Nonlinear Dynamics of Long-Range Protein-Helicoidal Dna Interactions

Abstract

The effects of long-range interactions between peptides on the protein–DNA dynamics in the long-wave limit are studied. The investigation, done at the physiological temperature, is based on a coupled spin system of DNA molecule which includes the helicoidal geometry of DNA molecule and the Kac–Baker long-range interaction between the peptides of the protein molecule. By using the Holstein–Primakoff bosonic representation of the spin operators, we show that the original discrete equations for the protein–DNA interaction dynamics can be reduced to the nonlinear Schrödinger (NLS) equation of which the dispersive and the nonlinear coefficients depend among other things on the protein long-range interaction parameter and on the helicoidal coupling coefficient. Furthermore, we find that the amplitude and the width of the resulting breather solution, in the form of the bubble moving along the DNA molecule, are strongly influenced by the long-range and helicoidal interactions. This result shows a relevant length scale for real protein–DNA interaction.

Transcription's bubble under the influence of long-range interactions and helicoidal coupling

“…Since our purpose is to study the behavior of a DNA molecule when an RNAP molecule binds to it, we focus our attention on Eq. (44). In this equation, the nonlinearity is brought by the RNAP binding, and the coupling of the thermal phonons in the DNA strand, respectively, while the dispersion is brought by the intra-strand coupling and HC, respectively.…”

“…The parameters F 0 , L e , θ 0 , and A 0 remain the same as in Section 3, and as mentioned in Section 2, the numerical values of the model parameters are those used in Refs. [38], [44]- [51]. The numerical values of n 1 and n 2 are inspired from the results of Section 3 and are considered to be n 1 = 20 and n 2 = 31; this means that the initial breather is formed by 12 base pairs.…”

“…The numerical values of the model parameters are those used in Refs. [38], [44]- [51]. We usually use a classical approach or semi-classical approximation to understand the spin dynamics when the spin value is large in the case of Heisenberg spin systems.…”

Inhomogeneities and nonlinear dynamics of a helical DNA interacting with a RNA-polymerase

“…Inspired by the past works [2][3][4][5][6][7], and specially by those by Vasumathi et al [4] and by Agüero et al [7], we consider the spin-like model of the B-form of DNA molecule as represented in figure 1. R and ′ R represent the complementary strands in the DNA double helical chain.…”

“…Vasumathi and Daniel have studied the protein-DNA interaction dynamics in a viscous medium by modeling the DNA bases as classical spins in a coupled spin system [4]. We have recently used the same model to study the effects of long-range interactions on the bubble features during DNA transcription [5,6]. In the situation where a DNA molecule is subjected to the action of external agents, Agüero and coauthors have applied the generalized coherent states approach to averaging the quasi-spin Hamiltonian for DNA macromolecules [7].…”