The basic form of DNA is a double helix, consisting of two sugar-phosphate ... both types of double helices, the right-handed forms A and B with 11 and 10 ...
IL NUOVO CIMENTO
V OL . 20 D, N. 6
Giugno 1998
Non-linear mechanical model of DNA dynamics( ) T. L IPNIACKI Institute of Fundamental Technological Research, Polish Academy of Science ´ etokrzyska 21, Poland Warsaw, Swi¸ (ricevuto il 9 Giugno 1997; revisionato il 3 Dicembre 1997; approvato il 23 Gennaio 1998)
Summary. — A non-linear mechanical model is constructed in order to study the dynamics of DNA double helix. The potential energy of the system is found to be a double-well function with respect to the first derivative of twist angle of the helix. The Lagrangian contains also the second derivative of the twist angle. Due to these facts the evolution equations of the system have kink and pulse solitary-wave solutions. The physical interpretation of our solutions is an untwisting of DNA during transcription of messenger RNA. PACS 87.10 – General, theoretical, and mathematical biophysics (including logic of biosystems, quantum biology, and relevant aspects of thermodynamics, information theory, cybernetics and bionics). PACS 87.15.He – Molecular dynamics and conformational changes.
1. – Introduction The basic form of DNA is a double helix, consisting of two sugar-phosphate backbones and a base pairs chain inside. A schematic drawing of the duplex DNA in an unwound hypothetical state, which one can call planar ladder state, is shown in fig. 1 (from Calladine and Drew [1]). The distance B between adjacent sugars or phosphates in the DNA ˚ while the thickness of the flat part of the DNA base is A = 3:3 A ˚, chains is roughly 6 A ˚ which implies a gap of 2:7 A between the bases. Because the four DNA bases—guanine, adenine, cytosine and thymine—are hydrophobic substances, these bases tend to stay together rather than let the surrounding water fill gaps between them. If the bases are in contact the distance between their centers is A, which is smaller than the distance between the adjacent sugars B . This is because the sugar-phosphate back bones, in order to preserve their length, must wrap around the base pairs chain. Elementary geometric considerations enable one to calculate the angle by which each phosphate turns
( ) The author of this paper has agreed to not receive the proofs for correction. G
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Fig. 1. – The scheme of DNA, the hypothetical planar DNA ladder with its key dimensions. S: sugar, P: phosphate.
relative to its neighboring phosphate along the helix 2 2 1 2 = 2 arcsin (B ,2RA ) = 32:3; =
(1)
˚ is the distance between the corresponding phosphates from the oppowhere 2R = 18 A site chains. From eq. (1) it follows that there are roughly 360=32:3 11 phosphates per a complete turn of the DNA helix. This result closely agrees with experiment: almost all DNA double helices have between 10 and 12 phosphates per turn of helix. The above analysis (following Calladine and Drew [1]) suggests that both right-handed and left-handed duplex helical structures are expected. Although there are, in fact, known examples of both types of double helices, the right-handed forms A and B with 11 and 10 phosphates per helical turn are preferred to the left-handed Z form with 12 phosphates per turn. The base pairs sequence codes the genetic information, but from the mechanical point of view it is simply a nonperiodic chain. The four bases composing DNA have different masses. This poses a great problem for all models. Fortunately, the two base pairs composing DNA consist of one light and one heavy base, so their masses are almost equal. Namely the adenine-thymine (A-T) pair has mass 259 a.m.u. and guanine-cytosine (C-T) is only slightly heavier with mass 260 a.m.u. The total mass per base pair m (i.e. the mass with adjacent sugars and phosphate groups) is approximately 580 a.m.u. Moreover the Hbond interaction coupling G-C pair is more then twice stronger then for an A-T pair. Only in an approximation in which one neglects internal degrees of freedom of the base pairs and treats each base pair as a rigid body, DNA can be regarded as a periodic structure. The interest in the non-linear dynamics of DNA was started when Englander et al. [2] suggested that the existence of solitons propagating along the DNA molecule may be important in the process called RNA transcription. In the last decade several models were proposed in order to substantiate this idea in quantitative terms. The review of recently developed models may be found in the paper of Gaeta et al. [3]. Although simple models
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lost many of the real DNA properties, they may produce results describing the dynamics of the fundamental biological processes of DNA. This suggests that this approach is a way toward an understanding of the basic features of DNA dynamics. We recall two important models, the one proposed by Yakushevich [4] and improved by Gaeta [5] to describe the motion of the transcription bubble, and the one proposed by Peyrard and Bishop [6] to describe transversal openings. In the Yakushevich model the potential energy is a sum of the stacking interaction energy and the pairing interaction energy. The stacking interaction energy kept in the harmonic approximation depends on the relative angle between neighboring bases along the chain. The pairing interaction energy is considered also in the harmonic potential approximation, but depends on the distance between the endpoinds of the two bases composing the base pair, i.e. on the distance of the atoms bridged by Hbonds. This distance is the function of two rotational angles describing the turn of each base with respect to axis connecting bases’s centers. As a result, the pairing interaction is of sine-Gordon type. Probably the weakest point of this model is the assumption that the pairing force grows proportionally to the distance, while for larger amplitude motions (the solitary-wave solutions) the H-bonds must be destroyed. In the model of Peyrard and Bishop the Morse potential is used as a pairing potential, while the stacking potential is assumed in harmonic approximation. Like in Yakushevich model each base pair has one degree of freedom, but here it corresponds to the displacement of the base along the direction of the axis connecting the two bases in a pair. The model which includes the bases motion along the main axis of the DNA molecule was developed by Kosevich and Volkov [7] to describe the propagation of local A-form to B -form transitions. It was assumed that the potential energy is a double-well function of the shift u. The essentially different approach to the dynamics of DNA may be found in papers of Tobias et al. [8] where the Kirchoff ’s theory of elastic rods is used to mimic the DNA chain. Even under such simplifying assumptions some important properties of DNA double helix have been successfully studied. For instance, the supercoiling of the DNA molecule was connected with the twist density of helix.
2. – Description of the model In the presented model (fig. 2) base pairs are represented by rigid plates situated along the z -axis. The centers of every two subsequent plates are connected by a spring with free length A and spring constant 2k . Those springs are to represent the hydrophobic forces between base pairs. Sides of slabs are connected by two side springs representing the sugar-phosphate chains. The side springs segments connecting subsequent slabs have free length B and spring constant q . The distance between back-bone springs and the duplex axis going through the centers of the slabs is r0 . The mass and inertial momentum of each plate is equal to the mass and momentum per base pair, while all the springs are assumed to have no mass. It is assumed that every slab has two degrees of freedom: it can move along and turn around the z -axis and its position is described by the displacement w and the torsion angle . When A < B , as in real DNA, the system has two natural minima of energy (for which all the springs are in the natural state): a left-hand and a right-hand twisted ladder. To construct the Lagrangian L = T , or the Hamiltonian H = T + of the system, let us assume that all springs satisfy Hook’s law, then the potential energy i;i+1 of any
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Fig. 2. – The schematic picture of the mechanical model.
two adjacent slabs interaction is
+1 = k(w , A)2 + q
(2)
q
i;i
,
2
w2 + 4r02 sin2 =2 , B :
When is small, 2 sin(=2) may be replaced by (for = 32 the error is 1:3%). In the continuum limit
w = w0 A ;
(3)
= 0 A ;
with the primes denoting differentiation with respect to z . Then the potential energy of whole chain is taken in the form (4)
=A
Z h
k(w0 , 1)2 + q
2 002 i w02 + r02 02 , B=A + e0 2 dz :
q
The term (5)
Z 002 A e0 2 dz
has been added to eq. (4) in order to describe the energy associated with the additional curvature of the side springs when 00 differs from zero. This higher-order derivative
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term plays a key role both in the physical interpretation and the mathematical development of the present model. The two zero potential energy states are characterized by B 2 , A2 1 2 (6) w = const = 1 ; = const = : Ar0 Without the last term in eq. (4) the states in which 0 jumps from ,
0
=
0
B 2 , A2 1 2 + Ar 0 ,
(7)
B 2 , A2 1 2 , Ar0 ,
=
to
=
also will have the minimum zero energy, which seems unphysical because in those states the side springs (the sugar-phosphate chains) are highly distorted. The kinetic energy T is Z h 2 _ 2i 1 T = A m2w_ + J 2 dz ;
(8)
where m and J are the mass and moment of inertia per base pair, and the dot denotes time derivative. 3. – Equations of motion let
To write the Lagrangian in a simpler form, one may employ units in which A
(9)
c = (B , A)=A > 0 ;
u0 := w0 , 1 ;
= 1, and
:= r0 :
The potential energy of the system is now (10)
=
Z h
ku02 + q
2 002 i (1 + u0 )2 + 02 , (1 + c) + e 2 dz ;
p
with e = e0 =r02 . For the further analysis we have to expand the potential energy into a power series. Without loss of generality we can assume that u0 , with 0, c > 0, both 1 ; 2
(20)
1 = 2(km+ q) ;
(21)
2 = Ie K 2 :
are real and positive. For c
1 , 2 correspond now to compressive and torsional modes, respectively.
= 0 the
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4. – Pulse- and kink-like solitary-wave solutions Now we focus our attention on solitary-wave solutions to eqs. (15) and (16). The solitary waves propagate without changing their form, i.e. (22)
u(z; t) = u(z , vt) = u( ) ;
(z; t) = (z , vt) = ( ) :
Equations (15)–(22) yield (23)
Iv2 0 = q
000 2 (u , c) + , e q + C1 ; 0
0
03
mv2 u0 = 2(k + q)u0 + q 02 + C20 ; where the prime denotes differentiation with respect to . C1 and C20 = C2 , 2qc are the integration constants, with C2 and C1 , respectively, the force and the torsional moment of (24)
force applied to the chain at its ends. From equation (24) we obtain 02 , 2qc + C 2 u0 = q mv2 , 2(k + q) ;
(25) and for f
= 0 we have
(26)
ef 00 = af 3 + bf + C1 ;
where
(27)
2 q a = q 1 + mv2 , 2(k + q) ;
(28)
2q(C2 , 2qc) , (Iv2 + 2qc) : b = mv 2 , 2(k + q)
Equation (26) may have solutions with horizontal asymptotes only when its right-hand side has 3 zeros, i.e. when the coefficients a and b have opposite signs. We restrict our considerations to the case when a > 0 and b < 0. Let (29)
p
p = ,b=a;
and note that in new variables y , g (30)
p
y = ,e=b;
w = Cpb1 ; g(y) = fp
eq. (26) takes the form (31)
g00 = g3 , g + w ;
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and has solutions with horizontal asymptotes only when the right-hand side has 3 zeros, which is the case when (32)
p jwj < 2=3 3
with
b < 0;
a > 0:
These conditions of existence of soliton-like solutions restrict the values of the coefficients m; I , the integration constants C1 ; C2 and the traveling velocity v. The explicit form of these conditions is rather complicated. In general, a static solution (v = 0) exists when the stretching force ,C2 and the torsional moment C1 are not too large. On the other hand, if the stretching force and torsional moment applied to the chain are small enough, solutions exist for
v2 2mk :
(33)
Multiplying eq. (31) by 4g 0 and integrating, one obtains
2g02 = g4 , 2g2 + 4wg + d ;
(34)
where d is another integration constant. For w = 0, i.e. when no torsional moment is applied, eq. (31) has the kink-like solution g = th y p, C ; 2
(35)
where C is an arbitrary constant. Equation (35) yields r
(36)
"
"
##
1 2 , b 2 e 0 + C 00 ; = a ln ch ( , C ) 2e =
with C 0 ; C 00 constants. The variable u is given by eq. (25). The pulse-like solutions p exist when the torsional moment is applied to the chain, namely for 0 < jwj < 2=(3 3). More precisely, for positive w the solutions have asymptotes g = const ,> 0 andpfora negative value of w the asymptotic value of g has to be negative. For jwj 2 0; 2=(3 3) the right-hand side of eq. (31) has 3 zeros and eq. (34) has 4 zeros. For solutions with horizontal asymptotes both g 0 and g 00 vanish at infinity, and so c0 = g1 is the root of the right-hand sides of eqs. (31) and (34). This yields a condition for the integration constant d. Because all the zeros of the right-hand side of eq. (31) are the extremum points of the right-hand side of eq. (34), c0 is a double root of eq. (34). Thus, eq. (34) may be written in the form (37)
2g02 = (g , c0 )2 (g , c1 )(g , c2 ) ;
where c1 , c2 are the two other roots depending on w. In this form eq. (37) can be analytically integrated. The profiles g (y ) (or 0 (z , vt) in arbitrary units) of pulse-like solutions for various w are given in fig. 3. Two integrated profiles from fig. 3 are shown on fig. 4. For relatively small w = 0:05 the profile (z , vt) is not monotonic which means that a moving segment of the chain is twisted oppositely to the rest of the chain. The length of the solitary wave,
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( , )
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Fig. 3. – The g y , or 0 z vt in arbitrary units, profiles of the pulse-like solutions to eq. (31) for : ; : ; : ; : ; : —the pulse various values of w C1 =pb, with C1 torsional moment. w profile flattens out as w increases.
=
= 0 001 0 05 0 2 0 3 0 38
defined as the width of the pulse-like profile g (y ) (fig. 3) at half of its height, is shown in fig. 5 as a function of w. If w or simply C1 (the applied torsional moment) p tends to 0, the length of the solitary wave diverges to infinity. For w tending to 2=(3 3) the length of the solitary wave also diverges p to infinity but its height tends to 0. Because the original variable is = (z , vt) = y ,b=e, the real length of the solitary wave is proportional to e1=2 .
( , )
Fig. 4. – The angle z vt for a pulse-like solution in arbitrary units; w profile, w : : monotonic profile.
=03
= 0 05: non-monotonic :
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Fig. 5. – The length L of the pulse-like solution as a function of w.
5. – Conclusion The solitary-wave solutions for evolution equations (15) and (16) are the main result of this paper. The existence of these solutions is due to the fact that, within limits of our mechanical model, the potential energy of the DNA chain is a double-well function of the twist density 0 . In the relaxed state the chain has the uniform twist density 0 = , 00 = B 2 , A2 1=2 =Ar0 (eq. (6)) or 0 = ,00. Let us concentrate on the first case. Now, applying the torsional moment at the ends of the chain, one can add some positive twist to the system. This additional twist will spread uniformly over the chain making the twist density > 0 . On the other hand, applying the opposite torsional moment, one can remove the twist from the chain, making the average twist density < 0 . When the average twist density gets smaller the energy of the chain grows up to some critical point when the system can jump to the state in which a part of the chain has opposite twist to the rest. If the parameter e0 at 00 is small, then it is “energetically worth” to create the oppositely twisted segment of the chain even if the average twist density is close to 00 . The untwisted (or oppositely twisted) segment thus created can then move
Fig. 6. – Transcription of messenger RNA (from [1]).
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along the chain. Those pulse-like solutions obtained for the mechanical system may be helpful when describing the mechanical aspects of the transcription of the messenger RNA—one of the most important processes of DNA evolution. The aim of this process is to copy the DNA genetic information into RNA. During transcription the DNA must untwist locally to let one strand serve as a template for synthesis of new RNA strand (fig. 6, from Calladine and Drew [1]). The untwisted open region 15-20 base pairs long then moves along the DNA. Of course within the limits of our model we can not describe the opening of the DNA (separation of two DNA strands) because we have assumed that each base pair is a rigid body. To analyse the opening one has to include at least one more degree of freedom to describe the separation of the two bases forming a pair. This goes far away beyond this model. Nevertheless one can check that in the short region, in which twist changes its sign, the side springs are highly stressed and may tend to break base pairs.
The author would like to express his gratitude to Profs. H. Z ORSKI, B. D. C OLEMAN and Dr. J. H OLYST for valuable discussions and corrections of the manuscript. This work was supported by grant KBN PB 1124/P4/93/04.
REFERENCES [1] C ALLADINE C. R. and D REW H. R., Understanding DNA (Academic Press) 1992. [2] E NGLANDER S. W., K ALLENBACH N. R., H EEGER A. J., K RUMHANSL J. A. and L ITWIN S., Proc. Natl. Acad. Sci. USA, 777 (1980) 7222. [3] G AETA G., R EISS C., P EYRARD M. and D AUXOIS T., Riv. Nuovo Cimento, 17, No. 4 1994. [4] YAKUSHEVICH L. V., Phys. Lett. A., 136 (1989) 413. [5] G AETA G., Phys. Lett. A, 190 (1994) 301. [6] P EYRARD M. and B ISHOP A. R., Phys. Rev. Lett., 62 (1989) 2755. [7] KOSEVICH A. M. and VOLKOV S. M., Dynamic of Conformational Excitation in DNA Macromolecule (National Academy of Sciences of Ukraine, Kiev) 1994, preprint. [8] T OBIAS I., C OLEMAN B. D. and O LSON W. K., J. Chem. Phys., 101 (1994) 10990.
