Composite model for DNA torsion 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.

An effective mesoscopic model of double-stranded DNA

Watson and Crick's epochal presentation of the double helix structure in 1953 has paved the way to intense exploration of DNA's vital functions in cells. Also, recent advances of single molecule techniques have made it possible to probe structures and mechanics of constrained DNA at length scales ranging from nanometers to microns. There have been a number of atomistic scale quantum chemical calculations or molecular level simulations, but they are too computationally demanding or analytically unfeasible to describe the DNA conformation and mechanics at mesoscopic levels. At micron scales, on the other hand, the wormlike chain model has been very instrumental in describing analytically the DNA mechanics but lacks certain molecular details that are essential in describing the hybridization, nano-scale confinement, and local denaturation. To fill this fundamental gap, we present a workable and predictive mesoscopic model of double-stranded DNA where the nucleotides beads constitute the basic degrees of freedom. With the inter-strand stacking given by an interaction between diagonally opposed monomers, the model explains with analytical simplicity the helix formation and produces a generalized wormlike chain model with the concomitant large bending modulus given in terms of the helical structure and stiffness. It also explains how the helical conformation undergoes overstretch transition to the ladder-like conformation at a force plateau, in agreement with the experiment.

Anharmonic stacking in supercoiled DNA

Multistep denaturation in a short circular DNA molecule is analyzed by a mesoscopic Hamiltonian model which accounts for the helicoidal geometry. Computation of melting profiles by the path integral method suggests that stacking anharmonicity stabilizes the double helix against thermal disruption of the hydrogen bonds. Twisting is essential in the model to capture the importance of nonlinear effects on the thermodynamical properties. In a ladder model with zero twist, anharmonic stacking scarcely affects the thermodynamics. Moderately untwisted helices, with respect to the equilibrium conformation, show an energetic advantage against the overtwisted ones. Accordingly moderately untwisted helices better sustain local fluctuational openings and make more unlikely the thermally driven complete strand separation.

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.

Heat conductivity of DNA double helix

Thermal conductivity of isolated single molecule DNA fragments is of importance for nanotechnology, but has not yet been measured experimentally. Theoretical estimates based on simplified (1D) models predict anomalously high thermal conductivity. To investigate thermal properties of single molecule DNA we have developed a 3D coarse-grained (CG) model that retains the realism of the full all-atom description, but is significantly more efficient. Within the proposed model each nucleotide is represented by 6 particles or grains; the grains interact via effective potentials inferred from classical molecular dynamics (MD) trajectories based on a well-established all-atom potential function. Comparisons of 10 ns long MD trajectories between the CG and the corresponding all-atom model show similar root-mean-square deviations from the canonical B-form DNA, and similar structural fluctuations. At the same time, the CG model is 10 to 100 times faster depending on the length of the DNA fragment in the simulation. Analysis of dispersion curves derived from the CG model yields longitudinal sound velocity and torsional stiffness in close agreement with existing experiments. The computational efficiency of the CG model makes it possible to calculate thermal conductivity of a single DNA molecule not yet available experimentally. For a uniform (polyG-polyC) DNA, the estimated conductivity coefficient is 0.3 W/mK which is half the value of thermal conductivity for water. This result is in stark contrast with estimates of thermal conductivity for simplified, effectively 1D chains ("beads on a spring") that predict anomalous (infinite) thermal conductivity. Thus, full 3D character of DNA double-helix retained in the proposed model appears to be essential for describing its thermal properties at a single molecule level.

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.

Solitary waves in twist-opening models of DNA dynamics

We analyze traveling solitary wave solutions in the Barbi-Cocco-Peyrard twist-opening model of nonlinear DNA dynamics. We identify conditions, involving an interplay of physical parameters and asymptotic behavior, for such solutions to exist, and provide first-order ordinary differential equations whose solutions give the required solitary waves; these are not solvable in analytical terms, but are easily integrated numerically. The conditions for existence of solitary waves are not satisfied for trivial asymptotic behavior and physical values of the parameters, i.e., the Barbi-Cocco-Peyrard model admits only solitary wave solutions that entail a global modification of the molecule; this is compared with the situation met in another recently formulated class of DNA models with two degrees of freedom per site.

Simple combined model for nonlinear excitations in DNA

We propose a simple model for DNA denaturation bases on the pendulum model of Englander [Proc. Natl. Acad. Sci. U.S.A. 77, 7222 (1980)] and the microscopic model of Peyrard and Bishop [Phys. Rev. Lett. 62, 2755 (1989)], so-called "combined model." The main parameters of our model are the coupling constant k along each strand, the mean stretching y* of the hydrogen bonds, the ratio of the damping constant and driven force gamma/F. We show that both the length L of unpaired bases and the velocity v of kinks depend on not only the coupling constant k but also the temperature T. Our results are in good agreement with previous works.