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The analytical reduction of the Kirchhoff thin elastic rod model with asymmetric cross section
Abstract
The Kirchhoff thin elastic rod models and related systems are always the important basis to research the topology and stability of the flexible structures in not only the macroscopic but also microscopic scale. Firstly the initial Kirchhoff equations are rebuilt in a complex style to suit the character of obvious asymmetry embodied on the cross section by considering the mathematical background of DNA double helix. Then we introduce a complex form variable solution of the torque, and extend the knowledge of effective bending coefficients as well as its facility in the high dimensional system by using the complicated system. As the result, a simplified second order ordinary differential equation with single variable is obtained. Furthermore the periodically varying bending coefficients of the DNA molecular are considered as the appended components to the effective bending coefficients. The whole reduction process makes the numerical simulation become not solely the exclusively eligible approach, and produces adaptable channel to quantitative analysis.
... 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 2 Mathematical Problems in Engineering with single variable. However, in above work, the complex expression of 3 according to the complex normal form method is not accurate; we will correct this error in the following section. ...
... where F and M denote the elastic force and moment, respectively. As shown in [17,18], the complex vector bases D are used to substitute the real form vectors {e 1 , e 2 , e 3 }: Thus, the complex Kirchhoff equations in the case of zero external moment m can be written out in terms of D , such as ...
... In [17,18], a complex expression of 3 is brought, which is analogous to the complex normal form method. However, this expression is inaccurate; we revised the expression as follows: ...
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.
The classical Kirchhoff elastic–rod model applied to DNA is extended to account for sequence–dependent intrinsic twist an curvature, anisotropic bending rigidity, electrostatic force interactions, and overdamped Brownian motion in a solvent. Th zero–temperature equilibrium rod model is then applied to study the structural basis of the function of the lac repressor protein in the lac operon of Escherichia coli. The structure of a DNA loop induced by the clamping of two distant DNA operator sites by lac repressor is investigated and the optimal geometries for the loop of length 76 bp are predicted. Further, the mimicked bindin of catabolite gene activator protein (CAP) inside the loop provides solutions that might explain the experimentally observe synergy in DNA binding between the two proteins. Finally, a combined Monte Carlo and Brownian dynamics solver for a worm–lik chain model is described and a preliminary analysis of DNA loop–formation kinetics is presented.
The extended Schrödinger equation for the Kirchhoff elastic rod with noncircular cross section is derived using the concept of complex rigidity. As a mathematical model of supercoiled DNA, the Schrödinger equation for the rod with circular cross section is a special case of the equation derived in this paper. In the twistless case of the rod when the principal axes of the cross section are coincident with the Frenet coordinates of the centreline, the Schrödinger equation is transformed to the Duffing equation. The equilibrium and stability of the twistless rod are discussed and a bifurcation phenomenon is presented.
The effective elastic modulus and fracture toughness of the nanofilm were derived with the surface relaxation and the surface energy taken into consideration by means of the interatomic potential of an ideal crystal. The size effects of the effective elastic modulus and fracture toughness were discussed when the thickness of the nanofilm was reduced. And the dependence of the size effects on the surface relaxation and surface energy was also analyzed.
The dynamics of a super-long elastic slender rod plays an important role in analyzing many properties of a DNA such as its stability and equilibrium. In order to provide a convenient environment for the post processing of the numerical results in studying the dynamical properties of a DNA, a differential/integral equation system is given in this paper to describe the surface of a super-long elastic slender rod. Numerical algorithms are provided to solve the equations and an 1 example of application is given.
The special solutions of the Kirchhoff equations, which are those relative to fixed coordinate system, principal coordinate system of a cross section of the rod, and Frenet coordinate system of the central line of the rod, respectively, are derived in this paper. Lyapunov stability of these solutions is discussed by use of theory on the first-approximation stability, and at the same time stability area in parameter's plane is given.
A DNA polymer with thousands of base pairs is modeled as an elastic rod with the capability of treating each base pair independently. Elastic theory is used to develop a model of the double helix which incorporates intrinsic curvature as well as inhomogeneities in the bending, twisting, and stretching along the length of the polymer. Inhomogeneities in the elastic constants can also be dealt with; thus, sequence-dependent structure and deformability can be taken into account. Additionally, external forces have been included in the formalism, and since these forces can contain a repulsive force, DNA self-contact can be explicitly treated. Here the repulsive term takes the form of a modified Debye–Hückel force where screening can be varied to account for the effect of added salt. The supercoiling of a naturally straight, isotropic rod in 0.1M NaCl is investigated and compared with earlier treatments of supercoiled DNA modeled by a line of point charges subject to electrostatic interactions and an elastic potential. © 1997 American Institute of Physics.
The torsional energy of a DNA is the energy involved when the average angle between adjacent base pairs is modified under applied constraints. A relationship is established between the energy of superhelix formation of a upercoiled DNA and its torsional energy. This relationship, valid when the number of the tertiary turns of the supercoiled DNA is small, contains the ratio of the bending to the torsional elastic paramaters of the DNA. From this relationship, the fluctations of the number of tertiary turns of the supercoiled DNA may be compared to those of the DNA with only one sinle-strand scission. An expression for the shear modulus of the DNA is derived from the anisotropy of the fluorescnece of the intercalated ethidium [Wahl et al. (190) Proc. Natl. Acad. Sci. USA 65, 417–421.]. The value of te torsional elastic parameter deduced from these computations is very close to the parameter of the energy of the superhelix formation determined on supercoiled DNAs by different authors. The energy that must be given to twist it. Consequently, the number of crossovers determined using the EM technique should be much less than the value deduced from the amount of ethiium necessary words, the first ethidium cations intercalating into a relaxed, closed double-standard DNA change the twist of the DNA but not its tertiary structure.
An elastic model for the supercoiling of duplex DNA is developed. The simplest assumptions regarding the elastic properties of double-helical DNA (homogeneous, isotropic, of circular cross section, and remaining straight when unstressed) will generate two orders of superhelicity when stressed. Recent experimental results [Brady, G.W., Fein, D.B. & Brumberger, H. (1976) Nature 264, 231-234] suggest that in supercoiled DNA molecules there are regions where two distinct orders of supercoiling arise, as predicted by this model.
The existence of spatially chaotic deformations in an elastica and the analogous motions of a free spinning rigid body, an extension of the problem originally examined by Kirchhoff are investigated. It is shown that a spatially periodic variation in cross sectional area of the elastica results in spatially complex deformation patterns. The governing equations for the elastica were numerically integrated and Poincare maps were created for a number of different initial conditions. In addition, three dimensional computer images of the twisted elastica were generated to illustrate periodic, quasiperiodic, and stochastic deformation patterns in space. These pictures clearly show the existence of spatially chaotic deformations with stunning complexity. This finding is relevant to a wide variety of fields in which coiled structures are important, from the modeling of DNA chains to video and audio tape dynamics to the design of deployable space structures.
We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria arise as a twisting angle alpha is varied. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. We present several examples of such distinguished bifurcation diagrams involving plots of the energy E, linking number Lk, and a twist moment m3, along families of cyclized equilibria of both intrinsically straight and intrinsically curved DNA fragments.
We study the three-dimensional static configurations of nonhomogeneous Kirchhoff filaments with periodically varying Young's modulus. This type of variation may occur in long tandemly repeated sequences of DNA. We analyse the effects of the Young's modulus frequence and amplitude of oscillation in the stroboscopic maps, and in the regular (non chaotic) spatial configurations of the filaments. Our analysis shows that the tridimensional conformations of long filaments may depend critically on the Young's modulus frequence in case of resonance with other natural frequencies of the filament. As expected, far from resonance the shape of the solutions remain very close to that of the homogeneous case. In the case of biomolecules, it is well known that various other elements, besides sequence-dependent effects, combine to determine their conformation, like self-contact, salt concentration, thermal fluctuations, anisotropy and interaction with proteins. Our results show that sequence-dependent effects alone may have a significant influence on the shape of these molecules, including DNA. This could, therefore, be a possible mechanical function of the ``junk'' sequences. Comment: 18 pages (twocolumn), 5 figures Revised manuscript