The Investigation On The Symmetry And Stability Of Elastic Rod Model Of DNA Supercoil

The investigation on elastic rod model for the3-D geometrical structure of DNAsupercoil, which is fundamentally different from the traditional elastic rod mechanicsfor its ultra-slender characteristic and large deformation, has developed as a newresearch field for the interdiscipline of classical mechanics and molecular biology.The elastic rod model is nonlinear equations for the super long and deformationof DNA. Symmetry analysis can reduce the nonlinear equations and find conservedquantities of systems. So application the symmetry into elastic rod injects the newmethod to study the DNAelastic rod, and can also expand the definition of invariants.The stability of DNA supercoil has important affect on its biological function,especially, the dynamical stability analysis has important application in the DNApacking. In this paper, the symmetry and dynamical instability of elasic rod model arestudied.The first part is the investigation on the symmetries and conserved quantities ofelastic rod model of DNAsupercoil.(1) The Mei symmetry and conserved quantities of Kirchhoff elastic rod of DNAsuperhelix are studied. The criterion of Mei symmetry is given, and the formation ofthe conserved quantities are obtained. For a particular Lagrangian of elastic rod, threetypes of first integrals with physical meanings are obtained. All the above results cango back to the known results when the cross section of rod is circular and the initialtwist is omitted. The recent reports suggested that the initial twist and curvature haveapparent effects on the stability of rod, so the results got in this paper may providereference to the stability research.(2) The discrete Noether symmetry of perturbation and adiabatic invariants forelastic rod are studied. The discrete nonholonomic Lagrange equation of elastic rod isgiven, the criterion of perturbation to Noether symmetry in discrete analogy arededuced and the discrete Noether adiabatic invariants are established. An example toillustrate how to find any order adiabatic invariants is given. (3) The Noether symmetry is expanded to Cossrat rodof DNA supercoiling withthe arc coordinates and time as independent variables. The Noether conservedquantity of rod dynamics is given. Some of the conserved quantities extend thedomain of classical invariants. The conserved quantities in integral forms take newproblems for classical analytical mechanics.The second part is on the wave propogation analogy of Schr dinger particles ofDNAelastic rod statics.The Kirchhoff dynamics analogy by use of the similarity between the equilibriumdifferential equations of elastic rod and the equations of rotation of rigid body about afixed point, which lays the foundations of the study on the equilibrium of elastic rod,injects the definition and methods of dynamics into elastic rod statics. In this chapter,we give the analogy relation between Kirchhoff equation as well as Kovalevskayaequation and1-D nonlinear Schr dinger equation. Consequently, we transplant theJacobi elliptic solution of1-D nonlinear Schr dinger equation to elastic rod equation.Theprobability density distribution of particles correspond to the3-D configuration ofelastic rod is given. The configurations of elastic rod with different initial conditionsare given by numerical solving the curvature equations.The third part is the dynamical instability of elastic rod dynamics of DNAsuperhelix in viscous fluidsThe former studies usually are the analysis of the stability of elastic rod statics,which cannot describe the dynamical deformation of rod. As we all know, DNA lie insolution, so the viscous resistance should be considered in. In this section, we willanalyze the dynamical instability of elastic rod by using a new perturbation scheme.(1) Considering two situations, the first order perturbation equation is deducedby the new perturbation scheme. Take the DNA elastic ring as an example, thecriterion of stability is given. The results in the two situations are compared and theinfluences of the viscous to the elastic ring are discussed.(2) The stability of growing elastic rod is studied. The dynamical equation ofgrowing rod are given, the criterion of instability of elastic rod are obtained.(3) The study on DNA supercoiling structure follows the signal rod theory, with the symmetrical axis of dsDNA as the center line of rod, and the orientation of crosssection denotes the helix direction. However, the signal rod can not describe the DNAmelting process. So the bi-rod theory is developed. We expand the2-braidsequilibrium theory of van der Heijden to dynamics. The geometry theory of2-braidsand dynamical equations are established.