Solitons in the Yakushevich model of DNA beyond the contact approximation

Ideas and methods of nonlinear mathematics and theoretical physics in DNA science: the McLaughlin-Scott equation and its application to study the DNA open state dynamics

The review is devoted to a new and rapidly developing area related to the application of ideas and methods of nonlinear mathematics and theoretical physics to study the internal dynamics of DNA and, in particular, the behavior of the open states of DNA. There are two main competing approaches to this research. The first approach is based on the molecular dynamics method, which takes into account the motions of all structural elements of the DNA molecule and all interactions between them. The second approach is based on prior selection of the main (dominant) motions and their mathematical description using a small number of model equations. This review describes the results of the study of the open states of DNA performed within the framework of the second approach using the McLaughlin-Scott equation. We present the results obtained both in the case of homogeneous sequences: poly (A), poly (T), poly (G) and poly (C), and in the inhomogeneous case when the McLaughlin-Scott equation has been used for studying the dynamics of open states activated in the promoters A1, A2 and A3 of the bacteriophage T7 genome, in the genes IFNA17, ADRB2, NOS1 and IL-5, in the pBR322 and pTTQ18 plasmids. Particular attention is paid to the results concerning the effect of various external fields on the behavior of open states. In the concluding part of the review, new possibilities and prospects for the development of the considered approach and especially of the McLaughlin-Scott equation are discussed.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s12551-021-00801-0.

A review on nonlinear DNA physics

The study and the investigation of structural and dynamical properties of complex systems have attracted considerable interest among scientists in general and physicists and biologists in particular. The present review paper represents a broad overview of the research performed over the nonlinear dynamics of DNA, devoted to some different aspects of DNA physics and including analytical, quantum and computational tools to understand nonlinear DNA physics. We review in detail the semi-discrete approximation within helicoidal Peyrard-Bishop model and show that localized modulated solitary waves, usually called breathers, can emerge and move along the DNA. Since living processes occur at submolecular level, we then discuss a quantum treatment to address the problem of how charge and energy are transported on DNA and how they may play an important role for the functioning of living cells. While this problem has attracted the attention of researchers for a long time, it is still poorly understood how charge and energy transport can occur at distances comparable to the size of macromolecules. Here, we review a theory based on the mechanism of 'self-trapping' of electrons due to their interaction with mechanical (thermal) oscillation of the DNA structure. We also describe recent computational models that have been developed to capture nonlinear mechanics of DNA in vitro and in vivo, possibly under topological constraints. Finally, we provide some conjectures on potential future directions for this field.

Stationary solitary and kink solutions in the helicoidal Peyrard-Bishop model of DNA molecule

We study nonlinear dynamics of the DNA molecule relying on a helicoidal Peyrard-Bishop model. We look for traveling wave solutions and show that a continuum approximation brings about kink solitons moving along the chain. This statement is supported by the numerical solution of a relevant dynamical equation of motion. Finally, we argue that an existence of both kinks and localized modulated solitons (breathers) could be a useful tool to describe DNA-RNA transcription.

Ensemble of DNA Kinks

The DNA open states, which are locally unwound regions of the double helix within which hydrogen bonds between complementary nitrous bases are broken, are often modeled as quasiparticles – DNA kinks. Most of the works on the DNA kinks are devoted to the studies of their dynamic properties, as well as their role in the functioning of the molecule. However, if not one but N open states are formed in the DNA molecule it is reasonable to consider the problem of the statistics of the ensemble of N DNA kinks. The statistical properties of such an ensemble are still poorly understood. In the present work, we study these properties applying new data on the dynamic characteristics of DNA kinks.

The influence of the DNA torque on the dynamics of transcription bubbles in plasmid PTTQ18

In this work, we study numerically the influence of the DNA torque on the movement of transcription bubbles in the potential field formed by the sequence of plasmid PTTQ18. To imitate the movement, we apply a modified sine-Gordon equation with the two additional terms that describe the effects of dissipation and the action of the DNA torsion torque, and with the coefficients that depend on the sequence of bases. We obtain the trajectories of the transcription bubbles and investigate the dependence of the trajectories on the initial bubble velocity and the DNA torsion torque. It is shown that not the initial bubble velocity but the DNA torsion torque governs the trajectories of the transcription bubbles.

On the mechanical analogue of DNA

The creation of mechanical analogues of biological systems is known as a useful instrument that helps to understand better the dynamical mechanisms of the functioning of living organisms. Mechanical analogues of biomolecules are usually constructed for imitation of their internal mobility, which is one of the most important properties of the molecules. Among the different types of internal motions, angular oscillations of nitrous bases are of special interest because they make a substantial contribution to the base pairs opening that in turn is an important element of the process of the DNA-protein recognition. In this paper, we investigate the possibility to construct a mechanical analogue for imitation of angular oscillations of nitrous bases in inhomogeneous DNA. It is shown that the analogue has the form of a mechanical chain of non-identical pendulums that oscillate in the gravitational field of the Earth and coupled by identical springs. The masses and lengths of pendulums, as well as the distances between neighboring pendulums and the rigidity of springs are calculated. To illustrate the approach, we present the result of construction of the mechanical analogue of the fragment of the sequence of bacteriophage T7D.

Internal nonlinear dynamics of a short lattice DNA model in terms of propagating kink-antikink solitons

We study the internal nonlinear dynamics of an inhomogeneous short lattice DNA model by solving numerically the governing discrete perturbed sine-Gordon equations under the limits of a uniform and a nonuniform angular rotation of bases. The internal dynamics is expressed in terms of open-state configurations represented by kink and antikink solitons with fluctuations. The inhomogeneity in the strands and hydrogen bonds as well as nonuniformity in the rotation of bases introduce fluctuations in the profile of the solitons without affecting their robust nature and the propagation. These fluctuations spread into the tail regions of the soliton in the case of periodic inhomogeneity. However, the localized form of inhomogeneity generates amplified fluctuations in the profile of the soliton. The fluctuations are expected to enhance the denaturation process in the DNA molecule.

Bubble solitons in an inhomogeneous, helical DNA molecular chain with flexible strands

Base pair opening in an inhomogeneous, DNA double helical molecular chain with flexible strands is investigated by studying its internal dynamics. For the study, a generalized model which takes into account the energies involved in stacking and hydrogen bonds along with inhomogeneity, helicity, and phonons coupled to the stacking and hydrogen bonds is proposed. The internal dynamics of the proposed DNA model is governed by a perturbed nonlinear Schrödinger equation. The unperturbed, completely integrable nonlinear Schrödinger equation which admits soliton solutions and forming a bubble corresponds to DNA dynamics with homogeneous and rigid strands. The results of the soliton perturbation analysis show that the inhomogeneity in stacking and hydrogen bonds in localized and periodic forms and the helicity do not alter the amplitude under perturbation. However, the flexibility of the strands diminishes the perturbed amplitude. On the other hand, the velocity of the soliton and bubble are unaltered due to all the above effects. However, the position and phase of the soliton and the bubble vary linearly in time. While the position of the soliton depends on the initial velocity, the phase depends on both the initial velocity and the initial amplitude of the soliton. The above effects introduce small fluctuation in the tail of the soliton, without affecting the robust nature of the soliton and the bubble during propagation. The soliton and the bubble obtained as solutions of the internal dynamics of the DNA molecule represent an opening of the base pairs which is essential for the transcription process.

Transverse interaction in DNA molecule

Interaction between nucleotides at a same site belonging to different strands is studied. This interaction is modelled by a Morse potential which depends on two parameters. We study a relationship between the parameters characterizing AT and CG pairs. We show that certain circumstances, i.e. certain values of these parameters, bring about a negligible influence of inhomogeneity on the solitonic dynamics.

DNA-RNA transcription as an impact of viscosity

The impact of viscosity on DNA dynamics is studied both analytically and numerically. It is assumed that the viscosity exists at the segments where DNA molecule is surrounded by RNA polymerase. We demonstrate that the frictional forces destroy the modulation of the incoming solitonic wave. We show that viscosity, crucial for demodulation, is essential for DNA-RNA transcription.

Discrete instability in the DNA double helix

Modulational instability (MI) is explored in the framework of the base-rotor model of DNA dynamics. We show, in fact, that the helicoidal coupling introduced in the spin model of DNA reduces the system to a modified discrete sine-Gordon (sG) equation. The MI criterion is thus modified and displays interesting features because of the helicoidal coupling. In the simulations, we have found that a train of pulses is generated when the lattice is subjected to MI, in agreement with analytical results obtained in a modified discrete sG equation. Also, the competitive effects of the harmonic longitudinal and helicoidal constants on the dynamics of the system are notably pointed out. In the same way, it is shown that MI can lead to energy localization which becomes high for some values of the helicoidal coupling constant.

Solitonlike base pair opening in a helicoidal DNA: an analogy with a helimagnet and a cholesteric liquid crystal

We propose a model for DNA dynamics by introducing the helical structure through twist deformation in analogy with the structure of a helimagnet and a cholesteric liquid-crystal system. The dynamics in this case is found to be governed by the completely integrable sine Gordon equation, which admits kink-antikink solitons with increased width, representing a wide base-pair opening configuration in DNA. The results show that the helicity introduces a length-scale variation and thus provides a better representation of the base-pair opening in DNA.

Composite model for DNA torsion dynamics

DNA torsion dynamics is essential in the transcription process; a simple model for it, in reasonable agreement with experimental observations, has been proposed by Yakushevich (Y) and developed by several authors; in this, the nucleotides (the DNA subunits made of a sugar-phosphate group and the attached nitrogen base) are described by a single degree of freedom. In this paper we propose and investigate, both analytically and numerically, a "composite" version of the Y model, in which the sugar-phosphate group and the base are described by separate degrees of freedom. The model proposed here contains as a particular case the Y model and shares with it many features and results, but represents an improvement from both the conceptual and the phenomenological point of view. It provides a more realistic description of DNA and possibly a justification for the use of models which consider the DNA chain as uniform. It shows that the existence of solitons is a generic feature of the underlying nonlinear dynamics and is to a large extent independent of the detailed modeling of DNA. The model we consider supports solitonic solutions, qualitatively and quantitatively very similar to the Y solitons, in a fully realistic range of all the physical parameters characterizing the DNA.