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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.
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.
In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables: time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.
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.
We have derived a time‐independent, one‐dimensional nonlinear Schrödinger equation for the stationary state configurations of supercoiled DNA. The effect of DNA self‐contact has been included analytically. For the cases of non‐self‐contact and periodic boundary conditions, closed‐form solutions have been obtained which describe the stationary state configurations of supercoiled DNA.
In this paper we consider the anharmonic corrections to the anisotropic elastic rod model for DNA. Our model accounts for the difference between the bending energies of positive and negative rolls, which comes from the asymmetric structure of the DNA molecule. We will show that the model can explain the high flexibility of DNA at small length scales, as well as kink formation at high deformation limit.
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.
An elastic rod model of a protein-bound DNA loop is adapted for application in multi-scale simulations of protein-DNA complexes. The classical Kirchhoff system of equations which describes the equilibrium structure of the elastic loop is modified to account for the intrinsic twist and curvature, anisotropic bending properties, and electrostatic charge of DNA. The effects of bending anisotropy and electrostatics are studied for the DNA loop clamped by the lac repressor protein. For two possible lengths of the loop, several topologically different conformations are predicted and extensively analyzed over the broad range of model parameters describing DNA bending and electrostatic properties. The scope and applications of the model in already accomplished and in future multi-scale studies of protein-DNA complexes are discussed.
The available range of the homoclinic bifurcation criterions are extended from the weakly nonlinear oscillation system to the strongly nonlinear oscillation system. It combines the analysis method of the strongly nonlinear oscillation system with the former criterions based on the improved complex normal form method. The periodic solution of this kind of system with a single degree of freedom is obtained by introducing the fundamental frequency under determination into the complex normal form computation. Then two different analytical criteria to predict the critical values of homoclinic bifurcation are adapted to the new system. It includes the undertermined fundamental frequency approaching zero and the collision of the periodic orbit with the saddle point. The results derived from different methods are compared in the specific systems with numerical simulation to testify the correctness and efficiency of the theoretical results.
This paper studies the collective compactness of composition operator sequences between the Bergman and Hardy spaces. Some sufficient and necessary conditions involving the generalized Nevanlinna counting functions for composition operator sequences to be collectively compact between weighted Bergman spaces are 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.
DNA topoisomerases participate in nearly all events relating to DNA metabolism including replication, transcription, and chromosome segregation. Recent studies in eukaryotic cells have led to the discovery of several novel topoisomerases, and to new questions concerning the roles of these enzymes in cellular processes. Gene knockout studies are helping to delineate the roles of topoisomerases in mammalian cells, just as similar studies in yeast established paradigms concerning the functions of topoisomerases in lower eukaryotes. The application of new technologies for identifying interacting proteins has connected the studies on topoisomerases to other areas of human biology including genome stability and aging. These studies highlight the importance of understanding how topoisomerases participate in the normal processes of transcription, DNA replication, and genome stability.
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.
DNA is often modelled as an isotropic rod, but its chiral structure suggests the possible importance of anisotropic mechanical properties, including coupling between twisting and stretching degrees of freedom. Simple physical intuition predicts that DNA should unwind under tension, as it is pulled towards a denatured structure. We used rotor bead tracking to directly measure twist-stretch coupling in single DNA molecules. Here we show that for small distortions, contrary to intuition, DNA overwinds under tension, reaching a maximum twist at a tension of approximately 30 pN. As tension is increased above this critical value, the DNA begins to unwind. The observed twist-stretch coupling predicts that DNA should also lengthen when overwound under constant tension, an effect that we quantitatively confirm. We present a simple model that explains these unusual mechanical properties, and also suggests a possible origin for the anomalously large torsional rigidity of DNA. Our results have implications for the action of DNA-binding proteins that must stretch and twist DNA to compensate for variability in the lengths of their binding sites. The requisite coupled DNA distortions are favoured by the intrinsic mechanical properties of the double helix reported here.
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
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