Formation of localized structures in the Peyrard–Bishop–Dauxois model

Abstract

We explore in detail the properties of modulational instability (MI) and the generation of soliton-like excitations in DNA nucleotides. Based on the Peyrard–Bishop–Dauxois (PBD) model of DNA dynamics, which takes into account the interaction with neighbors in the structure, we derive through the semidiscrete approximation a modified discrete nonlinear Schrödinger (MDNLS) equation. From this equation, we predict the condition for the propagation of modulated waves through the system. To verify the validity of these results we have carried out numerical simulations of the PBD model and the initial conditions in the form of planar waves whose modulated amplitudes are given by the examples studied in the MDNLS equation. In the simulations we have found that a train of pulses are generated when the lattice is subjected to MI, in agreement with the analytical results obtained in an MDNLS equation. Also, the effects of the harmonic longitudinal and helicoidal constants on the dynamics of the system are notably pointed out. The process of energy localization from a nonsoliton initial condition is also explored.

The Transfer Integral Operator Method in the Study of DNA Unzipping and Bubble Formation

“…This idea was further developed by Salerno [31,32] who introduced nonhomogeneities in the model and studied the dynamics of DNA promoters in the context. His results have attracted attention [5,12] and other nonlinear excitations [1,34,35], such as discrete breathers, are emerging as important for DNA dynamics and function. These works take advantage of the large amount of research [3,4,13,14,21,23,24] that has been concerned with understanding discrete breathers in nonlinear systems in recent years.…”

Bubble Formation in Helicoidal DNA Molecules

“…Furthermore, many studies have been carried out, showing that it support solitonic structures and is rather suitable to observe the localization of the energy which drives the key dynamical processes known as replication and transcription. Over the years, the PB model has been improved to take into account other features such the helicoidal structure of DNA [4,5], the cooperativity among adjacent base pairs [6,7] and the roto-torsional behaviours of the molecule [8][9][10][11]. For example, modifications brought to the PB model have shown new features in the way the base pairs oscillate and its modulational instability has been extensively studied as well [5].…”

“…Over the years, the PB model has been improved to take into account other features such the helicoidal structure of DNA [4,5], the cooperativity among adjacent base pairs [6,7] and the roto-torsional behaviours of the molecule [8][9][10][11]. For example, modifications brought to the PB model have shown new features in the way the base pairs oscillate and its modulational instability has been extensively studied as well [5]. Analytical solutions of the helicoidal PB model have been studied [12], but no relationship has been established, to our knowledge, between fluctuating bubbles and soliton emergence in DNA nonlinear models.…”

Formation and Interaction of Bright Solitons with Shape Changing in a DNA Model

“…The nonlinear effects might focus the vibration energy of DNA into localized soliton-like excitations [10][11][12][13][14][15]. Indeed, the local opening can be analytically described as breather-like objects of small amplitude, which have nevertheless interesting properties; as long as their amplitude is small enough they can move along the chain.…”

“…Discrete solitons in nonlinear lattices have been the focus of considerable attention in diverse branches of science [16][17][18] and they are possible in several physical settings, such as biological systems [19][20][21][22][23], atomic chains [24,25], solid state physics [26], electrical lattices [27] and BoseEinstein condensates [28]. In DNA, such waves have been shown to carry the energy necessary for the initiation of the complex and key phenomena of replication and transcription [11][12][13][14]. In recent studies, on DNA models, the role of the enzymes such as RNA polymerase has been shown to be the collection of the thermal energy available in the system [13,15], an argument which has motivated the present work.…”