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Nonlinear DNA dynamics: A new model
Volume 136, number 7,8
PHYSICS LETTERS A
17 April 1989
NONLINEAR DNA DYNAMICS: A NEW MODEL LV. YAKUSHEVICH Institute of Biological Physics, Academy ofSciences ofthe USSR, Pushchino 142292, USSR Received 9 January 1989; accepted for publication 7 February 1989 Communicated by AR. Bishop
A new model ofthe internal DNA dynamics is proposed. The nonlinear dynamical equations are derived and some particular solutions ofthe equations are discussed.
1. Introduction Recently much theoretical work on the nonlinear dynamics of the deoxyribonucleic acid (DNA) molecule has been done [1—8]. This was stimulated by new ideas on the possible role of nonlinear waves in the regulation of biological processes at the molecular level [9—13].However, the question which of the theoretical models is preferable, remains open, and the search for an optimum model still continues, In this paper, a new model of the internal DNA dynamics has been proposed. We derive the nonlinear dynamical equations, discuss their solutions and define the place of the model in the general hierarchy of the available nonlinear dynamical models.
2. Description of the model The general picture of the internal DNA dynamics is very complicated: many various motions can be found in every range of the time scale [14]. However, the contributions of the motions are different, which enables us to develop a simple method for studying the DNA dynamics. It consists in (I) choosing a few motions which dominate in the given range of the time scale and (II) modeling them by corresponding differential equations. Here we shall consider the nanosecond range. The model of Barkley and Zimm [15] is the simplest model that describes the motions in this range of the time scale. According to this model, the DNA mol-
ecule is a long elastic homogeneous rod (or strand) with a circular section (fig. la). It is just the form of the DNA molecule that can be seen on electronic microphotographs [161. The model of the rod suggests only two dominating motions: twist and bend ones. Other motions as well as other details of the internal DNA structure are not taken into account. To improve the description ofthe DNA dynamics, we propose here another model consisting of two long elastic and weakly interacting rods that are wound around each other to produce a double helix (fig. lb). Every rod simulates one of the two polynucleotide chains of the DNA molecule. The model proposed takes the intermediate position between the model of Barkley and Zimm and the exact model (fig. lc) used in computer simulations. To simplify the calculations, we shall make two assumptions. First, we neglect the helical structure of the model. So, instead of the double helix we shall consider two parallel rods (or strands) each having the form of a straight line. In the cases when the effects of helicity are important, the helical structure of the DNA molecule can be taken into account by the method proposed in ref. [17]. Second, we assume that the twist and bend motions are independent, and consider a model that takes into account only the twist motions of the DNA strands. Such an approximation is known as a model of the DNA torsion dynamics.
413
Volume 136. number 7,8
PHYSICS LETTERS A z
17 April t 989
z
a
z
b
C
Fig. I. Three models of DNA: (a) a fragment of an elastic homogeneous rod (or strand) with circular section. (b) two elastic rods forming a double helix, (c) the model of B-DNA obtained by computer simulation.
3. Hamiltonian and nonlinear dynamical equations
To derive the Hamiltonian, it is convenient to begin with the discrete case and then to pass to the continuous limit. Since the discrete analog of the model of Barkley and Zimm [1 5] consists of one chain of discs (or beads) connected with each other by longitudinal springs (fig. 2a), we can suppose that the discrete analog of our model will consist of two chains of discs (or beads) connected with each other by longitudinal and transverse springs (fig. 2b), the rigidity of the longitudinal springs being greater than that of the transverse ones. The Hamiltonian of such a model has then the form (I) Tis the kinetic energy oftorsional vibrations of the discs, and V~I) is the potential energy of the longitudinal springs and V121 is the potential energy of the transverse ones. For T and V1 1) we have where
414
z
4/
/ /
/ a Fig. 2. The discrete analogs of the model of Barkley and Zimm (a) and of our model (b).
~‘=
~ ~
(2) 2
~
~
/(~‘~_,,~-1—c~,.fl)
.
(3)
Here i and n are the numbers of the chains and discs, respectively (i= 1, 2: n= 1. 2 ~), ca,, is the angle
Volume 136, number 7,8
PHYSICS LETTERS A
of rotation of the nth disc of the ith chain, K1 is the rigidity of longitudinal springs of the ith chain, and 1, is the moment of inertia ofform the discs ofthe ith chain. t-’1 we assume the For V ~ ~k(&,,)2, (4)
where k is the rigidity of the transverse springs,and z~l,,is the stretching of the nth transverse spring due to rotations of the discs (see fig. 3), = [(2R+10—R cos ca1.~—R cos ca2.~) 2]i/2_/~ (5) + (R sin ca.~—Rsin ca2.~) Here R is the radius of the discs, and I~is the length ,
of the transverse spring in the equilibrium state. The dynamical equations associated with the Hamiltonian ~ are 2cai.~)
I1~o’122=K1(ço1~+1 ~ 2+R1 [(2R 0) ~ 2 1,,sin(ca —R 1,,+ca,~)],
—
—
k
2caLrkj [(2R2+R1 =Kia 0) sin —R2 sin(ca +ca2)]
I
ca
2+R1 2~2=K2a2ca2~_k~ [(2R 2sin(ca —R 1+ca2)]
0) sin
ca2
(8)
,
where a is the distance between the nearest bases (a~3.4 A). The nonlinear equations (8) are rather complex because ables ca the andcoefficient ~I/Iis a function of the van~,
~~~‘/I=1 ~I
0 [ (2R + 1~—R cos
+ (R ~
—
R sin
)
ca2 2]
cat
—
R cos ca2 )2 (9)
/2
A simpler form of eqs. (8) can be obtained if we assume the rods shortest distance between of thethat elastic is negligibly small (1 the surfaces 0<
=K,(ca2 ,,+ I +~92•,,_I —2ca2.fl)
~
~
17 April 1989
[(2R2 +RI In
0) sin
ca 2,,,
2 sin(ca2,,+caI,)], (6) —R wherel,=1 0+&,,. Now we can pass to the continuous limit. Substituting ca1(z, t) for ca,.,,(t) and expanding cat. ,± (1) by the Taylor series up to ca/-,, 2±..., (1)
ca,(z, 7) ±ca~(z,t)a+ ~cai~(z,t)a
(7)
we have
The approximate associated with dynamical equations Hamiltonian (10) is ~= dz ~
J
2[2cosca —kR
1+2cosca,—cos(ca1+ca2)]}
+const
(11)
.
4. Linear approximation The solutions of eqs. (10) can be easily obtained in the linear approximation, where Iø~ —K
2ca 1a
2(ca 1+kR
1 —ca2)=O,
1s~2K2aca2,,+kR(ca2_cai)0
.
(12)
Assuming the solutions have the form of plane waves R
~1=~10exp[i(qz—wt)], ~22~r~2oeXp[i(qZ.Wt)],
(13)
Fig. 3. Cross-section of the model consisting of two strands.
415
Volume 136. number 7.8
PHYSICS LETTERS A
and inserting (1 3) into (12). we find the dispersion law ~
_1:C02)
—12w2)-—k2R4=0
(14)
,
where A,=K,aq~+kR (1=1, 2), q is the wave vector. w is the frequency, and Pt~ P20 are the amplitudes. Finally, from (14) we find the values of the frequencies co/2(q)=~I1,.2+I~i,~ 1 [(IA2 121.l )2 4]l/2)/2JJ +4I1I~kR In the “symmetrical” case. when
7 April 1989
5. Nonlinear problem Thegeneral solutionofproblem (10) hasnotbeen found yet. But the first integral can be easily obtamed by the following algorithm. (I) Let us assume that the solution of eqs. (10) has the form of running waves Pt =p
(:—t’l).
P2 =p
(18)
2L—z’/).
(15) 1~=!2w1,
(11) cSubstitute (18) of into where is the velocity the(10) waves. ~‘/‘p~’—kR2[2
Sm Pi —sin(p
1 +p~)] =0.
K1
=
K2 K. the result (15) takes the form
(u1=Ka~qjI, (=(Ka~q~+2kR-)/I.
(16)
So we can conclude, that the torsional vibrations with the are the ofacoustic type =0)frequency and thosew1with frequency w (i.e. lim~1 0w1 of 4). optical 2/1)~0)2 are (fig. tYpe lim,1 ,w2=(2kR The(i.e. same picture can be found in the general case. Indeed, if we insert q= 0 into eq. (15), we have 0 ~ ~‘~i )= (q=
O,I/
( q= 0) = kR 2(J~ + I~)/I~‘2
(17)
‘
Expression (17) shows that the vibrational spectrum of the model considered consists of two branches: acoustic and optical ones. Comparing this result with the spectrum of the model of Barkley and Zimm [15] which has only one (acoustic) branch, we can conclude that the appearance of the second branch in the spectrum of our model can be cxplained by the double-strand character of the model.
U’oca~’—kRi2sin p2—sin(p +p-)]=0
~
r 7
FIg. 4. Two branches of the DNA torsional vibrations.
416
(20)
Here
2p/d~2. ~w:—i’l p’wd ~= 1.2. (III) Multiply eq. (19) by p~and eq. (20) by .
(IV) Sum the results. (V) Intergrate the expression obtained. As a result, we have the first integral in the form
~fl’/pY
+ J’V/çf~—kR2[2
cos ~Pt+2 cos p~
—cos(Pi =const (21 Besides we can point out sonic particular cases, when problem (10) reduces to well known equalions, the solution of which has been already found. (a) Pt =0, P2O (or Pt 0. P2=O). Eqs. (10) reduce then to one equation +~7)
.
IØ—Ka2p~+kR2sin p=O.
/
.
(19)
(22)
which is known as the sine-Gordon equation [18]. It can be interpreted as a model describing the torsional dynamics of one of the DNA strands, the influence of the second strand being taken into account by some empiric constant field. (b) Pl=P2. Eqs. (10) reduce then to the double sine-Gordon equation [19] I~—Ka2p~+2kR2 sin p—kR2 sin 2p=O.
(23)
Equation (23) describes the torsion dynamics of the DNA strands that move in phase. (c) Pt = —P2. Eqs. (10) reduce then to the equanon of the sine-Gordon type
Volume 136. number 7,8
PHYSICS LETTERS A
+2kR2 sin P0• (24) 1(~—Ka2p~ It describes the torsion dynamics of the strands that move out of phase by it. The particular equations mentioned above have one feature in common: all of them have the solitonlike solutions named kinks and antikinks. So we can expect that the system of equations (10) might also have the soliton-like solutions of kink (and antikink) type.
6. Discussion Model (10) is one ofthe class of nonlinear models [1—8] simulating the internal DNA dynamics. Let us compare (10) with other models of the class. We begin with the model of Englander et al. [I]. It can be considered as the simplest model of the class. The model describes the torsion dynamics of only one DNA strand which is under the action of a constant field, formed by the second strand. The equation used by Englander et al. can be easily derived from (10) by substituting 0 for P2’ This particular case has been discussed above. We can mention also the models of Yomosa [2,3] and Chang-Ting Zhang [7], which are more accurate than the model of Englander et al. The models of refs. [2,3,7] differ from (10) only by the form of the nonlinear terms describing the weak interactions between the two DNA strands. The authors succeeded in finding some particular solitonlike solutions of kink (and antikink) type. Finally, we must note the model proposed by Fedyanin, Gochev. and Lisy [5,6]. Their model is almost equivalent to (10). (It becomes equivalent tO (10) when I = ‘2 and K 1 = K2.) The exact solutions of both
17 April 1989
the the found modelyet. of refs. [5,6] and eqs. (10)equations have notof been References [I] S.W. Englander. N.R Kallenbach. A.J. Heeger, J.A. Krumhansl and S. Litwin. Proc. NatI. Acad. Sci. 77 (1980) 7222. [215. Yomosa, Phys. Rev. A 27 (1983) 2120. [3 IS. Yomosa, Phys. Rev. A 30 (1984) 474.
[41J.A.
Krumhansl and W.M. Alexander. in: Structure and dynamics: nucleic acids and proteins. eds. E. Clementi and RH. Sarma (Adenine, New York. 1983) p.61. 151 V.K. Fedyanin, 1. Gechev and V. Lisy, Stud. Biophys. 116 (1986) 59. [6) V.K. Fedyanin and V. Lisy. Stud. Biophys. 116 (1986) 65. [7) Chung-Ting Zhang, Phys. Rev. A 35 (1987) 886. [8] V. Muto. J. Halding, P.L. Christiansen and A.C. Scott, J. Struct. (1988) [91Biomol. P. Jensen. MV. Dyn. Jakis 5and K.H. 873. Bannenham. Phys. Lett. A
95(1983) 204. [101S.C. Harvey, Nucleic Acids Res. 11(1983) 4867. [II]A. Khan, D. Bhaumic and B. Dutta-Roy. Bull. Math. Biol. 47(1985)783. [12] J.J. Ladik, in: Structure and motion: membranes, nucleic acids and proteins, eds. E. Clementi, G. Corongiu. M.H. Sarma and RH. Sarma (Adenine. New York, 1985) p. 553. [13] R.V. Polozov and LV. Yakushevich, J. Theor. Biol. 130 (1988) 423. [14]R.V. Polozov and L.V. Yakushevich, in: Intermolecular interaciions and conformations of molecules, eds. N.N. Petropavlov and P.M. Zorkii (Academy of Sciences of the USSR, Pushchino. 1987) p. 186. [151M.D. Barkley and B.H. Zimm. J. Chem. Phys. 70 (1979) [16] 2991. B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and iD. Watson, Molecular biology ofthe cell, Vol. 1 (Garland, New York, 1985). [17] V,K. Fedyanin and L.V. Yakushevich. Stud. Biophys. 103 (1984) 171. [181AC. Scott, F.Y. Chu and D,W, McLaughlin. Proc. IEEE 61 (1973) 1443 [191CA. Condat, R.A. Gauer and M.D. Miller. Phys. Rev. B 27 (1983) 474.
417
PHYSICS LETTERS A
17 April 1989
NONLINEAR DNA DYNAMICS: A NEW MODEL LV. YAKUSHEVICH Institute of Biological Physics, Academy ofSciences ofthe USSR, Pushchino 142292, USSR Received 9 January 1989; accepted for publication 7 February 1989 Communicated by AR. Bishop
A new model ofthe internal DNA dynamics is proposed. The nonlinear dynamical equations are derived and some particular solutions ofthe equations are discussed.
1. Introduction Recently much theoretical work on the nonlinear dynamics of the deoxyribonucleic acid (DNA) molecule has been done [1—8]. This was stimulated by new ideas on the possible role of nonlinear waves in the regulation of biological processes at the molecular level [9—13].However, the question which of the theoretical models is preferable, remains open, and the search for an optimum model still continues, In this paper, a new model of the internal DNA dynamics has been proposed. We derive the nonlinear dynamical equations, discuss their solutions and define the place of the model in the general hierarchy of the available nonlinear dynamical models.
2. Description of the model The general picture of the internal DNA dynamics is very complicated: many various motions can be found in every range of the time scale [14]. However, the contributions of the motions are different, which enables us to develop a simple method for studying the DNA dynamics. It consists in (I) choosing a few motions which dominate in the given range of the time scale and (II) modeling them by corresponding differential equations. Here we shall consider the nanosecond range. The model of Barkley and Zimm [15] is the simplest model that describes the motions in this range of the time scale. According to this model, the DNA mol-
ecule is a long elastic homogeneous rod (or strand) with a circular section (fig. la). It is just the form of the DNA molecule that can be seen on electronic microphotographs [161. The model of the rod suggests only two dominating motions: twist and bend ones. Other motions as well as other details of the internal DNA structure are not taken into account. To improve the description ofthe DNA dynamics, we propose here another model consisting of two long elastic and weakly interacting rods that are wound around each other to produce a double helix (fig. lb). Every rod simulates one of the two polynucleotide chains of the DNA molecule. The model proposed takes the intermediate position between the model of Barkley and Zimm and the exact model (fig. lc) used in computer simulations. To simplify the calculations, we shall make two assumptions. First, we neglect the helical structure of the model. So, instead of the double helix we shall consider two parallel rods (or strands) each having the form of a straight line. In the cases when the effects of helicity are important, the helical structure of the DNA molecule can be taken into account by the method proposed in ref. [17]. Second, we assume that the twist and bend motions are independent, and consider a model that takes into account only the twist motions of the DNA strands. Such an approximation is known as a model of the DNA torsion dynamics.
413
Volume 136. number 7,8
PHYSICS LETTERS A z
17 April t 989
z
a
z
b
C
Fig. I. Three models of DNA: (a) a fragment of an elastic homogeneous rod (or strand) with circular section. (b) two elastic rods forming a double helix, (c) the model of B-DNA obtained by computer simulation.
3. Hamiltonian and nonlinear dynamical equations
To derive the Hamiltonian, it is convenient to begin with the discrete case and then to pass to the continuous limit. Since the discrete analog of the model of Barkley and Zimm [1 5] consists of one chain of discs (or beads) connected with each other by longitudinal springs (fig. 2a), we can suppose that the discrete analog of our model will consist of two chains of discs (or beads) connected with each other by longitudinal and transverse springs (fig. 2b), the rigidity of the longitudinal springs being greater than that of the transverse ones. The Hamiltonian of such a model has then the form (I) Tis the kinetic energy oftorsional vibrations of the discs, and V~I) is the potential energy of the longitudinal springs and V121 is the potential energy of the transverse ones. For T and V1 1) we have where
414
z
4/
/ /
/ a Fig. 2. The discrete analogs of the model of Barkley and Zimm (a) and of our model (b).
~‘=
~ ~
(2) 2
~
~
/(~‘~_,,~-1—c~,.fl)
.
(3)
Here i and n are the numbers of the chains and discs, respectively (i= 1, 2: n= 1. 2 ~), ca,, is the angle
Volume 136, number 7,8
PHYSICS LETTERS A
of rotation of the nth disc of the ith chain, K1 is the rigidity of longitudinal springs of the ith chain, and 1, is the moment of inertia ofform the discs ofthe ith chain. t-’1 we assume the For V ~ ~k(&,,)2, (4)
where k is the rigidity of the transverse springs,and z~l,,is the stretching of the nth transverse spring due to rotations of the discs (see fig. 3), = [(2R+10—R cos ca1.~—R cos ca2.~) 2]i/2_/~ (5) + (R sin ca.~—Rsin ca2.~) Here R is the radius of the discs, and I~is the length ,
of the transverse spring in the equilibrium state. The dynamical equations associated with the Hamiltonian ~ are 2cai.~)
I1~o’122=K1(ço1~+1 ~ 2+R1 [(2R 0) ~ 2 1,,sin(ca —R 1,,+ca,~)],
—
—
k
2caLrkj [(2R2+R1 =Kia 0) sin —R2 sin(ca +ca2)]
I
ca
2+R1 2~2=K2a2ca2~_k~ [(2R 2sin(ca —R 1+ca2)]
0) sin
ca2
(8)
,
where a is the distance between the nearest bases (a~3.4 A). The nonlinear equations (8) are rather complex because ables ca the andcoefficient ~I/Iis a function of the van~,
~~~‘/I=1 ~I
0 [ (2R + 1~—R cos
+ (R ~
—
R sin
)
ca2 2]
cat
—
R cos ca2 )2 (9)
/2
A simpler form of eqs. (8) can be obtained if we assume the rods shortest distance between of thethat elastic is negligibly small (1 the surfaces 0<
=K,(ca2 ,,+ I +~92•,,_I —2ca2.fl)
~
~
17 April 1989
[(2R2 +RI In
0) sin
ca 2,,,
2 sin(ca2,,+caI,)], (6) —R wherel,=1 0+&,,. Now we can pass to the continuous limit. Substituting ca1(z, t) for ca,.,,(t) and expanding cat. ,± (1) by the Taylor series up to ca/-,, 2±..., (1)
ca,(z, 7) ±ca~(z,t)a+ ~cai~(z,t)a
(7)
we have
The approximate associated with dynamical equations Hamiltonian (10) is ~= dz ~
J
2[2cosca —kR
1+2cosca,—cos(ca1+ca2)]}
+const
(11)
.
4. Linear approximation The solutions of eqs. (10) can be easily obtained in the linear approximation, where Iø~ —K
2ca 1a
2(ca 1+kR
1 —ca2)=O,
1s~2K2aca2,,+kR(ca2_cai)0
.
(12)
Assuming the solutions have the form of plane waves R
~1=~10exp[i(qz—wt)], ~22~r~2oeXp[i(qZ.Wt)],
(13)
Fig. 3. Cross-section of the model consisting of two strands.
415
Volume 136. number 7.8
PHYSICS LETTERS A
and inserting (1 3) into (12). we find the dispersion law ~
_1:C02)
—12w2)-—k2R4=0
(14)
,
where A,=K,aq~+kR (1=1, 2), q is the wave vector. w is the frequency, and Pt~ P20 are the amplitudes. Finally, from (14) we find the values of the frequencies co/2(q)=~I1,.2+I~i,~ 1 [(IA2 121.l )2 4]l/2)/2JJ +4I1I~kR In the “symmetrical” case. when
7 April 1989
5. Nonlinear problem Thegeneral solutionofproblem (10) hasnotbeen found yet. But the first integral can be easily obtamed by the following algorithm. (I) Let us assume that the solution of eqs. (10) has the form of running waves Pt =p
(:—t’l).
P2 =p
(18)
2L—z’/).
(15) 1~=!2w1,
(11) cSubstitute (18) of into where is the velocity the(10) waves. ~‘/‘p~’—kR2[2
Sm Pi —sin(p
1 +p~)] =0.
K1
=
K2 K. the result (15) takes the form
(u1=Ka~qjI, (=(Ka~q~+2kR-)/I.
(16)
So we can conclude, that the torsional vibrations with the are the ofacoustic type =0)frequency and thosew1with frequency w (i.e. lim~1 0w1 of 4). optical 2/1)~0)2 are (fig. tYpe lim,1 ,w2=(2kR The(i.e. same picture can be found in the general case. Indeed, if we insert q= 0 into eq. (15), we have 0 ~ ~‘~i )= (q=
O,I/
( q= 0) = kR 2(J~ + I~)/I~‘2
(17)
‘
Expression (17) shows that the vibrational spectrum of the model considered consists of two branches: acoustic and optical ones. Comparing this result with the spectrum of the model of Barkley and Zimm [15] which has only one (acoustic) branch, we can conclude that the appearance of the second branch in the spectrum of our model can be cxplained by the double-strand character of the model.
U’oca~’—kRi2sin p2—sin(p +p-)]=0
~
r 7
FIg. 4. Two branches of the DNA torsional vibrations.
416
(20)
Here
2p/d~2. ~w:—i’l p’wd ~= 1.2. (III) Multiply eq. (19) by p~and eq. (20) by .
(IV) Sum the results. (V) Intergrate the expression obtained. As a result, we have the first integral in the form
~fl’/pY
+ J’V/çf~—kR2[2
cos ~Pt+2 cos p~
—cos(Pi =const (21 Besides we can point out sonic particular cases, when problem (10) reduces to well known equalions, the solution of which has been already found. (a) Pt =0, P2O (or Pt 0. P2=O). Eqs. (10) reduce then to one equation +~7)
.
IØ—Ka2p~+kR2sin p=O.
/
.
(19)
(22)
which is known as the sine-Gordon equation [18]. It can be interpreted as a model describing the torsional dynamics of one of the DNA strands, the influence of the second strand being taken into account by some empiric constant field. (b) Pl=P2. Eqs. (10) reduce then to the double sine-Gordon equation [19] I~—Ka2p~+2kR2 sin p—kR2 sin 2p=O.
(23)
Equation (23) describes the torsion dynamics of the DNA strands that move in phase. (c) Pt = —P2. Eqs. (10) reduce then to the equanon of the sine-Gordon type
Volume 136. number 7,8
PHYSICS LETTERS A
+2kR2 sin P0• (24) 1(~—Ka2p~ It describes the torsion dynamics of the strands that move out of phase by it. The particular equations mentioned above have one feature in common: all of them have the solitonlike solutions named kinks and antikinks. So we can expect that the system of equations (10) might also have the soliton-like solutions of kink (and antikink) type.
6. Discussion Model (10) is one ofthe class of nonlinear models [1—8] simulating the internal DNA dynamics. Let us compare (10) with other models of the class. We begin with the model of Englander et al. [I]. It can be considered as the simplest model of the class. The model describes the torsion dynamics of only one DNA strand which is under the action of a constant field, formed by the second strand. The equation used by Englander et al. can be easily derived from (10) by substituting 0 for P2’ This particular case has been discussed above. We can mention also the models of Yomosa [2,3] and Chang-Ting Zhang [7], which are more accurate than the model of Englander et al. The models of refs. [2,3,7] differ from (10) only by the form of the nonlinear terms describing the weak interactions between the two DNA strands. The authors succeeded in finding some particular solitonlike solutions of kink (and antikink) type. Finally, we must note the model proposed by Fedyanin, Gochev. and Lisy [5,6]. Their model is almost equivalent to (10). (It becomes equivalent tO (10) when I = ‘2 and K 1 = K2.) The exact solutions of both
17 April 1989
the the found modelyet. of refs. [5,6] and eqs. (10)equations have notof been References [I] S.W. Englander. N.R Kallenbach. A.J. Heeger, J.A. Krumhansl and S. Litwin. Proc. NatI. Acad. Sci. 77 (1980) 7222. [215. Yomosa, Phys. Rev. A 27 (1983) 2120. [3 IS. Yomosa, Phys. Rev. A 30 (1984) 474.
[41J.A.
Krumhansl and W.M. Alexander. in: Structure and dynamics: nucleic acids and proteins. eds. E. Clementi and RH. Sarma (Adenine, New York. 1983) p.61. 151 V.K. Fedyanin, 1. Gechev and V. Lisy, Stud. Biophys. 116 (1986) 59. [6) V.K. Fedyanin and V. Lisy. Stud. Biophys. 116 (1986) 65. [7) Chung-Ting Zhang, Phys. Rev. A 35 (1987) 886. [8] V. Muto. J. Halding, P.L. Christiansen and A.C. Scott, J. Struct. (1988) [91Biomol. P. Jensen. MV. Dyn. Jakis 5and K.H. 873. Bannenham. Phys. Lett. A
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