Study On The Micro Geometry Configuration Of DNA Elastic Thin Rod

The chromosome is the main carrier of genetic material in living cells, and is made up of DNA and protein combinations, where DNA is the carrier of genetic material. The changes of DNA configuration may affect many biological functions, such as the replication and transcription of DNA etc., and the misfolding of DNA can also cause many illnesses, paroxysmal nocturnal hemoglobinuria(PNH) is one of them, thus, the study of DNA configuration mechanism becomes a research hotspots in the new field which is the cross field of molecular biology and many other subjects. The experiments of molecular biology found that DNA molecules with extreme slender, as well as the characteristic of small disturbance can appear in great deformation. In this paper, based on the theory of Kirchhoff elastic rod, the configuration mechanism of DNA molecular in physiological salt solution is researched and the results can provide theoretical support for the research on the related molecular biology. The main research findings are discussed as follows:1. According to the characteristic of DNA molecular physiological conditions is the intracellular salt solution environment, one established the Kirchhoff elastic rod model under the action of the solid-fluid interfacial tension, and the interfacial distribution stress in the interfacial tension theory characterize the interactions between DNA molecules and salt solution. By numerical simulation of a simple 2 dimensional Kirchhoff equation, a group of toroidal configuration of DNA elastic rod is found and the configuration is accord with DNA geometry model which is constructed by the experimental data(Hun-Downing-Balhorn model). The Kirchhoff equation with the interfacial tension is simplified by the introduction of a complex variable form and the elastic rod curvature balance equation is obtained. One finds the relationship between the curvature of the elastic rod center line and the interfacial tension, and the energy balance method is used for solving the approximate analytic solution of the differential equation, the influences of the initial boundary conditions of elastic rod configuration are discussed.2. From the view of DNA molecular fine structure, the effect of base pair sequence on the flexural rigidity of DNA elastic rod is introduced. 18 dimensional control equations are established to describe the center line configuration of DNA elastic rod, and numerical simulation of three-dimensional geometry of elastic rod with the different bending stiffness ratio and periodic stiffness are done, the study of the condensed phenomenon of DNA molecules and the effect of the bending stiffness on the geometrical configurations are discussed. The configuration of the loop, which is connected DNA and the protein, is further numerical studied with the different salt solution concentration.3. According to the characteristics of biological polymers usually adopting the helical configuration, the static Kirchhoff configuration equation with the solid-liquid interface tension is regarded as an inverse problem, and the conditions of helical conformation and the corresponding energy density functions of elastic rod with the different behavior of the twist angle are found. By application of the above density function of elastic rod, the formula of B- to Z-DNA transition free energy is derived, in which the interfacial tension and young’s modulus are as variables. The results showed that the energy of B- to Z-DNA transition reduced when the salt concentration increased(the ionic strength increased), which is in good agreement with the experimental data and empirical formula.4. In order to study the effect of the asymmetric cross section on the topology configuration of elastic rod, the Kirchhoff equation thin elastic rod model with asymmetric cross section is rebuilt as 4 dimensional differential equations by introduction of the concept of the complex vector basis. A complex form variable solution of the torque is introduced, and the dimension is considered, then the reduction of the equation with asymmetric cross section is obtained by the dimension reduction method. One family of exact solutions is obtained in terms of rational expression by the Krylov-Bogoliubov elliptic function method, and the effects of the asymmetric cross section on the DNA topology configuration are discussed by numerical simulation.