| The super long thin elastic rod is an important mechanics model across manyfields, such as life sciences, engineering safety, etc. Because of its special characters,for example, extremely slender character and large deformation characteristics with asmall perturbation, this model are proverbially concerned by many domestic andforeign scholars in recent years. DNA ring shows vastly different spatialconfigurations in the role of external forces and torques, which not only implies alarge number of genetic information, but also bears the nature of the process the lifeevolution. So researching of circular DNA has become an important branch of thestudy of the elastic rod theory. At present, many domestic and foreign researchershave attempted to combine the classical nonlinear dynamics theory and the researchof configurations of circular DNA, and obtained a certain amount of achievements. Inmolecular biology, changes of DNA topology is the result of the role of variousenzymes, and this can be reflected to the elastic ring rod model, which means we canestablish a corresponding relationship between each parameter of the modal and thechanges of the role of external factors such as external forces and moments. For suchproblems are simplified or even ignored under the conditions of this relationship insome complicated situation which isn’t reasonable. So, in this paper, based on thenonlinear dynamics thinking, and considered a variety of external factors, both of theexisting model and a re-establish mechanical one were conducted a detailed analysisto reveal the impact of the main parameters of circular cross-section and asymmetriccross section flexible rod topology configuration, and the numerical analysis andcomparison validation, and have a deep consideration of the relationship between theexternal action of the enzyme and DNA topology configuration changes from amathematical point of view. The main work and the important conclusions of thispaper are in the following three aspects:(1). Considering the forces and torques acted on the Kirchhoff thin elastic ringrod with asymmetric cross section, equivalent initial torque was introduced as a newconcept to reveal the quantitative relation between the state properties and parametersof torque dynamic model, which determined the configuration of the ring rod. Asthere existed formally similarity between the potential energy density function and the Hamiltonian function, the quantitative relation was got which connected theequivalent initial torque, Hamiltonian function and initial link number. In the physicalbackground of DNA ring molecule, this paper studied the nonlinear dynamic analysisof torque model, and used Undecided Fundamental Frequency Method to obtain theasymptotic expressions for steady state periodic solutions around the practicalequilibrium point. When the equivalent initial torque took different values, the graphof ArcLength-Torque and phase portrait were showed. The curvature could beexpressed by the torque function with considering the equivalent bending stiffness,which provided a new methodological way to understand and describe the stabletopological configuration of DNA ring by the influences of biological enzymes.(2). In order to have an in-depth study on the complex configuration of Kirchhoffthin elastic rod with circular section, two-degree-of-freedom equilibrium equation wasestablished by regarding the twisting components as two independent variables.Internal resonance was introduced to reveal the reasons of complexity of elastic rodconfiguration, and its ratio here ranged from0to0.531.(3). The calculation of linking number plays an important role in this field. So, inthis paper, the Gaussian integration for linking number was reduced effectively byconsidering the constitutive relation and Darboux vector angles relation of elastic rodand that would enhance its practicality. At last, the efficiency of reduction was provedby comparing of above model and the available results from present literature. |
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