I excitonphonon coupling effect and the effect of anharmonicity of phonon. In most of previous theories, only one of the two effects has been taken into account. Expressions for both effects are taken so that they coincide with the previous theoretical results in limiting cases. In the general case we show that both effects should be taken into account.
1. INTRODUCTION It had been recognized for some time that proteins may be considered as instruments of energy manipulation in biological systems, in which structure and molecular vibrations are crucial to the concept of molecular machines. Many theories [1-4] have suggested that the energy of the hydrolysis of A T P can be transferred without losses along protein molecules as special collectively excited states, i.e. solitons. However, thus far no completely satisfying theory exists for the most essential mechanism of energy transfer in the protein :molecule. The relative flexibility of large protein molecules is reflected in the large n u m b e r of possible conformations which are transformed into one another on changing the external conditions. One of the most important and interesting secondary structures is the alpha-helical structure which is formed as the result of intramolecular hydrogen bonds between the peptide groups of the protein molecule. The energy of formation of one hydrogen bond between peptide groups is of the order of 0.21 eV. The A m i d e I vibration in peptide groups has an energy equal to 0.21 eV and a large electrical dipole m o m e n t . When the adenosine triphosphate (ATP) molecule is hydrolysed, an energy about 0.49 eV is released which is insufficient to excite the electronic states of the molecule. It is only possible to excite the intramolecular A m i d e I vibration in a peptide group of protein. The vibrational dipole m o m e n t (0.3D) is also sufficiently large to provide a strong resonance interaction leading to collective vibrational excitons. In general, the interaction effect of peptide groups is separated into the A m i d e I e x c i t o n - p h o n o n coupling effect and the effect of anharmonicity of phonon. Expressions for both effects are taken so that they coincide with theoretical results in limiting cases. What is essential here is that the effect depends on the soliton velocity V. The first effect is important when V < c2 (longitudinal sound velocity), while the second one is when V > c2. In most of previous theories, only one of the two effects has been taken into account. Theories [1-3], where the effect of A m i d e I e x c i t o n - p h o n o n interaction is considered, are valid only when V < c2. On the other hand, theories [4, 5] where only the anharmonicity of interaction between molecules is taken into account are appreciable when V > c2. Such difficulties should be r e m o v e d by the inclusion of both effects. 501
502
S. |CHINOSE
Quite recently, the present author [5] made an investigation of the trapped states of the Amide I exciton by an acoustic soliton in the chain of peptide groups, where it is assumed that Amide I exciton interacts with the phonon in the chain with an exponential interaction potential. He showed there that the non-linearity of the hydrogen bond generated both the self-trapping mechanism for Amide I vibrations and the acoustic soliton formation for lattice vibrations by neglecting the coupling among three chains for alpha-helical protein. But, to be precise, we cannot neglect such coupling in the real alpha-helical protein. Actually, in a more detailed theoretical description of excited states of the molecule it is necessary to take into account the resonance interaction between the peptide groups of different chains. From these observations we try to calculate the energy and momentum of the soliton moving with the velocity V. After presenting our model for alpha-helical protein and giving a brief review on basic equations of a model protein system (Section 2), we calculate the energy and momentum of the excitation based on the continuum approximation (Section 3). In Section 4 we give a summary and comments on some other works. 2. MODEL ALPHA-HELICAL PROTEIN SYSTEM We consider an exponential lattice coupled with the Amide I intramolecular excitation which is described by the following Hamiltonian H = Hlatt q- H A + Hint,
(2.1)
with H,att = ~ [gMu~. .2
+ K b - t { - (u.-1.~, - u.,,~)
n,ol
+ b-l[exp(b(u._l,~-
u . , ~ ) ) - 1]}1,
(2.2)
2 "~ 1 HA = 1 E (l'-tgl2a + I'-re qT,o~) -- ~L1 Z (q..q.+l,~ + tt,ol
n,o~
1 + q.,~q.-l.) - -~L2 Z (q.~,q..~+a + q..q.,~,-1),
(2.3)
nff
1
Hint ~-~~ ~ -~-(Un+lor -- u._,~,)q2..
(2.4)
Here Hlatt is the lattice (intermolecular) vibration Hamiltonian, in which u,~, M, K, and b are the displacement of the peptide group (n, er) from its equilibrium position, the molecular mass, a coefficient of longitudinal elasticity of the harmonic lattice, and a parameter of anharmonicity, respectively, where l is the length of a single unit cell along one chain. The Amide I exciton subsystem is described by the Hamiltonian HA, in which qn~, ~, and e are normal coordinate, effective mass, and frequency, respectively, of the relevent molecular vibration of the peptide group (n, o0. L~ represents a force constant describing the Amide I exciton transfer between a particular Amicle I bond and its next neighbors in the longitudinal direction. L2 represents the corresponding coupling to lateral neighbors. The quantity Hint is the Amide I exciton-lattice interaction Hamiltonian, in which )~ is an interaction constant. Equation (2.4) may be understood by considering an example in which the frequency of the relevant Amide I exciton depends on the instantaneous position x , , = nl + u,,~ of the molecules. The interaction constant X is readily obtained by expanding (x,~) to first order with respect to the u's. In general, the coupling between two subsystems gives rise to two effects, i.e. the polaron interaction at each site
Soliton excitations
503
and the modification of the transfer integral. For simplicity, we only take into account the polaron effect through the Z term which lowers the site energy of the Amide I exciton (e). Equations of motion for Wn, and q~,~ are readily obtained as follows; I 'd2wnc~ - dt 2
Kb-1 ( 2 sinh~ 1 92 )~ n M
exp(blw,~)
-
t~;~ 2 ~ (qn-La
-
2
q.+L~)
=
O,
(2.5)
L1 L2 d2qn---£a2dt + eZqn~ - --it (qn+l,~ + q~-L.) - ~ (q~,~+l + q~,.-1) X + -i (u"+l"" - u.-1,~)q.. = 0, where lwn~ = Un-l,ec
-
-
(2.6)
Una. We then put
(2.7)
qn,(t) = ( p n , ( t ) e x p [ i m * n V - i~ot] + c.c.
in equations (2.5) and (2.6) where m* is the effective mass of the vibrational Amide I exciton and to is the energy of the bound state of vibrational exciton in the field of the local deformation moving with the velocity V along the chain. We employ a rotating-wave approximation to neglect q~,,exp[+ 2im*nV-T-2o)t] and d26p.Jdt 2 in comparison with cod(pn./dt, thus arriving at the set of equations Kb-I M
l -d2w
dt 2
2sinh~-~n
exp(blwn~) -
(Iq,._,d
-
= o,
d0~ L1 (cp._l~ - dpn+1.)sin(m*lV), dt 2to/~ (-
2
+
-
L1
_ _
(2,8)
(2.9)
(~On+l,. + (pn_l,a)cos(m*lV) -_L2 _ (q~.,0,+l +qb.,.-1) + Z~ (u.+L. - u.-x,.)q~.,¢ = 0,
(2.10)
where m* = I~to/Lll 2. When the displacement fields q~.~, w.. change slowly with n we can employ the so-called continuum approximation under the assumption that the velocity of soliton V satisfies the inequality m ' I V << 1.
(2.11)
Keeping fourth-order derivatives with respect to the coordinate x, we can transform equation (2.8) to the form ~2Wo~ 3t 2
232Wa (1C2)2~4W0~ c23X----;
12
3x 4
]/K-lc2 32(W2) 4/tZ 9Z]q~,]2 _ 0, - - 3X - - T - + M ~ 3x
(2.12)
where c2 = ~/(K/M). In the absence of coupling (2' = 0), (2.12) is called the Boussinesq equation. The solution of such an equation was studied by Toda and Wadati [6]. Then equations (2.9) and (2.10) are transformed to equations involving partial derivatives,
Lll 2 32~ 3X 2
3q~ _ _V3_.a~ ~t 5x ' 3u~ L2 ]./ (qh~+l + qh,-l) + 23X-~x q~¢ = [092 -- m0(V)2JqG,
where too(V) 2 = 0)2 + (LI/21~)m*V 2 with o)~ = e 2 - L,/p.
(2.13)
(2.14)
504
S. ICHINOSE
To study stationary solutions, belonging to the class of rapidly decreasing functions, we introduce the following dimensionless variable ~ = (x - Vt)/l and assume that w,(x, t) = w~(~), q~a(X, t) = ~,(~). Then equation (2.13) becomes an identity, and equations (2.12) and (2.14) are transformed as follows 1 d2w, 12 d~ 2 + (1 - s2)wa + ~/w 2 -
2Z ~ Mc22 di)2.
d2~ 1 - - - ~ 1 - 2 "[" 2)~w,~, + 1--~2(~o:+1 + (~o<-1) "q- ~"~(~a<-----0,
(2.15) (2.16)
where if2 = w 2 - co0(V) 2, s = VIc2, L1 = till'g, L2 = L21ll, (~a = qba(2~) 1/2" We are now in a position to solve these equations for the most simple case which corresponds to that with w , = w,+l + W~_l = w,,
~ , = (~,+1 = ~ - 1 = ~s
(2.17)
In this case all of the three chains of peptide groups take part in the formation of these excitations. The soliton excitations of this type are symmetric. From the conditions for the w's and q~'s we obtain the system of equations 1 d 2 Ws 12 ~d~ + (1 - s2)Ws + ,~w~ = ~.-c2A'f2%2"r"~2
L1 - - ~ + 2Xw, dp, + f2q)s = 0,
(2.18) (2.19)
where ~ = f~ + 2L2. It is convenient to replace the set of equations (2.18) and (2.19) by an integral equation. Integrating them and taking account of the boundary conditions (the functions q% and w s and their derivatives vanish at ~--~ ~) and using the replacement ~--0~,(~) in the integrands, we obtain the following equations 1 /dwl2 2~ 3 4X ~2 8Z ~0~' --12 ~ d ~ ] = (s2 - 1)w~ - ~yw, + --Mc~ (p,w, - --Mc2 w(v)rdr, dq)
]
+ 4% f0
w(r)rdr + 0~
= 0.
(2.20) (2.21)
In the case of bell-like solitons with centers at ~ = 0 the equality
(ad~,t1~=0 = 0
(2.22)
holds. Using this equality, we find the magnitude of the spectral parameter from equation (2.21) 4Z (3,0 - w(v)~d~, ~so = $~(0). (2.23) =
J0
Substituting this value into equation (2.21), we find the measure ( L1/1/2 [(~)s12 =
[\$s0l ff,0 w (Ovdr _
ff, Ws(r)rdT;]-l/2d~bs"
(2.24)
The desired amplitude ~,0 is determined from the normalization condition for the function ~ ( ~ ) . Under the displacement ~ by q)~(~) this condition becomes as follows
2,0
= ½c,
Soliton excitations
505
where C is the constant. Substituting (2.24) into this equation, we find the following one
fo ~;s°rz[( f-~---]zJo w,(y)ydy - fr .Io w~(y)ydy
]-1/2dr ~ [ ~ ]1/2~__~. C,_1j
(2.25)
which allows us to evaluate the maximum value ~s0 of the envelope ~(~). Finally, integrating equation (2.20), we obtain the required integral equation Ws(~)s) = 12
Ju
[
(2.26) We note that this integral equation is exactly equivalent to the set of differential equations, (2.20) and (2.21). If the norm of the corresponding integral operator is less than unity, the solution of (2.26) can be sought by the method of successive approximations.
3. SOLITONS OF ALPHA-HELICAL PROTEIN MOLECULE
We now proceed with the analysis. As mentioned in the previous section, the solution of the integral equation, (2.26), can be obtained using the method of successive approximations. Integrating equation (2.24), we find the envelope ~,(~) = ~ , ( - ~ ) in the implicit form
=
.JgSs,~,,(\~sO]fo
Ws(y)ydy - f]w~(y)ydy] -1/2 dr.
(3.1)
As the zeroth approximation we choose the following function = ¢, ,
(3.2)
where o~ is a positive number which is to be determined. Equation (2.25), taking into account (3.2), defines the amplitude ~,0 of the function
= ~PsO= (\ - - ~ ] 1 .
(3.3)
If cr is, however, a positive root of the following cubic equation ?~z 3 +(1-s2)z 6L 1
2X _ 0 ,
Mc~
(3.4)
the integral is simplified and becomes easily evaluated. Then the first-order approximation function takes the following form 2
w~l)(~)
sech2(~e),
--
X
sech(~),
~(1)
__= - - E 1 ~ 2 .
(3.5) (3.6)
The dimensionless parameter 0cgx /~ = 2L1
(3.7)
depending on the velocity, characterizes the inverse width of the soliton. Here g is given by g = [Llf//6vy] 1/2 with v = 1/12. It is easily proved that if the relation g = 1, or ~-,1~' = 6vx,
(3.8)
506
S. ICrUNOSE
holds, the first-order approximation coincides with the zeroth one. Therefore, if the relation (3.8) is satisfied, the solution ws(~) -
sech2(goff),
(3.9)
Z ~s(~) =
sech(go~),
~ = - Llgg,
(3.10)
become the exact one, if go is defined by the expression go-
2E~'
(3.11)
where o~ is a positive root of equation (3.4). Using equation (3.11), we can transform equation (3.4) to the following one
+ (1 - s2)go
X2
F_,IMC
--
0.
(3.12)
It involves directly go, which enters into the exact solutions. At a fixed value of s, equation (3.12) has the single positive root go(s) = go is s 2 < 1 and s 2 > 1. This is shown in Fig. 1. It follows from Fig. 1 that the sound velocity c2(s = 1) is not in any way special. The solutions have meaning for subsonic (s 2 < 1) and for supersonic (s 2 > 1) velocities. When ~s(~) is known, we can determine the displacement field
ws(~) = ws[~,(~)]
(3.13)
and with its help we can obtain the energy and momentum of the soliton moving with the velocity V E,(V) = {m
V
+ wo(V) 2 - r , ~ g o ( V ) 2 + -~Mcz[~--ff}
1 + s2 +
go(V) 2
go(V) 3, (3.14)
+16
3 Lt 2
Such a soliton excitation is stable if its energy is less than that of the bottom of the energetic band of free Amide I exciton, i.e. if E ( V ) < ~o0(V)2.
0
2
~-
Fig. 1. The dependence of /a0 on the velocity. Letters attached to each curve denote: (A) the subsonic case, (B) the supersonic case, (C) the general case, respectively.
Solitonexcitations (i)
507
Subsonic soliton
If the dispersion and the anharmonicity are absent and the parameters v and y approach zero simultaneously in such a way that the relation (3.8) remains valid, equation (3.12) has a positive solution only at the subsonic velocities /~sub) = L, I M ( c ~ - V2) '
V 2 < c~.
(3.16)
Then the functions (3.9) and (3.10) with /~0 =/~sub) coincide with those studied in both theories of Davydov and Takeno [1-3]. Substituting (3.16) into (3.14) and (3.15) we obtain the energy and momentum carried by the soliton of this kind E~SUb)(v) = ½m*V 2 + wo(V) 2 - ~q[/t~sub)]2 +
8. 2//~1~2[ ] gJvlc2(-~y ) 1 + s 2 + g1 [~sub)]Z [~sub)]3 p!sub)(v)
[m* + 16 sub 3 LI 2 -~-M[/~ )] (~-Z) ]V.
(3.17) (3.18)
k
As shown in Figs 2 and 3, if the soliton velocity approaches cz, both the soliton energy and is momentum grow infinitely. We note that such difficulty should be removed by the inclusion of anharmonicity into the interaction between molecules. (ii)
Supersonic soliton
If equation (3.12) holds as the ratio X2/ffq approaches zero then in the chain only supersonic acoustic solitons are excitated which are described for v = 1/12 by the function [4, 6]
~/~super)_ [3(s 2 _ 1)]1/2, V ----SC2, S2 > 1.
(3.19)
In this case there occurs a mutual compensation of the effects of dispersion and anharmonicity (y) of displacements. Then the energy and momentum of the acoustic soliton are given by the analytical expression E~upe~)(V) = ~/3.Z3(s 2 - 1)3/2(1 + 9s2)/30y 2,
(3.20)
P~Super)(v) = ~/3.tcZ(s 2 - 1)3/2y-2MV.
(3.21)
With a decrease in the velocity V, the soliton energy and momentum are lowered. Hence, in the limit V---> c2 they vanish as shown in Figs 2 and 3. This means that the longitudinal sound velocity c2 is in some way special. Such difficulty should be also removed by the inclusion of the coupling between the Amide I excitation mode and the intermolecular lattice phonon. (iii)
General case
As the velocity increases, ~ ( V ) increases also and the sound velocity c2(s = 1) is not in any way special as shown in Fig. 1. In the course of g~o), the above mentioned difficulties have been removed. A restriction on the velocity is determined by the condition of validity of the continuum approach /z~a)
<<
1
(3.22)
and also the requirement that the solitary excitation energy should be below the bottom of
508
S. ICHINOSE
,v f II
I
I/
0
I
$
L
Fig. 2. The dependence of the soliton energy on the velocity. Letters attached to each curve denote: (A) the subsonic case, (B) the supersonic case, (C) the general case, respectively.
I
I
Q.
JVI 0
I
2
Fig. 3. The dependence of the momentum of soliton on the velocity. Letters attached to each curve denote: (A) the subsonic case, (B) the supersonic case, (C) the general case, respectively.
the energy band of an intramolecular Amide I exciton. The stabilization of those kinds of excitations studied is provided by the influence of the four following effects; (1) the resonance interaction L1; (2) the Amide I exciton-phonon interaction X; (3) the dispersion of phonons v; (4) the anharmonocity (7) of displacements of molecules in the chain. In this general framework, the effect of interaction is separated into the Amide I exciton-phonon coupling effect and the effect of anharmonicity of phonon. Expressions for both effects are taken so that they coincide with theoretical results in limiting cases. CONCLUDING
REMARKS
The results obtained in this paper may be summarized as follows; In the present framework, the interaction between peptide groups is separated into the Amide I exciton-phonon coupling effect and the effect of anharmonicity of phonon. Expressions for both effects are taken so that they coincide with theoretical results in limiting cases. The first effect is important when V << c2, while the second one is when V >> c2. In most of previous theories, only one of the two interaction effects has been taken into account. Theories [1-3] where the effect of Amide I exciton-phonon interaction is considered are valid only when V << c2. On the other hand, theories [4, 5] where only the
Soliton excitations
509
anharmonicity of interaction between molecules is taken into account are applicable when V << c2. In the intermediate region V ~ c2, we have no such theories. Separation of the interaction effect into two categories has no clear meaning in this region. In general cases we showed both effects should be taken account. Throughout this paper we have neglected the effect of discreteness of the lattice. In actual situations this approximation is not always allowed. This point will be investigated in future publications. REFERENCES 1. A. S. Davydov, The theory of contraction of proteins under their excitation, J, Theor. Biol. 38, 559 (1973); Studia Biophys. 47,221 (1974); Physica Scripta 20,387 (1979). 2. A. C. Scott, Dynamics of Davydov solitons, Phys. Rev. A26,578 (1982); The vibrational structures of Davydov solitons, Physica Scripta 25,651 (1982). 3. S. Takeno, Vibron solitons in one-dimensional molecular crystals, Progr. Theor. Phys. 71,395 (1984). 4. S. Yomosa, Toda-lattice solitons ~r-helical proteins, J. Phys. Soc. Japan 34, 3692 (1984). 5. S. Ichinose, Migration of energy in the hydrogen-bonded chain of peptide groups, Unpublished. 6. M. Toda and M. Wadati, A soliton and two solitons in an exponential lattice and related equations, J. Phys. Soc. Japan 34, 18 (1973).