The Analysis For The Equilibrium Configuration And Stability Of DNA Based On An Elastic Rod Model

Since the structure of the DNA double helixes were discovered by James Watson and Francis Crick with X-ray diffraction in 1953,the shapes and stabilities of DNA chains have been proven that play a significant role during processes such as replication and transcription.Although the study of the interior-molecular structure for DNA belongs to the category of quantum mechanics,an elastic rod model of classical mechanics can be applied to analyze the equilibr ium configuration and mechanical property of DNA chain.It seems that there is an interdisciplinary area of classical mechanics and molecular biology.In fac t,the elastic rod model is based on the theory of continuum mechanics,while the equilibr ium configuration and stability of DNA are analyzed via the method of nonlinear mechanics and analysis mechanics.In order to study the formation mechanism of DNA chain more deeply,a three-dimensional elastic rod model is established to investigate the equilibr ium configuration and stability of DNA chain under the action of interfacial interaction or axial force.At the beginning of this dissertation,the geometrical description of an elastic thin rod equilibrium pos ition is outlined.The fundamental hypothesis,application conditions and signficances of Kirchhoff theory are presented.In addition,the Kirchhoff equations of a thin elastic rod are given.By means of Young-Laplace equation,the expression of interfacial tension between the rod and the solution is derived.Then,integrating along the perimeter of the rod cross section,we get the formulas of the interfacial traction.Combined with the Kirchhoff equations,the equilibrium equations of DNA under the interfacial traction are obtained.Based on the adsorption principle of solid-liquid interface,the relation between interfacial tension and ionic concentration of the solution environment is built with the help of Gibbs and Langmuir adsorption equation.Meanwhile,by means of the theory of Poisson-Boltzmann,a model is established to characterize the change of elastic modulus of DNA with the ionic concentration.By use of the classical Runge–Kutta algor ithm,the condensed configurations of DNA in the solution with different concentrations are calculated,and the condensed configurations corresponding to differnet original configurations are proposed.The influences of the ionic concentration on the equilibrium configuration of DNA are discussed.Taking account of the boundry condions under the action of external force,the study on the elastic response of DNA molecules under the action of interfacial traction and stretching can be attributed to solve a boundary value problem of Kirchhoff differential equations in mathematics.The shooting method is used to calculate the equilibrium configurations of condensed DNA.The force end-to-end distance curve are fitted by calculating the change of end-to-end distance when every 10 p N increment is added at the loading end,and the effect of ion concentrations on the equilibrium configuration of DNA subjected to stretching is also discussed.By applying the theory and method of analytical mechanics to the model of a thin elastic rod,Hamilton’s principle and Lagrange equation of arc coordinate are obtained by means of the Kirchhoff dynamical analogy.Then,Euler-Lagrange equations of DNA helix,whose free energy density functional depends on the curvature,torsion and their first-order derivatives with respect to the arc length,are derived.Considering the difference between the curve and elastic rod,the Euler-Lagrange equations characterizing the equilibr ium configuration of a thin elastic rod are derived from a free energy functional associated with the curvature,torsion,twisting angle and its derivative with respect to the arc-length.With the different shapes of rod,we obtain the equilibr ium equations of DNA with circular and noncircular cross sections,which provide an approach to describe the physical behaviors of A-,B-,Z-DNA.Meanwhile,the effect of geometric properties of cross-section on equilibrium of helical ribbon is also dicussed.By introducing external potential energy into the free energy functional of a thin elastic rod model,the general equilibr ium shape equations of DNA chain under axial force are derived in detail.The effects of the initial curvature and torsion on the stretching stability of a thin elastic rod model for DNA are investigated.