Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 351362, 7 pages

http://dx.doi.org/10.1155/2015/351362

## Equilibrium Configurations of the Noncircular Cross-Section Elastic Rod Model with the Elliptic KB Method

^{1}Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China^{2}School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China

Received 3 November 2014; Accepted 21 February 2015

Academic Editor: Lakshmanan Shanmugam

Copyright © 2015 Yongzhao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The mechanical deformation of DNA is very important in many biological processes. In this paper, we consider the reduced Kirchhoff equations of the noncircular cross-section elastic rod characterized by the inequality of the bending rigidities. One family of exact solutions is obtained in terms of rational expressions for classical Jacobi elliptic functions. The present solutions allow the investigation of the dynamical behavior of the system in response to changes in physical parameters that concern asymmetry. The effects of the factor on the DNA conformation are discussed. A qualitative analysis is also conducted to provide valuable insight into the topological configuration of DNA segments.

#### 1. Introduction

DNA is a long polymer made of millions (or even hundreds of millions) of nucleotides arranged in two complementary strands forming a double helix. Genetic information in living cells is carried in the linear sequence of nucleotides in DNA. Conformational features and mechanical properties of DNA in vivo (such as supercoil formation and bend/twist rigidity) play an important role in its packing, gene expression, protein synthesis [1, 2], protein transport [3], and so forth, as misfolding of DNA has become the major cause of many illnesses, such as paroxysmal nocturnal hemoglobinuria (PNH) disease [4]. Thus, it is necessary to understand the basic mechanisms of DNA folding that leads to new ways for preventing such diseases. In recent years, geometrical configuration of a DNA chain has attracted considerable attentions.

The elastic properties of ds-DNA molecules are believed to play an important role in many biological functions [5, 6]. Over the past two decades, the elastic properties have been extensively studied with the development of single-molecule manipulation techniques [7–11] and experiment capabilities. Analytical models based on classical elasticity theory [12–14] have no spatial/temporal limitations and have widely been used to study the DNA configurations. The elastic rod model to research the flexible structures is to assume that they are made of an elastic material obeying the appropriate laws of elasticity. The well-known Kirchhoff models for rods are widely used to describe the stationary states of elastic filaments within the approximation of linear elasticity theory through a system of six coupled ordinary differential equations [12]. In 1859, Kirchhoff discovered that the equations that describe the thin elastic rod in equilibrium are mathematically identical to those used to describe the dynamics of the heavy top. Shi and Hearst [15] derived a time-independent, one-dimensional nonlinear Schrödinger equation for the stationary state configurations of supercoiled DNA. Xue et al. [16] extended the Schrödinger equation to fit the noncircular Kirchhoff elastic rod by using the complex rigidity. Wang et al. [17, 18] rebuilt the initial Kirchhoff equations in a complex style to suit the character of obvious asymmetry and the periodically varying bending coefficients, which is embodied on the cross-section by considering the mathematical background of DNA double helix, and introduced a complex form variable solution of the torque to obtain a simplified second ordinary differential equation with single variable. However, in above work, the complex expression of according to the complex normal form method is not accurate; we will correct this error in the following section.

In recent years, the analysis of static and dynamic configurations of elastic rod has drawn great attentions. In this paper, we will consider the revised reduced Kirchhoff equations of the noncircular cross-section elastic rod characterized by the inequality of the bending rigidities. It is crucial to find the exact or approximate solutions for the revised simplified second ordinary differential equation in order to investigate the configurations of DNA segments. Reference [19] applied the enhanced cubication method to develop approximate solutions for the most common nonlinear oscillators and leads to amplitude-time response curves and angular frequency values. Reference [20] developed a nonlinear transformation approach to obtain the equivalent representation form of conservative two-degree-of-freedom nonlinear oscillators. Lai and Chow [21] used Jacobi elliptic Krylov–Bogoliubov (KB) method to find two families of exact solutions for oscillators with quadratic damping and mixed-parity nonlinearity. Motivated by the above literatures review, this paper focuses on accurate solutions for the reduced Kirchhoff equations and undertakes a qualitative analysis of the topological configuration of DNA segments.

The paper is divided into four parts. In the next section, the reduced form of Kirchhoff’s equations is revised. In the third section, the periodic solutions of the equations are found and the effects of anisotropic on configuration of DNA are discussed. Finally, some conclusions are drawn and the paper is closed.

#### 2. The Reduced Form of Kirchhoff Equations

As a coarse-grained description, a DNA can be approximately regarded as a thin flexible and inextensible rod or string [12, 15, 16]. The classical theory of elasticity describes the geometry of an elastic rod in terms of its center line , three-dimensional curve parameterized by its arc-length . In presence of external moment and external load which are distributed along the central axis (as show in Figure 1), the static Kirchhoff equations in body fixed frame are as follows:where and denote the elastic force and moment, respectively.