基于空间弹性细杆模型的DNA平衡几何构型与稳定性研究

发布时间：2018-06-13 14:37

本文选题：DNA凝聚 + 弹性细杆模型　；参考：《南京航空航天大学》2016年博士论文

【摘要】：近年来,随着分子生物学的飞速发展,人们将越来越多的目光集中于生物大分子这一研究领域。而DNA作为一种典型的生物大分子,其所具备的储存和传递生命遗传信息的能力使得它逐渐成为了分子生物领域的一个研究热点。由于DNA几何构型的平衡与稳定直接影响着遗传信息的表达,因此,对于DNA平衡几何构型及其稳定性的研究是了解并应用DNA分子链的基础。自1953年Watson和Crick借助X射线晶体衍射技术推测出DNA的双螺旋结构以来,关于DNA分子链的理论基础研究不断取得突破。尽管对于DNA分子链的内部分子结构的探索属于量子力学的研究范畴,但其在生物细胞中的宏观几何构型与外力势能作用下的力学性能的研究可以借助于经典力学的弹性细杆模型来实现。由此产生了一个经典弹性力学与分子生物学相互交叉的新的研究领域。其在本质上采用连续介质力学的概念与思路研究曲杆的几何形态与变形,而在方法上又可运用非线性力学和分析力学对DNA分子链的平衡构型与稳定性进行分析。因此,本文在连续介质力学的基础上,通过建立空间弹性细杆模型,来模拟计算DNA分子链在溶液中的平衡几何构型,并对其外力作用下的弹性响应与几何构型的稳定性做了研究分析。本文的第一部分简要介绍了DNA弹性细杆的几何描述,概述了Kirchhoff弹性杆理论的基本假定与其适用的前提条件和物理意义,给出了弹性细杆的Kirchhoff平衡方程;阐述了物质的界面层与界面张力、界面自由能的概念,通过Young-Laplace方程推导出了固-液界面张力的表达式,将其引入Kirchhoff方程,并沿细杆截面周长对弧坐标进行积分得到了受界面牵引力作用的DNA分子链平衡控制方程。第二部分则基于固-液界面的吸附原理,通过Gibbs吸附方程与Langmuir吸附方程,建立起溶液浓度与固-液界面张力的关系,并推导出其具体表达式;与此同时,借助Poisson-Boltzmann理论将溶液浓度引入长分子链的熵弹性,得到了溶液浓度与DNA分子链弹性模量的直接关系式。将上述两式代入DNA弹性细杆的平衡控制方程,应用龙格-库塔数值算法模拟计算出以圆柱形螺旋线与椭圆柱形螺旋线为初始构型的DNA链段在不同溶度浓度中的凝聚构型,并对其进行了初步的分析。第三部分针对DNA在外力载荷作用下的边界条件,将端部力与界面张力共同作用下的DNA链段的几何构型与弹性响应的研究归结为求解Kirchhoff微分方程组的边值问题,并选用打靶法给出在其不同端部力作用下的数值解,从而确定DNA链段的几何构型。同时,利用DNA链段两端在主轴坐标系中的坐标计算得到不同端部力对应的末端距数值,拟合出力-末端距曲线,并分析了不同溶液浓度对DNA链段的力-末端距曲线的影响。第四部分从分析力学的角度出发,通过Kirchhoff动力学比拟,将动力学的时间变量t置换为一维空间变量即弧坐标s,从而得到了基于弧坐标的哈密顿原理与Lagrange方程。并基于此推导出能量密度函数依赖于曲率、挠率及它们一阶导数的DNA螺旋线的Euler-Lagrange方程。第五部分则通过引入截面的扭转角及其一阶导数,将曲线的Euler-Lagrange方程推广至曲杆的Euler-Lagrange方程组。在此基础上,通过与实验数据进行了对比,分析了采用圆截面弹性细杆模型与椭圆截面弹性细杆模型分别模拟A-,B-,Z-DNA几何构型的可行性,并拟合了两种模型的r0-h曲线。同时,分析了弹性细杆截面的几何特性对螺旋带模型几何构型的影响。第六部分通过将外力势能项引入弹性细杆的能量密度函数,在弹性曲杆模型的Euler-Lagrange方程组基础上,推导出DNA弹性细杆在端部拉伸力作用下的Euler-Lagrange方程组,并基于此对不同初始曲率与初始扭率的DNA弹性细杆模型的拉伸稳定性进行了初步分析。
[Abstract]:In recent years, with the rapid development of molecular biology, people will focus more and more attention on the research field of biological macromolecules. As a typical biological macromolecule, DNA has the ability to store and transmit the genetic information of life, which has gradually become a hot research topic in the division of biological fields. Because of DNA geometry, it has become a research hotspot. The balance and stability of the configuration directly affect the expression of genetic information. Therefore, the study of the geometric configuration and stability of DNA is the basis for understanding and applying the DNA molecular chain. Since Watson and Crick have deduced the double helix structure of DNA with the X ray crystal diffraction technology in 1953, the theoretical foundation of the DNA molecular chain has been continuously taken. Although the exploration of the internal molecular structure of the DNA molecular chain belongs to the field of quantum mechanics, the study of the mechanical properties of the macrogeometries and the external force potential in the biological cells can be realized by the elastic thin rod model of the classical mechanics. The new research field of biology intersecting with each other. It uses the concept and thought of continuous medium mechanics to study the geometric shape and deformation of the curved rod in essence, and the method of nonlinear mechanics and analytical mechanics can be used to analyze the equilibrium configuration and stability of the DNA molecular chain. The space elastic thin bar model is established to simulate the equilibrium geometric configuration of the DNA molecular chain in the solution. The elastic response and the stability of the geometric configuration under its external force are studied and analyzed. The first part of this paper briefly introduces the geometric description of the DNA elastic rod, and summarizes the basic assumptions of the theory of the Kirchhoff elastic rod. The Kirchhoff equilibrium equation of the elastic thin rod is given, and the concept of interfacial tension and interfacial free energy is expounded. The expression of the solid liquid interfacial tension is derived through the Young-Laplace equation, which is introduced into the Kirchhoff equation, and the arc coordinates are integrated along the circumference of the section of the fine rod. The equilibrium control equation of DNA molecular chain is obtained by the interface traction. The second part is based on the adsorption principle of solid liquid interface. Through the Gibbs adsorption equation and the Langmuir adsorption equation, the relationship between solution concentration and solid liquid interfacial tension is established and its specific expression is derived. At the same time, the solution with the aid of Poisson-Boltzmann theory is used. The direct relation between the concentration of the solution and the elastic modulus of the DNA molecular chain is obtained by introducing the entropy elasticity of the long molecular chain. The above two formula is substituted for the equilibrium control equation of the DNA elastic thin rod, and the DNA chain of the initial configuration with the cylindrical spiral and the elliptical columnar spiral is simulated and calculated by the Runge Kutta numerical algorithm. The third part aims at the boundary conditions of DNA under the force of external force. The study of the geometric configuration and elastic response of the DNA segment under the joint action of the end force and the interfacial tension is reduced to the solution of the boundary value problem of the Kirchhoff differential equations. The numerical solution of the same end force is used to determine the geometric configuration of the DNA segment. At the same time, the end distance values corresponding to different end forces are obtained by calculating the coordinates of the two ends of the DNA segment in the spindle coordinate system, and the force - end distance curve is fitted, and the effect of different solution concentration on the force and end distance curve of the DNA segment is analyzed. The fourth part is from the analysis of the effect of the force to the end distance curve of the chain segment. Based on the analysis of mechanics, the time variable t of dynamics is replaced by a one-dimensional space variable, the arc coordinate s by Kirchhoff dynamics analogy, and the Hamilton principle and the Lagrange equation based on the arc coordinate are obtained. Based on this, the Euler of the energy density function is dependent on the curvature, the torsion and the Euler of the DNA spiral of their first derivative. In the fifth part, the fifth part, by introducing the torsional angle of the cross section and its first derivative, generalizes the Euler-Lagrange equation of the curve to the Euler-Lagrange equation set of the curved bar. On this basis, through the comparison with the experimental data, it is analyzed that the elastic thin rod model of the circular section and the elastic thin rod model of the elliptical cross section are respectively simulated, B-, and B-. The feasibility of the Z-DNA geometric configuration and the fitting of the r0-h curves of the two models are fitted. Meanwhile, the influence of the geometric characteristics of the section of the elastic thin section on the geometric configuration of the spiral belt model is analyzed. The sixth part is derived from the energy density function of the external force potential energy term to the elastic thin rod, and on the basis of the Euler-Lagrange equation set of the elastic curved bar model. The Euler-Lagrange equations of the DNA elastic thin rod under the end tensile force are given, and the tensile stability of the DNA elastic thin bar model with different initial curvature and initial torsion is preliminarily analyzed.
【学位授予单位】：南京航空航天大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：Q523

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