Folding and Unfolding of Polygonal Linkages, with Applications to Structural Biology

摘要我们正在研究多边形连杆机构(以及其他具有多个自由度的部分刚性结构)在二维和三维的折叠和展开过程，重点是在分子生物学中的应用。我们的主要关注点是基本的数学性质(拓扑、几何、代数)和新的算法方法，长期目标是了解蛋白质折叠模拟问题的基本规律。我们提出了一种新的蒙特卡罗方法，用于二维能量最小化模拟中的移动生成器，该方法基于同时改变多个角度并且保证无碰撞的机构，并研究了将该方法扩展到三维的可行性。这项研究涉及的数学包括代数拓扑、代数几何和刚性理论的技术，而计算工作有望导致开发有效的数据结构和算法来规划、分析、逼近和跟踪此类运动。除了其内在的科学价值，我们预计这个项目将通过一种整合研究、教学、机构间和跨学科合作的新颖方式产生更广泛的影响。这笔补助金将允许在一所女子学院任教的一名专业人士建立一个包括(除本科生以外的)研究生在内的综合研究小组。通过将顶尖的研究资源和强大的合作带到一所致力于在文科环境中促进女性技术教育的本科生机构，我们预计将影响未来的跨学科课程，加强唯一一所设有工程项目的女子学院的研究和教育基础设施，并为培养多样化的学生群体提供研究生工作、研究和教学的模式。

ABSTRACTWe are studying folding and unfolding processes for polygonal linkages (and other partially rigid structures with many degrees of freedom) in dimensions two and three, with particular emphasis on applications in molecular biology. Our primary focus is on fundamental mathematical properties (topologic, geometric, algebraic) and novel algorithmic approaches, with the long run goal of understanding the fundamental laws of the protein folding simulation problem. We are proposing a novel Monte Carlo approach for move generators in energy minimization simulations in two dimensions, based on mechanisms which simultaneously change many angles and are guaranteed to be collision free, and we investigate the feasibility of extending the approach in three dimensions. The mathematics involved in this research includes techniques from algebraic topology, algebraic geometry and rigidity theory, while the computational effort is expected to lead to developing efficient data structures and algorithms for planning, analyzing, approximating, and tracking such motions. Besides its intrinsic scientific merit, we anticipate this project to have broader impact through a novel way of integrating research, teaching, inter-institutional and interdisciplinary collaborations. The grant will allow one of the PIs, who teaches at the undergraduate level at a women's college, to establish an integrated research group including (besides undergraduates) graduate students. By bringing top research resources and powerful collaborations to an undergraduate institution devoted to promoting the technical education of women in a liberal arts environment, we expect to influence the interdisciplinary curriculum of the future, enhance the infrastructure for research and education at the only women's college with an engineering program, and generate models for training a diverse body of students for graduate work, research and teaching.