Geometric mechanics of charged ribbons

PutkaradzeDMS-0908755 研究者建立了弹性弦的运动方程，弦上有刚性的“束”装药，这些装药可以绕着弹性基座旋转。 除了考虑弯曲的弹性能外，还考虑了电荷间的任意非局域相互作用。 这些研究是完全一般性的，可以包括静电相互作用，防止弦自相交的Lennard-Jones相互作用，或任何其他非局部相互作用。 研究者和他的同事推导出了非局部相互作用弹性弦运动的精确几何理论，并将其应用于各种问题。 这个理论是基于精确弹性杆的Simo-Marsden-Krishnaprasad理论。 在理论推导之后，找到了计算方程精确螺旋解的显式方法，并指出由于几何原因，可以导出螺旋解上线性波传播和稳定性的显式色散关系。 这种方法还提供了一种方法，用于计算线性组合的弹性常数，如果螺旋分子的形状和电荷之间的相互作用是已知的。 调查员还推导出一个离散的模拟theequations允许广泛的一类精确的解决方案，calledn-螺旋。 这些是离散的螺旋，在n次移位后图案重复。 研究了这些离散螺旋的分叉结构。 最后，它显示了如何使用helicesas的基本积木的建设更复杂的分子构象。 有机分子的长链，如蛋白质，构成了地球上所有生命的基础。 这些分子的形状对其在生物体内的功能起着决定性的作用，因此了解这些分子的内部结构是一项重要的任务。 已经发现，许多蛋白质的内部结构由紧密缠绕的螺旋组成，并且这些螺旋以复杂的方式堆积在一起。 螺旋是由静电力和弹性力（以及其他）的复杂相互作用而结合在一起的。 如何描述这些复杂形状的形成？ 为什么螺旋在自然界如此普遍？ 研究者和他的同事们通过推导分子链演化理论来接近这些方程，该理论可以包括单个原子之间许多可能的相互作用。研究者指出，由于几何原因，螺旋是简单分子的稳定态，不管原子之间的相互作用多么复杂。 这一理论构成了理解生物分子内部结构及其动力学的基础。 更好地理解分子的空间结构及其形成有助于为化学工业设计更好的聚合物，为医学应用设计更好的药物，以及为未来的高科技工程设计具有新的和改进的性能的材料。

PutkaradzeDMS-0908755 The investigator develops equations of motion for elasticstrings with rigid "bouquets" of charges that may swivel aroundthe elastic base. In addition to elastic energy of the bend,arbitrary nonlocal interactions between the charges areconsidered. The studies are kept completely general and caninclude electrostatic interactions, the Lennard-Jonesinteractions preventing self-intersection of the string, or anyother nonlocal interactions. The investigator and his colleaguesderive an exact geometric theory of the motion of the nonlocallyinteracting elastic strings and apply it to a variety ofproblems. This theory is based on the Simo-Marsden-Krishnaprasadtheory of exact elastic rods. After the theory is derived, anexplicit method for computing exact helical solutions of theequations is found. The investigator also shows that forgeometric reasons, an explicit dispersion relation for thepropagation and stability of linear waves on helical solutionscan be derived. This technique also provides a method forcomputing linear combinations of elastic constants if the shapeof helical molecules and the interactions between the charges areknown. The investigator also derives a discrete analogue of theequations allowing a wide class of exact solutions, calledn-helices. These are discrete helices with the pattern repeatingafter n shifts. The bifurcation structure of these discreten-helices is examined. Finally, it is shown how to use helicesas basic building blocks for construction of more complexmolecular conformations. Long chains of organic molecules, like proteins, form thebasis of all life on Earth. The shape of these molecules plays acrucial role for their function in a living body, and thusunderstanding of the inner structure of these molecules is animportant task. It has been found that the inner structure ofmany proteins consists of tightly wound helices, and thosehelices are packed together in a complex fashion. The helicesare held together by a complex interplay of electrostatic andelastic (and other) forces. How can one describe the formationof these complex shapes? Why are helices so ubiquitous inNature? The investigator and his colleagues approach thesequestions by deriving a theory of molecular chain evolution thatcan include many possible interactions between individual atoms. The investigator shows that, because of geometric reasons,helices are stationary states for simple molecules, no matter howcomplex the interaction between the atoms may be. This theoryforms a basis for understanding of the inner structure ofbiological molecules and their dynamics. Better understanding ofthe spatial structure of molecules and their formation helps indesigning better polymers for the chemical industry, better drugsfor medical applications, and materials with novel and improvedproperties for high-tech engineering of the future.