Biophysical Implications of the Peyrard-Bishop-Dauxois model of

DNA dynamics

S. Zdravkovic,1 M. Sataric,2 and J. Tuszynski3

1Faculty of Electrical Engineering, University of Pristina,

Kosovska Mitrovica, F. R. Yugoslavia

2Faculty of Technical Sciences, University of Novi Sad, F. R. Yugoslavia

3Department of Physics, University of Alberta, Edmonton, T6G 2J1, Canada

(Dated: November 14, 2003)

In this paper we expand and elaborate on one of the most widely used nonlinear

models of DNA dynamics. A part of this work is dedicated to the parameter of

helical interaction within this simple, but not trivial model of a DNA molecule. We

also propose an interesting role for the so-called breather solitons emerging from

the model and being involved in protein regulation of nonlinear dynamics of DNA.

Finally, we examined the possible role of harmonic electric fields in the unzipping of

complementary strands of DNA.

Keywords: DNA; PBD model; breather modes; regulatory proteins

I. INTRODUCTION

Double-helical DNA is a unique polymer. Unlike singly-bonded main-chain polymers, as

for example polystyrene, along which the backbone may be appreciably reoriented at a single

bond, DNA is very stiff. This is due to the tight constraint of successive pairs of nucleic

acids, the so-called base pairs (bp), by chemical and hydrogen bonds [1]. DNA has a thermal

bending persistence length of about 150 bp (50 nm). The double-helical structure also ex-

hibits twisting rigidity [2], with a thermal twist persistence length of about 220 bp (75 nm).

These elastic properties are stated for “standard” aqueous solutions with 0.14 M univalent

salt concentrations. We are exclusively concerned here with the ubiquitous B-double-helix

DNA found in vivo. Although there is some variation in the helix repeat distance with base

pair sequence, the unstressed B-DNA makes one turn about every 10.5 base pairs or roughly

every H = 34 A. The spacial angular frequency of an unstressed helix is 2πH

= 1.85 nm−1.

2

The diameter of the double helix is about 20 A, see Fig. 1.

The persistence lengths for bending and twisting are the natural length scales for thermal

fluctuations and, as mentioned above, are 15–20 times H. The comparatively large energetic

cost of disrupting the hydrogen bonded structure of base pairs and their helical stacking

means that the backbone or central axis of the molecule can be considered inextensible and

its distances can be given in either nm or bp. We assume that on the length scales of interest,

small irregularities of structures due to specific base pairs are averaged out and we do not

consider the effects of strong intrinsic bonds due to phased adenine–thymine (A–T) pairs.

The method of fluorescence depolarization is widely used for measurements of the tor-

sional constants of biopolymers. In 1992 Selvin et al. [3] used this method for measurements

of the torsional rigidity of relaxed, positively- and negatively-supercoiled DNA. To this end

they used time-correlated single-photon counting of an intercalated dye, ethidium bromide.

The main result of the measurements was rather unusual, namely the torsional rigidity of

the DNA molecule was not constant at all. They found out that at physiological ionic

strengths (0.175 M), the torsional rigidity monotonically increases from 1.76× 10−26 J m for

the most positively supercoiled DNA, to 2.28×10−26 J m for the most negatively supercoiled

DNA. At different ionic strengths the tendencies were similar. From these data the authors

drew the conclusion that a DNA molecule is not a linear system. According to realistic

estimates, the anharmonic term in the model Hamiltonian for DNA dynamics should ac-

count for approximately 15 % of twist fluctuations at room temperature. Moreover many

mechanical models dealing with overall DNA dynamics under high tension including over-

stretching, undertwisting, and supercoiled DNA show without doubt that DNA dynamics

have an intrinsically nonlinear character [4].

Therefore, we begin here with persuasive arguments of the nonlinear characteristics of

internal DNA dynamics. The possibility that nonlinear effects might focus the vibrational

energy of DNA into localized soliton-like objects was first contemplated by Englander et al.

[5]. Although several authors [6–12] have suggested that either topological kink solitons

or bell-shaped breathers would be good candidates to play a basic role in DNA nonlinear

dynamics, there are still several unresolved questions in this regard. A hierarchy of the most

important models for nonlinear DNA dynamics was presented by Yakushevich [13]. In the

present paper we strongly rely on the extended model for DNA dynamics first proposed by

Peyrard and Bishop [8] and later extended by Dauxois [14].

3

The present paper is organized as follows:

In Section 2 we first outline the main features of the model discussed which will be

henceforth referred to as the PBD model for short. We carefully analyse the necessary

conditions for the existence and stability of breather solitons in an idealized homogeneous

model considering the role of model parameters, especially the helical spring constant of the

DNA chain. We discover that a “window” of favourable values of this parameter exists. The

possible role of water molecules regarding this constant is also discussed. The role of water

viscosity regarding breather dynamics is also considered.

In Section 3 we outline the results of our application of the PBD model to the transmission

of regulatory signals along the DNA molecule in the process of regulation of gene expression.

We consider the role of regulatory proteins that possess only hydrogen bonds with the DNA.

In Section 4 we scrutinise the possible role of endogenous AC fields in the nonlinear

dynamics of DNA and try to connect the local fluctuational openings or unzipping of DNA

strands predicted by the PBD model to some still unpredictable biological effects in the

functioning of living cells.

We close this paper with summary and concluding remarks which are reported in Sec-

tion 5.

II. THE PBD MODEL

The B-form DNA in the Watson-Crick model is a double helix which consists of two

strands labelled s1 and s2 in Fig. 1, linked by nearest-neighbour harmonic interactions along

the chains. The strands are coupled to each other through hydrogen bonds, which are

supposed to be responsible for transverse displacements of base pairs. Molecular masses

of nucleosides (with the attached sugar groups) range from 227 (C) to 267 (G). Thus,

the four different base nucleotides (deoxyadenosine, deoxycytidine, deoxyguanosine, and

deoxythymidine) differ in mass by less than 18 %, hence the inhomogeneities due to the

base sequences are neglected here. Therefore, a common mass parameter m is used for

all the bases and the same coupling constant k along each strand is assumed. This is a

simplification made in the model and it means that the DNA chain is treated as a perfectly

periodic structure. However, it should be noticed that in the discrete PBD model, computer

simulations of breather dynamics reveal a very important role of mass inhomogeneities in

4

reflecting or even capturing breathers leading to their growth in fusion processes [15, 16].

The potential energy for the hydrogen bonds connecting A–T or C–G base pairs is mod-

elled by a Morse potential which describes not only the hydrogen bonds but the repulsive

interactions of the phosphate groups, partly screened by the surrounding solvent as well.

The helicoidal three-dimensional structure of a DNA chain, which is responsible for the

stability of the DNA architecture is such that neighbouring base nucleotides from different

strands are close enough so that they interact in terms of filaments of solvent. This means

that a base at the site n of one strand interacts with both the (n + h) th and (n − h) th

bases of the other strand (h = 4), see Fig. 2. Introducing the transverse displacements un,

vn of the bases from their equilibrium positions, the Hamiltonian for the DNA chains in the

context of the PBD model [14] has the form

H =∑n

m

2(u2

n + v2n) +

k

2[(un − un−1)2 + (vn − vn−1)2] +

+K

2[(un − vn+h)

2 + (un − vn−h)2] +D[e−a(un−vn) − 1]2

(1)

where k (respectively K) represents the harmonic constant of longitudinal (respectively

helicoidal) springs, see Fig. 2. The Morse potential depicted in Fig. 3 has depth D and its

inverse width is a.

Criticism of this model could be raised regarding the fact that despite experimentally

verified complex motions of DNA chains (even under the action of thermal fluctuations)

including stretching, bending and twisting degrees of freedom, the model restricts motions

to only two transverse degrees of freedom. We expect that due to long thermal persistence

lengths (150–200 bp) at physiological temperatures, conditions prevail that at least within

such distances the above restricted model can be justified.

The idea is that the complex motion of a DNA chain as a whole can be decoupled into

almost independent motions of several degrees of freedom, and the physical expectation is

that due to the conspicuous nonlinearity of hydrogen bonds this specific degree of freedom

would be favourable for sustaining the expected (and desirable) nonlinear excitations. The

latter are good candidates for information transfer along DNA and for local openings of base

pairs (fluctuational openings).

The elegance of the PBD model is that by using the centre-of-mass coordinates repre-

5

senting the in-phase and out-of-phase transverse motions

xn = (un + vn)/√

2, yn = (un − vn)/√

2 (2)

the corresponding dynamical equations of motion derived from the Hamiltonian in Equation

(1) are perfectly decoupled leading to out-of-phase nonlinear equations of motion of the form

myn = k(yn+1 + yn−1 − 2yn)−K(yn+h + yn−h + 2yn) + 2√

2aD(e−a√

2yn − 1)e−a√

2yn . (3)

The PBD model was later [17] amended by a different choice of sign for the coordinates

involved. On the basis of this choice, the decoupling of in-phase and out-of-phase transverse

coordinates is still possible. But, nevertheless the physical picture is thus also inverted. The

out-of-phase coordinate obeys a linear dynamical equation of motion while the equation

describing the in-phase motion is nonlinear as it includes a Morse potential. This could

imply that the breather-like out-of-phase openings are absent within this approach. In our

opinion, the breather model is more physically tractable regarding fluctuational openings

and unzipping of DNA.

According to the original approach of Dauxois [14], it is assumed that the oscillations of

bases are large enough to be essentially anharmonic, but still insufficient to break the bonds

since the plateau of the Morse potential is not reached.

In this respect, the following two-time-scale expansion in the semi-discrete approximation

could be relevant

yn = εΦn(ε) (ε 1)

Φn(t) = εΨ1(εn`, εt)eiθn + ε2[Ψ0(εn`, εt) + Ψ2(εn`, εt)ei2θn ] + cc+O(ε3) (4)

θn = nq`− ωt

where ` = 0.34 nm is the distance between neighbouring base pairs, ω is the optical frequency

of the linear approximation of the base pair vibrations and q is the wave number of a carrier

wave whose role will be discussed later.

Equating the coefficients for corresponding harmonics of the expansion in Equation (4)

the main result is that the leading terms obey the nonlinear Schrodinger equation below

(subscript τ and S mean derivatives.):

iΨ1τ + PΨ1SS +Q|Ψ1|2Ψ1 = 0 (5)

6

where the scaled travelling coordinate S and time τ are S = ε(z − Vgt) and τ = ε2t,

respectively, while the dispersion coefficient P and the coefficient of nonlinearity Q are

given by

P =1

ω

`2

m[k cos(q`)−Kh2 cos(qh`)]− V 2

g

, (6a)

Q = −ω2g

2ω[2α(µ+ δ) + 3β]. (6b)

The remaining parameters (group velocity Vg, ωg, α, µ, β) all arise from the expansions of

Equation (4) as follows:

Vg =1

mω[k sin(q`)−Kh sin(qh`)], (7a)

ω2g =

4a2D

m, (7b)

ω2 = ω2g −

2k

m(cos(q`)− 1) +

2K

m(cos(qh`) + 1), (7c)

α =−3a√

2, (7d)

µ = −2α

[1 +

4K

mω2g

]−1

, (7e)

β =7a2

3. (7f)

The parameter δ will be defined in this paper in the new form so the physical significance

of this step will be discussed shortly.

For PQ > 0, Equation (5) has an envelope soliton solution representing a small-amplitude

breather for out-of-phase base pair displacement yn of the form (replacing n` with z − z0,

where z0 is the position of the centre of a breather)

yn = ε2A sech

[ε

Le(z − z0 − Vet)

]×

×

cos(Θz − Ωt) + εA sech

[ε

Le(z − z0 − Vet)

](µ2

+ δ cos[2(Θz − Ωt)])

+O(ε3), (8)

where the breather amplitude A, the width Le, the envelope velocity Ve, the carrier wave

vector Θ and its frequency Ω are coupled with scaled carrier and envelope velocities (uc, ue)

7

as follows:

A =

√u2e − 2ueuc

2PQ, (9a)

Le =2P√

u2e − 2ueuc

, (9b)

Ve = Vg + εue, (9c)

Θ = q +εue2P

, (9d)

Ω = ω +εue2P

(Vg + εuc). (9e)

Our analysis of the conditions of existence and stability for breather solitons in Equation

(5) reveals the following important features of the PBD model:

Numerical analysis of the necessary condition PQ > 0 for breather existence [18] shows

the dispersion coefficient P and the group velocity Vg from Equation (6a) and Equation

(7a), depicted respectively in Fig. 4 and Fig. 5. It is readily recognized that the conditions

Vg > 0 and P > 0 are satisfied on several intervals of values q`. Thus it follows that the

wavelength of a carrier wave q = 2πλ

should be an integer multiple of the spacing `. This

allowed “window” of carrier wave numbers is a narrow “Brillouin zone” and contains four

values λ = (6`, 7`, 8`, 9`). The breather corresponding to λ = 8` (q` = 0.78 rad) is depicted

in Fig. 6. It reveals that the breather is represented by a modulated soliton so that the

envelope covers approximately twenty oscillations of the carrier wave.

A quite new point regarding the PBD model pertains to the parameter β contained in the

parameter of nonlinearity Q, Equation (6b). So far this parameter arises from the expansion

in Equation (4), connecting the first and second harmonics of this expansion via Ψ2 = δ ·Ψ21.

This was done by equating the coefficients for terms exp(i3θn) or exp(i4θn) giving constant

values for δ. We concluded that a more physically-based approach should be to equate terms

for the more influential second harmonic, exp(i2θn), yielding a new relation between Ψ2 and

Ψ1 as follows:[4ω2 +

2k

m(cos(2q`)− 1)− 2K

m(cos(2hq`) + 1)− ω2

g

]Ψ2 = ω2

gαΨ21 (10a)

or δ = αω2g

[4ω2 +

2k

m(cos(2q`)− 1)− 2K

m(cos(2hq`) + 1)− ω2

g

]−1

. (10b)

This means that δ is not constant, but rather a function of q`. Thus, besides the requirement

PQ > 0 that should be satisfied for the breather solution of Equation (5), the adjustable

8

coefficient δ must now satisfy Q > 0 from Equation (6b). In our paper [18] we adopted the

set of PBD model parameters from Dauxois [14] and this set reads:

k = 3K = 24 N/m, (11a)

a = 2× 1010 m−1, (11b)

D = 0.1 eV, (11c)

` = 3.4× 10−10 m, (11d)

m = 5.4× 10−25 kg, (11e)

ue = 105 m/s, (11f)

uc = 0, (11g)

ε = 0.007. (11h)

We should emphasize that the values of these parameters are still controversial. For example,

in Dauxois et al. [19] the Morse well was represented as remarkably shallower and narrower

(D = 0.04 eV and a = 4.45 × 1010 m−1). Nevertheless we relied on Dauxois’ first choice,

Equation (11). With the set of parameters in Equation (11) and the nonconstant δ given

by Equation (10), the parameter of nonlinearity Q is positive only for a rather large q`,

see Fig. 7, i.e. for q` > 1.36 rad. However, these values for q` do not satisfy the other

requirements (P > 0, Vg > 0, λ = N`, N integer) seen in Fig. 4 and Fig. 5. A possible

solution of this controversy is in the assumption that the commonly accepted value of the

helicoidal spring constant K = 8 N/m is overestimated. Consequently, we studied the impact

of different, smaller values of K. From Fig. 8 we see that Q > 0 holds only for K ≤ 4.6 N/m

if q` = 0.78 rad (λ = 8`) is picked. More details about this interesting analysis will be

published elsewhere [20], but the general conclusion is that there exists a “window” of values

of K leading to the emergence of a breather (0 < K < 4.6) N/m which is the consequence

of a straightforward introduction of the parameter δ, in Equation (10). A similar “window”

of allowed values of the nonlinearity parameter is known in the theory of Davydov solitons

and was the object of controversy about the existence of Davydov solitons at physiological

temperatures [21].

A few additional words are in order concerning the helicoidal spring constant K. Recently

Peyrard [22] stressed the importance of hydrating water for the biochemical activities of

proteins and DNA. Since water molecules are highly polar they form an ordered network

9

on the surface of a protein along which protons can be transferred. The formation of this

network with long-range connectivity has been detected as a percolation transition when the

water content approaches 0.5 g per gram of protein [23]. We suggest that the helicoidal spring

constant in the PBD model may reflect the mediating role of the of the water molecules.

Moreover, the helicoidal structure itself arises as the result of optimization of the interplay

between hydrogen bonding, hydrophobicity and long-range connectivity.

If the helicity is more strongly expressed by the tension of supercoiling, Fig. 9, it should

increase the value of K by decreasing the distance of the helicoidally adjacent bases on

opposite strands. Increasing the value of K, Fig. 8, would lead to the decrease of Q and thus

to an increase in the breather’s amplitudes, see Equation (9a), beyond the Morse plateau

causing DNA unzipping.

On the other hand, we must take into account the fact that the solvating water does act

as a viscous medium that damps out DNA dynamics, favouring energy expenditure. We

have also analysed this important biophysical situation in the context of the PBD model

[20]. It appears that a viscous force proportional to the velocity of nucleotide oscillations

does not impact the out-of-phase base pair dynamics, so we took into consideration a viscous

force of the Newtonian kind, proportional to the square of the velocity. The perturbation

procedure of “slowly varying coefficients” [24] gives important information about damped

dynamics including the slow decrease in the wave’s amplitude and the deceleration of the

breather according to a linear function of time:

Ve(t) = Ve0(1− Ve0βt) (12)

Here Ve0 is the initial envelope velocity of the breather and β is a constant dependant on

the viscosity and on other parameters of the PBD model.

III. THE POSSIBLE IMPACT OF REGULATORY PROTEINS ON PBD

DYNAMICS

The idea of regulatory signal transmission along DNA came from the results of experi-

ments in which to so-called long-range effects were discovered in DNA [25, 26]. Some reg-

ulatory proteins in transcription processes were seen to bind with high efficiency to special

segments of a DNA molecule. Numerous experimental data show that even if the distances

10

between attached proteins can reach hundreds or thousands of base pairs, they exhibit mu-

tual influence [27]. To explain this effect many alternative models of the action at a long

distance have been proposed.

Our preferred assumption is that binding one regulatory protein molecule to DNA pro-

duces an excited state which is accompanied by a local conformational distortion of a base

pair generating a breather excitation or enhancing the amplitude and the speed of an al-

ready existing decelerated breather. Such a breather propagates along the DNA chain and

upon reaching the target site changes the conformational state of the site, which in turn,

changes the binding constants for a second protein. This enables it to be attached via the

lock-and-key mechanism to the same DNA chain. Specific interactions of regulatory pro-

teins with DNA are usually defined through hydrogen bond interactions between functional

groups of amino acid side chains or peptide bonds and groups of base pairs in the major

or minor grooves of the DNA chain. We here restrict our consideration to those regulatory

proteins that possess only hydrogen bonds with the DNA. Fig. 10 represents the amino acid

glutamine bound by two hydrogen bridges to a A–T base pair in the minor groove.

We here recall that every protein has a peptide group which contains a double-bonded

carbon-oxygen complex (or amide-I bond) with a characteristic quantum of energy of

0.205 eV (corresponding to a spectral peak at 1650 cm−1). The amide-I bond appears to

be of great interest here as a potential “basket” for storage and transport of biological en-

ergy. This part of an glutamine molecule is indicated in Fig. 10 by a circle. The amide-I

exciton mode was given a prominent place in the theory of Davydov molecular solitons that

was also applied to α-helical chains [20]. However, a problem arises when we realize that

at a single peptide group, the lifetime of an amide-I vibration is of the order of 10−12 s [20].

We conjecture that the energy of this mode which is comparable to the depth of the Morse

potential well of neighbouring base pairs (D = 0.2 eV) could be utilized in producing a con-

formational change of the respective A–T base pair. This energy transfer is supposed to be

mediated by hydrogen bonds of an glutamine attachment to DNA depicted by a rectangle

in Fig. 10. Finally, the hydrogen bonds within a A–T base pair are in the ellipsoidal area.

In order to describe the possible mechanisms of long-range signal propagation induced

by regulatory proteins, we start from the nonequilibrium statistical approach presented by

Zubarev [28], and straightforwardly performed in our recent paper [29]. Here, we only outline

the main features and results of this method.

11

Let us first introduce the extended Hamiltonian in an attempt to model the above regu-

latory process in DNA. The Hamiltonian should consist of two parts as follows:

H = H0 +Hint, (13)

where the Hamiltonian H0 consists of two terms, the first of which represents the part of

the PBD Hamiltonian Equation (1) containing the separated coordinates yn and momenta

Pyn = myn, while the second one corresponds to the amide-I mode in the regulatory protein,

considered here. Hence,

H0 = Hy +HC=0, (14)

Hy =∑

n

P 2yn

2m+ k

2(yn − yn−1)2 + K

2[(yn − yn−4)2

+(yn − yn+4)2] +D(e−√

2ayn − 1)2, (15)

and

HC=0 = EκC+κ Cκ, Eκ = hΩκ = 0.205 eV, (16)

where Eκ is the energy of the amide-I mode, while C+κ , Cκ represent the creation and

annihilation operators, respectively, of an excited state possessing the wave vector κ.

Finally, Hint describes the interaction between the amide-I mode and the nearest base

paris of DNA. It could be conveniently written in the form

Hint =∑n

Hopintfn(t) (17)

where the operator part of the interaction is given as follows

Hopint = (Cκ + C+

κ )yn (18)

while the Heaviside-type interaction switching function has the form

fn(t) = F0e−σ2(m−`)2θ(t− τ(n− 1))− θ(t− τ(n+ 1)). (19)

Since we already used τ for a variable in Equation (5) we now use a modified symbol

τ in order to avoid confusion. We assumed here that due to its hydrogen character, the

interaction term drops exponentially with distance from its original magnitude F0. It is also

12

assumed here that the protein molecule is located at site ` of the DNA chain. The Fourier

transform of the time-dependent part of the interation in Equation (19)

fn(t) =1

2u

∫ +∞

−∞dω e−iωtfn(ω) (20)

yields

fn(ω) = 2F0e−σ2(n−`)2

[sin(τω)

ω

]eiωτn. (21)

Since the breather-solitons in DNA can be generated in different ways, various causes

that have been suggested include the thermal fluctuations as well as local ligand-protein

interactions considered here, in addition to the chemical energy released during ATP hy-

drolysis. It is apparent to us that in a very long DNA chain an ideal gas of breathers can

be generated via one or several of these mechanisms. Consequently, we need to develop a

statistical approach in order to compute the average value of the base pair displacement

resulting in the process. For this purpose we use the well-known method of nonequilibrium

statistical mechanics developed by Zubarev [28] according to which the average value of an

arbitrary physical operator A can be evaluated as

〈A〉 = 〈A〉0 +∞∑m=1

(1

ih

)m ∫ t

−∞dt1

∫ t1

−∞dt2

×∫ t2

−∞dt3 · · ·

∫ tn−1

−∞dtn−1TrA(t)[Hint(t1) (22)

× [Hint(t2) · · · [Hint(tn), ρ0]],

where 〈A〉0 is the average value with respect to the density matrix ρ0 pertaining to the

system (Equation (15) and Equation (16) unperturbed by the interaction, Equation (17).

The square brackets above stand for the corresponding commutators, and Tr means the

trace.

If we retain only the two leading terms, Equation (22) then yields

〈A〉 = 〈A〉0 +1

ih

∫ t

−∞dt1TrA(t)[Hint(t1), ρ0]

− 1

h2

∫ t

−∞dt1

∫ dt1

−∞dt2TrA(t)[Hint(t1), [Hint(t2), ρ0]] (23)

13

Substituting A = yn, into Equation (23) gives for the average displacement at lattice site n:

〈yn〉 = 〈yn〉0 +

∫ +∞

−∞dt1 〈〈yn(t)|Hop

int(t1)〉〉 f(t1) +

∫ +∞

−∞dτ1

×∫ +∞

−∞dτ2 〈〈yn(0)|Hop

int(−τ1)‖Hopint(−τ1 − τ2)〉〉 (24)

× f(t− τ1)f(t− τ1 − τ2) · · · ,

where the Green’s functions have been introduced as follows:

〈〈yn(0)|Hint(−τ1)‖Hint(−τ1 − τ2)〉〉

=1

(ih)2θ(τ1)θ(τ2)Tryn(0) (25)

×[Hint(−τ1), [Hint(−τ1 − τ2), ρ0]] · · · .

Calculating the average out-of-phase base pair displacements along the DNA chain, after

tedious algebraic steps one obtains [29]:

〈yn〉 = 〈yn〉0

1 +8F 2

0Dτ2

mh2Ω2κ

sin2(τΩκ)

[2⟨

exp(−2√

2ayn)⟩

0−⟨

exp(−√

2ayn)⟩

0

](26)

where 〈· · · 〉0 stands for the corresponding averages along DNA in the absence of the amide-I

trigger mode; the parameters a, D and m are listed in Equation (11), F0 stands for the

magnitude of the force attributed to the hydrogen bonding amide-I mode–base pair (G–C)

interaction, and τ is supposed to be equal to the amide-I mode lifetime (τ = 10−12 s). The

most recent estimates of the hydrogen bonding force reveal that F0 ' 8 pN.

From Equation (26) it follows that for resonantly coupled energy and lifetime in the

“quantized” form

τΩκ = (2k + 1)π

2; k = 0, 1, 2, . . . (27)

the amide-I triggering would contribute maximally to the process considered. The average

〈yn〉 can be estimated taking 〈yn〉0 = 10−11 m, see Dauxois et al. [19]. The estimate reveals

that due to the presence of glutamine the average “breathing” of the out-of-phase mode

could reach as much as 〈yn〉 = 0.5 A which is on the order of the fluctuational openings of

DNA. It could therefore be inferred that the regulatory proteins may remarkably support the

increase of the average amplitude of breather excitations in DNA enabling them to sustain

over longer distances and to reach target sites in order to switch them on or off, as required

by biological processes.

14

IV. ENDOGENOUS AC FIELDS AND PBD DYNAMICS

We have recently investigated thoroughly the possible role of electromagnetic fields in

DNA dynamics in the context of the PBD model [30] discussed in the present paper. Our

interest in this topic was piqued by the experimental evidence [31] that in yeast cells signals

within the range (50–70) MHz were detectable. These fields could perhaps be attributed to

coherent oscillations of dipolar membranes of the cells or their nuclei. On the other hand,

it was reported that switching electric fields may exist within and around microtubules due

to their ferroelectric properties. The amplitude of such fields in microtubules was estimated

[32] to be on the order of E0 = 2.6× 106 V/m.

We analysed statistically, using the Kubo linear response method [30], the possible reso-

nant effect of the harmonic electric field

E = E0 sin(ω0t) (28)

with magnitude E0 and frequency ω0 transversally polarized with respect to the linear DNA

chain.

The corresponding Hamiltonian representing the interaction of an “ideal gas” of breathers

with the applied AC field can be introduced starting from the leading term of Equation (8),

(with z = 0 and t = 0) and from Equation (16), as follows

W (t) = qeffE0 sin(ω0t)

Nb∑j=1

2εAcos(Θz0j)

cosh(

εLez0j

) (29)

where qeff is the effective protonic charge in the displaced base pair, and the summation

with respect to j represents the initial distribution of centres of Nb breathers along the

DNA chain.

In order to estimate the average stretching of base pairs involved when breathers are

driven by an external electric field, we introduced the average transveral displacement 〈yn〉

as follows:

〈yn〉 = 〈yn〉0 + ∆ 〈yn〉field . (30)

Here the equilibrium average 〈yn〉0 at temperature T is calculated with respect to the

equilibrium canonical ensemble as

〈yn〉0 = (ZhN)−1

∫ +∞

−∞

N∏n=1

dPyndyn(yne−βHy

); β = (kBT )−1 (31)

15

where Z stands for the corresponding partition function, h is Planck’s constant, Pyn = myn,

and Hy is the Hamiltonian given by Equation (15).

For physiological conditions we estimated [30] that

〈yn〉0 ' 1× 10−11 m, (32)

which is highly in accordance with Dauxois et al. [19].

This is significantly beyond average thermal fluctations, which are on the order of 0.2×

10−11 m but still insufficient for local unzipping of a DNA chain.

The linear response term of Equation (30) has the form

∆ 〈yn〉field = β

∫ ∞0

dt

⟨∂yn∂t

W (t)

⟩0

, (33)

where the time derivative ∂yn∂t

follows from the classical Poisson bracket for a Hamiltonian

and Equation (15);

∂yn∂t

= yn, Hy =∂Hy

∂Pyn. (34)

Replacing W (t) and ∂yn∂t

in Equation (33) one gets

∆ 〈yn〉field =B

h

∫ ∞0

dVem∗ exp

(−βm∗V 2

e

2

)(εA)2

×∫ +∞

−∞

dz0 cos(Θz0)

cosh((ε/Le)z0)

∫ ∞0

dt[cos(ω0t)]1

`(35)

×∫ +∞

−∞dz

sin(Θz − Ωt)

cosh[(ε/Le)(z − z0 − Vet)].

where

B = 4E0qeffβΩ exp(−βE0), m∗ =4

15

m

`

(A2ε2

Le

), (36)

Ve is the envelope velocity, see Equation (9c), and E0 is the bound energy of a single breather,

E0 ' 0.1 eV.

After making the appropriate calculations of Equation (35) and by using the set of param-

eters proposed by Dauxois, see Equation (11), but instead using the new value K = 4 N/m,

16

we estimated

qeff = 2× 10−19 C, (37a)

A = 3× 10−8 m, (37b)

Le = 1× 10−11 m, (37c)

Ω = 0.5× 1012 s−1, (37d)

Θ = 1.5× 109 m−1, (37e)

q = 2.3× 109 m−1, (37f)

Ve = 1.6× 103 m/J, (37g)

m∗ = 3× 10−26 kg, (37h)

which enables us to get the field induced nonequilibrium displacement in the following form

∆ 〈yn〉field = 2× 10−15(E0ω0) A. (38)

Thus, under extremely favourable conditions, a strong local electric fields with a magnitude

on the order of 106 V/m [32], and with a frequency of 7× 107 Hz [31], gives on the basis of

Equation (38) very significant average displacements of

∆ 〈yn〉field ' 0.9 A. (39)

This number is close to the threshold of local unzipping which is about 1 A. It could be thus

concluded that extremely high intrinsic endogenous AC fields could have effects on DNA

quite similar to those of high energy photons [33]. In this context the possible influences on

living cells of “electromagnetic smog” produced by massive use of ever increasing frequencies

should be considered seriously. Cellular telephones with working frequencies of roughly

2 GHz are particularly interesting in this respect.

V. CONCLUSIONS AND DISCUSSION

In this paper we have revisited the Peyrard-Bishop-Dauxois model of DNA dynamics

and showed some of our applications of this interesting model to important biological con-

sequences of nonlinear DNA dynamics.

17

In Section II we briefly outlined the physical establishment of the PBD model and a

corresponding algorithm leading to a nonlinear Schrodinger equation and its breather mode

solution.

We introduced a new choice of the parameter δ, see Equation (10), which is incorporated

in the coefficient of nonlinearity Q, see Equation (6b). By numerical analysis we have shown

that with this choice of δ the value of the helicoidal spring constant K must be revised.

We found the the “window” of possible values is such that 0 N/m < K < 4.6 N/m. The

role of hydrating water in sustaining the interaction which is responsible for the helicoidal

spring was stressed by Peyrard [22]. We took the next step, considering the viscosity of the

hydrating water and its impact on the breather’s dynamics. The perturbation technique

used shows that breathers persist with stable but linearly decelerated motion.

In Section III we examined a very interesting biological application of PBD breathers the

mediators for regulatory proteins’ interactions with DNA. We analysed the specific example

of the amino acid glutamine attached by hydrogen bonds to an A–T base pair in the major

groove of a B-DNA helix. Our basic assumption is that amide-I bond excitation plays the

role of a trigger for high amplitude breather formation.

Section IV starts from the breather function, Equation (8), and examines the interaction

of an “ideal gas” of breathers with a harmonic AC field. Kubo formalism leads to the

expression in Equation (38) which indicates the possible role of strong fields with high

frequencies in unzipping of DNA chains with profound physiological consequences.

Finally, we give a few general remarks regarding the PBD model. It is clear that this

simple version for an ideal homogeneous DNA chain does not pertain to real circumstances

where inhomogeneities either with respect to mass or to hydrogen bond number (two or

three) could play an important role in DNA dynamics.

Nevertheless, on the basis of exhaustive numerical results [34–36] it was concluded that

this additional interplay between nonlinearity, discreteness, and inherent inhomogeneity

within the PBD model sustains the stability and higher localization of breather excitations,

making them candidates for diverse DNA functions.

Recently, substantial progress was made in experiments where single DNA strands within

a genome are manipulated and the forces necessary for separation of complementary strands

are measured [37]. Establishing the connections between the parameters of theoretical mod-

els, such as the PBD model, and experimental results is the next stage in this particularly

18

important field of biophysics.

Acknowledgments

The authors thank Prof. John Marko of the University of Illinois, Chicago for very stim-

ulating discussions during his visit to Edmonton. We are also grateful to Prof. Michel

Peyrard of the Ecole Normale Superieure de Lyon for inspiration. This project has been

supported by funds from NSERC, MITACS-MMPD and the Theoretical Physics Institute

at the University of Alberta.

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[17] G. Gaeta, Phys. Lett. A 172 (1993) 365–372

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dynamics (????), submitted

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11 935–11 940

20

FIG. 1: The Watson-Crick double helix model of the DNA molecule with characteristic dimensions.

Figures

21

! "#

$!%

&!'()

*+ ,

-.0/2143506214.07,8:9<;1=.0/

>0?A@CBDAE<BF2GA@IHKJ<LMB4N0OP0QP0RCS=P0TVUXWZY[QW\T]\:^<WMS4P0_

FIG. 2: Illustration of the PDB model including elastic constants introduced and the assumed

transverse displacements of each base pair.

22

- -1

üï

ýï

þ

ïï

ïï

1- -

-

! " #$

% &

'

(

)

*

+,-/.10 23 46587:9

;< =>@?BACD/EGF

HIKJ:L1MONPIQ@MSRTQ8U8VXW

Y[Z

\^]

FIG. 3: Representation of the Morse potential with depth D and inverse length a.

q` (rad)

P(1

0−7

m2/s)

6543210

40

20

0

-20

FIG. 4: Dispersion coefficient P of the NSE versus q` for ` = 3.4 × 10−10 m and q = 2πλ , from

Equation (6a).

23

q` (rad)Vg

(m/s

)6543210

4000

2000

0

-2000

-4000

FIG. 5: Group velocity Vg of a breather versus q` (` = 3.4× 10−10 m, q = 2πλ ), see Equation (7a).

t (ps)

Φ(n

m)

6260585654525048

2

1

0

-1

-2

FIG. 6: The breather corresponding to the chosen λ = 8` (q` = 0.78 rad).

q` (rad)

Q(1

032

m−

2s−

1)

2.01.51.00.50.0

0

-10

-20

-30

-40

FIG. 7: Nonlinear coefficient Q of the NSE as a function of q` (K = 8 N/m), from Equation (6b).

24

K (N/m)Q

(103

2m−

2s−

1)

76543210

100

50

0

FIG. 8: Nonlinear coefficient Q of the NSE as a function of K (q` = 0.78 rad), from Equation (6b)

and Equation (10).

FIG. 9: Supercoiling of DNA.

25

!#"

$

%'&&'(

)+*,.-0/2143657

8:9.;=<?>A@CBD<E;

FHGJIJK4LDMEN

O

FIG. 10: The amino acid glutamine bound as if in the major grove by two hydrogen bonds (rect-

angle) to a adenosine–thymine base pair. The amide-I bond is circled; note the hydrogens bonding

between the two bases are in the elipse.