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用限制在球体内的随机多边形来模拟，而提取出来的（即松弛的）圆形分子则用不受约束的松弛随机多边形来模拟。pi提出探索多边形结空间从约束到松弛过渡过程中发生的几何变化，并建立这些几何变化与多边形拓扑复杂性之间的相关性。本研究项目的结果将为某些结复杂性度量和某些几何度量之间的关系提供基准数据，其中所有数量都是作为随机多边形家族在放松之前和之后的平均值来测量的。该结果可指导诸如噬菌体P4病毒等实验数据的评价。为了实现本研究的目标，必须实现以下几个关键目标：a)开发一种快速、可靠和无偏的算法，以在限定体积内生成大量长等边随机多边形；b)发展了等边随机多边形的松弛方案及其相应算法；c)量化随机多边形从约束过渡到松弛时拓扑对几何变化的影响。d)利用松弛前后多边形的平均几何性质对随机多边形的拓扑性质进行推断。本研究将系统地研究当所考虑的结经历从体积约束到弛豫的过渡变化时，各种几何测度与平均意义上的结拓扑性质之间的关系。拟议的研究将揭示这些数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发生在每个生物体中复杂的相互作用网络中的关键角色。其中一种自组装过程是将遗传物质包装在病毒的衣壳中。人们对包装过程的细节所知甚少，因为在有限的小体积内，DNA通常以难以用实验量化的方式浓缩和折叠。从噬菌体P4衣壳上强行移除的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携带着关于DNA如何被包装在衣壳内的重要信息。如何破译这些信息的问题是提出研究的主要动机。限制在小体积内的圆形分子通常用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用不加约束的松弛随机多边形来建模。本研究将探讨多边形结空间从约束到松弛的转变过程中发生的几何变化，并建立这些几何变化与多边形拓扑复杂性之间的相关性。该结果将为某些结复杂度度量与某些几何度量之间的关系提供一些重要的基准数据，这对于我们全面了解DNA在紧密空间中的包装机制具有重要意义。他们的学生（从非常有天赋的高中生，到本科生、研究生和博士生）将开发数学工具和计算模型，这些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或紧密排列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.