Hu Kai
Department of Applied Mathematics, National Dong Hwa University, Shoufeng, Hualien 97401, Taiwan, Republic of China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Sep;76(3 Pt 1):031910. doi: 10.1103/PhysRevE.76.031910. Epub 2007 Sep 12.
The recent development of DNA structure, brought by the elastic rod model, revives the study of the so-called Michell-Zajac instability for isotropic naturally straight elastic rings. The instability states that when subjected to a terminal twist, a manipulation which cuts, rotates, and then seals closed rods, an elastic ring does not writhe until the amount of rotation exceeds a rod-dependent threshold. From the data generated by a finite element method, Bauer, Lund, and White [Proc. Natl. Acad. Sci. USA. 90, 833 (1993)] concluded that the instability becomes extreme for isotropic naturally singly bent, doubly bent, and O -ring elastic rings since they writhe immediately as subject to a terminal twist. This paper continues their study for other closed rods. In order to understand DNA structure in DNA-protein interactions, this paper also extends the study to open rods with clamped ends; for such rods, a terminal twist is a manipulation which releases, rotates, and then reclamps one end of the rods. Moreover, the rods under consideration need not be isotropic or may violate Kirchhoff-Clebsch conservation law of total energy. By linearizing the Euler-Lagrange equations which govern equilibrium rods and analyzing the linearized equations, this paper establishes an inequality such that if the initial values of the bending curvatures, their first derivatives, and the twisting density of an equilibrium rod satisfy the inequality, the rod axis deforms immediately as the rod is subject to a terminal twist. Since the initial data dissatisfying the inequality form a hypersurface in the five-dimensional Euclidean space, this paper asserts that a terminal twist makes the axis deformed instantly for almost every equilibrium rod.
由弹性杆模型带来的DNA结构的最新进展,重振了对各向同性自然直弹性环所谓的米切尔 - 扎亚克不稳定性的研究。该不稳定性表明,当受到末端扭转时,即一种对杆进行切割、旋转然后封闭的操作时,弹性环在旋转量超过与杆相关的阈值之前不会发生缠绕。根据有限元方法生成的数据,鲍尔、伦德和怀特[《美国国家科学院院刊》90, 833 (1993)]得出结论,对于各向同性自然单弯、双弯和O形环弹性环,不稳定性变得极为明显,因为它们在受到末端扭转时会立即发生缠绕。本文延续他们对其他封闭杆的研究。为了理解DNA - 蛋白质相互作用中的DNA结构,本文还将研究扩展到两端夹紧的开口杆;对于这种杆,末端扭转是一种对杆的一端进行松开、旋转然后重新夹紧的操作。此外,所考虑的杆不必是各向同性的,或者可能违反基尔霍夫 - 克莱布施总能量守恒定律。通过将控制平衡杆的欧拉 - 拉格朗日方程线性化并分析线性化方程,本文建立了一个不等式,使得如果平衡杆的弯曲曲率、其一阶导数和扭转密度的初始值满足该不等式,当杆受到末端扭转时,杆轴会立即变形。由于不满足该不等式的初始数据在五维欧几里得空间中形成一个超曲面,本文断言对于几乎每一个平衡杆,末端扭转都会使轴立即变形。