Solitary waves in twist-opening models of DNA dynamics

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.

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.

Nonlinear breathing modes at a defect site in DNA

Molecular-dynamics simulations of a normal DNA duplex show that breathing events typically occur on the microsecond time scale. This paper analyzes a 12 base pairs DNA duplex containing the "rogue" base difluorotoluene (F) in place of a thymine base (T), for which the breathing events occur on the nanosecond time scale. Starting from a nonlinear Klein-Gordon lattice model and adding noise and damping, we obtain a mesoscopic model of the DNA duplex close to that observed in experiments and all-atom molecular dynamics simulations. The mesoscopic model is calibrated to data from the all-atom molecular dynamics package AMBER for a variety of twist angles of the DNA duplex. Defects are considered in the interchain interactions as well as in the along-chain interactions. This paper also discusses the role of the fluctuation-dissipation relations in the derivation of reduced (mesoscopic) models, the differences between the potential of mean force and the potential energies used in Klein-Gordon lattices, and how breathing can be viewed as competition between the along-chain elastic energy and the interchain binding energy.