A fast method for solving time-dependent nonlinear convection diffusion problems

Qian He; Wenxin Du; Feng Shi; Jiaping Yu; Qian He; Wenxin Du; Feng Shi; Jiaping Yu

doi:10.3934/era.2022109

Electronic Research Archive

2022, Volume 30, Issue 6: 2165-2182. doi: 10.3934/era.2022109

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Research article Special Issues

A fast method for solving time-dependent nonlinear convection diffusion problems

1.
College of Science, Harbin Institute of Technology, Shenzhen 518055, China
2.
College of Science, Donghua University, Shanghai 201620, China

Academic Editor: Jingzhi Li

Received: 31 December 2021 Revised: 07 April 2022 Accepted: 12 April 2022 Published: 18 April 2022

In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.

Keywords:

nonlinear convection diffusion problems,
operator splitting,
multistep technique,
finite element method,
Burgers type equation,
Buckley-Leverett type equation

Citation: Qian He, Wenxin Du, Feng Shi, Jiaping Yu. A fast method for solving time-dependent nonlinear convection diffusion problems[J]. Electronic Research Archive, 2022, 30(6): 2165-2182. doi: 10.3934/era.2022109

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Abstract

1. Introduction

It is generally accepted that the transcription bubbles, which are small, locally untwisted regions (or distortions) of the double helix [1]–[4], are formed at the initial stage of the transcription and then move along the DNA molecule. On the other hand, the DNA molecule is considered by many researchers as a medium in which nonlinear conformational distortions (or solitary waves) can arise and propagate [5],[6]. This amazing property of DNA associates such a purely biological object as DNA with numerous nonlinear, mostly physical systems. Mathematically the movement of the nonlinear distortions are modeled by nonlinear differential equations, in particular, by the nonlinear sine-Gordon equation:

$φ_{τ τ} - φ_{ξ ξ} + \sin φ = 0,$ (1)

having, among others, one-soliton solutions (kinks) [7],[8]:

$φ_{k} (ξ, τ) = 4 arctg {exp [\cdot (ξ - υ_{k} \cdot τ - ξ_{0})]},$ (2)

where $υ_{k}$ is the kink velocity; $= {(1 - υ_{k}^{2})}^{- 1 / 2}$ and ξ₀ is an arbitrary constant.

Beginning with the work of Englander et al. [9], kinks have been actively used to model nonlinear conformational distortions in DNA, of which transcription bubbles are a special case. Currently, to model the dynamics of transcription bubbles, different modifications of the Englander model are used. Usually, they consist of a system of nonlinear differential equations with kink-like solutions. These equations contain many DNA dynamic parameters, such as the moments of inertia of nitrous bases, the distances between base pairs, the distances from the centers of mass of bases to sugar-phosphate chains, the rigidity of the sugar-phosphate backbone and the coefficients characterizing interactions between bases within pairs. Taken together, the equations and their parameters constitute the so-called dimensional model.

To reduce the number of parameters, it is much more convenient and efficient to use a dimensionless analog of a dimensional model. It is believed that the dimensionless analog not only makes it possible to noticeably reduce the number of parameters, but it also significantly simplifies the analysis of equations and methods for finding their solutions. In addition, the results obtained within the dimensionless model are valid not only for DNA, but also for other similar nonlinear media.

In this work, we have constructed a dimensionless analog of the nonlinear differential equations simulating the dynamics of transcription bubbles. We show that carrying out the dimensionless procedure really leads to a decrease in the number of model parameters, as well as to justification of the validity of applying the perturbation theory and the McLaughlin-Scott method [10],[11] based on it, which greatly facilitates finding solutions to model equations, their analysis and understanding the nature of the motion of transcription bubbles.

Methods for reducing model equations to a dimensionless form are varied [12]–[15]. To simplify calculations, we limited ourselves to the case of homogeneous (synthetic) DNA. In this case, the desired analog can be obtained using a simple transformation from the variables z and t to the new variables ξ and τ:

$ξ = σ z, τ = η t,$ (3)

which is accompanied by additional requirements for the transformation coefficients. We obtained dimensionless model equations, estimated their parameters, found one-soliton solutions (kinks) imitating transcription bubbles and justified the McLaughlin-Scott method, which made it possible to

calculate the dimensionless velocity of transcription bubbles and to plot their trajectories in the {ξ, τ} plane.

2. Dimensional DNA model and kink characteristics

Let us consider a homogeneous case when one of the two polynucleotide chains contains only one type of nitrous bases, such as adenines, and the second chain contains only thymines (Figure 1).

Figure 1. A schematic picture of the double stranded homogeneous DNA. Adenines are shown in red, thymines in blue, sugar-phosphate chains in black-white and hydrogen bonds as dotted lines.

DownLoad: Full-Size Img PowerPoint

In this case, more general model equations proposed in [16] to describe transcription bubbles dynamics take the following form:

$\begin{array}{l} I_{A} \frac{d^{2} φ_{n, A} (t)}{d t^{2}} - K_{A}^{'} [φ_{n + 1, A} (t) - 2 φ_{n,, A} (t) + φ_{n - 1, A} (t)] + \\ + k_{A - Т} R_{A} (R_{A} + R_{T}) \sin φ_{n, A} - k_{A - T} R_{A} R_{T} \sin (φ_{n, A} - φ_{n, T}) = \\ = - β_{A} \frac{d φ_{n, A} (t)}{d t} + M_{0}, \end{array}$ (4)

$\begin{array}{l} I_{T} \frac{d^{2} φ_{n, T} (t)}{d t^{2}} - K_{T}^{'} [φ_{n + 1, T} (t) - 2 φ_{n, T} (t) + φ_{n - 1, T} (t)] + \\ + k_{A - T} R_{T} (R_{A} + R_{T}) \sin φ_{n, T} - k_{A - T} R_{A} R_{T} \sin (φ_{n, T} - φ_{n, A}) = \\ = - β_{T} \frac{d φ_{n, T} (t)}{d t} + M_{0} . \end{array}$ (5)

Here $φ_{n, A} (t)$ and $φ_{n, T} (t)$ are the angular displacements of the n-th nitrous base in the poly(A) and poly(T) chains, respectively; I_A and I_T are the moments of inertia of the nitrous base in the poly(A) and poly(T) chains, respectively; R_A is the distance from the center of mass of the nitrous base in the poly(A) chain to the sugar-phosphate backbone; R_T is the distance from the center of mass of the nitrous base in the poly(T) chain to the sugar-phosphate backbone; $K_{A}^{'} = K R_{A}^{2}$ ; $K_{T}^{'} = K R_{T}^{2}$ ; K is the rigidity of the sugar-phosphate backbone; $β_{A} = α R_{A}^{2}; β_{T} = α R_{T}^{2}$ ; α is the dissipation coefficient; k_A_−T is a constant characterizing the interaction between bases within pairs; M₀ is a constant torsion moment.

Equations (4)–(5) take into account only one type of internal DNA motion, namely, angular displacements of the nitrous bases, which is believed to make the main contribution to the opening base pairs and formation of the transcription babbles. An alternative opinion developed in [17] is that the transverse displacements are more important. The model taking into account only transverse displacements is known as a BP model. In general, a more accurate model imitating the formation dynamics of the transcription bubbles should include both the transverse and angular displacements of the bases, as well as their longitudinal displacements [18].

Despite the limitations of the homogeneous case, equations (4)–(5) contain a fairly large number of dynamic parameters. However, they are only a part of the vast and complex space of parameters used in DNA melting and deformation models [19]–[21]. Therefore, the question of finding methods to reduce the number of parameters seems to be very relevant.

Restricting themselves to the continuum approximation and taking into account the features of the distribution of interactions within the DNA molecule results in the following: the presence of «weak» hydrogen bonds between nitrous bases inside complementary pairs and «strong» valence interactions along the sugar-phosphate chains; equations (4)–(5) can be reduced (in the first approximation) to two independent equations:

$I_{A} φ_{A, t t} - K_{A}^{'} a^{2} φ_{A, z z} + V_{A} \sin φ_{A} = - β_{A} φ_{A, t} + M_{0},$ (6)

$I_{T} φ_{T, t t} - K_{T}^{'} a^{2} φ_{T, z z} + V_{T} \sin φ_{T} = - β_{T} φ_{T, t} + M_{0} .$ (7)

Here $V_{A} = k_{A - T} R_{A}^{2}$ and $V_{T} = k_{A - T}$ . The first of these two equations describes the angular displacements of the nitrous bases in the poly(A) chain. The second is the angular displacements of the bases in the complementary chain poly(T). The parameters of equations (6)–(7) are presented in Table 1. The values of the parameters were collected in [22] and then refined in [23].

Table 1. Parameters of the dimensional DNA model.

Homogeneous sequence type	I×10⁻⁴⁴ (kg·m²)	K′×10⁻¹⁸ (N·m)	V×10⁻²⁰ (J)	β×10⁻³⁴ (J·s)	M₀×10⁻²² (J)
poly(A)	7.61	2.35	2.09	4.25	3.12
poly(T)	4.86	1.61	1.43	2.91	3.12

| Show Table

DownLoad: CSV

2.1. Case βA≅0, βT≅0 and M0≅0

In a particular case, when the effects of dissipation and the action of a constant torsion moment are small ( $β_{A} ≅ 0, β_{T} ≅ 0$ and $M_{0} ≅ 0$ ), equations (6)–(7) take the form of classical sine-Gordon equations, with coefficients depending on the DNA parameters:

$I_{A} φ_{A, t t} - K_{A}^{'} a^{2} φ_{A, z z} + V_{A} \sin φ_{A} = 0,$ (8)

$I_{T} φ_{T, t t} - K_{T}^{'} a^{2} φ_{T, z z} + V_{T} \sin φ_{T} = 0.$ (9)

Let us write Hamiltonians corresponding to equations (8)–(9):

$H_{A} = \int^{} (I_{A} \frac{φ_{A, t}^{2}}{2} + K_{A}^{'} a^{2} \frac{φ_{A, z}^{2}}{2} + V_{A} (1 - \cos φ_{A})) \frac{d z}{a},$ (10)

$H_{T} = \int^{} (I_{T} \frac{φ_{T, t}^{2}}{2} + K_{T}^{'} a^{2} \frac{φ_{T, z}^{2}}{2} + V_{T} (1 - \cos φ_{T})) \frac{d z}{a},$ (11)

and exact one-soliton solutions of equations (8)–(9) – kinks:

$φ_{k, A} (z, t) = 4 arctg {\exp [(_{A} / d_{A}) (z - υ_{k, A} t - z_{0, A})]},$ (12)

$φ_{k, T} (z, t) = 4 arctg {\exp [(_{T} / d_{T}) (z - υ_{k, T} t - z_{0, T})]} .$ (13)

Here, $υ_{k, A}$ and $υ_{k, T}$ are the kink velocities in the poly(A) and poly(T) chains, respectively; $γ_{A} = {(1 - υ_{k, A}^{2} / С_{A}^{2})}^{- 1 / 2}$ and $γ_{T} = {(1 - υ_{k, T}^{2} / С_{T}^{2})}^{- 1 / 2}$ are Lorentz factors; $C_{A} = {(K_{A}^{'} a^{2} / I_{A})}^{1 / 2}$ and $C_{T} = {(K_{T}^{'} a^{2} / I_{T})}^{1 / 2}$ are the sound velocities in the chains poly(A) and poly(T), respectively; $d_{A} = {(K_{A}^{'} a^{2} / V_{A})}^{1 / 2}$ and $d_{T} = {(K_{T}^{'} a^{2} / V_{T})}^{1 / 2}$ are the kink sizes; $z_{0, A}$ and $z_{0, T}$ are the kink coordinates at the initial moment of time.

The 3D graphs of the two DNA kinks (12)–(13) are presented in Figure 2. They have a canonical kink-like shape, which has been observed in a variety of media, including the mechanical chains of coupled pendulums [24], optical media [25], the chains of Josephson junctions [26]–[29], crystals [30], superfluid media [27], the Earth's crust [31], ferromagnetic and antiferromagnetic materials [32],[33] and biological molecules [34]–[36].

Figure 2. Two DNA kinks moving along the main (red) and complementary (blue) chains. The calculations were carried out with the help of equations (12)–(13) and parameters presented in Table 1.

DownLoad: Full-Size Img PowerPoint

Substituting equation (12) into equation (10) and equation (13) into equation (11), we find formulas for the kink total energies E_A and E_T:

$E_{A} = E_{0, A} \cdot γ_{A},$ (14)

$E_{T} = E_{0, T} \cdot γ_{T},$ (15)

where E_A and E_T are the the kink rest energies:

$E_{0, A} = 8 \sqrt{K_{A}^{'} V_{A}},$ (16)

$E_{0, T} = 8 \sqrt{K_{T}^{'} V_{T}} .$ (17)

In the case of low velocities ( $υ_{k, A} ≪ C_{A}$ , $υ_{k, T} ≪ C_{T}$ ), equations (14) and (15) are transformed into the following form:

$E_{A} ≅ E_{0, A} + \frac{m_{A} υ_{k, A}^{2}}{2},$ (18)

$E_{T} ≅ E_{0, T} + \frac{m_{T} υ_{k, T}^{2}}{2},$ (19)

where $m_{A} = \frac{E_{0, A}}{2 C_{A}^{2}}$ and $m_{T} = \frac{E_{0, T}}{2 C_{T}^{2}}$ are the masses of kinks propagating along homogeneous poly(A) and poly(Т) sequences, respectively. The form of equations (14) and (15) gives reason to consider kinks as quasi-particles having a certain mass, velocity and energy. Using the parameters from Table 1, we calculated the size, rest energy and mass of the kinks activated in the homogeneous poly(A) and poly(T) sequences. The results are presented in Table 2.

Table 2. Physical characteristics of the kinks in the dimensional DNA model.

Homogeneous sequence type	d (nm)	E₀×10⁻¹⁸ (J)	m×10⁻²⁵ (kg)
poly(A)	3.61	1.77	2.48
poly(T)	3.61	1.21	1.58

| Show Table

DownLoad: CSV

2.2. Case βA≠0, βT≠0 and M0≠0

For the general case, when $β_{A} \neq 0, β_{T} \neq 0$ and $M_{0} \neq 0$ , exact solutions of equations (6)–(7) have not yet been found. Approximate solutions of these equations, obtained by the method of McLaughlin and Scott [10],[11], have a form similar to equations (12)–(13):

$φ_{k, A} (z, t) = 4 arctg {\exp [(_{A} / d_{A}) (z - υ_{k, A} (t) t - z_{0, A})]},$ (20)

$φ_{k, T} (z, t) = 4 a r c t g {\exp [(_{T} / d_{T}) (z - υ_{k, T} (t) t - z_{0, T})]} .$ (21)

However, unlike equations (12)–(13), in equations (20)–(21), the kink velocities $υ_{k, A} (t)$ and $υ_{k, T} (t)$ are the functions of time, which are determined by the following equations:

$\frac{d υ'_{k, A} (t)}{d t} = - \frac{β_{A}}{I_{A}} υ'_{k, A} (t) (1 - υ'_{k, A}^{2} (t)) + \frac{М_{0} π}{4 \sqrt{I_{A} V_{A}}} {(1 - υ'_{k, A}^{2} (t))}^{3 / 2},$ (22)

$\frac{d υ'_{k, T} (t)}{d t} = - \frac{β_{T}}{I_{T}} υ'_{k, T} (t) (1 - υ'_{k, T}^{2} (t)) + \frac{М_{0} π}{4 \sqrt{I_{T} V I_{T}}} {(1 - υ'_{k, T}^{2} (t))}^{3 / 2} .$ (23)

Here, $υ'_{k, A} (t) = \frac{υ_{k, A} (t)}{C_{A}}$ ) and $υ'_{k, T} (t) = \frac{υ_{k, T} (t)}{C_{T}}$ are the relative kink velocities.

Having determined the kink coordinates from the relations $υ_{k, A} (t) = \frac{d z_{k, А}}{d t}$ and $υ_{k, T} (t) = \frac{d z_{k, Т}}{d t}$ , we have constructed the kinks trajectories on the plane {z, t} by using the method proposed in [37], where, however, only one of the two DNA polynucleotide chains was considered. Here, we extended this method to the model (4) – (5), which took into account two DNA polynucleotide chains.

3. Dimensionless DNA model and kink characteristics

Let us consider the transformation from variables z and t to new variables ξ and τ according to equation (3). To find the transformation coefficients σ and η, substitute equation (22) into equations (6) and (7), respectively:

$I_{A} η^{2} φ_{A, τ τ} - K_{A}^{'} a^{2} σ^{2} φ_{A, ξ ξ} + V_{A} \sin φ_{A} = - β_{A} η φ_{A, τ} + M_{0},$ (24)

$I_{T} η^{2} φ_{T, τ τ} - K_{T}^{'} a^{2} σ^{2} φ_{T, ξ ξ} + V_{T} \sin φ_{T} = - β_{T} η φ_{T, τ} + M_{0} .$ (25)

Next, let us multiply and divide equations (24) and (25) by $(I_{A} η^{2})$ :

$(I_{A} η^{2}) (φ_{A, τ τ} - \frac{K_{A} a^{2} σ^{2}}{I_{A} η^{2}} φ_{A, ξ ξ} + \frac{V_{A}}{I_{A} η^{2}} \sin φ_{A}) = (I_{A} η^{2}) (- \frac{β_{A} η}{(I_{A} η^{2})} φ_{A, τ} + \frac{M_{0}}{(I_{A} η^{2})}),$ (26)

$(I_{A} η^{2}) (\frac{I_{T}}{I_{A}} φ_{T, τ τ} - (\frac{K_{A} a^{2} σ^{2}}{I_{A} η^{2}}) \frac{K_{T}}{K_{A}} φ_{T, ξ ξ} + (\frac{V_{A}}{I_{A} η^{2}}) \frac{V_{T}}{V_{A}} \sin φ_{T}) = (I_{A} η^{2}) (- \frac{β_{A} η}{(I_{A} η^{2})} \frac{β_{T}}{β_{A}} φ_{T, τ} + \frac{M_{0}}{(I_{A} η^{2})}),$ (27)

and require the fulfillment of two conditions:

$\frac{K_{A}^{'} a^{2} σ^{2}}{I_{A} η^{2}} = 1, \frac{V_{A}}{I_{A} η^{2}} = 1,$ (28)

from which we find the transformation coefficients σ and η:

$η = \sqrt{\frac{V_{A}}{I_{A}}}, σ = \sqrt{\frac{V_{A}}{K_{1}^{'} α^{2}}} .$ (29)

In the new variables, equations (26) and (27) take the following form:

$φ_{A, τ τ} - φ_{A, ξ ξ} + \sin φ_{A} + {\tilde{β}}_{A} φ_{A, τ} - {\tilde{M}}_{0, A} = 0,$ (30)

$i_{A T} φ_{T, τ τ} - k_{A T} φ_{T, ξ ξ} + v_{A T} \sin φ_{T} + {\tilde{β}}_{T} φ_{T, τ} - {\tilde{M}}_{0, A} = 0,$ (31)

where ${\tilde{β}}_{A} = \frac{β_{A}}{I_{A} η} = \frac{β_{A}}{\sqrt{I_{A} V_{A}}}$ , ${\tilde{β}}_{T} = \frac{β_{T}}{I_{A} η} = \frac{β_{T}}{\sqrt{I_{A} V_{A}}}$ , ${\tilde{M}}_{0, A} = \frac{M_{0}}{I_{A} η^{2}} = \frac{M_{0}}{V_{A}}$ . $i_{A T} = \frac{I_{T}}{I_{A}}$ , $k_{A T} = \frac{K_{T}}{K_{A}}$ , $v_{A T} = \frac{V_{T}}{V_{A}} .$

It can be seen that the number of parameters in the first of the two dimensionless model equations has decreased from five to two. In the second equation, the number of parameters has not changed. Taking into account the fact that the torsion moment in both equations is the same, the total number of parameters became equal to six, which is less than the nine parameters of the dimensional model (6)–(7).

We estimated the values of the coefficients of dimensionless model equations (30)–(31) and presented them in Table 3.

Table 3. Values of the coefficients of dimensionless model equations (30)–(31).

Coefficients	Estimated values
$i_{A T}$	0.64
$k_{A T}$	0.68
$v_{A T}$	0.68
${\tilde{β}}_{A}$	0.01
${\tilde{β}}_{T}$	0.007
${\tilde{M}}_{0, A}$	0.01

| Show Table

DownLoad: CSV

Note that, in the case of dimensional equations (4)–(5), it was difficult to judge which coefficients were large and which were small. In the dimensionless case, the difference became obvious. The dimensionless coefficients in equation (29), ${\tilde{β}}_{A}$ and ${\tilde{M}}_{0, A}$ , are much less than unity. And, in equation (30), the coefficients ${\tilde{β}}_{T}$ and ${\tilde{M}}_{0, T}$ are less than the coefficients $i_{A T}$ , $k_{A T}$ and $v_{A T}$ . And, this indicates that the effects of dissipation and the impact of the torsion moment are really small. This means that the use of perturbation theory in deriving the McLaughlin-Scott equation is quite reasonable.

3.1. Case β ˜A=0, β ˜T=0 and M ˜0,A=0

In the particular case when the effects of dissipation and external action are negligibly small equation (30) takes the form of the classical canonical sine-Gordon equation (1), which has no coefficients:

$φ_{A, τ τ} - φ_{A, ξ ξ} + \sin φ_{A} = 0.$ (32)

At the same time equation (31) retains all three coefficients:

$i_{A T} φ_{T, τ τ} - k_{A T} φ_{T, ξ ξ} + v_{A T} \sin φ_{T} = 0.$ (33)

Hamiltonians corresponding to equations (32) and (33) have the following form:

${\tilde{H}}_{A} = \int^{} (\frac{φ_{A, τ}^{2}}{2} + \frac{φ_{A, ξ}^{2}}{2} + (1 - \cos φ_{A})) d ξ,$ (34)

${\tilde{H}}_{T} = \int (i_{A T} \frac{φ_{T, τ}^{2}}{2} + k \frac{φ_{T, ξ}^{2}}{2} + v_{A T} (1 - \cos φ_{T})) d ξ .$ (35)

To obtain one-soliton solutions of equations (32), (33), we take dimensional solutions (11) and (12) and rewrite them with new variables:

${\tilde{φ}}_{k, A} (ξ, τ) = 4 arctg {exp [(γ_{A}) (ξ - \frac{υ_{k, A}}{C_{A}} \cdot τ - ξ_{0, A})]},$ (36)

${\tilde{φ}}_{k, T} (ξ, τ) = 4 arctg {exp [(γ_{T}) \frac{d_{A}}{d_{T}} (ξ - \frac{υ_{k, T}}{C_{A}} \cdot τ - ξ_{0, T})]} .$ (37)

Substituting the solutions into the Hamiltonians (34), (35) we find the total kink energies:

${\tilde{E}}_{A} = {\tilde{E}}_{0 A} γ_{A},$ (38)

${\tilde{E}}_{T} = {\tilde{E}}_{0 T} γ_{T},$ (39)

where ${\tilde{E}}_{0 A}, {\tilde{E}}_{0 T}$ are the kink rest energies:

${\tilde{E}}_{0 A} = 8,$ (40)

${\tilde{E}}_{0 T} = 8 \sqrt{k_{A T} v_{A T}} .$ (41)

3.2. Case β ˜A≠0, β ˜T≠0 and M ˜0,A≠0

In the general case, when ${\tilde{β}}_{A} \neq 0$ , ${\tilde{β}}_{T} \neq 0$ and ${\tilde{M}}_{0, A} \neq 0$ the solutions of equations (30) and (31) have the following form:

${\tilde{φ}}_{k, A} (ξ, τ) = 4 arctg {e x p [(γ_{A}) (ξ - {\tilde{υ}}_{k, A} (τ) \cdot τ - ξ_{0, A})]},$ (42)

${\tilde{φ}}_{k, T} (ξ, τ) = 4 arctg {exp [(γ_{T}) \frac{d_{A}}{d_{T}} (ξ - {\tilde{υ}}_{k, T} (τ) \cdot τ - ξ_{0, T})]},$ (43)

where ${\tilde{υ}}_{k, A} = \frac{υ_{k, A} (τ)}{C_{A}}$ and ${\tilde{υ}}_{k, T} = \frac{υ_{k, T} (τ)}{C_{A}}$ are dimensionless kink velocities which are respectively determined by the following equations:

$\frac{d}{d τ} {\tilde{υ}}_{k, A} = - {\tilde{β}}_{A} {\tilde{υ}}_{k, A} (1 - {\tilde{υ}}_{k, A}^{2}) + \frac{M_{0} π}{4 V_{A}} {(1 - {\tilde{υ}}_{k, A}^{2})}^{3 / 2},$ (44)

$\frac{d}{d τ} {\tilde{υ}}_{k, T} = - {\tilde{β}}_{T} ({\tilde{υ}}_{k, T}) (1 - {\tilde{υ}}_{k, T}^{2} {(\frac{C_{A}}{C_{T}})}^{2}) + \frac{M_{0} π}{4 V_{A}} \frac{1}{\sqrt{i_{A T} v_{A T}}} (\frac{C_{T}}{C_{A}}) {(1 - {\tilde{υ}}_{k, T}^{2} {(\frac{C_{A}}{C_{T}})}^{2})}^{3 / 2} .$ (45)

Equations (44) and (45) were solved numerically using the parameter values from Table 1. Figure 3a and 3b present the results obtained for the dimensionless kink velocities ( ${\tilde{υ}}_{k, A}, {\tilde{υ}}_{k, T}$ ) and coordinates ( ${\tilde{ξ}}_{k, A}, {\tilde{ξ}}_{k, T}$ ). The latter were determined by the following relations:

${\tilde{υ}}_{k, A} = \frac{d}{d τ} {\tilde{ξ}}_{k, A}, {\tilde{υ}}_{k, T} = \frac{d}{d τ} {\tilde{ξ}}_{k, T} .$ (46)

Figure 3. Time dependence of the kink dimensionless velocities (a) and coordinates (b). Red curves refer to the kinks propagating in the poly(A) chain and blue curves refer to the kinks propagating in the poly(T) chain. The calculations were carried out with the help of equations (44)–(46) and the parameters presented in Table 3. The initial kink velocities were suggested to be zero.

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Figure 3 shows that as, the time τ increases, the kink velocities and coordinates increase monotonically. As τ→∞, the velocities tend to the stationary values 0.68 ${\tilde{υ}}_{s t, A} =$ and ${\tilde{υ}}_{s t, T} =$ 0.85, and the coordinates tend to infinity.

4. Discussion and conclusions

In this work, we have constructed the dimensionless analog of the model simulating the dynamics of transcription bubbles and demonstrated the method of constructing in detail. As a basic dimensional model, we used a system of nonlinear differential equations proposed in [16], the one-soliton solutions of which (kinks) were interpreted as mathematical images of transcription bubbles. The main results, including equations of motion, kink-like solutions, kink rest energy, and total kink energy obtained for the dimensionless model, are presented in Table 4.

Table 4. Basic formulas calculated for dimensionless model of homogeneous DNA.

Parameters	Model characteristics	Kink movement in the poly(A) chain	Kink movement in the poly(T) chain
${\tilde{β}}_{A} = 0$ , ${\tilde{β}}_{T} = 0$ , ${\tilde{M}}_{0, A} = 0$	Equations of motion	$φ_{A, τ τ} - φ_{A, ξ ξ} + \sin φ_{A} = 0$	$i_{A T} φ_{T, τ τ} - k_{A T} φ_{T, ξ ξ} + v_{A T} \sin φ_{T} = 0$
	Hamiltonian	${\tilde{H}}_{A} = \int^{} (\frac{φ_{A, τ}^{2}}{2} + \frac{φ_{A, ξ}^{2}}{2} + (1 - \cos φ_{A})) d ξ,$	${\tilde{H}}_{T} = \int^{} (i_{A T} \frac{φ_{T, τ}^{2}}{2} + k \frac{φ_{T, ξ}^{2}}{2} + v_{A T} (1 - \cos φ_{T})) d ξ$
	Kink-like solution	${\tilde{φ}}_{k, A} (ξ, τ) = 4 arctg {exp [γ_{A} (ξ - {\tilde{υ}}_{k, A} (τ) \cdot τ - ξ_{0, A})]}$	${\tilde{φ}}_{k, T} (ξ, τ) = 4 arctg {exp [γ_{T} (\frac{d_{A}}{d_{T}}) (ξ - {\tilde{υ}}_{k, T} (τ) \cdot τ - ξ_{0, T}]}$
	Total energy	${\tilde{E}}_{A} = 8 γ_{A}$	${\tilde{E}}_{T} = {\tilde{E}}_{0 T} γ_{T}$
	Rest energy	${\tilde{E}}_{0 A} = 8$	${\tilde{E}}_{0 T} = 8 \sqrt{k_{A T} v_{A T}}$

${\tilde{β}}_{A} \neq 0,$ ${\tilde{β}}_{T} \neq 0,$ ${\tilde{M}}_{0, A} \neq 0$	Equations of motion	$φ_{A, τ τ} - φ_{A, ξ ξ} + \sin φ_{A} = - {\tilde{β}}_{A} φ_{A, τ} + {\tilde{M}}_{0, A}$	$i_{A T} φ_{T, τ τ} - k_{A T} φ_{T, ξ ξ} + v_{A T} \sin φ_{T} = - {\tilde{β}}_{T} φ_{T, τ} + {\tilde{M}}_{0, T}$
	Kink-like solution	${\tilde{φ}}_{k, A} (ξ, τ) = 4 arctg {exp [γ_{A} (ξ - {\tilde{υ}}_{k, A} \cdot τ - ξ_{0, A})]}$	${\tilde{φ}}_{k, T} (ξ, τ) = 4 arctg {exp [γ_{T} (\frac{d_{A}}{d_{T}}) (ξ - {\tilde{υ}}_{k, T}) \cdot τ - ξ_{0, T}]}$
	Equation for kink velocity	$\frac{d {\tilde{υ}}_{k, A}}{d τ} = - {\tilde{β}}_{A} {\tilde{υ}}_{k, A} (1 - {\tilde{υ}}_{k, A}^{2})$ $+ \frac{M_{0} π}{4 V_{A}} {(1 - {\tilde{υ}}_{k, A}^{2})}^{3 / 2}$	$\frac{d {\tilde{υ}}_{k, T}}{d τ}$ $= - {\tilde{β}}_{T} ({\tilde{υ}}_{k, T}) (1 - {\tilde{υ}}_{k, T}^{2} {(\frac{C_{A}}{C_{T}})}^{2})$ $+ \frac{M_{0} π}{4 V_{A}} \frac{1}{\sqrt{i_{A T} v_{A T}}} (\frac{C_{T}}{C_{A}})$ $(1 - {\tilde{υ}}_{k, T}^{2} {(\frac{C_{A}}{C_{T}})}^{2})$

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For comparison, we present in Table 5, similar results obtained for the corresponding dimensional model.

Table 5. Basic formulas calculated for dimensional model of homogeneous DNA.

Parameters	Model characteristics	Kink movement in the poly(A) chain	Kink movement in the poly(T) chain
$β_{A} = 0$ , $β_{T} = 0$ , $M_{0} = 0$	Equations of motion	$I_{A} φ_{A, t t} - K_{A}^{'} a^{2} φ_{A, z z}$ $+ V_{A} sin φ_{A} = 0$	$I_{T} φ_{T, t t} - K_{T}^{'} a^{2} φ_{T, z z}$ $+ V_{T} sin φ_{T} = 0$
	Hamiltonian	$H_{A} = \int^{} (I_{A} \frac{φ_{A, t}^{2}}{2} + K_{A}^{'} a^{2} \frac{φ_{A, t}^{2}}{2}$ $+ V_{A} (1 - cos φ_{A})) \frac{d z}{a}$	$H_{T} = \int^{} (I_{T} \frac{φ_{T, t}^{2}}{2} + K_{T}^{'} a^{2} \frac{φ_{T, t}^{2}}{2}$ $+ V_{T} (1 - cos φ_{T})) \frac{d z}{a}$
	Kink-like solution	$φ_{k, A} (z, t)$ $= 4 arctg {exp [(γ_{A}' d_{A}) (z -$ $- υ_{k, A} \cdot t - z_{0. A})]}$	$φ_{k, T} (z, t)$ $= 4 arctg {exp [(γ_{T}' d_{T}) (z -$ $- υ_{k, T} \cdot t - z_{0. T})]}$
	Total energy	$E_{A} = E_{0, A} \cdot_{A}$	$E_{T} = E_{0, T} \cdot_{T}$
	Rest energy	$E_{0, A} = 8 \sqrt{K_{A}^{'} V_{A}}$	$E_{0, T} = 8 \sqrt{K_{T}^{'} V_{T}}$

$β_{A} \neq 0$ , $β_{T} \neq 0$ , $M_{0} \neq 0$	Equations of motion	$I_{A} φ_{A, t t} - K_{A}^{'} a^{2} φ_{A, z z}$ $+ V_{A} sin φ_{A} = - β_{A} φ_{A, t} + M_{0}$	$I_{T} φ_{T, t t} - K_{T}^{'} a^{2} φ_{T, z z}$ $+ V_{T} sin φ_{T} = - β_{T} φ_{T, t} + M_{0}$
	Kink-like solution	$φ_{k, A} (z, t)$ $= 4 arctg {exp [(γ_{A}' d_{A}) (z -$ $- υ_{k, A} (t) \cdot t - z_{0. A})]}$	$φ_{k, T} (z, t)$ $= 4 arctg {exp [(γ_{T}' d_{T}) (z -$ $- υ_{k, T} (t) \cdot t - z_{0. T})]}$
	Equation for kink velocity	$\frac{d υ_{k, A}^{'} (t)}{d t} = - \frac{β_{A}}{I_{A}} υ_{k, A}^{'} (t) (1 - υ_{k, A}^{' 2} (t)) + \frac{M_{0} π}{4 \sqrt{I_{A} V_{A}}} {(1 - υ_{k, A}^{' 2} (t))}^{3 / 2}$	$\frac{d υ_{k, T}^{'} (t)}{d t} = - \frac{β_{T}}{I_{T}} υ_{k, T}^{'} (t) (1 - υ_{k, T}^{' 2} (t)) + \frac{M_{0} π}{4 \sqrt{I_{T} V_{T}}} {(1 - υ_{k, T}^{' 2} (t))}^{3 / 2}$

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It can be seen that the dimensionless model has advantages over the dimensional one. Indeed, carrying out the procedure of transformation of the dimensional model to the dimensionless one leads to the decrease in the number of model parameters from 9 to 6. Moreover, in the framework of the dimensionless analog, it became obvious that the coefficients in the terms simulating the dissipation effects and the action of a constant torsion moment are really small, which proves the validity of the application of the perturbation theory and the method of McLaughlin and Scott.

It should be noted, however, that in order to simplify calculations and to present the procedure of transition to dimensionless form more clear, we limited ourselves to the case of homogeneous (synthetic) DNA. Obviously, the next step in the development of this direction can be the construction of a dimensionless analog for the case of inhomogeneous DNA. This will provide an answer to the question of whether the advantages of the dimensionless model will be preserved in the inhomogeneous case.

One more step could be the improvement if the dimensional model itself by removing the limitations and simplifications detailed in Section 2. Finally, one could consider the issue of DNA kinks being created by mutual DNA-DNA interactions, which were considered theoretically in [38],[39], and experimentally in [40].

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