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Moving breather collisions in the

Peyrard-Bishop DNA model

A Alvarez1, FR Romero1, J Cuevas2, and JFR Archilla2

1Grupo de F´ ısica No Lineal.´Area de F´ ısica Te´ orica. Facultad de F´ ısica. Universidad

de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain),

azucena@us.es,

WWW home page: http://www.grupo.us.es/gfnl

2Grupo de F´ ısica No Lineal. Departamento de Fisica Aplicada I. ETSI Inform´ atica.

Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain)

Abstract. We consider collisions of moving breathers (MBs) in the

Peyrard-Bishop DNA model. Two identical stationary breathers, sep-

arated by a fixed number of pair-bases, are perturbed and begin to move

approaching to each other with the same module of velocity. The outcome

is strongly dependent of both the velocity of the MBs and the number

of pair-bases that initially separates the stationary breathers. Some col-

lisions result in the generation of a new stationary trapped breather of

larger energy. Other collisions result in the generation of two new MBs.

In the DNA molecule, the trapping phenomenon could be part of the

complex mechanisms involved in the initiation of the transcription pro-

cesses.

Key words: Discrete breathers, intrinsic localized modes, moving breathers,

breather collisions, Peyrard-Bishop model.

1 Introduction and model set-up

The DNA molecule is a discrete system consisting of many atoms having a quasi-

one-dimensional structure. It can be considered as a complex dynamical system,

and, in order to investigate some aspects of the dynamics and the thermodynam-

ics of DNA, several mathematical models have been proposed. Among them, it

is worth remarking the Peyrard–Bishop model [1] introduced for the study of

DNA thermal denaturation. This model, as well as some variations of it, have

also been used extensively for the study of some dynamical properties of DNA.

The study of discrete breathers (DBs) in chains of oscillators is an active re-

search field in nonlinear physics [2, 3, 4, 5]. Under certain conditions, stationary

breathers can be put in motion if they experience appropriate perturbations [6],

and they are called moving breathers (MBs). There are no exact solutions for

MBs, but they can be obtained by means of numerical calculations.

In the Peyrard–Bishop model, the existence of DBs has been demonstrated [7,

8], and DBs are thought to be the precursors of the bubbles that appear prior

to the transcription processes in which large fluctuations of energy have been

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2A Alvarez et al.

experimentally observed. Some studies about the existence and properties of

MBs in the Peyrard–Bishop model including dipole-dipole dispersive interaction

are carried out in [9, 10].

In this work, we consider the Peyrard-Bishop DNA model, which Hamiltonian

can be written as

H =

N

?

nrepresents the kinetic energy of the nucleotide of mass m

at the nthsite of the chain, and unis the variable representing the transverse

stretching of the hydrogen bond connecting the base at the nthsite. The Morse

potential, i.e., D(e−bun− 1)2, represents the interaction energy due to the hy-

drogen bonds within the base pairs, being D the well depth, which corresponds

to the dissociation energy of a base pair, and b−1is related to the width of the

well. The stacking energy is1

constant.

In scaled variables this Hamiltonian can be writing as:

?1

where unrepresents the displacement of the nthpair-base from the equilib-

rium position, ε is the coupling parameter and V (un) is:

n=1

?1

2m˙ u2

n+ D(e−bun− 1)2+1

2ε0(un+1− un)2

?

,

(1)

the term

1

2m˙ u2

2ε0(un+1− un)2, where ε0is the stacking coupling

H =

?

n

2˙ u2

n+ V (un) +1

2ε(un− un+1)2

?

,

(2)

V (un) =1

2(exp(−un) − 1)2.

(3)

Time-reversible, stationary breathers can be obtained using methods based

on the anti-continuous limit [11]. At t = 0, ˙ un= 0,∀n, and the displacements

of a breather centered at n0are denoted by {uSB,n}. A moving breather {ut,n}

can be obtained with the following initial displacements and velocities:

u0

˙ u0

MB,n= uSB,ncos(α(n − n0))

MB,n= ±uSB,nsin(α(n − n0)) .

(4)

The plus-sign corresponds to a breather moving towards the positive direction

and the minus one, the opposite. This procedure works as well as the marginal-

mode method [6] and gives good mobility for a large range of ε. The translational

velocity and the translational kinetic energy of the MB increase with α. We

use Eqs. (4) as initial conditions to integrate the dynamical equations using a

symplectic algorithm [13].

The study begins generating two identical stationary breathers, with the same

frequency, separated by a fixed number of pair-bases between their centers. We

call Ncthe number of pair-bases separating initially the centers of the two DBs.

Both breathers are in phase, that is, before the perturbation, each breather is

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Moving breather collisions3

always like the mirror image of the other one. The perturbation should be given

simultaneously to both breathers and the initial conditions of each breather given

by Eqs. (4), with the plus sign for one breather and the minus sign for the other

one. In this way the MBs travel with the same modulus of velocity, but opposite

directions, and they are in phase.

2 Results and conclusions

We can analyze collisions with a fixed value of the parameter α and different

values of Ncso that the colliding MBs keep unchanged. Also, we can analyze

collisions varying the parameter α maintaining fixed the number Nc, thus the

colliding MBs change for each value of α. We write

Nc= No+ jj,

(5)

where Nois a fixed number to guarantee that the breathers are initially far

apart, and jj is a positive even number.

In the first approach we fix the parameter α and perturb the DBs varying

their separation Nc, thus the only difference between two collisions is the time

passed between the initial perturbation and the initiation of the collision.

We consider collisions where the DBs are in phase and perturbed simultane-

ously. We have taken No= 40 and jj varies in the interval [0,100] with step size

2. Then, up to fifty different collisions can be analyzed for a fixed value of α and

ε.

We have performed an extensive numerical simulations considering different

values of the coupling parameter ε, and MBs with different values of the wave

number α. The values of ε have been taken in the interval [0.13,0.35] with step

size 0.01. For each value of ε the values of α have been taken in the interval

[0.030,0.200] with step size 0.002. We present the results obtained with ε = 0.32

and α = 0.048; α = 0.138; α = 0.18, which correspond to MBs with increasing

velocities. These values are representative of the different scenarios that can be

found. Fig. 1 represents the trapped energy versus jj for these three cases. The

qualitative results are similar for other values of the parameters (ε,α).

Fig. 1(left) corresponds to the case with the smallest velocity, i.e., α = 0.048,

the distribution of points appears in a narrow band and there are no points

with trapped energy close to zero. When the MBs have small enough kinetic

energy, most of the energy gets trapped after the collision and two small MBs

are generated traveling with opposite directions, they transport the remaining

energy except a small part that is lost in the form of phonon radiation. Notice

that for jj up to 30, the points oscillate following a repetitive regular pattern,

and this regularity begin to change as jj increases.

For intermediate values of α the phenomenon of non-trapping, or breather

generation, appears for some values of jj. This can be appreciated in Fig. 1(central),

obtained with α = 0.138, where some points appear with trapped energy close

to zero, this means that after the collision almost all the energy is transported

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4A Alvarez et al.

by two emerging MBs with the same velocities that the incoming MBs’. For this

value of α two points in the upper band is followed by one point that fall down

close to zero.

For α = 0.18, the upper band is divided in pieces and another fragmented

lower band appears. There are alternating intervals of Ncvalues corresponding

to the upper band and other corresponding to the lower band. This means that

there are some successive values of jj associated with trapping, followed by other

ones associated with breather generation, see Fig. 1(right).

0 50

jj

100

0

0.5

1

1.5

trapped energy

α=0.048

050

jj

100

0

0.5

1

1.5

trapped energy

α=0.138

0 50

jj

100

0

0.5

1

1.5

trapped energy

α=0.18

Fig. 1. Three distributions of points representing the trapped energy versus jj, for

α = 0.048, α = 0.138, and α = 0.18, respectively. Coupling parameter ε = 0.32 and

breather frequency ωb= 0.8.

Fig. 2 shows the displacements versus time for eight collisions of Fig. 1(central),

corresponding to jj = 34,....,48, with the fixed value α = 0.138, ε = 0.32 and

breather frequency ωb= 0.8.

Fig. 3 shows the evolution of the trapped energy for the collisions with jj =

34,...40 of Fig. 2. For jj = 34 and jj = 40 two new breathers are generated.

The other cases correspond to breather trapping with breather generation.

It is interesting to study the collisions maintaining fixed the number Ncand

varying α for fixed values of ωband ε. In real DNA the MBs could be generated at

fixed points of the chain by the action of proteins. Obviously, the phenomenology

is similar to the previous case and the study has permitted to observe a great

sensitivity of the outcomes with respect to the parameter α ( Ref.[14]). To see

this, let us consider the results for three nearness values of α with Nc = 40,

ε = 0.32 and ωb= 0.8:

For α = 0.1370, the collision produces three new breathers, a trapped

breather containing most of the initial energy and two new MBs.

For α = 0.1372, there is a noticeable attenuation of the amplitude of the

trapped breather, which anticipates an entirely new outcome. The emerging

MBs contain most of the initial energy.

For α = 0.1374, there is no trapping and two new MBs emerge with almost

the same velocity that the colliding breathers’.

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Moving breather collisions5

01000

80

100

120

140

160

n

jj=34

0 1000

80

100

120

140

160

n

jj=36

01000

80

100

120

140

160

n

jj=38

01000

80

100

120

140

160

n

jj=40

0 1000

time

80

100

120

140

160

n

jj=42

01000

time

80

100

120

140

160

n

jj=44

01000

time

80

100

120

140

160

n

jj=46

0 1000

time

80

100

120

140

160

n

jj=48

Fig. 2. Displacements versus time for eight collisions corresponding to jj = 34,...,48,

with the fixed value α = 0.138. Coupling parameter ε = 0.32 and breather frequency

ωb= 0.8.

02000

time

0

0.5

1

1.5

2

2.5

trapped energy

jj=34

0 2000

time

0

0.5

1

1.5

2

2.5

trapped energy

jj=36

02000

time

0

0.5

1

1.5

2

2.5

trapped energy

jj=38

02000

time

0

0.5

1

1.5

2

2.5

trapped energy

jj=40

Fig. 3. Trapped energy versus time corresponding to the first four collisions of Fig. 2,

respectively.

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6A Alvarez et al.

The previous studies let us to conclude that for a given values of ε and ωb,

the relevant parameters to determine the outcomes of the collisions are both α

and the number Nc.

The simulations of MB collisions in the Peyrard-Bishop DNA model show

a new mechanism for concentrating energy in DNA. When two MBs collide, it

is possible, in some favorable cases, to get stationary trapped breathers with

more energy than the colliding breathers. These breathers are also movable and

after colliding with other ones, could give rise to even more energetic stationary

breathers. This mechanism could be part of the complex mechanisms involved

in the initiation of the transcription processes.

We are performing extensive numerical simulations of other types of collisions

that can appear in the DNA molecule, which will be published elsewhere.

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