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The nonlinear stretching model of hydrogen bonds and local self-fluctuation of base rotation in DNA
Chemical Physics ELSEVIER
Chemical Physics 191 (1995) 17-23
The nonlinear stretching model of hydrogen bonds and local self-fluctuation of base rotation in DNA Chun-Ting Zhang a, Kuo-Chen Chou b " Department of Physics, Tianjin University, Tianjin 300072, China h Computer-Aided Drug Discovery, Upjohn Laboratories, Kalamazoo. Mi 49001, USA Rec,,'ived 20 June 1994; in final form 23 September 1994
Abstract b
A new model in which the stretching of hydrogen bonds causes the rotation of bases around an axis being parallel to the helical axis in DNA has been proposed. When the angular displacement tends to zero, it is shown that a plane wave may be excited. When the angular displacement is in the range of several degrees, a cubic nonlinear term in the hydrogen bonds potential has been taken into account. Thus the following new results have been derived. ( 1) A solitary wave with narrow width (about one base pair), small amplitude (5 ° ), low energy (0.38 kcal/mol) and short life-time (about 10- ~3s) may be excited by the collision of water molecules. (2) The nonlinearity of hydrogen bonds would ensure the rotation of bases toward the major groove. Accordingly, a local and quick self-fluctuation of bases rotation toward the major groove with small amplitude seems to occur in a DNA chain. This new prediction has been compared with the relevant experimental results available.
1. Introduction The vibration modes of hydrogen bona stretching in B-DNA at 10-120 c m - ~ have been observed in both low frequency Raman scattering [ 1,2] and Fouriertransform infrared absorption [ 3 ]. In the double helix structure the base A (adenine) is bound to the base T (thymine) with two hydrogen bonds, and the base G (guanine) is bound to the base C (cytosine) with three hydrogen bonds. Therefore, the stretching of hydrogen bond causes the vibration of bases. Chou has developed a linear vibration theory of H-bond stretching in DNA and discussed some relevant biological functions [4]. On the other hand, many investigators have pointed out that the hydrogen bond is essentially nonlinear [5,6]. Prohofsky has found that the modes of hydrogen bond stretching at 10-120 c m - ~are highly nonlinear [7]. A model of hydrogen-bond stretching in which the hydrogen bond potential is approximately expressed by a 0301-0104/95/$09.50 © 1995 Elsevier Science B.V. All fights reserved SSD1030 ! - 0 1 0 4 ( 9 4 ) 0 0 3 6 2 - 9
Morse potential has been studied by Techera et al. [ 8 ]. A similar study has been carried out by Zhang by using a 2-3 power potential instead of the Morse potential [9]. Recently, Muto et al. have proposed a two-dimensional discrete model in which the hydrogen bond is represented by a Lennard-Jones potential [10]. It should be pointed out that in the above studies of hydrogen-bond stretching, each base pair is modeled by a pair of free point masses connected by a nonlinear massless spring. However, each such base in DNA is actually bounded to the C 1' atom of the double helix through a C l'-N-glucoside bond. Therefore, the stretching of hydrogen bond should cause the rotation of bases. To consider this fact a new model of nonlinear hydrogen-bond stretching has been proposed in this paper. It is called a "chest expander-like" model in which the bases are permitted to rotate along an axis parallel to the helical axis of the double helix. The
18
C.-T. Zhang, K.-C. Chou / Chemical Physics 191 (1995) 17-23
nonlinearity of hydrogen bond ensures that the angular displacement of bases mainly opens towards the major groove. A similar rotation of bases has also been studied by Ramstein and Lavery [ 11 ], Briki et al. [ 12], and Volkov and Kosevich [ 13]. But in all of these studies the effect of nonlinearity of hydrogen bond has not yet been taken into account. The current work is closely related to the studies by Yakushevich [ 14] and Gaeta [I5]. However, Yakushevich's treatment only dealt with the stable soliton excitation•
J
(a)
rV'vx.~
/p"
2. The chest expander.like model
S" According to the above description, the model may be illustrated in Fig. I, where B and B' represent a pair of complementary bases, which are connected by a rigid massless rod with points P and P', respectively. The latter, however, are attached to the two double helical strands of DNA, S and S', respectively. The zigzag line connecting B and B' is a nonlinear massless spring, representing the hydrogen bonds between the complementary bases. As mentioned above, each base has the freedom to rotate around the axis passing through the point P (P') being parallel to the DNA helical axis (see Fig. 1). Therefore, the stretching of the H-bond is associated with the rotation of the bases. In other words, when each base in a base pair rotates outward or inward, the corresponding H-bond would be stretched or compressed accordingly as shown in Fig. 1, where PB and P'B' can be likened to the two arms o f a person engaging in the chest-expanding exercise, and the zigzag line to the spring of a chest expander. Therefore, the stretching of the H-bonds associated with the rotation of the corresponding base pair can be actually compared to the chest expanderlike motion. The model based on such a physical picture is called the "chest expander-like model". For convenience, like all the previous treatments [8-10], the difference between pyrimidine and purine bases is ignored. Thus the following symmetrical conditions are valid: L B o P P ' = / _ B ~ P ' P = 0o, where 0o is the angle corresponding to the equilibrium position of the base pair. LBPBo = LB'P'B~ = ok, where ck represents the angle displacement of each base. When ck> 0, the Hbonds are stretched, when ~ < 0, the H-bonds are compressed, l = l' is the length of the rigid massless rod connecting P and B or P' and B'.
(b) /
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Fig. I. The chest expander-like model for a DNA chain with N base pairs. Each base is represented by a point mass, and the hydrogen bonds betweentwo complementarybases are simulated by a nonlinear massless spring. The rigid rods PB and P'B' reflect the fact that each base in DNA is boundedto C 1' atomof the double helix through a Cl'-N-glucoside bond. Therefore, the stretching motion of the complementaryhydrogenbonds is associatedwith the rotationof the respectivebases around the axis passing through the point P (or P') and being parallel :o the DNA axis. Apparently, such an internal motionof DNA can be likened to the "chest expander-likemotion". (a) A sketchof the DNA double helix; (b) a view perpendicularto the base pair plane. For simplification,only one base pair is shown althoughthe model is used to describea DNA double helix segment with N complementarybase pairs. For an angle displacement ~, the length variation of the spring can be expressed as 8 r = 2/[cos 0o - cos( 0o + ~) ] • The H-bond is represented by a Morse potential
(2. I )
C.-Z Zhang, K.-C. Chou / Chemical Physics 191 (1995) 17-23
V(Sr) = Uo( 1 - e -b~) 2,
(2.2)
where Uo and b are two parameters. Now consider a DNA chain. Let 4'. be the angle displacement associated with the nth base pair. As usual, the stacking energy takes the form [ 8-I0,16,17 ] ½k(4", - 4"._, )2 + ½k(4',+, - 4'.)2,
(2.3)
where k is a parameter related to the stacking energy between two adjoining base pairs. The Hamiltonian of DNA chain reads H-- ~
I "2 [2X~14'~ + 2 X ~k(4".-4"#_l)2+V(4"~)] ,
n
(2.4) where I = M l 2 is the mor~ent of inertia of the base (Fig. l ), and M is the mass of each base. In Eq. (2.4) V(qb,) is the H-bond potential as a function of the angle displacement 4',: V(4',,) = Uo( 1 - e -bs~,,)2,
(3.4)
Setting too = 1 / ~ = 2
sin 00 ~/-Uob2/M,
(2.6)
The equation of motion is soon obtained by Eq. (2.4), 10V(4'.) 2 04',
(2.7)
We shall find the various solutions of Eq. (2.7) in the following sections.
(3.5)
Eq. (3.3) reduces to k ~, = - too2~b,,+ ~ ( 4 ' , + t - 2 4 ' , + 4 ' , - t ) •
(3.6)
Suppose that 4', = 4',o exp[ - i( ma - tot) ] ,
(3.7)
where 4',o is a constant, v is the wave number and a is the base spacing, a = 3.4/~. Substituting Eq. (3.7) into Eq. (3.6) we find the dispersion relation O.i2
= to2 + ~sin2(½ m) .
(3.8)
Hereafter we shall take the continuum approximation 4',,(0 ~ 4"(z, t),
8r,, = 2/[cos 0o - cos( 00 + 4',) ] •
I~. = k ( 4 ' . , + , - 2 4 ' . + 4 ' . _ , )
A =4Uob212 sin20o.
(2.5)
where
19
1I
y" ~ -a
dz,
(3.9)
where z is the coordinate along the helical axis..Then Eq. (3.6) reduces to the famous Klein-Gordon equation 4,,,-c24'=
= - tog4',
(3.1o)
where (3.11)
c=vr~a
is the velocity of wave. Eq. (3.10) has a plane-wave solution
3. Phonon mode solution
4'otexp[ - i ( ~z ' t )t) ]
Let us first study an extreme case, in which 4', ~ 0. In other words, 8rn ~ O, too. In this case the nonlinear term in Eq. (2.7) may be neglected. So,
with the following dispersion relation:
8r,, =21 sin 0o 4'n.
Since
(3.1)
Accordingly, only the first term of the Taylor expansion of Eq. (2.5) remains, V( 4'~) =4UobZl 2 sin20o 4',~.
(3.2)
The equation of motion in this case is I~, =k(4',+ ! - 2 4 ' , + ~b,_,) - a 4 ' , , where
(3.3)
to2=to2 +c2v2.
sin2(½va)=~v2a 2,
(3.12)
(3.13)
m < < 1,
Eq. (3.8) reduces to Eq. (3.13), when va << 1 or a << A, i.e. for the wave with large wave-length the continuum approximation is a good one. According to the phonon mode solution Eq. (3.7) or Eq. (3.12), a plane wave can be propagated along the DNA chain with velocity c. Under some boundary conditions, the standing wave may be formed. Basing on these stand-
C.-T. Zhang, K.-C. Chou / Chemical Physics 191 (1995) 17-23
20
ing waves the resonant coupling in DNA chain has been studied by Chou and co-workers [ 18-20].
4. Nonlinear wave solution
by the phonons. We shall analyze this problem briefly in the following. Without losing generality we assume that the velocity of the solitary wave is equal to zero. Then the wave takes the form d~o(Z) = tkma~ sech 2( z/ 2d) .
Now let us study the solution in the case of small amplitude. In this case the nonlinear term in Eq. (2.7) cannot be neglected. Then instead of Eq. (3.2) the cubic term in the Taylor expansion of Eq. (2.5) has also to be taken into account V(~.) =A~b2 -B@~,
(4.1)
where
Now suppose that a small perturbation term ~bp(Z,t) is added to ~bo(Z). That is $(z, t ) = tko(Z)+ ~bp(Z, t ) .
(4.2)
The equation of motion is I~. -- k(q~.+, - 2 4 , . + 4~.- ,) -aq~. + B ~ 2 .
We hope to study the evolution of ~bp(Z,t) with respect to the time. To do this, we further assume a product solution form (~p(z, t) = ¢(z)e s' ,
(4.4)
where o-2=B/I.
(4.5)
Eq. (4.4) is one of the nonlinear Klein-Gordon equations. Sometimes we call Eq. (4.4) the square nonlinear Klein-Gordon (2NKG) equation. Eq. (4.4) has the solitary wave solution [9] O(z, t) = Sm~, seeh2[ y ( z - vt)/2d] ,
(4.13)
(4.3)
Taking the continuum approximation, we can reduce Eq. (4.3) to Su -- C 2 ~ = = -- tO02~ + O-2~b2 ,
(4.12)
Substituting Eq. (4.12) into Eq. (4.4), a linear equation about ~bp(Z, t) is obtained approximately ~)p,tt -- C2~p.zz = ( -- O302+ 2o-2t~0) t~p "
B = 12Uob313 sin30o.
(4.11 )
(4.6)
(4.14)
where s is a parameter. According to Scott et al. [21 ], "The basic problem of linear stability is to determine whether or not for any such product solution, with reasonable boundary conditions on ¢ as z tends to infinite, the real part ofs is greater than zero". Substituting Eq. (4.14) into Eq. (4.13) we obtain - ~b= - (2o-2/c 2) tko~ = _ [ (s 2 + &,)
(4.15)
Ic 2] ¢.
Eq. (4.15) is the famous Schr6dingcc equation and we have to solve an eigenvalue problem. The potential in this equation is
where 2 o -2
V(z) =
1 ~bma,,= 2b/sin 0o'
(4.7)
d=c/too =adp=~ kX/~o ,
(4.8)
c 2 (ko(Z)
and the eigenvahe term is - (s2+tO2o)/C2 "
1
~-" 1/1 -
(v/c)
2
(4.9)
and v is the velocity of solitary wave. The width of the solitary wave is approximately W--4d=4aCmax k f ~ o .
(4.10)
It has been shown that the solitary wave solution Eq. (4.6) is unstable in the sense that it tan be destroyed
(4.16)
(4.17)
The ground-state eigenfunction of the Schr6dinger equation takes the form [9] constant × sech3(z/2d).
(4.18)
with an eigenvalue -- (S2 + 002)/C2--~- -9Uob212sin2Oo/ka 2 which corresponds to
(4.19)
C-T. Zhang, K.-C Chou / Chemical Physics 191 (1995) 17-23
(4.20)
s z = 5Uo b2 sin2Oo/M.
Since s is a real number in our case, we have at least one solution in which the real part of s is greater than zero. We thus conclude that the solitary wave solution is unstable. Define 1
r-- - = ~ s b sin 0o
Uo
,
B
M = 2 . 0 X 10 -28 kg,
a-- 1.6 k m / s .
(4.22)
Vo=70.3 cm - i ,
1 =5.0 ° 2bl sin 0o
Es
and its width is
where the first and second term on the right-hand side correspond to the kinetic and potential energies of the solitary wave, respectively.
= 0 " 7 × 10-~3s
(5.8)
The amplitude of the solitary wave is ~bm,~=
(4.23)
(5.7)
which is in the range of 10-120 cm -t. According to Eq. (4.21) the life-time of the solitary wave is
Substituting Eq. (4.6) into Eq. (4.22) and completing the integral, we find k ffoo(V/C) 2 + r~m,~ ,6 kl~o
(5.6)
The wave number associated with ~Oois
1 "/~o "r= b sin--'-"~o
O~
(5.5)
we find c=
(I1~2 ..~_ka2~2 +A~b2_ ~B~b 2 3) dz.
E~= a
It was found that k=0.08 eV [22]. Therefore, all the five parameters are determined completely. Taking the average mass of base as [ 10]
(4.21)
when t > "r, ~bp becomes divergent quickly. It means that the solitary wave is destroyed by the small perturbation, or by the plane wave discussed in the above section. The time constant "r may be regarded as the life-time of the solitary wave. In terms of the continuum approximation, the energy of the solitary wave Eq. (4.6) may be calculated as
21
W=aa6ma~
kVrkf'~o= a .
(5.9)
(5.10)
The potential energy of the solitary wave is es =~Ckm~ kv/~o=0.017eV =0.38 kcal/mol.
5. Estimate of parameters There are five parameters in this theory. Two parameters b and Uo are associated with the H-bond energy. Another two parameters l and 0o are related to the structure of DNA double helix. The last one k is related to the stacking energy of bases. According to Prohofsky
[7] b = 2 . 5 ,~,-t,
(5.1)
Uo = 0.4 eV,
(5.2)
where the value of Uo is considered already to represent two to three H-bonds. Volkov and Kosevich have calculated the values of l and 0o based on the X-ray data for B-DNA [ 13 ] /=4.9 A ,
(5.3)
0o = 2 8 ° •
(5.4)
(5.11 )
6. Discussion and conclusion There are two kinds of solutions that we have found in this study, both for the cases of small amplitude. When the angular displacement tends to zero, a plane wave can be excited and propagated along the DNA chain with a velocity of approximately 1 km/s. The frequency of the wave is in the low frequency region, about 70 cm-~. When the angular displacement is larger than in the former case, however, a cubic nonlinear term in the hydrogen bonds potential has been taken into ~-=.ount. Consequently, we have to solve a nonlinear Kleiu-Gordon equation. Fortunately, one of the analytical solutions has been found. The solution is a solitary wave. The potential energy of this solitary
22
C. - T. Zhang, K.- C. Chou / Chemical Physics 191 ( ! 995) 17-23
wave is only 0.38 kcal/mol. Since this quantity < kBT, where T= 300 K, it implies that such a solitary wave is easily excited by the collision of the water molecules in the cell. The width of the solitary wave is only about one base pair. It means that the opening for only one base is possible. The amplitude of the nonlinear wave is +5 ° by our calculation. That is to say, the opening of the bases is toward the major groove. In the previous studies [ 11,12], the rotation of bases to the minor groove is prohibited by introducing an infinite potential of stearic hindrance. In the current model, however, such a constraint is imposed by the nonlinearity of the hydrogen bonds. Furthermore, the solitary wave in this system is found to be unstable. See also the discussion of Dauxois et al. [ 23 ]. Its life-time is roughly estimated to be 10-t3 s. By summarizing the result discussed above, the physical picture may be described as follows. Owing to the collision of water molecule~, one or two base pairs may be opened toward the/major groove with an amplitude of 5 o. After that, the excitation decays into the plane wave quickly. It is estimated that an angle of 15 ° corresponds to the opening leading to the breaking of the hydrogen bonds in base pair [ 12]. So, in our case the bases are far from being opened to break the hydrogen bonds. The bases undergo only the quick fluctuation of a small amplitude. It is a dynamical self-fluctuation, i.e., the fluctuation is due to the dynamics described by the differential equation. Such fluctuation of base rotation may be used to explain the experimental results of the fluorescence depolarization studies [ 24 ], see also Ref. [ 12] and the references cited therein. Recently, Georghiou and co-workers [24] studied the conformational flexibility of poly(dA)poly(dT) and (dA)2o-(dT)2o by using the intrinsic fluorescence. They excited selectively the T bases by using a vertically polarized exciting laser pulse and observed the vertical Iv and the horizontal IH fluorescence components as a function of time. Actually they looked at the fluorescence anisotropy r(t) defined as r(t) = [Iv(t) - In(t) ] / [Iv(t) + 2Iv(t) ]. If the bases of DNA do not move considerably during the lifetime of the excited state, one would expect the anisotropy to have a high value and not to change with time. However, the experimental results show that the anisotropy exhibits a large drop within one nanosecond (ns) and then tends gradually to a constant over the time scale of 4 ns [24]. This implies that the DNA double helix possesses considerable flexibility which allows the
bases to undergo motions of large amplitudes. We think that the local self-fluctuation of base rotation in DNA helix studied in this paper may be one of the possible origins, which causes the large drop of the anisotropy within 1 ns. In conclusion, a new model called the chest expander-like model has been proposed in this paper. In this model the bases are permitted to rotate along an axis being parallel to the helical axis of the double helix. The rotations are assumed to be caused by the stretching of the hydrogen bonds. If the angular displacement tends to zero, then a plane wave can be excited and propagated along the DNA chain. If the angular displacement is in the range of several degrees, a cubic nonlinear term of the hydrogen bonds potential has been taken into account. Consequently, a solitary wave of small amplitude (5 o ), narrow width (about one base pair), short life-time (about 10-13 s) and low energy (0.38 kcal/mol) may be excited. This theory predicts a qaick fluctuation of the base rotation toward the major groove. The nonlinearity of the hydrogen bonds ensures the rotation of bases is toward the major groove. The dynamical self-fluctuation of bases is also due to the characteristic of hydrogen bonds. The predicted motion of bases may be used to explain the recent experimental results of the picosecond anisotropy study using the intrinsic fluorescence [24].
Acknowledgement Valuable discussions with Dr. B. Mao and Dr. G.M. Maggiora are gratefully acknowledged. This study was supported in part by grant no. 19325002 from the China Natural Science Foundation.
References Ill H. Urabe and Y. Tominaga, J. Phys. Soe. Japan 50 (1981) 3543. [2] P.C. Painter, L.E. Mosher and C. Rhoads, Biopolymers 20 (1981) 243. [3] J.W. Powell, G.S. Edwards, L. Genzel and A. Wittlin, Phys. Rev. A 35 (1987) 3929. 14l K.C. Chou, Bioehem. J. 221 (1984) 27. [5l A.C. Scott, in, Nonlinear phenomena in physics and biology, ed. R.H. Enns (Plenum Press, New York, 1973) p. 9. [61 S. Yomosa, Phys. Rev. A 32 (1985) 1752.
C.-T. Zhang, K.-C. Chou /Chemtcal Physics 191 (1995) 17-23 [71 E.W. Prohofsky, Phys. Rev. A 38 (1988) 1538. [81 M. Techera, L.L. Daemen and E.W. Prohofsky, Phys. Re.:. A 40 (1989) 6636. [ 9 ] C.T. Zhang, J. Phys. Condensed Matter 2 (1990) 8259. [ I 0 ] V. Muto, P.S. Lomdahl and P.L. Christiansen, Phys. Rev. A 42 (1990) 7452. [11 ] J. Ramstein and R. Lavery, L Biomol. Struct. Dyn. 7 (1990) 915. [ 121 F. Briki, J. Ramstein, R. Lavery and D -~nest, J. Am. Chem. Soe, 113 (1991) 2490. [13l S.N. Volkov and A.M. Kosevich, J . . . . Struct, Dyn. 8 (1991) 1069, [ 14] L. Yakushevich, Phys. Letters A 136 (1989) 413. [ 15 ! G. Gaeta, Phys. Letters A 172 (1993) 365.
23
i 161 C.T. Zhang, Phys. Rev. A 35 (1987) 886. [ 171 C.T. Zhang, Phys. Rev. A 40 (1989) 2148. 118] K.C. Chou and G.M. Maggiora, British Polym. J. 20 (1988) 143. 1191 K.C. Chou and B. Mao, Biopolymers 27 (1988) 1795. i201 K.C. Chou, Biophys. Chem. 30 (1988) 3. (211 A.C. Scott, F.Y. Chu and D.W. McLanghlin, Proc. IEEE 61 (1973) 1443. 1221 C.T. Zhang and S.X. Shang, J. Theoret. Biol. 149 ( 1991 ) 257. 1231 T. Dauxois, M. Peyrard and C.R. Willis, Physica D 57 (1992) 267. 1241 S. Georghiou, J.M. Beechem, T.D. Bradrick and A. Philippetis, preprint, University of Tennessee, USA (1994).
Chemical Physics 191 (1995) 17-23
The nonlinear stretching model of hydrogen bonds and local self-fluctuation of base rotation in DNA Chun-Ting Zhang a, Kuo-Chen Chou b " Department of Physics, Tianjin University, Tianjin 300072, China h Computer-Aided Drug Discovery, Upjohn Laboratories, Kalamazoo. Mi 49001, USA Rec,,'ived 20 June 1994; in final form 23 September 1994
Abstract b
A new model in which the stretching of hydrogen bonds causes the rotation of bases around an axis being parallel to the helical axis in DNA has been proposed. When the angular displacement tends to zero, it is shown that a plane wave may be excited. When the angular displacement is in the range of several degrees, a cubic nonlinear term in the hydrogen bonds potential has been taken into account. Thus the following new results have been derived. ( 1) A solitary wave with narrow width (about one base pair), small amplitude (5 ° ), low energy (0.38 kcal/mol) and short life-time (about 10- ~3s) may be excited by the collision of water molecules. (2) The nonlinearity of hydrogen bonds would ensure the rotation of bases toward the major groove. Accordingly, a local and quick self-fluctuation of bases rotation toward the major groove with small amplitude seems to occur in a DNA chain. This new prediction has been compared with the relevant experimental results available.
1. Introduction The vibration modes of hydrogen bona stretching in B-DNA at 10-120 c m - ~ have been observed in both low frequency Raman scattering [ 1,2] and Fouriertransform infrared absorption [ 3 ]. In the double helix structure the base A (adenine) is bound to the base T (thymine) with two hydrogen bonds, and the base G (guanine) is bound to the base C (cytosine) with three hydrogen bonds. Therefore, the stretching of hydrogen bond causes the vibration of bases. Chou has developed a linear vibration theory of H-bond stretching in DNA and discussed some relevant biological functions [4]. On the other hand, many investigators have pointed out that the hydrogen bond is essentially nonlinear [5,6]. Prohofsky has found that the modes of hydrogen bond stretching at 10-120 c m - ~are highly nonlinear [7]. A model of hydrogen-bond stretching in which the hydrogen bond potential is approximately expressed by a 0301-0104/95/$09.50 © 1995 Elsevier Science B.V. All fights reserved SSD1030 ! - 0 1 0 4 ( 9 4 ) 0 0 3 6 2 - 9
Morse potential has been studied by Techera et al. [ 8 ]. A similar study has been carried out by Zhang by using a 2-3 power potential instead of the Morse potential [9]. Recently, Muto et al. have proposed a two-dimensional discrete model in which the hydrogen bond is represented by a Lennard-Jones potential [10]. It should be pointed out that in the above studies of hydrogen-bond stretching, each base pair is modeled by a pair of free point masses connected by a nonlinear massless spring. However, each such base in DNA is actually bounded to the C 1' atom of the double helix through a C l'-N-glucoside bond. Therefore, the stretching of hydrogen bond should cause the rotation of bases. To consider this fact a new model of nonlinear hydrogen-bond stretching has been proposed in this paper. It is called a "chest expander-like" model in which the bases are permitted to rotate along an axis parallel to the helical axis of the double helix. The
18
C.-T. Zhang, K.-C. Chou / Chemical Physics 191 (1995) 17-23
nonlinearity of hydrogen bond ensures that the angular displacement of bases mainly opens towards the major groove. A similar rotation of bases has also been studied by Ramstein and Lavery [ 11 ], Briki et al. [ 12], and Volkov and Kosevich [ 13]. But in all of these studies the effect of nonlinearity of hydrogen bond has not yet been taken into account. The current work is closely related to the studies by Yakushevich [ 14] and Gaeta [I5]. However, Yakushevich's treatment only dealt with the stable soliton excitation•
J
(a)
rV'vx.~
/p"
2. The chest expander.like model
S" According to the above description, the model may be illustrated in Fig. I, where B and B' represent a pair of complementary bases, which are connected by a rigid massless rod with points P and P', respectively. The latter, however, are attached to the two double helical strands of DNA, S and S', respectively. The zigzag line connecting B and B' is a nonlinear massless spring, representing the hydrogen bonds between the complementary bases. As mentioned above, each base has the freedom to rotate around the axis passing through the point P (P') being parallel to the DNA helical axis (see Fig. 1). Therefore, the stretching of the H-bond is associated with the rotation of the bases. In other words, when each base in a base pair rotates outward or inward, the corresponding H-bond would be stretched or compressed accordingly as shown in Fig. 1, where PB and P'B' can be likened to the two arms o f a person engaging in the chest-expanding exercise, and the zigzag line to the spring of a chest expander. Therefore, the stretching of the H-bonds associated with the rotation of the corresponding base pair can be actually compared to the chest expanderlike motion. The model based on such a physical picture is called the "chest expander-like model". For convenience, like all the previous treatments [8-10], the difference between pyrimidine and purine bases is ignored. Thus the following symmetrical conditions are valid: L B o P P ' = / _ B ~ P ' P = 0o, where 0o is the angle corresponding to the equilibrium position of the base pair. LBPBo = LB'P'B~ = ok, where ck represents the angle displacement of each base. When ck> 0, the Hbonds are stretched, when ~ < 0, the H-bonds are compressed, l = l' is the length of the rigid massless rod connecting P and B or P' and B'.
(b) /
./
\.
majorgroove
/ i
.
%. \
o"
\
•
•
"% "%
minororoove
"%.~
1.f"
/
•
/
Fig. I. The chest expander-like model for a DNA chain with N base pairs. Each base is represented by a point mass, and the hydrogen bonds betweentwo complementarybases are simulated by a nonlinear massless spring. The rigid rods PB and P'B' reflect the fact that each base in DNA is boundedto C 1' atomof the double helix through a Cl'-N-glucoside bond. Therefore, the stretching motion of the complementaryhydrogenbonds is associatedwith the rotationof the respectivebases around the axis passing through the point P (or P') and being parallel :o the DNA axis. Apparently, such an internal motionof DNA can be likened to the "chest expander-likemotion". (a) A sketchof the DNA double helix; (b) a view perpendicularto the base pair plane. For simplification,only one base pair is shown althoughthe model is used to describea DNA double helix segment with N complementarybase pairs. For an angle displacement ~, the length variation of the spring can be expressed as 8 r = 2/[cos 0o - cos( 0o + ~) ] • The H-bond is represented by a Morse potential
(2. I )
C.-Z Zhang, K.-C. Chou / Chemical Physics 191 (1995) 17-23
V(Sr) = Uo( 1 - e -b~) 2,
(2.2)
where Uo and b are two parameters. Now consider a DNA chain. Let 4'. be the angle displacement associated with the nth base pair. As usual, the stacking energy takes the form [ 8-I0,16,17 ] ½k(4", - 4"._, )2 + ½k(4',+, - 4'.)2,
(2.3)
where k is a parameter related to the stacking energy between two adjoining base pairs. The Hamiltonian of DNA chain reads H-- ~
I "2 [2X~14'~ + 2 X ~k(4".-4"#_l)2+V(4"~)] ,
n
(2.4) where I = M l 2 is the mor~ent of inertia of the base (Fig. l ), and M is the mass of each base. In Eq. (2.4) V(qb,) is the H-bond potential as a function of the angle displacement 4',: V(4',,) = Uo( 1 - e -bs~,,)2,
(3.4)
Setting too = 1 / ~ = 2
sin 00 ~/-Uob2/M,
(2.6)
The equation of motion is soon obtained by Eq. (2.4), 10V(4'.) 2 04',
(2.7)
We shall find the various solutions of Eq. (2.7) in the following sections.
(3.5)
Eq. (3.3) reduces to k ~, = - too2~b,,+ ~ ( 4 ' , + t - 2 4 ' , + 4 ' , - t ) •
(3.6)
Suppose that 4', = 4',o exp[ - i( ma - tot) ] ,
(3.7)
where 4',o is a constant, v is the wave number and a is the base spacing, a = 3.4/~. Substituting Eq. (3.7) into Eq. (3.6) we find the dispersion relation O.i2
= to2 + ~sin2(½ m) .
(3.8)
Hereafter we shall take the continuum approximation 4',,(0 ~ 4"(z, t),
8r,, = 2/[cos 0o - cos( 00 + 4',) ] •
I~. = k ( 4 ' . , + , - 2 4 ' . + 4 ' . _ , )
A =4Uob212 sin20o.
(2.5)
where
19
1I
y" ~ -a
dz,
(3.9)
where z is the coordinate along the helical axis..Then Eq. (3.6) reduces to the famous Klein-Gordon equation 4,,,-c24'=
= - tog4',
(3.1o)
where (3.11)
c=vr~a
is the velocity of wave. Eq. (3.10) has a plane-wave solution
3. Phonon mode solution
4'otexp[ - i ( ~z ' t )t) ]
Let us first study an extreme case, in which 4', ~ 0. In other words, 8rn ~ O, too. In this case the nonlinear term in Eq. (2.7) may be neglected. So,
with the following dispersion relation:
8r,, =21 sin 0o 4'n.
Since
(3.1)
Accordingly, only the first term of the Taylor expansion of Eq. (2.5) remains, V( 4'~) =4UobZl 2 sin20o 4',~.
(3.2)
The equation of motion in this case is I~, =k(4',+ ! - 2 4 ' , + ~b,_,) - a 4 ' , , where
(3.3)
to2=to2 +c2v2.
sin2(½va)=~v2a 2,
(3.12)
(3.13)
m < < 1,
Eq. (3.8) reduces to Eq. (3.13), when va << 1 or a << A, i.e. for the wave with large wave-length the continuum approximation is a good one. According to the phonon mode solution Eq. (3.7) or Eq. (3.12), a plane wave can be propagated along the DNA chain with velocity c. Under some boundary conditions, the standing wave may be formed. Basing on these stand-
C.-T. Zhang, K.-C. Chou / Chemical Physics 191 (1995) 17-23
20
ing waves the resonant coupling in DNA chain has been studied by Chou and co-workers [ 18-20].
4. Nonlinear wave solution
by the phonons. We shall analyze this problem briefly in the following. Without losing generality we assume that the velocity of the solitary wave is equal to zero. Then the wave takes the form d~o(Z) = tkma~ sech 2( z/ 2d) .
Now let us study the solution in the case of small amplitude. In this case the nonlinear term in Eq. (2.7) cannot be neglected. Then instead of Eq. (3.2) the cubic term in the Taylor expansion of Eq. (2.5) has also to be taken into account V(~.) =A~b2 -B@~,
(4.1)
where
Now suppose that a small perturbation term ~bp(Z,t) is added to ~bo(Z). That is $(z, t ) = tko(Z)+ ~bp(Z, t ) .
(4.2)
The equation of motion is I~. -- k(q~.+, - 2 4 , . + 4~.- ,) -aq~. + B ~ 2 .
We hope to study the evolution of ~bp(Z,t) with respect to the time. To do this, we further assume a product solution form (~p(z, t) = ¢(z)e s' ,
(4.4)
where o-2=B/I.
(4.5)
Eq. (4.4) is one of the nonlinear Klein-Gordon equations. Sometimes we call Eq. (4.4) the square nonlinear Klein-Gordon (2NKG) equation. Eq. (4.4) has the solitary wave solution [9] O(z, t) = Sm~, seeh2[ y ( z - vt)/2d] ,
(4.13)
(4.3)
Taking the continuum approximation, we can reduce Eq. (4.3) to Su -- C 2 ~ = = -- tO02~ + O-2~b2 ,
(4.12)
Substituting Eq. (4.12) into Eq. (4.4), a linear equation about ~bp(Z, t) is obtained approximately ~)p,tt -- C2~p.zz = ( -- O302+ 2o-2t~0) t~p "
B = 12Uob313 sin30o.
(4.11 )
(4.6)
(4.14)
where s is a parameter. According to Scott et al. [21 ], "The basic problem of linear stability is to determine whether or not for any such product solution, with reasonable boundary conditions on ¢ as z tends to infinite, the real part ofs is greater than zero". Substituting Eq. (4.14) into Eq. (4.13) we obtain - ~b= - (2o-2/c 2) tko~ = _ [ (s 2 + &,)
(4.15)
Ic 2] ¢.
Eq. (4.15) is the famous Schr6dingcc equation and we have to solve an eigenvalue problem. The potential in this equation is
where 2 o -2
V(z) =
1 ~bma,,= 2b/sin 0o'
(4.7)
d=c/too =adp=~ kX/~o ,
(4.8)
c 2 (ko(Z)
and the eigenvahe term is - (s2+tO2o)/C2 "
1
~-" 1/1 -
(v/c)
2
(4.9)
and v is the velocity of solitary wave. The width of the solitary wave is approximately W--4d=4aCmax k f ~ o .
(4.10)
It has been shown that the solitary wave solution Eq. (4.6) is unstable in the sense that it tan be destroyed
(4.16)
(4.17)
The ground-state eigenfunction of the Schr6dinger equation takes the form [9] constant × sech3(z/2d).
(4.18)
with an eigenvalue -- (S2 + 002)/C2--~- -9Uob212sin2Oo/ka 2 which corresponds to
(4.19)
C-T. Zhang, K.-C Chou / Chemical Physics 191 (1995) 17-23
(4.20)
s z = 5Uo b2 sin2Oo/M.
Since s is a real number in our case, we have at least one solution in which the real part of s is greater than zero. We thus conclude that the solitary wave solution is unstable. Define 1
r-- - = ~ s b sin 0o
Uo
,
B
M = 2 . 0 X 10 -28 kg,
a-- 1.6 k m / s .
(4.22)
Vo=70.3 cm - i ,
1 =5.0 ° 2bl sin 0o
Es
and its width is
where the first and second term on the right-hand side correspond to the kinetic and potential energies of the solitary wave, respectively.
= 0 " 7 × 10-~3s
(5.8)
The amplitude of the solitary wave is ~bm,~=
(4.23)
(5.7)
which is in the range of 10-120 cm -t. According to Eq. (4.21) the life-time of the solitary wave is
Substituting Eq. (4.6) into Eq. (4.22) and completing the integral, we find k ffoo(V/C) 2 + r~m,~ ,6 kl~o
(5.6)
The wave number associated with ~Oois
1 "/~o "r= b sin--'-"~o
O~
(5.5)
we find c=
(I1~2 ..~_ka2~2 +A~b2_ ~B~b 2 3) dz.
E~= a
It was found that k=0.08 eV [22]. Therefore, all the five parameters are determined completely. Taking the average mass of base as [ 10]
(4.21)
when t > "r, ~bp becomes divergent quickly. It means that the solitary wave is destroyed by the small perturbation, or by the plane wave discussed in the above section. The time constant "r may be regarded as the life-time of the solitary wave. In terms of the continuum approximation, the energy of the solitary wave Eq. (4.6) may be calculated as
21
W=aa6ma~
kVrkf'~o= a .
(5.9)
(5.10)
The potential energy of the solitary wave is es =~Ckm~ kv/~o=0.017eV =0.38 kcal/mol.
5. Estimate of parameters There are five parameters in this theory. Two parameters b and Uo are associated with the H-bond energy. Another two parameters l and 0o are related to the structure of DNA double helix. The last one k is related to the stacking energy of bases. According to Prohofsky
[7] b = 2 . 5 ,~,-t,
(5.1)
Uo = 0.4 eV,
(5.2)
where the value of Uo is considered already to represent two to three H-bonds. Volkov and Kosevich have calculated the values of l and 0o based on the X-ray data for B-DNA [ 13 ] /=4.9 A ,
(5.3)
0o = 2 8 ° •
(5.4)
(5.11 )
6. Discussion and conclusion There are two kinds of solutions that we have found in this study, both for the cases of small amplitude. When the angular displacement tends to zero, a plane wave can be excited and propagated along the DNA chain with a velocity of approximately 1 km/s. The frequency of the wave is in the low frequency region, about 70 cm-~. When the angular displacement is larger than in the former case, however, a cubic nonlinear term in the hydrogen bonds potential has been taken into ~-=.ount. Consequently, we have to solve a nonlinear Kleiu-Gordon equation. Fortunately, one of the analytical solutions has been found. The solution is a solitary wave. The potential energy of this solitary
22
C. - T. Zhang, K.- C. Chou / Chemical Physics 191 ( ! 995) 17-23
wave is only 0.38 kcal/mol. Since this quantity < kBT, where T= 300 K, it implies that such a solitary wave is easily excited by the collision of the water molecules in the cell. The width of the solitary wave is only about one base pair. It means that the opening for only one base is possible. The amplitude of the nonlinear wave is +5 ° by our calculation. That is to say, the opening of the bases is toward the major groove. In the previous studies [ 11,12], the rotation of bases to the minor groove is prohibited by introducing an infinite potential of stearic hindrance. In the current model, however, such a constraint is imposed by the nonlinearity of the hydrogen bonds. Furthermore, the solitary wave in this system is found to be unstable. See also the discussion of Dauxois et al. [ 23 ]. Its life-time is roughly estimated to be 10-t3 s. By summarizing the result discussed above, the physical picture may be described as follows. Owing to the collision of water molecule~, one or two base pairs may be opened toward the/major groove with an amplitude of 5 o. After that, the excitation decays into the plane wave quickly. It is estimated that an angle of 15 ° corresponds to the opening leading to the breaking of the hydrogen bonds in base pair [ 12]. So, in our case the bases are far from being opened to break the hydrogen bonds. The bases undergo only the quick fluctuation of a small amplitude. It is a dynamical self-fluctuation, i.e., the fluctuation is due to the dynamics described by the differential equation. Such fluctuation of base rotation may be used to explain the experimental results of the fluorescence depolarization studies [ 24 ], see also Ref. [ 12] and the references cited therein. Recently, Georghiou and co-workers [24] studied the conformational flexibility of poly(dA)poly(dT) and (dA)2o-(dT)2o by using the intrinsic fluorescence. They excited selectively the T bases by using a vertically polarized exciting laser pulse and observed the vertical Iv and the horizontal IH fluorescence components as a function of time. Actually they looked at the fluorescence anisotropy r(t) defined as r(t) = [Iv(t) - In(t) ] / [Iv(t) + 2Iv(t) ]. If the bases of DNA do not move considerably during the lifetime of the excited state, one would expect the anisotropy to have a high value and not to change with time. However, the experimental results show that the anisotropy exhibits a large drop within one nanosecond (ns) and then tends gradually to a constant over the time scale of 4 ns [24]. This implies that the DNA double helix possesses considerable flexibility which allows the
bases to undergo motions of large amplitudes. We think that the local self-fluctuation of base rotation in DNA helix studied in this paper may be one of the possible origins, which causes the large drop of the anisotropy within 1 ns. In conclusion, a new model called the chest expander-like model has been proposed in this paper. In this model the bases are permitted to rotate along an axis being parallel to the helical axis of the double helix. The rotations are assumed to be caused by the stretching of the hydrogen bonds. If the angular displacement tends to zero, then a plane wave can be excited and propagated along the DNA chain. If the angular displacement is in the range of several degrees, a cubic nonlinear term of the hydrogen bonds potential has been taken into account. Consequently, a solitary wave of small amplitude (5 o ), narrow width (about one base pair), short life-time (about 10-13 s) and low energy (0.38 kcal/mol) may be excited. This theory predicts a qaick fluctuation of the base rotation toward the major groove. The nonlinearity of the hydrogen bonds ensures the rotation of bases is toward the major groove. The dynamical self-fluctuation of bases is also due to the characteristic of hydrogen bonds. The predicted motion of bases may be used to explain the recent experimental results of the picosecond anisotropy study using the intrinsic fluorescence [24].
Acknowledgement Valuable discussions with Dr. B. Mao and Dr. G.M. Maggiora are gratefully acknowledged. This study was supported in part by grant no. 19325002 from the China Natural Science Foundation.
References Ill H. Urabe and Y. Tominaga, J. Phys. Soe. Japan 50 (1981) 3543. [2] P.C. Painter, L.E. Mosher and C. Rhoads, Biopolymers 20 (1981) 243. [3] J.W. Powell, G.S. Edwards, L. Genzel and A. Wittlin, Phys. Rev. A 35 (1987) 3929. 14l K.C. Chou, Bioehem. J. 221 (1984) 27. [5l A.C. Scott, in, Nonlinear phenomena in physics and biology, ed. R.H. Enns (Plenum Press, New York, 1973) p. 9. [61 S. Yomosa, Phys. Rev. A 32 (1985) 1752.
C.-T. Zhang, K.-C. Chou /Chemtcal Physics 191 (1995) 17-23 [71 E.W. Prohofsky, Phys. Rev. A 38 (1988) 1538. [81 M. Techera, L.L. Daemen and E.W. Prohofsky, Phys. Re.:. A 40 (1989) 6636. [ 9 ] C.T. Zhang, J. Phys. Condensed Matter 2 (1990) 8259. [ I 0 ] V. Muto, P.S. Lomdahl and P.L. Christiansen, Phys. Rev. A 42 (1990) 7452. [11 ] J. Ramstein and R. Lavery, L Biomol. Struct. Dyn. 7 (1990) 915. [ 121 F. Briki, J. Ramstein, R. Lavery and D -~nest, J. Am. Chem. Soe, 113 (1991) 2490. [13l S.N. Volkov and A.M. Kosevich, J . . . . Struct, Dyn. 8 (1991) 1069, [ 14] L. Yakushevich, Phys. Letters A 136 (1989) 413. [ 15 ! G. Gaeta, Phys. Letters A 172 (1993) 365.
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i 161 C.T. Zhang, Phys. Rev. A 35 (1987) 886. [ 171 C.T. Zhang, Phys. Rev. A 40 (1989) 2148. 118] K.C. Chou and G.M. Maggiora, British Polym. J. 20 (1988) 143. 1191 K.C. Chou and B. Mao, Biopolymers 27 (1988) 1795. i201 K.C. Chou, Biophys. Chem. 30 (1988) 3. (211 A.C. Scott, F.Y. Chu and D.W. McLanghlin, Proc. IEEE 61 (1973) 1443. 1221 C.T. Zhang and S.X. Shang, J. Theoret. Biol. 149 ( 1991 ) 257. 1231 T. Dauxois, M. Peyrard and C.R. Willis, Physica D 57 (1992) 267. 1241 S. Georghiou, J.M. Beechem, T.D. Bradrick and A. Philippetis, preprint, University of Tennessee, USA (1994).