Lattice Models of Polymers: Entanglement Complexity and Confined Geometries

聚合物是由被称为单体的重复分子单元组成的任何大分子。统计力学的晶格模型已被证明是研究聚合物在溶液中的相变行为和平衡性质的简单而有力的方法。为此，聚合物链由格子或晶格上的行走来表示，因此链中的每个单体都与其相邻的单体相距一步。有利的是，得到的晶格聚合物具有柔韧性(它可以呈现各种形状)，并且两个单体不能在同一位置的要求(排除的体积效应)很容易合并。尽管高度简化，但它对于预测和理解由于聚合物是大的柔性分子这一事实而产生的任何性质是有用的。与此同时，由于大量具有挑战性的开放数学问题，这些模型具有内在的数学兴趣，其中许多问题来自聚合物物理，最近来自分子生物学。例如，酶作用于DNA以消除缠结，从而使正常的细胞过程继续进行。为了研究这一点，可以用晶格高分子链两端连接成环的模型来描述DNA分子的大尺度结构，并发展了酶对DNA的作用模型。与此同时，人们有兴趣了解限制在纳米通道或衣壳中的聚合物(如病毒衣壳中的DNA)或在单分子原子力显微镜实验中拉伸时的纠缠复杂性。关于聚合物如何拉伸或堆积与其纠缠复杂性之间的联系，所有这些情况都存在悬而未决的问题。用来研究这些问题的格子模型已经发展起来。 我计划利用组合分析和计算机模拟继续我对聚合物晶格模型的研究，以便更好地了解酶对DNA的作用和受限聚合物的纠缠复杂性。在这样做的过程中，我致力于在我的研究项目中促进公平、多样性和包容性。我知道，妇女和少数群体在科学界的代表性历来偏低，因此将鼓励妇女、明显的少数群体成员、土著人和残疾人申请加入我的研究团队。这一研究项目的结果将引起分子生物学家和聚合物化学家的兴趣。就酶对DNA的作用而言，对其作用的更好理解有可能通过改进癌症治疗来影响加拿大人。同时，这些结果将增加我们对聚合物的晶格模型和聚合物体系中的相变的一般理解。参与研究计划的学员将受益于批判性思维和解决问题能力的发展，以及技术、合作和沟通技能的发展。

A polymer is any large molecule made up of repeated molecular units called monomers. The lattice models of Statistical Mechanics have proved to be simple but powerful for studying phase-change behaviour and equilibrium properties for polymers in solution. For this, a polymer chain is represented by a walk on a grid or lattice so that each monomer in the chain is one step apart from its neighbours. Advantageously, the resulting lattice polymer has flexibility (it can take on a variety of shapes) and the requirement that two monomers cannot be in the same location (the excluded volume effect) is easily incorporated. Although highly simplified, it is useful for predicting and understanding any properties which are a consequence of the fact that a polymer is a large flexible molecule. At the same time, these models are of intrinsic mathematical interest due to the wealth of challenging open mathematical questions, many of which have arisen from polymer physics and more recently from molecular biology. For example, enzymes act on DNA to remove entanglements in order for normal cellular processes to proceed. To study this, a model in which two ends of a lattice polymer chain are joined into a ring can be used to represent the large-scale structure of the DNA molecule, and models of enzyme action on DNA have been developed. At the same time, there is interest in understanding the entanglement complexity of polymers confined in nanochannels or capsids (such as DNA in a viral capsid) or when stretched as in single-molecule atomic force microscopy experiments. There are open questions for all these cases regarding the connection between how the polymer is stretched or packed and its entanglement complexity. Lattice models for investigating these questions have been developed. I plan to continue my research on lattice models of polymers using combinatorial analysis and computer simulations in order to better understand enzyme action on DNA and the entanglement complexity of confined polymers. In doing so, I am committed to promoting equity, diversity, and inclusion in my research program. I am aware that women and minorities are historically underrepresented in the sciences and will therefore encourage applications from women, members of visible minorities, indigenous persons, and persons with disabilities to join my research team. The results of this research program will be of interest to molecular biologists and polymer chemists. In the case of enzyme action on DNA, an improved understanding of their action has the potential to impact Canadians by leading to improved cancer treatments. At the same time, the results will add to our general understanding of lattice models of polymers and phase transitions in polymer systems. Trainees involved in the research program will benefit from the development of critical thinking and problem-solving abilities, as well as technical, collaborative and communication skills.