Contact Problems in Kirchhoff's Nonlinear Theory of Rods

主要研究者将对基尔霍夫弹性杆理论中的非线性问题进行研究，其目标是（i）为受终端力和扭矩的打结和未打结封闭杆和开放杆找到平衡构型的精确解析表示，该平衡构型显示了孤立点和自接触间隔;（ii）推导出平衡构形在给出弹性能的严格局部最小值的意义上是稳定的实际必要和充分条件;（iii）深入了解分叉图的依赖结类型和存在的固有曲率;（iv）了解plectonemic循环的发生导致滞后的方式在扭转拉伸实验的弹性棒和DNA的单分子操作实验。平衡构型的解析表示将被用来开发一个新的大都会蒙特卡罗程序评估分配函数的热起伏的DNA分子受到指定的限制和结束条件。 研究生将参与这项研究，如下所述，在现代连续介质力学和分子生物学之间的接口。每个人类细胞在直径小于1微米的细胞核中有一米的DNA。在细胞的整个生命过程中，其紧凑的DNA处于快速但受控的活性状态，因为生命功能的调节需要将遗传密码的适当部分重复转录成RNA链。本研究项目解决了理论力学中的问题，这些问题必须在人们能够充分理解高度紧凑的DNA可用于转录，复制和重组过程的方式之前得到解决。

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.