s. zdravkovic, j.a. tuszynski and m.v. sataric: peyrard-bishop-dauxois model of dna dynamics and...

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Copyright © 2005 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 2, 1–9, 2005

Peyrard-Bishop-Dauxois Model ofDNA Dynamics and Impact of Viscosity

S. Zdravkovic,1�∗ J. A. Tuszynski,2 and M. V. Sataric31Faculty of Technical Sciences, University of Pritina, Kosovska Mitrovica, Yugoslavia

2Department of Physics, University of Alberta, T6G 2J1, Canada3Faculty of Technical Sciences, 21000 Novi Sad, Yugoslavia

In this paper we try to elucidate the physical significance of the analytical solutions in the Peyrard-Bishop-Dauxois (extended Peyrard-Bishop) model of DNA dynamics. We discuss the impact ofsome parameters of the model, especially the harmonic constant of the helicoidal springs. We studyboth the case of DNA dynamics when viscosity is ignored and when it is taken into consideration.

Keywords: DNA, PBD Model, Solitons, Viscosity.

1. INTRODUCTION

Deoxyribonucleic acid (DNA) is doubtlessly one of themost important biomolecules. Its double standard heli-cal structure undergoes a very complex dynamics and theknowledge of that dynamics provides insights into var-ious related biological phenomena such as transcription,translation and mutation. The key problem in DNA bio-physics is how to relate functional properties of DNA withits structural and physical dynamical characteristics. Thepossibility that nonlinear effects might focus the vibrationenergy of DNA into localized soliton-like excitations wasfirst contemplated by Englander et al.1 Although severalauthors2–9 have suggested that either topological kink soli-tons or bell-shaped breathers would be good candidates toplay a basic role in DNA nonlinear dynamics, there arestill several unresolved questions in this regard. A hier-archy of the most important models for nonlinear DNAdynamics was presented by Yakushevich.10

The local openings can be analytically described asbreather-like objects of small amplitude, which have nev-ertheless interesting properties; as long as their amplitudeis small enough they can move along the chain. This isunidimensional, which allows local openings of the hydro-gen bonds and formation of denaturation bubbles.

In the present paper we deal with the extended model forDNA dynamics, first proposed by Peyrard and Bishop7 and

∗Author to whom correspondence should be addressed.

further developed by Dauxois.11�12 This Peyrard-Bishop-Dauxois model will be henceforth referred to as the PBD-model for short.

The paper is organized in the following way. In Sec-tion 2 we outline the PBD model primarily intended todescribe the process of local opening of base pairs (orlocal melting of the double helix). Then we attempt to shednew light on the still somewhat vague parameter values ofthe PBD model, especially discussing the harmonic con-stant of helicoidal springs. This section represents a certainextension of two of our previous papers.13�14 However, weintroduce a new procedure which yields qualitatively newresults and solves a couple of problems.

In Section 3 we present our consideration of DNAbreather dynamics in the context of the PBD model whenviscosity is taken into consideration. Section 4 puts for-ward our discussion and concluding remarks.

2. THE PBD MODEL OF DNA

The B-form DNA in the Watson-Crick model is a doublehelix, which consists of two strands s1 and s2 (Fig. 1),linked by the nearest-neighbour harmonic interactionsalong the chains. The strands are coupled to each otherthrough hydrogen bonds, which are supposed to be respon-sible for transversal displacements of nucleotides.

It was argued7�11�12 that we could safely assume a com-mon mass m for all the nucleotides and the same couplingconstant k along each strand. We will get back to this

J. Comput. Theor. Nanosci. 2005, Vol. 2, No. 2 1546-198X/2005/2/001/009/$17.00+.25 doi:10.1166/jctn.2005.110 1

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Peyrard-Bishop-Dauxois Model of DNA Dynamics and Impact of Viscosity Zdravkovic et al.

s1

s2

k

k

m

m

vn–1 vn+1vn

un–1 un+1un

MorsePotential

Fig. 1. Graphical representation of the simple model for DNA strands.

point later. As was pointed out, we rely on the PBD model,which takes into account the fact that the DNA moleculeis twisted. This helicoidal structure of the DNA chainimplies that nucleotides from different strands becomeclose enough so that they interact through water filaments.This means that a nucleotide at the site n of one strandinteracts with both (n+h)th and (n−h)th nucleotides ofthe other strand. In Refs. [13, 14] we used h = 411�12

for all calculations. However, h = 5 might be better.15�16

This comes from the fact that there are approximately tennucleotides per one turn. Introducing the transversal dis-placements un, vn of the nucleotides from their equilibriumpositions along the direction of the hydrogen bonds, theHamiltonian for the DNA chain becomes11�12

H = ∑{m

2u2

n+ v2n�+

k

2

[un−un−1�

2 + vn−vn−1�2]

+ K

2

[un−vn+h�

2 + un−vn−h�2]

+D�e−aun−vn�−1�2

}(1)

Here k (respectively K) is the harmonic constant of thelongitudinal (respectively helicoidal) spring. The last termin the Hamiltonian represents a Morse potential approxi-mating the potential of the hydrogen bonds while D and aare the depth and the inverse width of the Morse potentialwell, respectively.

It is more convenient to describe the motion of twostrands by making a transformation to the centre-of-masscoordinates representing the in-phase and out-of-phasetransversal motions, namely

xn = un+vn�/√

2� yn = un−vn�/√

2 (2)

From relations (1) and (2) one can obtain the follow-ing dynamical equations describing linear waves (phonons)and nonlinear waves (breathers):

mxn = kxn+1 +xn−1 −2xn�+Kxn+h+xn−h−2xn� (3)

and

myn = kyn+1 +yn−1 −2yn�−Kyn+h+yn−h+2yn�

+2√

2aD(e−a

√2yn −1

)e−a

√2yn (4)

As was explained in a couple of articles11–13�17, we canapply the transformation

y = �� 1� (5)

This means that we assume that oscillations of nucleo-tides are large enough to be anharmonic but still smallenough so that the particles oscillate around the bottom ofthe Morse potential. Equations (4) and (5), and the expan-sion of exponential terms in Eq. (4), yield

�n = k

m�n+1 +�n−1 −2�n�−

K

m�n+h+�n−h+2�n�

−�2g�n+��2

n+�2��3n� (6)

where

�2g =

4a2D

m� �= −3a√

2and �= 7a2

3(7)

To solve Eq. (6) we use the semi-discrete approxi-mation.17 This means that we look for wave solutions ofthe form

�nt� = F1�nl� �t�ei n +��F0�nl� �t�+F2�nl� �t�e

i2 n �

+ cc+O(�2� (8)

and ≡ n = nql−�t (9)

Here, l is the distance between two neighbouring nucleo-tides in the same strand, � is the optical frequency ofthe linear approximation of their vibrations, q is thewave number whose role will be discussed later, cc areconjugate-complex terms and the function F0 is real.

Before we proceed with solving Eq. (6) we give someexplanations for Eq. (8). If there were not the last term inEq. (6), the one with �2

g , which comes from the nonlinearterm in Eq. (4), we would expect the solution in the formF1e

i n + cc instead of Eq. (8). This would be a modulatedwave with a carrier component ei n and an envelope F1. Wewill see later that the modulation factor F1 will be treatedin a continuum limit while the carrier wave will not. Inother words, the carrier component of the modulated waveincludes the discreteness and the procedure is called semi-discrete approximation. As if there are terms with �2

n and�3

n in Eq. (6) we can not expect solution of this equation inthe simple form F1e

i n and nonexponential term as well asterms with ei2 n should be incorporated into the expressionfor the solution. A whole method is explained in muchmore details in Ref. [18], while its mathematical basis isgiven in Refs. [19, 20].

Now, we are ready for solving Eq. (6). It was alreadypointed out that the functions Fi would be treated in thecontinuum limit. So, taking this limit (nl→ z) and apply-ing the transformations

Z = �z� T = �t (10)

2 J. Comput. Theor. Nanosci. 2, 1–9, 2005

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Zdravkovic et al. Peyrard-Bishop-Dauxois Model of DNA Dynamics and Impact of Viscosity

yields the following continuum approximation

F �n±h�l� �t� → F Z�T �±FZZ�T ��lh

+ 12FZZZ�T ��

2l2h2 (11)

where FZ and FZZ mean corresponding derivatives withrespect to Z. This allows us to obtain a new expressionfor the function �nt�:

�nt�→ F1Z�T �ei +��F0Z�T �+F2Z�T �e

i2 �+ cc

= F1ei +��F0 +F2e

i2 �+F ∗1 e

−i +�F ∗2 e

−i2 (12)

where ∗ stands for complex conjugate and Fi ≡ FiZ�T �.From Eqs. (9)–(12) one can straightforwardly obtain

expressions for all the terms in Eq. (6).18 From the con-tinuum version of Eq. (6), equating the coefficients forthe various harmonics, we can get a set of importantrelations.11–13 For example, equating the coefficients for ei

one obtains a dispersion relation

�2 = �2g +

2km

1− cosql��+ 2Km

cosqhl�+1� (13)

This can be used to find the corresponding group velocityd�/dq as

Vg =l

m��k sinql�−Kh sinqhl�� (14)

In the same way, equating the coefficients for ei0 = 1,we can easily obtain

F0 = '�F1�2 (15)

where

'=−2�

[1+ 4K

m�2g

]−1

(16)

However, for the next three harmonics (ei2 , ei3 andei4 ) the matter becomes more complicated. All of theseharmonics give a relation

F2 = (F 21 (17)

but with different values for the parameter (. For example,equating the coefficients for ei2 one can get[

4�2 + 2km

cos2ql�−1�

− 2Km

cos2hql�+1�−�2g

]F2 = �2

g�F2

1 (18)

which means that ( is not constant but a function of ql.However, coefficients for ei3 and ei4 give constant values(, − �

2� and − 3��

respectively. We will return to this pointlater.

As if we can express functions F0 and F2 through thefunction F1 we can derive an equation for F1. Using newcoordinates again:

S = Z−VgT � * = �T (19)

we can finally obtain the nonlinear Schrödinger equation(NSE) for the function F 11–14

1

iF1* +PF1SS +Q�F1�2F1 = 0 (20)

where the dispersion coefficient P and the coefficient ofnonlinearity Q are given by

P = 12�

{l2

m�k cosql�−Kh2 cosqhl��−V 2

g

}(21)

and

Q =−�2g

2��2�'+(�+3�� (22)

Those derivations can be found in some more details inRef. [18].

For PQ > 0 the solution of the NSE (20) is11�12�21

F1S� *�= A sech(S−ue*

Le

)exp

iueS−uc*�

2P(23)

where the envelope amplitude A and its width Le have theforms

A=√u2e −2ueuc

2PQ(24)

andLe =

2P√u2e −2ueuc

(25)

and ue and uc being the velocities of the envelope and thecarrier waves, respectively. Using Eqs. (23), (8), (9), (15),(17), (10), and (19) we obtain11–14

�nt� = 2Asech[�

Le

nl−Vet�

]

×{

cos1nl−2t�+�Asech[�

Le

nl−Vet�

]

×('

2+(cos�21nl−2t��

)}+O�2� (26)

where

Ve = Vg +�ue (27)

1 = q+ �ue

2P(28)

and2= �+ �ue

2PVg +�uc� (29)

In Ref. [13] we suggested discrete values for a wavenumber q. For h= 4 we obtained the following four valuesfor the wave length 3: 6l, 7l, 8l, and 9l or, as if q = 24/3,

J. Comput. Theor. Nanosci. 2, 1–9, 2005 3

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the following values for ql: 1505 rad, 0590 rad, 0.78 rad =4/4 and 0570 rad. We gave an argument to reject the firstvalue (3= 6l�. For h= 5, using the same procedure,13 oneobtains the following values for 3: 7l, 8l, 9l, 10l, and 11land the following values for ql: 0590 rad, 0578 rad, 0570 rad,0.63 rad=4/5 and 0557 rad. For the most favourable modewe suggested the one for which ql = 4/4 for h = 4. Forh= 5 that would be ql = 4/5.

For all the calculations we chose the following set ofvalues characterizing the DNA molecule:

k = 3K = 24 N/m (30a)

l = 354 ·10−10 m (30b)

m= 551 ·10−25 kg (30c)

a= 2 ·1010 m−1 (30d)

D = 051 eV (30e)

The value for the mass is an average value for all fournucleotides. Namely, molecular masses of nucleotides arefrom 340 g/mol (Cytosine) to 380 g/mol (Guanine). There-fore, for the average value, after subtraction the masses ofthree water molecules, one obtains 305.75 g/mol, whichgives the above value for the mass of the single nucleotide.All other constants were taken from the papers.11�12

It was pointed out that there were two possibilities forthe parameter (, constant and nonconstant. In Ref. [13] weaccepted the constant ( coming from ei3 . Therefore, wedid not use the lowest possible harmonic ei2 ! In Ref. [14],however, we made a choice having more physical sence.This is nonconstant ( given by Eqs. (17) and (18). In thatarticle we suggested that the value of the parameter Kwas overestimated. It was shown that there should be K ≤456 N/m for ql = 0578 rad and h = 4 and all the otherconstants given by (30). For h= 5 and ql = 0563 rad oneobtains K ≤ 454 N/m.

Of course, one can argue that those values for ql,0578 rad and 0563 rad, were obtained for K = 8 N/m inRef. [13]. This is correct. However, using the same proce-dure for the possible ql values, explained in Ref. [13], weobtain, for h= 4, 3= 6l, 7l, 8l, and 9l, i.e., ql= 1505 rad,059 rad, 0578 rad and 057 rad for both K = 8 N/m and K =6 N/m. The value K = 4 N/m gives the same four valuesfor ql and two more (3= 10l and 11l, or ql= 0563 rad andql = 0557 rad) in addition. For K = 3 N/m we obtain 3=6l�7l� 5 5 5 �13l while K = 2 N/m gives 3= 6l�7l� 5 5 5 �19l.Hence, the value ql = 0578 rad, as well as ql = 0563 radfor h = 5, corresponds to both large and small values ofthe parameter K. This means that ql = 0578 rad can besuitable choice for all acceptable values of K, but does notmean that this is the optional value for our calculations.Thus, we need one more criterion for ql to be determinedand we show in the following that this exists.

The breather-type solution in Eq. (26) represents a sortof a modulated solitonic wave. From hyperbolic and cosine

terms in Eq. (26) we can recognize wave numbers of boththe envelope and the carrier wave. In other words, we cansee that the length of the envelope, 6, and the wavelengthof the carrier wave, 3c, are

6= 24Le

�(31)

and3c =

24q+ �ue

2P

(32)

This allows us to calculate the number of wavelengthsof the carrier wave contained within the length of the enve-lope as

Do ≡6

3c

(33)

In what follows, we call Do a density of internal oscilla-tions (density of carrier wave oscillations) for short eventhough, strictly speaking, this is not density, i.e., the num-ber of wavelengths per unit length. From Eqs. (31)–(33)and (25) we can easily obtain

Do =ue√

u2e −2ueuc

(1+ 2qlP

�lue

)(34)

We should keep in mind that the dispersion coefficient Pis defined by Eq. (21) and is a function of ql.

If we choose the following set of values for parameterscharacterizing a traveling wave solution11�12

ue = 105 m/s (35a)

uc = 0 (35b)

�= 05007 (35c)

we can plot the function Do vs. ql and this is done inFigure 2 for h = 5 and three values for the parameter K.From this figure we see that the density of the inter-nal oscillations of the soliton Do reaches a maximum for

10

8

6

40.4 0.5 0.6 0.7 0.8 0.9

ql (rad)

a

b

c

D0

Fig. 2. Density of the carrier wave oscillations Do as a function of ql.(a: K = 455 N/m, b: K = 4 N/m, c: K = 355 N/m).


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ql = 05636 rad, which is extremely close to the value of0.628 rad = 4/5 that was earlier selected as the mostfavourable one. For h = 4 the highest value of the func-tion occurs at about ql = 0577 rad, which is very close to0.78 rad = 4/4. Should we accept this value, correspond-ing to the highest Do, for our calculations? Certainly yes,since a higher Do probably means a more efficient modu-lation. It is required for modulation, when both signals arecosine functions, that the frequency of the carrier compo-nent be much higher then the frequency of the envelope.In our case, however, there are no two frequencies, butwe introduced Do instead. Our intuition strongly suggeststhat nature wants modulation with Do as high as possi-ble, which one might call the most efficient modulation.According to Figure 2, the maximum value of the functionDo is around 8 to 10. We should emphasize that this doesnot have too profound meaning since this value simplydepends on the choice of the somewhat arbitrary parame-ters � and ue. In fact, we are looking for the maximum ofthe product function ql ·Pql�.

So, we can make two important conclusions based onthe analysis of how Do depends on ql. One can see fromFigure 2 that the highest values of the density of the internaloscillations Do increases when the parameter K increases.What is also important is the fact that an optimal ql, for anyreasonable K, does not differ significantly from our previ-ously accepted values of ql= 0563 rad and ql= 0578 rad forh= 5 and h= 4. Otherwise, for K = 0 one obtains Dom ≈ 2for ql = 0553 rad for both values of h. This means that,from the criterion of modulation, the PBD model11�12 is bet-ter then the original PB model,7 which could be obtainedfrom the former one by letting K = 0.

Now, we can plot our solitonic function �nt� given byEq. (26), or the function y defined by Eq. (5). Before wedo this, however, we want to study how the amplitude Adepends on ql for various K. We should keep in mind thatthe value A is the amplitude of the function F1S� *� butis not the amplitude of the soliton �nt�. From Figure 3

0.3 0.4

0.6

0.9

1.2

1.5

1.8

0.5 0.6 0.7 0.8 0.9

ql (rad)

A (

nm)

Fig. 3. Amplitude A as a function of ql for (= const (K = 4 N/m).

0.4 0.5 0.6 0.7 0.8ql (rad)

A (

nm)

a

b

c

1

2

3

4

5

Fig. 4. Amplitude A as a function of ql for (= (ql�. (a: K = 455 N/m,b: K = 453 N/m, c: K = 358 N/m).

we can see how A depends on ql for constant (, i.e., for( = −�/2�, the value that was previously accepted forour calculations,13 and for h= 5. We selected K = 4 N/m,but for different K, both larger and smaller, the curves arealmost the same.

However, for ( given by Eqs. (17) and (18) the situationlooks completely different. Figure 4 shows how the ampli-tude A depends on ql for the three values of K and forh= 5. Instead of minimum (Fig. 3) the function Aql� nowhas a maximum. The highest values are 4575 nm at ql ≈0562 rad for K = 453 N/m, 2531 nm at ql ≈ 0562 rad forK = 4 N/m and 1578 nm at ql≈ 0564 rad for K = 357 N/m.One can see that those maxima decrease with smaller K.For K = 4558 N/m the maximum of the amplitude A iseven 16 nm, which probably does not make physical sense,while for K = 354 N/m the function Aql� does not haveany maximum at all.

Therefore, we can state three facts concerning theparameter K. First, there is a maximum value Km =4539 N/m for ql= 0563 rad. Second, higher K means largerD0, i.e., a better transmission of a signal through the DNAchain, as was explained above. Third, K should be smallenough to ensure that the amplitude A be small enough sothat the oscillating nucleotide does not exceed the depthof Morse potential well.

Finally, we can plot our soliton �nt� given by Eq. (26).In Figures 5–7 we show the elongation �nt� vs. timefor K = 453 N/m, 4 N/m and 255 N/m, respectively. Wechose h = 5 for all of them. According to the shape ofthe curve in Figure 5, and the maximum of the function�nt�, we can conclude that this value for K, i.e., K =453 N/m, would not be acceptable. For K = 4539 N/m,�nt� has only positive values with the maximum reach-ing up to 360 nm which is totally unacceptable becausethe expansion of the Morse potential, Eq. (6), presumesthat the amplitude may not reach the Morse plateau! For�n = 360 nm, from Eq. (5) it follows that yn = 255 nmwhich is far beyond the plateau. However, Figures 6 and 7


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15

10

5

0

–5

48 52 56 60 64

t (ps)

Φ (

nm)

Fig. 5. Elongation of the out-of-phase motion as a function of time(n= 300, ql = 05628 rad, h= 5, K = 455 N/m, (= (ql�).

–4

–2

0

2

4

6

48 52 56 60 64t (ps)

Φ (

nm)


–2

–1

0

1

2

3

52 565450 58 60 62

t (ps)

Φ (

nm)


suggest that the values K = 4 N/m and K = 255 N/m mightbe acceptable. Of course, we should keep in mind thata higher K means more oscillations of the carrier wavein one envelope, i.e., a higher D0. For K = 0 only threeinternal oscillations can hardly be discerned which shows,as we stated above, the superiority of the PBD model11�12

over the original PB version.7

At this point we can speak of a certain range foraccepted values for the parameter K. Also, as was pointedout earlier, K should be big enough to ensure a large D0,but still not too large to bring about a very large amplitude.

Finally it might be interesting to compare solitonic solu-tions �nt� given in Figures 5–7 with the same functionobtained for constant (.13 In Figure 8 we show the soli-tonic wave �nt� for K = 4 N/m and constant (, i.e., (=− �

2� . Comparing Figures 6 and 8, both plotted with thesame K, we see that the amplitude for the non-constant( (Fig. 6) is about three times larger than that for theconstant ( (Fig. 8). In both cases positive amplitudes arehigher than negative, which is a result of the term withthe parameter ( in Eq. (26). One can see that the shapesof both curves are almost the same. This is not surprisingsince D0, the density of the carrier wave oscillations of thebreather, does not depend on (.

Finally, we need to point out a couple of advantageous-ness of this procedure (nonconstant () over the one when (was constant. First, the amplitude Aql� has the maximumhaving very big values in a limit when the parameter Kapproaches its critical value, which suggests a certain res-onance behaviour. This procedure solves the problem of asmall amplitude discussed in Ref. [13].

Also, the existence of the upper limit of the parame-ter K, which we explained in this chapter and in Ref. [14],solves one more problem. Namely, in Ref. [11] opticaland acoustical frequencies were compared. There are fourcrossing points in the first Brillouin zone for h= 4. How-ever, for K < Km those crossing points do not exist and�o > �a in the whole zone.22 In that article,22 a resonance

48 52 56 60 64t (ps)

Φ (

nm)

1.5

1.0

–1.0

0.5

–0.5

0.0

Fig. 8. Elongation of the out-of-phase motion as a function of time(n= 300, ql = 05628 rad, h= 5, K = 4 N/m, (= const).


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mode was defined. It might be interested to point outthat the resonance mode occurs at ql = 05785 rad/s and05628 rad for h= 4 and h= 5 respectively. Those are wellknown values first obtained in Ref. [13] and confirmed inthis section by maximization of the function D0ql�.

3. THE INFLUENCE OF VISCOSITY

In the previous paper on this topic13 we attempted to dealwith a more realistic description of DNA dynamics, whichtakes into account the impact of the medium, surround-ing a DNA molecule. It is well known the importance ofhydrating water for the biochemical activities of proteinsand especially DNA. Since water molecules are highlypolar they form an ordered network on the surface of aprotein along which protons can be transferred. The forma-tion of this network with long-range connectivity has beendetected as a percolation transition when the water con-tent approaches 055 g per 1 g of protein.23 We suggest thatthe helicoidal spring parameter K in the model consideredhere may reflect the mediating role of the water molecules.Moreover, the helicoidal structure itself arises as the resultof optimization of the interplay between hydrogen bond-ing, hydrophobicity and long-range connectivity. On theother hand, we must take into consideration the fact thatthe solvating water does act as a viscous medium thatdamps out DNA dynamics, favouring energy expenditure.

We took this effect in Ref. [13] by adding the viscousforce on the nucleotide pairs, i.e., in Eq. (4) of this paper.In that treatment13�24 viscous force was considered as com-petitive with other forces arising from Hamiltonian (1).The consequence of that approach was the outcome whichshowed the impact of viscosity being so strong that thebreather solution (26) decays almost instantaneously intoits asymptotic form which is localized bell-shaped modegiven by expression:

ynt�= �2A29 ' sech2

[�

L9

nl−Ve9t�

](36)

where A9 , L9 , Ve9 are parameters defined by Eqs. (24),(25), and (14), respectively, renormalized by viscosityimpact. This procedure was shown in Refs. [13, 24] wherewe defined so called “big” and “small” viscosities. So, thefirst one, big viscosity, was explained in Refs. [13, 24],while the second one will be in what follows.

Here we start from probably more realistic and favour-able approach that viscous force has features of small per-turbation. The viscous forces exerted on the bases withina pair n are −�9un and −�9vn. The small �, the same asin Eq. (5), indicates that viscous force has the character ofsmall perturbation. It leads to the effective damping forceacting on the out of phase base pair motion as follows:

Fv =−�9yn (37)

Now starting from the perturbed equation of motion

myn = kyn+1 +yn−1 −2yn�−Kyn+h+yn−h+2yn�

+2√

2aDe−a√

2yn −1�e−a√

2yn −�9yn (38)

and performing the expansion according to Eqs. (5), (8),(10) and (19) one finds

iF1* +PF1SS +Q�F1�2F1 =−�9

2m�F1* (39)

This equation could be solved by the method of “slowlyvarying parameters” developed in Ref. [25]. The essenceof the method is that the carrier wave number 1= q+� ue

2Pslowly changes with time through change of ue.

The breather solution, Eq. (23), can be written in moresuitable form:

F1S�:�=Asech[

1Le

S−:�

]exp

{i

[;

(S− uc

ue

:

)+<

]}(40)

where ; = ue2P is the carrier wave number shift, which is

expected to depend of viscosity, : = ue* is the position ofcentre of the breather, and < is time phase.

According to Ref. [25], the above parameters shouldslightly depend on time on the following ways:

;*�= ;0 + f ;�* (41)

:*�= :0 +ue0

;0

;* (42)

<*� = ue0

2;0

[1L2e

+;2

]*

= ue0

2;0

(1L2e

+;20

)*+ [ue0f ;�*+>;�

]* (43)

where ;0, :0, and ue0 are corresponding unperturbedvalues, while >;� denotes possible phase shift of thebreather caused by viscosity.

The wave function of the breather perturbed by viscosityF1v could be expressed by expansion with respect to smallparameter �:

F1v = F1S� :�+�

[(?F1

?<

)+Le

(?F1

?:

)](ue0

2;0

)f ;�*2

(44)The above approximation holds for time * smaller thancharacteristic time scale *0 = ;0

�f ;0�. It can be shown on

the basis of Eqs. (41)–(43) that this expansion divergeswhen �→ 0 except if the secular sum in square bracketsvanishes. If we use definitions, Eqs. (41)–(43), and takeinto account that the derivative F1* on the right hand sideof Eq. (39) could be transformed as follows

?F1

?*=(?F1

?;

)d;

d*+(?F1

?:

)d:

d*= ?F1

?:

(?F1?;?F1?:

d;

d*+ d:

d*

)(45)


RESEARCHARTICLE


the effective perturbation caused by viscosity could bewritten as

�;� :� S� = −i

[?F1

?;f ;�+ ?F1

?<>;�

]

− ue0

�;0

[9

m;+ f ;�

]?F1

?:(46)

The unknown functions from Eqs. (41), (43) f ;� and>;� now play the role of Lagrange multipliers, whichcould be determined by orthogonality conditions:∫ �

−�dS�;� :� S�

(?F1

?:

)∗= 0 (47a)

∫ �

−�dS�;� :� S�

(?F1

?<

)∗= 0 (47b)

where ∗ stands for complex conjugates. This removes thementioned divergence of Eq. (44). Taking into account thederivatives ?F1

?;= iSF1, for uc = 0, and ?F1

?<= iF1, contained

in Eq. (46), the second condition, Eq. (47b), gives that thephase shift >;� is zero in this approximation, while thefirst condition yields the rate of change of carrier wavenumber in the form:

d;

d*= f ;�=− 9

m

(D

1+D

); (48)

where

D = 13ue0

�Le

= 16u2e0

P�(49)

From Eq. (48) easily follows

;= ;0 exp[− 9

m

D

1+D*

](50)

Having in mind the definition ;= ue2P we get

ue = ue0 exp[− 9

m

D

1+D*

](51a)

ue0 = 2P;0 (51b)

If we estimate P and � from Eqs. (21) and (13) takingK = 4 N/m and ql = 0578 rad, we readily get P = 0589 ·10−6 m2/s and �= 856 ·1012 rad/s.

If we choose, as before, ue0 = 105 m/s, dimensionlessparameter D, Eq. (49), becomes

D = 217 � 1 (52)

Under such circumstances of high launching breather’svelocity, the envelope velocity, Eq. (27), by using Eqs. (10)and (19), becomes

Ve = Vg +�ue0 exp(− 9

m�2t

)(53)

From the absolute viscosity of water (T ≈ 300 K), @ =7 ·10−4 kg/ms, and considering a base as thin rigid rode, it

could be roughly estimated 9 ∼ 10−12 kg/s. With �= 05007the breather’s decay time is td = m

9�2 ≈ 10−8 s. Starting withVe ≈ 159 ·103 m/s the path passed by the breather for td =10−8 s reaches approximately 20 · 10−6 m or 6 · 104 basepairs along DNA chain. This is quite favourable regardingexpected role of breathers as long-range effects mediatorsin DNA.

Finally, on the basis of Eqs. (24) and (25) it followsthat the breather’s amplitude decays with the same rate asenvelope velocity, and the width of the breather spreadsexponentially.

4. CONCLUSION AND DISCUSSION

In this paper we have dealt with the PBD model appliedto an idealized DNA chain. The hydrogen bonds in thisDNA chain are represented by the Morse potential, whichresults in the nonlinear Schrödinger equation describingthe breather excitations.

In Section 2, we briefly described the model proposed.The nonlinear dynamics in the context of this modeldepends on several parameters and the values of some ofthem are still rather vague and uncertain. We first consid-ered our attention on the issue of choosing the parameter(, which relates the amplitudes of the first and second har-monics in the expansion in Eq. (8). Usually, this parameterhas been taken to be a constant, but a richer physical situ-ation arises if we take the condition in Eqs. (17) and (18),which seems more significant emerging from a lower har-monic in the perturbation procedure. This brought aboutthe qualitatively different dependence of the amplitude Afrom ql (Figs. 3 and 4). Our second concern was the dis-cussion of the helicoidal spring constant K.

In Section 3 we introduced viscosity term, i.e., the inter-action of DNA nucleotides with the surrounding watermolecules. Here, we have expanded our earlier approachto this problem13 by considering viscous term as small per-turbation, Eq. (43). We found that the envelope velocityof breather exponentially decays eventually reaching thegroup velocity, Eq. (14). We estimated that the decay timefor breather with parameters used in this paper is of theorder of td ≈ 10−8 s. This leads to the path of about 20 'm.If the electric field of appropriate frequency is applied onDNA26 the energy loss due to viscous dissipation shouldbe balanced by the parametric resonance effect preventingbreather’s decay.

It should be stressed that a great deal of attentionwas given earlier to the numerical analysis of the PBDmodel but this was done without viscous perturbation.27–30

The so-called discrete breathers arising in these numericaltreatment exhibit a number of very interesting properties,especially regarding their mutual interactions.

For example, the modulation instability of the wave putin the discrete PBD Hamiltonian splits it into wave pack-ets, which are small breathers. Subsequently, the interaction


RESEARCHARTICLE


of these breathers tends to favour the largest excitations,which grow at the expense of others. This process satu-rates when the breathers, that get narrower as they growin amplitude, are so narrow that they are trapped by latticediscreteness and no longer propagate in the lattice. It stillremains to be demonstrated numerically how the helicos-ity and viscosity affect the dynamics behaviour of discretebreathers. It is expectable that the viscosity which showsopposite tendency of above described growing of ampli-tude and narrowing of discrete breather could establishsubtle balance which enables breather to self sustain sta-bility and maintain capability of reaching long way alongDNA chain.

Acknowledgments: This project was supported byfunds from NSERC, MITACS, the Institute of Theoreti-cal Physics at the University of Alberta and from SerbianMinistry of Sciences, project No H1822.

References and Notes

1. S. W. Englander, N. R. Kalenbach, A. J. Heeger, J. A. Krumhansl,and S. Litwin, Proc. Natl. Acad. Sci. USA 77, 7222 (1980).

2. S. Yomosa, Phys. Rev. A 30, 474 (1984).3. S. Homma and S. Takeno, Prog. Theor. Phys. 72, 679 (1984).4. Ch. T. Zhang, Phys. Rev. A 35, 886 (1987).5. V. Muto, A. C. Scott, and P. L. Christiansen, Phys. Lett. A 13, (1989).6. S. N. Volkov, J. Theor. Biol. 143, 485 (1989).7. M. Peyrard and A. R. Bishop, Phys. Rev. Lett. 62, 2755 (1989).8. Ch. T. Zhang, Phys. Rev. A 40, 2148 (1989).9. L. V. Yakushevich, Phys. Lett. A 136, 413 (1989).10. L. V. Yakushevich, Nonlinear Physics of DNA, Wiley Series in Non-

linear Science, John Wiley, Chichester (1998).

11. T. Dauxois, Phys. Lett. A 159, 390 (1991).12. T. Dauxois and M. Peyrard, Dynamics of Breather Modes in a Non-

linear Helicoidal Model of DNA, Lecture Notes in Physics 393,Dijon (1991), p. 79.

13. S. Zdravkovic and M. V. Sataric, Phys. Scripta 64, 612 (2001).14. S. Zdravkovic, J. A. Tuszynski, and M. V. Sataric, J. Comput. Theor.

Nanosci. 1, 171 (2004).15. Peyrard, private communication.16. G. Gaeta, Phys. Lett. A 172, 365 (1993).17. M. Remoissenet, Phys. Rev. B 33, 2386 (1986).18. S. Zdravkovic, Nonlinear Dynamics of DNA Chain, will be pub-

lished in Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering, Edited by A. Spasic and Jyh-Ping Hsu, Marcel Dekker,Inc., New York (2005).

19. T. Kawahara, J. Phys. Soc. Japan 35, 1537 (1973).20. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Soli-

tons and Nonlinear Wave Equations, Academic Press, Inc., London(1982).

21. A. C. Scot, F. Y. F. Chu, and D. W. McLaughlin, Proc. IEEE 61,1443 (1973).

22. S. Zdravkovic and M. V. Sataric, About optical and acoustical fre-quencies in a nonlinear helicoidal model of DNA molecule, submit-ted to Chin. Phys. Lett.

23. G. Carreri, M. Geraci, A. Ginansanti, and J. A. Ruply, Proc. Natl.Acad. Sci. USA 82, 5342 (1985).

24. S. Zdravkovic and M. V. Sataric, Int. J. Mod. Phys. B 17, 5911(2003).

25. N. N. Bogoliubov and Y. A. Mitropolskii, The Asymptotic Methodin the Theory of Nonlinear Vibrations, Nauka, Moscow (1974), (inRussian).

26. M. V. Sataric, Physica D 126, 60 (1999).27. T. Dauxois, M. Peyrard, and C. R. Willis, Phys. Rev. E 48, 4768

(1993).28. O. Bang and M. Peyrard, Phys. Rev. E 53, 4143 (1996).29. K. Forinach, T. Cretegny, and M. Peyrard, Phys. Rev. E 55, 4740

(1997).30. M. Peyrard, Physica D 119, 184 (1998).

Received: 7 December 2004. Accepted: 5 January 2005.


s. zdravkovic, j.a. tuszynski and m.v. sataric: peyrard-bishop-dauxois model of dna dynamics and...

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