eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

Is it possible to govern the movement of the transcription bubbles of DNA by constant and periodic external fields?

Ludmila V Yakushevich,1 Valentina N Balashova,2 Farit K Zakiryanov2
1Institute of Cell Biophysics RAS, Russia
2Bashkir State University, Russia
Received: May 04, 2018 | Published: June 06, 2018

Correspondence: Farit K Zakiryanov, Institute of Physics and Technology, Bashkir State University, Zaki Validi 32, Ufa, Russia, Tel +791-7406-5235, Email

Citation: Yakushevich LV, Balashova VN, Zakiryanov FK. Is it possible to govern the movement of the transcription bubbles of DNA by constant and periodic external fields? Biom Biostat Int J. 2018;7(3):218‒224. DOI: 10.15406/bbij.2018.07.00212

Abstract

The influence of constant and periodic external fields on the living systems can lead to various changes in their functional properties. At the molecular level, these changes can be modeled as the changes in the dynamic properties of biomolecules. In this paper we investigate the effect of constant and periodic fields on the movement of transcription bubbles being small unwound regions (~10bp) in the DNA double helix, which result from the interaction of the RNA polymerase with the DNA promoter region at the initiation stage of the process of transcription. The bubbles are modeled mathematically by kinks being one–soliton solutions of the modified sine–Gordon equation. To obtain these solutions and calculate the dynamic characteristics of the bubbles including their velocity, coordinate, phase portrait, total energy and size, we apply the method of McLaughlin and Scott and quasi–homogeneous approximation. For definiteness all of the calculations were made for the pTTQ18 sequence. The results were used to analyze the bubbles dynamic behavior and to answer the question: is it possible to govern the movement of the transcription bubbles of DNA by constant and periodic external fields? Possible consequences on the gene expression and biological activity of cells are discussed.

Keywords: transcription bubble, DNA torque, periodic fields, sine–Gordon equation, method of McLaughlin and Scott, quasi–homogeneous approximation, nonlinear DNA dynamics

Abbreviations

DNA, deoxyribo nucleic acid; AOT, angular optical trap; MTT, magnetic tweezers technique

Introduction

The action of constant and periodic fields on the living systems is one of the hotly discussed themes of modern biophysics because of the growing amount and variety of electronic devices and their influence on the basic, fundamental life processes and, consequently, on human health. It is assumed, in particular, that external periodic fields with the frequency of terahertz diapason can cause the changes in the physicochemical properties of DNA, which in turn can lead to the changes in the transcription and thus to the changes in the gene expression and cell differentiation.1,2 However, until now the mechanisms of the action of these fields remain unclear. Special attention is also paid to the influence of constant fields because of significant progress in the experimental studies of the dynamics of single molecules of DNA and by the development of new techniques that allows to measure directly the DNA torque.3 The method of the angular optical trap (AOT)4,5 and the method of magnetic tweezers technique (MTT)6–8 are among them. However, these and other currently existing experimental methods do not allow measure the DNA torque directly in the biological processes in which the DNA molecule is involved.

In this paper we apply the methods of mathematical modeling to investigate the action of constant and periodic fields on the movement of transcription bubbles of DNA which are small unwound regions (~10bp) in the DNA double helix resulting from the interaction of the RNA polymerase with the DNA promoter region at the initiation stage of the process of transcription (Figure 1).9 Our aim is to clarify the mechanisms of the influence of the fields on the bubbles dynamics, to find the relationship between the DNA torque and the bubble velocity, to estimate the value of the DNA torque necessary for the moving of the bubbles with the velocity of the process of transcription, to find the relation between the dynamic behavior of the bubbles and their initial velocities, and to give an answer to the question is it possible to govern the movement of the transcription bubbles of DNA by constant and periodic external fields (Figure 1).

Figure 1 Locally unwound region (bubble) in the DNA double strand, which is formed at the initial stage of transcription.

In physics, the DNA molecule is considered as a complex dynamic system consisting of a large number of coupled atoms and atomic groups which are arranged in a certain way in space. Such a system is not static, but movable. It has a large amount of internal motions caused by the effect of temperature, collisions with the molecules of the solution, and interactions with proteins. Among the variety of the internal motions of DNA there are the translational movements of transcription bubbles that can be considered as translational movements of quasi–particles in the potential field of the DNA.

Mathematically the transcription bubbles can be described as the soliton–like solutions of the nonlinear differential equations imitating the internal DNA mobility. In the case of homogeneous synthetic DNA having the sequence of identical bases, Englander and co–authors10 showed that the bubble movement can be modeled by the sine–Gordon equation with constant coefficients, this equation having the exact one–soliton solutions in the form of kinks. In the work of Englander and co–authors10 as well as in the works of other authors developing the Englander’s idea,11–20 just these solutions were used to simulate the DNA open states or bubbles. To take into account effects of dissipation and the action of some external field, they modified the sine–Gordon equation by adding the following two terms: one to model effect of dissipation and the other to model influence of an external field. With the help of the method of McLaughlin and Scott21 the kink–like solutions of the modified sine–Gordon equation were found in the two particular cases: when the external field is constant and when the external field is periodic.22,23

In the case of inhomogeneous DNA, the coefficients of the modified sine–Gordon equation are no longer constants, but depend on the sequence of bases (Table 1). However, in the quasi–homogeneous approximation, this equation can be reduced to the homogeneous one but with the changed coefficients the values of which depend on the concentrations of different types of bases. In this paper we apply the quasi–homogeneous approximation to solve the equation of that type and to calculate the main dynamic characteristics of the bubbles necessary to analyze the influence of constant and periodical fields on the bubble behavior.

For definiteness we take a sequence of small circular DNA molecule–plasmid pTTQ18 (Figure 2) which is widely used in genetic engineering to transfer genetic information and for genetic manipulations.24 The plasmid sequence contains four functionally important regions: promoter (54–333), terminator (611–8011), and two coding regions CDS–1 (1585–2544) and CDS–2 (2762–3622). The sequence length is 4563 bases. Of these, the sequence has 1105 adenines, 1090 thymines, 1193 guanines and 1175 cytosines. In the second section, we describe the model and methods used. In the further two sections the results on the velocity, coordinate, phase portrait, total energy and size of the bubbles moving under the action of the constant and periodic fields are presented. In the final section the obtained results are discussed and the main conclusions are made (Figure 2).

Figure 2 Schematic picture of plasmid pTTQ18. Four functional areas are shown: promoter (Pr), terminator (Term) and two coding regions (CDS-1, CDS-2).

Models and methods

Let us begin with the discrete version of the modified sine–Gordon equation:25

I n d 2 φ n (t) d t 2 + α n d φ n (t) dt K R n ( R n+1 φ n+1 (t)2 R n φ n (t)+ R n1 φ n1 (t))+ V n sin φ n (t)= = M 0 + M 1 cos(Ωt). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGjbWaaSbaaKqbGeaacaWGUbaajuaGbeaadaWcaaqaaiaadsgadaah aaqabKqbGeaacaaIYaaaaKqbakabeA8aQnaaBaaabaGaamOBaaqaba GaaiikaiaadshacaGGPaaabaGaamizaiaadshadaahaaqabKqbGeaa caaIYaaaaaaajuaGcqGHRaWkcqaHXoqydaWgaaqcfasaaiaad6gaaK qbagqaamaalaaabaGaamizaiabeA8aQnaaBaaajuaibaGaamOBaaqc fayabaGaaiikaiaadshacaGGPaaabaGaamizaiaadshaaaGaeyOeI0 Iaam4saiaadkfadaWgaaqcfasaaiaad6gaaKqbagqaaiaacIcacaWG sbWaaSbaaKqbGeaacaWGUbGaey4kaSIaaGymaaqcfayabaGaeqOXdO 2aaSbaaKqbGeaacaWGUbGaey4kaSIaaGymaaqcfayabaGaaiikaiaa dshacaGGPaGaeyOeI0IaaGOmaiaadkfadaWgaaqcfasaaiaad6gaaK qbagqaaiabeA8aQnaaBaaajuaibaGaamOBaaqcfayabaGaaiikaiaa dshacaGGPaGaey4kaSIaamOuamaaBaaajuaibaGaamOBaiabgkHiTi aaigdaaKqbagqaaiabeA8aQnaaBaaajuaibaGaamOBaiabgkHiTiaa igdaaKqbagqaaiaacIcacaWG0bGaaiykaiaacMcacqGHRaWkcaWGwb WaaSbaaKqbGeaacaWGUbaajuaGbeaaciGGZbGaaiyAaiaac6gacqaH gpGAdaWgaaqcfauaaiaad6gaaKqbagqaaiaacIcacaWG0bGaaiykai abg2da9aGcbaqcfaOaeyypa0JaamytamaaBaaajuaibaGaaGimaaqc fayabaGaey4kaSIaamytamaaBaaajuaibaGaaGymaaqcfayabaGaci 4yaiaac+gacaGGZbGaaiikaiabfM6axjaadshacaGGPaGaaiOlaaaa aa@95EC@    (1)

Here φ n (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOXdO 2aaSbaaKqbGeaacaWGUbaajuaGbeaacaGGOaGaamiDaiaacMcaaaa@3C63@  is the angular displacement of the n–th base, I n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamysam aaBaaajuaibaGaamOBaaqcfayabaaaaa@3922@  is the moment of inertia of the n–th base, K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4saa aa@3754@  is the stiffness of the sugar–phosphate chain, a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyaa aa@376A@ is the distance between adjacent base pairs, V n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvam aaBaaajuaibaGaamOBaaqcfayabaaaaa@392F@  is the factor characterizing the interaction between the complementary bases inside the n–th pair, n = 1, 2, ... N, N is the number of bases in the sequence, α n = R n 2 λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWGUbaajuaGbeaacqGH9aqpcaWGsbWaa0baaKqb GeaacaWGUbaabaGaaGOmaaaajuaGcqaH7oaBaaa@4011@ , λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@  is the dissipation factor, M 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38ED@  is the constant torque, M 1 cos(Ωt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGymaaqabaqcfaOaci4yaiaac+gacaGGZbGaaiik aiabfM6axjaadshacaGGPaaaaa@3FA1@ is a periodic field.

The values of the coefficients in the left side of the system of equations (1) are shown in Table 1. The values of the parameters of the external fields ( M 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38ED@ , M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaqbaGaaGymaaqabaaaaa@3880@  and Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuyQdC faaa@3812@  ) are yet arbitrary.

Type of the
n-th base

I n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamysam aaBaaajuaibaGaamOBaaqcfayabaaaaa@3922@
(10-44 kg×m2)

K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4saa aa@3754@
(J/m2)

R n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuam aaBaaajuaibaGaamOBaaqcfayabaaaaa@392B@
(10-10 m)

V n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvam aaBaaajuaibaGaamOBaaqcfayabaaaaa@392F@
(10-20 J)

a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyaa aa@376A@
(10-10 m)

α n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@39F3@
(J×с)

adenine

7.61

6.75

5.8

2.09

3.4

4.25

thymine

4.86

6.75

4.8

1.43

3.4

2.91

guanine

8.22

6.75

5.7

3.12

3.4

4.10

cytosine

4.11

6.75

4.7

2.12

3.4

2.79

Table 1 Coefficients of Eq. (1)25,26

Let us assume that the desired solutions of Eqs. (1) are sufficiently smooth functions. Then, we can apply the continuum approximation:

a0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyai abgkziUkaaicdaaaa@3A11@ , z n =naz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaamOBaaqcfayabaGaeyypa0JaamOBaiaadggacqGH sgIRcaWG6baaaa@3F1E@ ,

ϕ n (t)ϕ( z n ,t)ϕ(z,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqy1dy 2aaSbaaKqbGeaacaWGUbaajuaGbeaacaGGOaGaamiDaiaacMcacqGH HjIUcqaHvpGzcaGGOaGaamOEamaaBaaajuaibaGaamOBaaqabaqcfa OaaiilaiaadshacaGGPaGaeyOKH4Qaeqy1dyMaaiikaiaadQhacaGG SaGaamiDaiaacMcaaaa@4D86@ ,

I n I( z n )I(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamysam aaBaaajuaibaGaamOBaaqcfayabaGaeyyyIORaamysaiaacIcacaWG 6bWaaSbaaKqbGeaacaWGUbaajuaGbeaacaGGPaGaeyOKH4Qaamysai aacIcacaWG6bGaaiykaaaa@44F4@ , V n V( z n )V(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvam aaBaaajuaibaGaamOBaaqcfayabaGaeyyyIORaamOvaiaacIcacaWG 6bWaaSbaaKqbGeaacaWGUbaajuaGbeaacaGGPaGaeyOKH4QaamOvai aacIcacaWG6bGaaiykaaaa@451B@ ,

R n R( z n )R(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuam aaBaaajuaibaGaamOBaaqcfayabaGaeyyyIORaamOuaiaacIcacaWG 6bWaaSbaaKqbGeaacaWGUbaajuaGbeaacaGGPaGaeyOKH4QaamOuai aacIcacaWG6bGaaiykaaaa@450F@ , α n α( z n )α(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWGUbaajuaGbeaacqGHHjIUcqaHXoqycaGGOaGa amOEamaaBaaajuaibaGaamOBaaqcfayabaGaaiykaiabgkziUkabeg 7aHjaacIcacaWG6bGaaiykaaaa@4767@ .

As a result Eqs. (1) are transformed to:

I(z) d 2 φ(z,t) d t 2 +α(z) dφ(z,t) dt KR(z) a 2 d 2 [R(z)φ(z,t)] d z 2 +V(z)sinφ(z,t)= = M 0 + M 1 cos(Ωt). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGjbGaaiikaiaadQhacaGGPaWaaSaaaeaacaWGKbWaaWbaaeqajuai baGaaGOmaaaajuaGcqaHgpGAcaGGOaGaamOEaiaacYcacaWG0bGaai ykaaqaaiaadsgacaWG0bWaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOa ey4kaSIaeqySdeMaaiikaiaadQhacaGGPaWaaSaaaeaacaWGKbGaeq OXdOMaaiikaiaadQhacaGGSaGaamiDaiaacMcaaeaacaWGKbGaamiD aaaacqGHsislcaWGlbGaamOuaiaacIcacaWG6bGaaiykaiaadggada ahaaqabKqbGeaacaaIYaaaaKqbaoaalaaabaGaamizamaaCaaabeqc fasaaiaaikdaaaqcfaOaai4waiaadkfacaGGOaGaamOEaiaacMcacq aHgpGAcaGGOaGaamOEaiaacYcacaWG0bGaaiykaiaac2faaeaacaWG KbGaamOEamaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgUcaRiaadA facaGGOaGaamOEaiaacMcaciGGZbGaaiyAaiaac6gacqaHgpGAcaGG OaGaamOEaiaacYcacaWG0bGaaiykaiabg2da9aGcbaqcfaOaeyypa0 JaamytamaaBaaajuaibaGaaGimaaqcfayabaGaey4kaSIaamytamaa BaaajuaibaGaaGymaaqabaqcfaOaci4yaiaac+gacaGGZbGaaiikai abfM6axjaadshacaGGPaGaaiOlaaaaaa@8787@        (2)

To simplify further calculations, we use the quasi–homogeneous approximation according to which we can replace the coefficients of the left–hand side of Eq. (2) by the averaged values:27

I(z) I ¯ = I A N A N + I T N T N + I G N G N + I C N C N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamysai aacIcacaWG6bGaaiykaiabgkziUkqadMeagaqeaiabg2da9iaadMea daWgaaqcfasaaiaadgeaaKqbagqaamaalaaabaGaamOtamaaBaaaju aibaGaamyqaaqcfayabaaabaGaamOtaaaacqGHRaWkcaWGjbWaaSba aKqbGeaacaWGubaajuaGbeaadaWcaaqaaiaad6eadaWgaaqcfasaai aadsfaaKqbagqaaaqaaiaad6eaaaGaey4kaSIaamysamaaBaaajuai baGaam4raaqcfayabaWaaSaaaeaacaWGobWaaSbaaKqbGeaacaWGhb aajuaGbeaaaeaacaWGobaaaiabgUcaRiaadMeadaWgaaqcfasaaiaa doeaaKqbagqaamaalaaabaGaamOtamaaBaaajuaibaGaam4qaaqcfa yabaaabaGaamOtaaaaaaa@5787@ ,

R(z) R ¯ = R A N A N + R T N T N + R G N G N + R C N C N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai aacIcacaWG6bGaaiykaiabgkziUkqadkfagaqeaiabg2da9iaadkfa daWgaaqcfasaaiaadgeaaKqbagqaamaalaaabaGaamOtamaaBaaaju aibaGaamyqaaqcfayabaaabaGaamOtaaaacqGHRaWkcaWGsbWaaSba aKqbGeaacaWGubaajuaGbeaadaWcaaqaaiaad6eadaWgaaqcfasaai aadsfaaKqbagqaaaqaaiaad6eaaaGaey4kaSIaamOuamaaBaaajuai baGaam4raaqcfayabaWaaSaaaeaacaWGobWaaSbaaKqbGeaacaWGhb aajuaGbeaaaeaacaWGobaaaiabgUcaRiaadkfadaWgaaqcfasaaiaa doeaaKqbagqaamaalaaabaGaamOtamaaBaaajuaibaGaam4qaaqcfa yabaaabaGaamOtaaaaaaa@57BD@ , (3)

V(z) V ¯ = V A N A N + V T N T N + V G N G N + V C N C N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aacIcacaWG6bGaaiykaiabgkziUkqadAfagaqeaiabg2da9iaadAfa daWgaaqcfasaaiaadgeaaKqbagqaamaalaaabaGaamOtamaaBaaaju aibaGaamyqaaqcfayabaaabaGaamOtaaaacqGHRaWkcaWGwbWaaSba aKqbGeaacaWGubaajuaGbeaadaWcaaqaaiaad6eadaWgaaqcfasaai aadsfaaKqbagqaaaqaaiaad6eaaaGaey4kaSIaamOvamaaBaaajuai baGaam4raaqcfayabaWaaSaaaeaacaWGobWaaSbaaKqbGeaacaWGhb aajuaGbeaaaeaacaWGobaaaiabgUcaRiaadAfadaWgaaqcfasaaiaa doeaaKqbagqaamaalaaabaGaamOtamaaBaaajuaibaGaam4qaaqcfa yabaaabaGaamOtaaaaaaa@57D5@ ,

α(z) α ¯ = α A N A N + α T N T N + α G N G N + α C N C N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde MaaiikaiaadQhacaGGPaGaeyOKH4QafqySdeMbaebacqGH9aqpcqaH XoqydaWgaaqcKvaG=haacaWGbbaajuaGbeaadaWcaaqaaiaad6eada WgaaqcfasaaiaadgeaaKqbagqaaaqaaiaad6eaaaGaey4kaSIaeqyS de2aaSbaaeaacaWGubaabeaadaWcaaqaaiaad6eadaWgaaqcfasaai aadsfaaKqbagqaaaqaaiaad6eaaaGaey4kaSIaeqySde2aaSbaaKqb GeaacaWGhbaajuaGbeaadaWcaaqaaiaad6eadaWgaaqcfasaaiaadE eaaKqbagqaaaqaaiaad6eaaaGaey4kaSIaeqySde2aaSbaaKqbGeaa caWGdbaajuaGbeaadaWcaaqaaiaad6eadaWgaaqcfasaaiaadoeaaK qbagqaaaqaaiaad6eaaaaaaa@5D74@ ,

where N A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaajuaibaGaamyqaaqcfayabaaaaa@38FA@ is the number of adenines, N T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaajuaibaGaamivaaqcfayabaaaaa@390D@  is the number of thymines, N G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaajuaibaGaam4raaqcfayabaaaaa@3900@ is the number of guanines, N C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaajuaibaGaam4qaaqcfayabaaaaa@38FC@  is the number of cytosines, and N=( N A + N T + N G + N C ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtai abg2da9iaacIcacaWGobWaaSbaaKqbGeaacaWGbbaajuaGbeaacqGH RaWkcaWGobWaaSbaaKqbGeaacaWGubaajuaGbeaacqGHRaWkcaWGob WaaSbaaKqbGeaacaWGhbaajuaGbeaacqGHRaWkcaWGobWaaSbaaKqb GeaacaWGdbaajuaGbeaacaGGPaaaaa@464F@  is the total number of bases in the sequence.

After the averaging procedure, Eq. (2) takes the form similar to the sine–Gordon equation, but with the coefficients recalculated by formulas (3):

I ¯ 2 φ(z,t) t 2 + α ¯ φ(z,t) t K ¯ ' a 2 2 φ(z,t) z 2 + V ¯ sinφ(z,t)= M 0 + M 1 cos(Ωt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmysay aaraWaaSaaaeaacqGHciITdaahaaqabKqbGeaacaaIYaaaaKqbakab eA8aQjaacIcacaWG6bGaaiilaiaadshacaGGPaaabaGaeyOaIyRaam iDamaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgUcaRiqbeg7aHzaa raWaaSaaaeaacqGHciITcqaHgpGAcaGGOaGaamOEaiaacYcacaWG0b GaaiykaaqaaiabgkGi2kaadshaaaGaeyOeI0Iabm4sayaaraGaai4j aiaadggadaahaaqabeaacaaIYaaaamaalaaabaGaeyOaIy7aaWbaaK qbGeqabaGaaGOmaaaajuaGcqaHgpGAcaGGOaGaamOEaiaacYcacaWG 0bGaaiykaaqaaiabgkGi2kaadQhadaahaaqabKqbGeaacaaIYaaaaa aajuaGcqGHRaWkceWGwbGbaebaciGGZbGaaiyAaiaac6gacqaHgpGA caGGOaGaamOEaiaacYcacaWG0bGaaiykaiabg2da9iaad2eadaWgaa qcfasaaiaaicdaaKqbagqaaiabgUcaRiaad2eadaWgaaqcfasaaiaa igdaaeqaaKqbakGacogacaGGVbGaai4CaiaacIcacqqHPoWvcaWG0b Gaaiykaaaa@7947@ ,               (4)

where K ¯ '=K R ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4say aaraGaai4jaiabg2da9iaadUeaceWGsbGbaebadaahaaqcfasabeaa caaIYaaaaaaa@3BE8@ . Numerical values of the coefficients that are averaged over the entire sequence of plasmid pTTQ18 are given in Table 2.

Type of the sequence

I ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmysay aaraaaaa@376A@
(10-44 kg×m2)

K ¯ ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4say aaraGaai4jaaaa@3817@
(10-18 J)

V ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmOvay aaraaaaa@3777@
(10-20 J)

α ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqySde Mbaebaaaa@383B@
(10-34 J×c)

pTTQ18

6.21

1.88

2.21

3.51

Table 2 Coefficients of Eq. (4)

For convenience, let us introduce new (dimensionless) variables:

τ=σt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq Naeyypa0Jaeq4WdmNaamiDaaaa@3C0B@ , ζ=μz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOTdO Naeyypa0JaeqiVd0MaamOEaaaa@3BFC@ , where σ= ( V ¯ / I ¯ ) 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm Naeyypa0JaaiikaiqadAfagaqeaiaac+caceWGjbGbaebacaGGPaWa aWbaaeqajuaibaGaaGymaiaac+cacaaIYaaaaaaa@3FAC@ , μ= a 1 ( V ¯ / K ¯ ') 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyyamaaCaaabeqcfasaaiabgkHiTiaaigdaaaqcfaOa aiikaiqadAfagaqeaiaac+caceWGlbGbaebacaGGNaGaaiykamaaCa aabeqcfasaaiaaigdacaGGVaGaaGOmaaaaaaa@43B8@ . In these variables, Eq. (4) takes the form:

φ ττ +β φ τ φ ζζ +sinφ= f 0 + f 1 cos(ωτ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOXdO 2aaSbaaKqbGeaacqaHepaDcqaHepaDaKqbagqaaiabgUcaRiabek7a IjabeA8aQnaaBaaajuaibaGaeqiXdqhajuaGbeaacqGHsislcqaHgp GAdaWgaaqaaiabeA7a6jabeA7a6bqabaGaey4kaSIaci4CaiaacMga caGGUbGaeqOXdOMaeyypa0JaamOzamaaBaaajuaibaGaaGimaaqcfa yabaGaey4kaSIaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaci4y aiaac+gacaGGZbGaaiikaiabeM8a3jabes8a0jaacMcaaaa@5DF1@ ,          (5)

where β= α ¯ / ( I ¯ V ¯ ) 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JafqySdeMbaebacaGGVaGaaiikaiqadMeagaqeaiqadAfa gaqeaiaacMcadaahaaqabKqbGeaacaaIXaGaai4laiaaikdaaaaaaa@4141@ , f 0 = M 0 / V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaaqcfayabaGaeyypa0JaamytamaaBaaajqwb a+FaaiaaicdaaKqbagqaaiaac+caceWGwbGbaSaaaaa@3FD8@ , f 1 = M 1 / V ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqabaqcfaOaeyypa0JaamytamaaBaaajuai baGaaGymaaqabaqcfaOaai4laiqadAfagaqeaaaa@3E1D@ , ω=Ω ( I ¯ / V ¯ ) 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC Naeyypa0JaeuyQdCLaaiikaiqadMeagaqeaiaac+caceWGwbGbaeba caGGPaWaaWbaaeqajuaibaGaaGymaiaac+cacaaIYaaaaaaa@4144@ .

If the dimensionless coefficient of dissipation is small ( β<<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi MaeyipaWJaeyipaWJaaGymaaaa@3AE8@ ), Eq. (5) has approximate one–soliton solution in the form of kink:

φ=4arctan[exp(γ(ζυ(τ)τ ζ 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOXdO Maeyypa0JaaGinaiGacggacaGGYbGaai4yaiaacshacaGGHbGaaiOB aiaacUfaciGGLbGaaiiEaiaacchacaGGOaGaeq4SdCMaaiikaiabeA 7a6jabgkHiTiabew8a1jaacIcacqaHepaDcaGGPaGaeqiXdqNaeyOe I0IaeqOTdO3aaSbaaKqbGeaacaaIWaaajuaGbeaacaGGDbaaaa@54C7@      (6)

The velocity of which is defined by the equation of McLaughlin and Scott:21

dυ dτ =βυ(1 υ 2 )+ (1 υ 2 ) 3/2 + π 4 [ f 0 + f 1 cos(ωτ)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaeqyXduhabaGaamizaiabes8a0baacqGH9aqpcqGHsisl cqaHYoGycqaHfpqDcaGGOaGaaGymaiabgkHiTiabew8a1naaCaaabe qcfasaaiaaikdaaaqcfaOaaiykaiabgUcaRiaacIcacaaIXaGaeyOe I0IaeqyXdu3aaWbaaKqbGeqabaGaaGOmaaaajuaGcaGGPaWaaWbaae qajuaibaGaaG4maiaac+cacaaIYaaaaKqbakabgUcaRmaalaaabaGa eqiWdahabaGaaGinaaaacaGGBbGaamOzamaaBaaajuaibaGaaGimaa qcfayabaGaey4kaSIaamOzamaaBaaajuaibaGaaGymaaqabaqcfaOa ci4yaiaac+gacaGGZbGaaiikaiabeM8a3jabes8a0jaacMcacaGGDb aaaa@64CF@            (7)

Here γ=1/ [1 (υ(τ)) 2 ] 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0JaaGymaiaac+cacaGGBbGaaGymaiabgkHiTiaacIcacqaH fpqDcaGGOaGaeqiXdqNaaiykaiaacMcadaahaaqabKqbGeaacaaIYa aaaKqbakaac2fadaahaaqabKqbGeaacaaIXaGaai4laiaaikdaaaaa aa@4859@ , ζ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOTdO 3aaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39D8@  is an arbitrary constant.

Let us determine the coordinates of the kink x by formula:

dξ dτ =υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaeqOVdGhabaGaamizaiabes8a0baacqGH9aqpcqaHfpqD aaa@3EBB@ ,      (8)

and the total energy and the size of the kink by the formulas:

e(τ)= 8 1 υ 2 (τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzai aacIcacqaHepaDcaGGPaGaeyypa0ZaaSaaaeaacaaI4aaabaWaaOaa aeaacaaIXaGaeyOeI0IaeqyXdu3aaWbaaeqajuaibaGaaGOmaaaaju aGcaGGOaGaeqiXdqNaaiykaaqabaaaaaaa@449B@ , D(τ)= 1 1 υ 2 (τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacqaHepaDcaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaa aeGabaGwhiaaigdacqGHsislcqaHfpqDdaahaaqabKqbGeaacaaIYa aaaKqbakaacIcacqaHepaDcaGGPaaabeaaaaaaaa@44D7@ .    (9)

Differentiating (9) with respect to τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq haaa@3849@ , we get two additional differential equations:

de dτ = 8υ ( 1 υ 2 ) 3/2 dυ(τ) dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaamyzaaqaaiaadsgacqaHepaDaaGaeyypa0ZaaSaaaeaa caaI4aGaeqyXduhabaWaaeWaaeaacaaIXaGaeyOeI0IaeqyXdu3aaW baaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaadaahaaqabKqb GeaacaaIZaGaai4laiaaikdaaaaaaKqbaoaalaaabaGaamizaiabew 8a1jaacIcacqaHepaDcaGGPaaabaGaamizaiabes8a0baaaaa@50DC@ ,    (10)

dD dτ = υ ( 1 υ 2 ) 3/2 dυ(τ) dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaamiraaqaaiaadsgacqaHepaDaaGaeyypa0ZaaSaaaeaa cqaHfpqDaeaadaqadaqaaiaaigdacqGHsislcqaHfpqDdaahaaqabK qbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaamaaCaaabeqcfasaaiaa iodacaGGVaGaaGOmaaaaaaqcfa4aaSaaaeaacaWGKbGaeqyXduNaai ikaiabes8a0jaacMcaaeaacaWGKbGaeqiXdqhaaaaa@4FF9@ .            (11)

Eqs. (7), (8) and (10), (11) are a complete set of equations which are necessary to calculate the time dependencies of the velocity, coordinate, size and total energy of the transcription bubbles.

Results and discussion

Here we present the results of numerical calculations of the velocity, coordinate, size and total energy of the transcription bubbles. To better understand the dynamic behavior of the bubbles, we made for three different values of the external torsion moment and for three different values of the initial velocities of the bubbles.

Bubbles dynamics under the action of constant external field

This case corresponds to f 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaeimaaqcfayabaGaeyiyIKRaaGimaaaa@3B80@ , f 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaeymaaqcfayabaGaeyypa0JaaGimaaaa@3AC0@ . Dynamic characteristics of the bubbles, obtained for three different values of the torsion moment f 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIWaaabeaaaaa@37C7@ , are shown in Figure 3A & 3B. The bubbles trajectories in the phase plane ( υ,ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu Naaiilaiabe67a4baa@3ABE@ ) are shown in Figure 3C. The bubble total energy and size are shown in Figures 3D & 3E. The time required for calculation of the curves in Figures 3A &3B and of the curve 1 in Figure 3(C), is equal to 2.5×103. The time required for calculation of the curves 2 and 3 in Figure 3C is equal to 5×103 and 5×104, respectively (Figure 3).

The values f 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaaaa@39C1@ and f 03 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaaaa@39C3@  are chosen arbitrarily. The value f 02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaGimaiaaikdaaKqbagaaaaaaaa@39C3@ is assumed to be equal to f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaa8ajuaGca WGMbWaa0baaKqbGeaacaaIWaaabaGaam4yaiaadkhacaWGPbGaamiD aaaaaaa@3D05@  which in turn is determined from Eq. (7) at dυ dτ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaeqyXduhabaGaamizaiabes8a0baacqGH9aqpcaaIWaaa aa@3DB2@ :

f 0 crit = 4β υ 0 π 1 υ 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaaqc faOaeyypa0ZaaSaaaeaacaaI0aGaeqOSdiMaeqyXdu3aaSbaaKqbaf aacaaIWaaajuaGbeaaaeaacqaHapaCdaGcaaqaaiaaigdacqGHsisl cqaHfpqDdaqhaaqcfasaaiaaicdaaeaacaaIYaaaaaqcfayabaaaaa aa@4B50@          (12)

From Figure 3A it is seen that at f 02 = f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaikdaaKqbagqaaiabg2da9iaadAgadaqh aaqcfasaceaawbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaa aaaa@40DC@ (curve 2) the bubble velocity is constant and equal to the initial velocity υ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS baaSqaaiaaicdaaeqaaaaa@38A3@ . At f 03 < f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIWaGaaG4maaqabaGccqGH8aapcaWGMbWaa0baaSqaceaa wbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaaaaaa@3F83@  (curve 3), the bubble kink velocity initially decreases and after a time period (T), equal approximately 500 (0.84ns), reaches the stationary value:

υ 03 st = [ 1+ ( 4β π f 03 ) 2 ] 1/2 =0.010 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aa0baaKqbGeaacaaIWaGaaG4maaqaaiaadohacaWG0baaaKqbakab g2da9maadmaabaGaaGymaiabgUcaRmaabmaabaWaaSaaaeaacaaI0a GaeqOSdigabaGaeqiWdaNaamOzamaaBaaajuaibaGaaGimaiaaioda aKqbagqaaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaa qcfaOaay5waiaaw2faamaaCaaabeqcfasaaiabgkHiTiaaigdacaGG VaGaaGOmaaaajuaGcqGH9aqpcaaIWaGaaiOlaiaaicdacaaIXaGaaG imaaaa@544A@        (13)

At f 01 > f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg6da+iaadAgadaqh aaqcfasaceaawbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaa aaaa@40DD@ (curve 1), the kink velocity initially increases and after approximately the same time period T, reaches the stationary value:

υ 01 st = [ 1+ ( 4β π f 0 1 ) 2 ] 1/2 =0,207 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aa0baaKqbGeaacaaIWaGaaGymaaqaaiaadohacaWG0baaaKqbakab g2da9maadmaabaGaaGymaiabgUcaRmaabmaabaWaaSaaaeaacaaI0a GaeqOSdigabaGaeqiWdaNaamOzamaaBaaajuaibaGaaGimaaqabaqc fa4aaSbaaKqbGeaacaaIXaaabeaaaaaajuaGcaGLOaGaayzkaaWaaW baaeqajuaibaGaaGOmaaaaaKqbakaawUfacaGLDbaadaahaaqabKqb GeaacqGHsislcaaIXaGaai4laiaaikdaaaqcfaOaeyypa0JaaGimai aacYcacaaIYaGaaGimaiaaiEdaaaa@5529@        (14)

From Figure 3B it is seen that at f 02 = f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaikdaaKqbagqaaiabg2da9iaadAgadaqh aaqcfasaceaawbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaa aaaa@40DC@  (line2) the bubble coordinate is a completely straight line. At f 03 < f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaiabgYda8iaadAgadaqh aaqcfasaceaawbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaa aaaa@40DB@  (curve 3), the coordinate initially forms a small bend, and then after a certain period of time it transforms into a straight line. Similarly, at f 01 > f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWPbjuaGca WGMbWaaSbaaKqbGeaacaaIWaGaaGymaaqcfayabaGaeyOpa4JaamOz amaaDaaajuaibiqaayfacaaIWaaabaGaam4yaiaadkhacaWGPbGaam iDaaaaaaa@4182@  (curve 1), the coordinate initially forms a small bend (in opposing direction), and then it also transforms into a straight line.

Figure 3 (A) Velocity υ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu Naaiikaiabes8a0jaacMcaaaa@3B69@ , (B) coordinate ξ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOVdG Naaiikaiabes8a0jaacMcaaaa@3B65@ , (C) phase trajectory, (D) total energy e(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzai aacIcacqaHepaDcaGGPaaaaa@3A8C@ and (E) size D(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacqaHepaDcaGGPaaaaa@3A6B@ of the transcription bubble. The curves with number 1 correspond to . The lines with number 2 correspond to f 01 =2.429 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg2da9iaaikdacaGG UaGaaGinaiaaikdacaaI5aGaeyyXICTaaGymaiaaicdadaahaaqabK qbGeaacqGHsislcaaIZaaaaaaa@442B@ . The curves with number 3 correspond to f 02 =1.215 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaikdaaeqaaKqbakabg2da9iaaigdacaGG UaGaaGOmaiaaigdacaaI1aGaeyyXICTaaGymaiaaicdadaahaaqabK qbGeaacqGHsislcaaIZaaaaaaa@4424@ . Initial bubble velocity υ 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaaaaa@3BA2@ . Dissipation factor β=0.009 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGimaiaac6cacaaIWaGaaGimaiaaiMdaaaa@3CCE@ .

From Figures 3D & 3E it is seen that at f 02 = f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaikdaaKqbagqaaiabg2da9iaadAgadaqh aaqcfasaceaawbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaa aaaa@40DC@ (curve 2) the bubble energy and size are constants. At f 03 < f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaiabgYda8iaadAgadaqh aaqcfasaceaawbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaa aaaa@40DB@  (curves with number 3), the bubble energy before reaching the stationary value is reduced from the value e 01 =8.04 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzam aaBaaajuaibaGaaGimaiaaigdaaeqaaKqbakabg2da9iaaiIdacaGG UaGaaGimaiaaisdaaaa@3DB2@ , and the bubble size is reduced from D 01 =0.995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaaGimaiaaigdaaeqaaKqbakabg2da9iaaicdacaGG UaGaaGyoaiaaiMdacaaI1aaaaa@3E56@  till D st,1 =0.99 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4CaiaadshacaGGSaGaaGymaaqcfayabaGaeyyp a0JaaGimaiaac6cacaaI5aGaaGyoaaaa@3F7E@ . At f 01 > f 0 crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIWaGaaGymaaqabaGccqGH+aGpcaWGMbWaa0baaSqaceaa wbGaaGimaaqaaiaadogacaWGYbGaamyAaiaadshaaaaaaa@3F85@  (curves with number 1), the bubble energy before reaching the stationary value is increased from the value e 01 =8.04 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg2da9iaaiIdacaGG UaGaaGimaiaaisdaaaa@3DB2@  till the value e st,1 =8.16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzam aaBaaajuaibaGaam4CaiaadshacaGGSaGaaGymaaqcfayabaGaeyyp a0JaaGioaiaac6cacaaIXaGaaGOnaaaa@3F9C@ , and the bubble size is increased from D 01 =0.995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg2da9iaaicdacaGG UaGaaGyoaiaaiMdacaaI1aaaaa@3E56@  till D st,1 =1.01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4CaiaadshacaGGSaGaaGymaaqcfayabaGaeyyp a0JaaGymaiaac6cacaaIWaGaaGymaaaa@3F6E@ . Hence, by setting different values of constant external torsion field we can force the bubble to move with a certain fixed velocity. The opposite is true. If we know that bubble is moving at a given velocity, we can calculate the value of the torsion moment, which will ensure the movement of bubble at that velocity.

The above studies make it possible to estimate the value of the constant torsion moment f 03 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIWaGaaG4maaqabaaaaa@3884@ , necessary for the movement of the transcription bubble at the velocity equal to the velocity of the process of transcription. To do this, we replace υ 03 st MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aa0baaKqbGeaacaaIWaGaaG4maaqaaiaadohacaWG0baaaaaa@3C03@  in formula (13) with the transcription velocity υ tr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aa0baaeaaaKqbGeaacaWG0bGaamOCaaaaaaa@3A8B@ and replace the constant field f 03 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaaaa@39C3@  with the value f 0 tr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaGimaaqaaiaadshacaWGYbaaaaaa@3A69@ . As a result, we obtain the ratio between the transcription velocity and the sought value of the constant external field:

υ tr = [ 1+ ( 4β π f 0 tr ) 2 ] 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aa0baaeaaaKqbGeaacaWG0bGaamOCaaaajuaGcqGH9aqpdaWadaqa aiaaigdacqGHRaWkdaqadaqaamaalaaabaGaaGinaiabek7aIbqaai abec8aWjaadAgadaqhaaqcfasaaiaaicdaaeaacaWG0bGaamOCaaaa aaaajuaGcaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaaaKqbak aawUfacaGLDbaadaahaaqabKqbGeaacqGHsislcaaIXaGaai4laiaa ikdaaaaaaa@4ED7@     (15)

We rewrite the formula (15) in dimensional variables:

U tr = С 0 1+ ( 4 α ¯ π M 0 tr ) 2 V ¯ I ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaOaamyvam aaCaaabeqcfasaaiaadshacaWGYbaaaKqbakabg2da9maalaaabaGa amyiemaaBaaajuaibaGaaGimaaqcfayabaaabaWaaOaaaeaacaaIXa Gaey4kaSYaaeWaaeaadaWcaaqaaiaaisdacuaHXoqygaqeaaqaaiab ec8aWjaad2eadaqhaaqcfasaaiaaicdaaeaacaWG0bGaamOCaaaaaa aajuaGcaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaajuaGdaWc aaqaaiqadAfagaqeaaqaaiqadMeagaqeaaaaaeqaaaaaaaa@4CD6@        (16)

where U tr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamiDaiaadkhaaaaaaa@38EB@ is the dimensional transcription velocity, M 0 tr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaDaaajuaibaGaaGimaaqaaiaadshacaWGYbaaaaaa@3A4E@ is the unknown constant external field, С 0 = K ¯ ' a 2 / I ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaOaamyiem aaBaaajuaibaGaaGimaaqcfayabaGaeyypa0ZaaOaaaeaaceWGlbGb aebacaGGNaGaamyyamaaCaaabeqcfasaaiaaikdaaaqcfaOaai4lai qadMeagaqeaaqabaaaaa@3F85@  is the sound velocity in the plasmid pTTQ18. From (16) we find the sought torsion moment 0.49.10 31 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaaiOlaiaaisdacaaI5aGaaiOlaiaaigdacaaIWaWd amaaCaaabeqcfasaa8qacaGGtaIaaG4maiaaigdaaaaaaa@3E54@ :

M 0 tr = 4 α ¯ π V ¯ I ¯ U tr С 0 1 ( U tr С 0 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaDaaajuaibaGaaGimaaqaaiaadshacaWGYbaaaKqbakabg2da9maa laaabaGaaGinaiqbeg7aHzaaraaabaGaeqiWdahaamaakaaabaWaaS aaaeaaceWGwbGbaebaaeaaceWGjbGbaebaaaaabeaadaWcaaqaamaa laaabaGaamyvamaaCaaabeqcfasaaiaadshacaWGYbaaaaqcfayaai aadgcbdaWgaaqcfasaaiaaicdaaKqbagqaaaaaaeaadaGcaaqaaiaa igdacqGHsisldaqadaqaamaalaaabaGaamyvamaaCaaabeqcfasaai aadshacaWGYbaaaaqcfayaaiaadgcbdaWgaaqcfasaaiaaicdaaeqa aaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaaqcfa yabaaaaaaa@52FA@            (17)

If we assume that the velocity of transcription is equal to 100 base pairs per second U tr = 0.34.10 7 m/s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaOaamyvam aaCaaabeqcfasaaiaadshacaWGYbaaaKqbacbaaaaaaaaapeGaeyyp a0JaaGimaiaac6cacaaIZaGaaGinaiaac6cacaaIXaGaaGima8aada ahaaqabKqbGeaapeGaeyOeI0IaaG4naaaajuaGcaWGTbGaai4laiaa dohaaaa@45A6@ , then from formula (17) we find the desired estimate:   M 0 tr =  0.49.10 31 J. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGGcWdaiaad2eadaqhaaqcfasaaiaaicdaaeaacaWG0bGa amOCaaaajuaGpeGaeyypa0JaaeiiaiaaicdacaGGUaGaaGinaiaaiM dacaGGUaGaaGymaiaaicdapaWaaWbaaeqajuaibaWdbiaacobicaaI ZaGaaGymaaaajuaGcaWGkbGaaiOlaaaa@47A9@  Figure 4 shows the results of numerical calculations of the dynamic characteristics of the bubbles, made for three different values of the initial velocity υ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39E2@  (Figure 4).

Figure 4A shows that for any value of the initial bubble velocity υ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39E2@  the values of stationary velocity are the same. Figure 4B shows that the slopes of the bubbles trajectories reach the same value. Figure 4C & 4D shows that for different values of initial velocities the bubbles energy and size also reach the same values. Thus, we can conclude that the stationary dynamic characteristics of the bubbles do not depend on the initial velocities. On the contrary, they depend only on the value of the torsion moment.

Figure 4 (a) Velocity υ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDcaGGOaGaeqiXdqNaaiykaaaa@3B6A@ , (b) coordinate ξ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEcaGGOaGaeqiXdqNaaiykaaaa@3B66@ , (c) phase trajectory, (d) total energy e(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb Gaaiikaiabes8a0jaacMcaaaa@3A8D@ and (e) size D(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaaiikaiabes8a0jaacMcaaaa@3A6C@ of the transcription bubble. Initial velocities: υ 01 =0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaiaaigdaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaigdaaaa@3F8F@ , υ 02 =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaiaaikdaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaiwdaaaa@3F94@ , υ 03 =0.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaiaaiodaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaiIdaaaa@3F98@ . Torsion moment f 01 =2.429 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaaicdacaaIXaaaleqaaKqzGeGaeyyp a0JaaGOmaiaac6cacaaI0aGaaGOmaiaaiMdacqGHflY1caaIXaGaaG imaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaIZaaaaaaa@47B3@ . Dissipation factor β=0.009 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaicdacaaIWaGaaGyoaaaa@3CCF@ .

Bubbles dynamics under the action of periodic external field

This case corresponds to f 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaaqcfayabaGaeyypa0JaaGimaaaa@3AC6@ , f 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqabaqcfaOaeyiyIKRaaGimaaaa@3B88@ , ω0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC NaeyiyIKRaaGimaaaa@3AD2@ . It could be realized, for example, in an experiment with a single molecule at the work of atomic force microscope in an oscillating manner.28 Figure 5A–5C show the dynamic characteristics of the bubbles, obtained for three different values of the amplitude of the external periodic field f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqabaaaaa@3879@ (Figure 5).

Analyzing the graphs of coordinate and velocity presented in Figure 5A & 5B one can notice that at the beginning of the bubbles movement there exists a short period T~500 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivai aac6hacaaI1aGaaGimaiaaicdaaaa@3A92@  where oscillations are establishing. After the end of the period T the bubbles continues to oscillate with the constant frequency ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC haaa@3851@ . Figure 5D & 5E show the time dependence of the total energy and the size of the bubbles. It can be seen the doubling of the frequency of the stationary oscillations. Figure 6 shows the dynamic characteristics of the bubbles, obtained for three different values of the initial velocity: υ 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaGaaGymaaqcfayabaaaaa@3A9D@ , υ 02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaGaaGOmaaqcfayabaaaaa@3A9E@ , υ 03 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaGaaG4maaqcfayabaaaaa@3A9F@ (Figure 6).

Figure 5 (a) Velocity υ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDcaGGOaGaeqiXdqNaaiykaaaa@3B6A@ , (b) coordinate ξ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEcaGGOaGaeqiXdqNaaiykaaaa@3B66@ , (c) phase portrait, (d) total energy e(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb Gaaiikaiabes8a0jaacMcaaaa@3A8D@ and (e) size D(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaaiikaiabes8a0jaacMcaaaa@3A6C@ of the kink. The curves with number 1 correspond to f 11 =2.429 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaaigdacaaIXaaaleqaaKqzGeGaeyyp a0JaaGOmaiaac6cacaaI0aGaaGOmaiaaiMdacqGHflY1caaIXaGaaG imaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaIZaaaaaaa@47B4@ . The curves with number 2 correspond to f 12 =1.215 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaaigdacaaIYaaaleqaaKqzGeGaeyyp a0JaaGymaiaac6cacaaIYaGaaGymaiaaiwdacqGHflY1caaIXaGaaG imaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaIZaaaaaaa@47AD@ , the curves with number 3 correspond to f 13 =1.215 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaaigdacaaIZaaaleqaaKqzGeGaeyyp a0JaaGymaiaac6cacaaIYaGaaGymaiaaiwdacqGHflY1caaIXaGaaG imaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaI0aaaaaaa@47AF@ . Frequency ω=0.02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHjp WDcqGH9aqpcaaIWaGaaiOlaiaaicdacaaIYaaaaa@3C3A@ . Initial bubble velocity υ 0 =0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaaWcbeaajugibiabg2da9iaa icdacaGGUaGaaGymaaaa@3ED4@ .
Figure 6 (a) Velocity υ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDcaGGOaGaeqiXdqNaaiykaaaa@3B6A@ , (b) coordinate ξ(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEcaGGOaGaeqiXdqNaaiykaaaa@3B66@ , (c) phase portrait, (d) total energy e(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb Gaaiikaiabes8a0jaacMcaaaa@3A8D@ and (e) size D(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaaiikaiabes8a0jaacMcaaaa@3A6C@ of the bubble. Initial velocities: υ 01 =0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaiaaigdaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaigdaaaa@3F8F@ , υ 02 =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaiaaikdaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaiwdaaaa@3F94@ , υ 03 =0.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDjuaGdaWgaaqcbasaaKqzadGaaGimaiaaiodaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaiIdaaaa@3F98@ . Amplitude f 11 =2.429 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaaigdacaaIXaaaleqaaKqzGeGaeyyp a0JaaGOmaiaac6cacaaI0aGaaGOmaiaaiMdacqGHflY1caaIXaGaaG imaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaIZaaaaaaa@47B4@ . Frequency ω=0.02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHjp WDcqGH9aqpcaaIWaGaaiOlaiaaicdacaaIYaaaaa@3C3A@ . Dissipation factor β=0.009 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaicdacaaIWaGaaGyoaaaa@3CCF@ .

From Figure 6 it is seen that after some short period T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivaa aa@375D@  the velocity and coordinate of the bubbles having different initial velocities at the "start–up, begin to oscillate with constant amplitude and frequency. It can be seen also the doubling of the frequency of the oscillations of the bubbles energy and size.

Conclusion

Transcription bubbles have been modeled as quasi–particles – kinks, moving in the potential field of DNA. The influence of constant and periodic fields on the movement of the kinks was studied by the method McLaughlin and Scott. The time dependences of the kinks velocity, coordinate, total energy and size were obtained.

When calculating we used the following dimensionless values of the initial bubbles velocity: υ 01 =0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaGaaGymaaqcfayabaGaeyypa0JaaGimaiaa c6cacaaIXaaaaa@3DCA@ , υ 02 =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaGaaGOmaaqcfayabaGaeyypa0JaaGimaiaa c6cacaaI1aaaaa@3DCF@ , υ 03 =0.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaSbaaKqbGeaacaaIWaGaaG4maaqcfayabaGaeyypa0JaaGimaiaa c6cacaaI4aaaaa@3DD3@ . In the dimensional units these values correspond to: v 01 =187 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamODam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg2da9iaaigdacaaI 4aGaaG4naaaa@3D15@ m/s, v 02 =935 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamODam aaBaaajuaibaGaaGimaiaaikdaaKqbagqaaiabg2da9iaaiMdacaaI ZaGaaGynaaaa@3D17@  m/s, v 03 =1494 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamODam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaiabg2da9iaaigdacaaI 0aGaaGyoaiaaisdaaaa@3DD3@ m/s. To model the constant torsion moment, we used the dimensionless values: f 01 =2.429 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg2da9iaaikdacaGG UaGaaGinaiaaikdacaaI5aGaeyyXICTaaGymaiaaicdadaahaaqabK qbGeaacqGHsislcaaIZaaaaaaa@442B@ , f 02 =1.215 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaGimaiaaikdaaKqbagaaaaGaeyypa0JaaGymaiaa c6cacaaIYaGaaGymaiaaiwdacqGHflY1caaIXaGaaGimamaaCaaabe qcfasaaiabgkHiTiaaiodaaaaaaa@4425@ , f 03 =1.215 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaiabg2da9iaaigdacaGG UaGaaGOmaiaaigdacaaI1aGaeyyXICTaaGymaiaaicdadaahaaqcfa sabeaacqGHsislcaaI0aaaaaaa@4426@ , corresponding to the dimensional values: M 01 =5.37 10 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaiaaigdaaKqbagqaaiabg2da9iaaiwdacaGG UaGaaG4maiaaiEdacqGHflY1caaIXaGaaGimamaaCaaabeqcfasaai abgkHiTiaaikdacaaIZaaaaaaa@4412@  J, M 02 =2.68 10 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaiaaikdaaKqbagqaaiabg2da9iaaikdacaGG UaGaaGOnaiaaiIdacqGHflY1caaIXaGaaGimamaaCaaajuaibeqaai abgkHiTiaaikdacaaIZaaaaaaa@4414@  J, M 03 =2.68 10 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaiaaiodaaKqbagqaaiabg2da9iaaikdacaGG UaGaaGOnaiaaiIdacqGHflY1caaIXaGaaGimamaaCaaabeqcfasaai abgkHiTiaaikdacaaI0aaaaaaa@4416@ J. To simulate the amplitude and frequency of the periodic external field, we used dimensionless values: f 11 =2.429 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiabg2da9iaaikdacaGG UaGaaGinaiaaikdacaaI5aGaeyyXICTaaGymaiaaicdadaahaaqabK qbGeaacqGHsislcaaIZaaaaaaa@442C@ , f 12 =1.215 10 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaiaaikdaaKqbagqaaiabg2da9iaaigdacaGG UaGaaGOmaiaaigdacaaI1aGaeyyXICTaaGymaiaaicdadaahaaqabK qbafaacaGGSaaaaaaa@434B@ , f 13 =1.215 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaiaaiodaaKqbagqaaiabg2da9iaaigdacaGG UaGaaGOmaiaaigdacaaI1aGaeyyXICTaaGymaiaaicdadaahaaqabK qbGeaacqGHsislcaaI0aaaaaaa@4427@ , ω=0.02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC Naeyypa0JaaGimaiaac6cacaaIWaGaaGOmaaaa@3C39@ , corresponding to the dimensional values: M 11 =5.37 10 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiabg2da9iaaiwdacaGG UaGaaG4maiaaiEdacqGHflY1caaIXaGaaGimamaaCaaabeqcfasaai abgkHiTiaaikdacaaIZaaaaaaa@4413@ J, M 12 =0.268 10 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGymaiaaikdaaKqbagqaaiabg2da9iaaicdacaGG UaGaaGOmaiaaiAdacaaI4aGaeyyXICTaaGymaiaaicdadaahaaqabK qbGeaacqGHsislcaaIYaGaaG4maaaaaaa@44CF@ J, M 13 =2.68 10 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGymaiaaiodaaKqbagqaaiabg2da9iaaikdacaGG UaGaaGOnaiaaiIdacqGHflY1caaIXaGaaGimamaaCaaabeqcfasaai abgkHiTiaaikdacaaI0aaaaaaa@4417@ J, Ω 0 =0.119 10 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuyQdC 1aaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaGaaiOlaiaa igdacaaIXaGaaGyoaiabgwSixlaaigdacaaIWaWaaWbaaeqajuaiba GaaGymaiaaigdaaaaaaa@43D9@  s–1.

It was shown that in the case of constant torque, there is a small time period T~500 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivai aac6hacaaI1aGaaGimaiaaicdaaaa@3A92@  (0.84 ns), after which the bubble velocity reaches the stationary value. This stationary value does not depend on the initial bubble velocity and is completely determined by the value of constant torque. By setting this or that value of the constant torsion moment we can force the bubble to move at a predetermined velocity. We used this result to estimate theoretically the value of the DNA torque which can not be measured by currently existing experimental methods. We obtained that this value is equal to 0.49∙10–31 J if the velocity of transcription process is equal to 100 base pairs per second. Moreover we derived formula relating DNA torque and the velocity of transcription.

It was also shown that in the case of periodic field there is almost the same time period after which the bubble begins to oscillate with constant amplitude and frequency the value of which does not depend on the initial bubble velocity. This can lead to the destruction of the transcription bubble.

It is necessary to note, however, that all these results has been obtained in the frameworks of a rather simple model, which simulates angular oscillations of nitrous bases in one of the two polynucleotide chains, the other chain being modeled as an averaged external field. But we expect that further improvement of the model due to taking into account the transverse and longitudinal displacements of nitrous bases, the mobility of bases in the other polynucleotide chain, the helicity of the DNA structure will not change the main conclusions about the role of the constant and the periodic field in the bubble dynamics

Acknowledgements

The authors thank Dr. Dana Flavin for the valuable discussions which helped to broaden and deepen our understanding of the importance of the considered problems and its relationship to the fundamental problems of vital activity and human health.

Conflict of interest

Author declares that there is no conflict of interest.

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