Collaborative Research: A Study of the Transition of Knot Space from Confinement to Relaxation

限制在小体积内的圆形分子通常由限制在球体中的随机多边形建模，并且提取的（即松弛的）圆形分子由没有限制的松弛随机多边形建模。PI建议探索多边形knotspace从限制到松弛的过渡期间发生的几何变化，并建立这些几何变化和多边形的拓扑复杂性之间的相关性。本研究项目的结果将提供基准数据之间的关系，某些结的复杂性措施和一些几何措施，其中所有的数量被测量为家庭的随机多边形之前和之后，他们放松的平均值。结果可以指导实验数据的评估，例如噬菌体P4病毒的情况下可用的数据。为了达到本研究的目的，必须实现以下几个关键目标：a）开发一种快速、可靠、无偏的算法，以在有限体积内生成大量的长等边随机多边形：B）开发等边随机多边形的松弛格式及其相应的算法; c）当从限制过渡到松弛时，拓扑对随机多边形的几何变化的影响的量化，以及利用松弛前后随机多边形的平均几何性质，识别关于随机多边形拓扑性质的推论。拟议的研究将提供一个系统的研究之间的关系，各种几何措施和拓扑性质的平均意义上的结时，考虑中的结经历了从体积限制到松弛的过渡变化。拟议的研究将揭示这些量之间潜在的重要和有趣的关系，以及限制在这些关系中的作用。众所周知，大分子自组装过程是每个生物体中发生的复杂相互作用网络中的关键角色。这些自我组装过程之一是将遗传物质包装在病毒衣壳中。关于包装过程的细节知之甚少，因为在有限的小体积中，DNA通常以难以实验量化的方式浓缩和折叠。从噬菌体P4衣壳中被强行移除的DNA分子通常会形成复杂的结，这是包装过程的结果。因此，提取的DNA携带关于DNA如何包装在衣壳内的重要信息。如何破译这些信息的问题是拟议研究的主要动机。被限制在小体积内的圆形分子通常由被限制在球体内的随机多边形来模拟。另一方面，提取的圆形分子通常由松弛的无约束随机多边形建模。拟议的研究将探讨多边形结空间从限制到松弛的过渡期间发生的几何变化，并建立这些几何变化和多边形的拓扑复杂性之间的相关性。这些结果将提供一些基本的基准数据之间的关系，某些结的复杂性措施和一些几何措施，这是重要的，以便我们充分了解DNA包装在一个紧凑的空间的机制。PI的学生（从特别有才华的高中生，到本科生，研究生和博士生）。学生）将开发数学工具和计算模型，将免费提供给科学界和/或感兴趣的教育工作者。这项工作的结果可以用于生物学和物理学等领域，以检查高度浓缩的DNA或紧密堆积的聚合物模型的有效性。

Circular molecules confined to a small volume are often modeled by random polygons confined in a sphere and extracted (that is relaxed) circular molecules are modeled by relaxed random polygons without confinement. The PIs propose to explore the geometric changes that occur during the transition of the polygonal knotspace from confinement to relaxation and to establish correlations between these geometric changes and the topological complexity of the polygons. The results of this research project will provide benchmark data on the relationships between certain knot complexity measures and some geometric measures, where all quantities are measured as averages over families of random polygons before and after they are relaxed. The results can guide the evaluation of experimental data such as the data available in the case of the bacteriophage P4 virus. To reach the goal of the proposed research, several critical objectives must be achieved: a) The development of a fast, reliable, and unbiased algorithm to generate large sets of long equilateral random polygons within a confining volume; b) The development of relaxation schemes for equilateral random polygons and their corresponding algorithms; c) Quantification of the effect of topology on geometric changes of random polygons when transitioning from confinement to relaxation and d) Identification of inferences about topological properties of the random polygons using the average geometric properties of the polygons before and after relaxation. The proposed research will provide a systematic study between the relationships between various geometric measures and topological properties of knots in the average sense when the knots under consideration undergo a transition change from volume confinement to relaxation. The proposed research will reveal potentially important and interesting relationships among these quantities and the role of confinement in these relationships. It is well known that macromolecular self-assembly processes are key players in the complex network of interactions that take place in every organism. One of these self-assembly processes is the packing of the genetic material in the capsids of viruses. Little is know about the details of the packing processes, because in a confined small volume DNA is usually condensed and folds in ways that are difficult to quantify experimentally. DNA molecules that are forcefully removed from bacteriophage P4 capsids often form complicated knots that are a result of the packing process. Thus, the extracted DNA carries important information about how the DNA is packed inside the capsids. The question of how to decipher such information is a main motivation of the proposed research. Circular molecules confined to a small volume are often modeled by random polygons confined in a sphere. On the other hand, extracted circular molecules are usually modeled by relaxed random polygons without confinement. The proposed research will explore the geometric changes that occur during the transition of the polygonal knot space from confinement to relaxation and to establish correlations between these geometric changes and the topological complexity of the polygons. The results will provide some essential benchmark data on the relationships between certain knot complexity measures and some geometric measures, which are important in order for us to fully understand the mechanism of DNA packing in a tight space. The PIs their students (ranging from exceptionally talented high-school students, to undergraduates, graduates, and Ph. D. students) will develop mathematical tools and computational models that will be made freely available to the scientific community and/or interested educators. The results of the work can be used in areas such as biology and physics to check the validity of models of highly condensed DNA or tightly packed polymers.