用于分析DNA构型的分叉弹性杆的平衡稳定性和动力学研究

DNA replication is a fundamental process,that occurs in all living organisms that copies their DNA.This process is replication,in that each strand of the original double-stranded DNA molecule serves as template for the reproduction of the complementary strand.Therefore,following DNA replication,two identical DNA molecules have been produced from a single double-stranded DNA molecule.The replication fork is a structure that forms within the nucleus during DNA replication.It is created by helicases,which break the hydrogen bonds holding the two DNA strands together.With DNA replication in the background,this project lays emphasis on studies on nonlinear mechanics of branched elastic rods macroscopically,in order to describe and forecast microcosmic mechanics behaviour in process of DNA replication.We plan to adopt methods of analitical mechanics and nonlinear dyanmics to build its nonlinear mechanics model.One hand,carry out the analytise of mechanics of branched point and branched elastic rods and build the mathematical model. Using analytical method solves the governing equation,judges the stability of solution, studies the equilibria stability of branched point and branched elastic rods,and finds the influence of branched point on the equilibria stability of branched elastic rods and their interaction.The other hand,by simulating numerically branched point and branched elastic rods to study their dynamical characteristics,influnce of branched point on the equilibria stability of branched elastic rods and simulate geometric shape and motion state.To seek a way of studing microcosmic mechanics behaviour in process of DNA replication via macroscopic mechanics,which will contribute to the development of cross fields of mechanics and molecular biology.

DNA复制是指DNA一条双链变成两条一样的双链的过程。DNA在复制时，其双链首先解开，形成复制叉。本项目以DNA复制为研究背景，注重对宏观上分叉结构弹性杆的非线性力学进行研究，来描述和预测微观中DNA复制过程的力学行为。拟采用分析力学和非线性动力学的研究方法，建立其非线性力学模型。一方面，通过对分叉点和分叉结构弹性杆进行力学分析，并分别建立数学模型，使用近似解析方法求解控制方程并判断解的稳定性，进而研究分叉点和分叉结构弹性杆的平衡稳定性，及其分叉点对分叉结构弹性杆平衡稳定性的影响及关联；另一方面，通过对分叉点和分叉结构弹性杆数值模拟，研究它们的动力学特性、分叉点对分叉结构弹性杆的影响，模拟分叉结构弹性杆的几何形态和运动状态。探求通过宏观力学来研究微观DNA复制过程力学行为的一种途径，有助于力学与分子生物学交叉领域的发展。

基于变分原理，探讨了Green-Naghdi弹性杆理论和弹性体的Euler理论对分叉弹性杆的应用：将分叉弹性杆看作一个点连接三个弦线的情况，建立分叉弹性杆的动力学方程。针对分叉点的影响：考虑了具有有限拉伸粘附并连接组合形成的杆的模型。.非分叉弹性杆模型的对称性与守恒量研究。基于Kirchhoff动力学比拟，细长弹性杆静力学可比拟刚体动力学。由积分变分原理导出弹性杆的Lagrange运动方程。讨论了弹性杆Mei对称性的共形不变性与Mei 对称性的关系。构造了弹性杆Mei对称性的共形不变性导致守恒量的结构方程，并给出了守恒量的形式。结果表明：守恒量有助于弹性杆精确解和稳定性分析，以及后续的数值计算。.使用微分求积法(DQM) 直接数值模拟了所建立的动力学方程。通过整理数值计算的数据分析了分叉点对分叉结构弹性杆整体的平衡稳定性和动力学特性的影响，并对DNA复制过程进行了动态演化。