Contact Problems in Kirchhoff's Nonlinear Theory of Rods

主要研究人员将对Kirchhoff弹性杆理论中的非线性问题进行研究，目标是：(I)对于有结和无结的闭杆和开杆，在终端力和力矩作用下，找到平衡构形的精确解析表示，表明孤立点和自接触间隔；(Ii)推导出平衡构形在给出严格的局部最小值的意义下稳定的实用充要条件；(Iii)深入了解分叉图对结型和本征曲率的依赖；(4)了解在弹性棒的扭转拉伸实验和DNA单分子操纵实验中，血小板增多环的出现导致迟滞的方式。平衡构型的解析表示将被用来开发一种新的Metropolis蒙特卡罗方法，用于计算受特定约束和末端条件约束的热波动DNA分子的配分函数。研究生将参与这项位于现代连续介质力学和分子生物学之间的研究，如下所述。每个人类细胞在直径不到1微米的细胞核中都有一米长的DNA。在细胞的整个生命过程中，其紧凑的DNA处于快速但可控的活动状态，因为生命功能的调节需要将适当的遗传密码部分重复转录到RNA链中。目前的研究项目解决了理论力学中必须解决的问题，然后才能充分了解高度紧凑的DNA是如何用于转录、复制和重组过程的。

The principal investigator will do research on non-linear problems in Kirchhoff's theory of elastic rods with the goals of (i) finding, for knotted and unknotted closed rods and open rods subject to terminal forces and torques, precise analytic representations of equilibrium configurations that show both isolated points and intervals of self-contact; (ii) deriving practical necessary and sufficient conditions for an equilibrium configuration to be stable in the sense that it gives a strict local minimum to elastic energy; (iii) obtaining insight into the dependence of bifurcation diagrams on knot type and the presence of intrinsic curvature; (iv) understanding the way in which the occurrence of plectonemic loops leads to hysteresis in torsion-stretching experiments for elastic rods and in single molecule manipulation experiments on DNA. The analytical representations of equilibrium configurations will be employed to develop a new Metropolis Monte Carlo procedure for evaluating partition functions for thermally fluctuating DNA molecules subject to specified constraints and end conditions. Graduate students will participate in this research which lies, as described below, at the interface between modern continuum mechanics and molecular biology. Each human cell has a meter of DNA in a nucleus that is less than 1 micron in diameter. Throughout the life of the cell, its compacted DNA is in a state of rapid, yet controlled, activity, because the regulation of life functions requires repeated transcription of appropriate portions of the genetic code into strands of RNA. The present research project addresses issues in theoretical mechanics that must be resolved before one can attain full understanding of the way in which highly compacted DNA is made available for the processes of transcription, replication, and recombination.