New Algorithms for Modeling Flexibility in Proteins

这项拟议的研究通过开发和应用刚性理论来描述蛋白质中的刚性区域和灵活性，从而将数学和生物学联系起来。刚性理论是以前应用于工程和材料科学的数学分支。这些刚性区域可以具有受力部分，其约束比稳定所需的约束更多，也可以具有只是刚性的区域。刚性区域之间的灵活连接允许发生功能上重要的运动。框架的刚性作为数学的一部分已经被研究了一个多世纪。有人建议使用这一理论来开发准确和自然的蛋白质表示，并结合现代计算技术来模拟和可视化蛋白质运动。这种新颖的合作，包括一位数学家、一位物理学家和一位生物化学家，将使基本的数学问题和重要的生物学应用得以解决。将有四个涉及数学、物理和生物的重点领域。准确地表示蛋白质中的相互作用以预测它们的柔性和刚性区域，这涉及将分子框架猜想应用于三维蛋白质键网络，以及发展张力整体理论来更真实地表示蛋白质。这种约束网络方法将被用来识别蛋白质内部以及跨进化的特定相互作用，特别是稳定或破坏结构的相互作用。柔性预测、蒙特卡罗方法和多边形链展开理论将被用来模拟蛋白质折叠和展开的过程，并检验其相变行为。这些动态方法也将被用来探索蛋白质及其分子伙伴(配体)如何弯曲，以优化结合在一起。刚性理论的这些应用将与现实势结合起来，以可视化蛋白质及其与配体的络合物的运动。这项研究的结果，以新算法和蛋白质灵活性的三维地图和动画的形式，将在网络上公布和提供。因此，数学的发展将集中在两个领域。一是发展刚性理论，通过严格证明和应用分子框架猜想，并通过张拉整体框架扩展分子框架的现实主义(它将不等式作为收缩和扩张的约束)。其次，三维多边形链展开，这是一个新兴的数学和计算几何领域，将在给定的约束集下进行探索。在生物学方面，正在解决三个具有挑战性的领域：模拟蛋白质展开途径，包括展开相变；确定蛋白质框架内关键的稳定相互作用；以及在蛋白质-配体结合的基础上模拟重要的功能灵活性。因此，这项研究解决了数学和生物学中的几个主要挑战，同时将数学进步与生物学应用相结合。

The proposed research bridges mathematics and biology by developing and applying rigidity theory, a branch of mathematics previously applied to engineering and material science, to characterize rigid regions and flexibility in proteins. These rigid regions can have stressed parts, with more constraints than are required for stability, as well as regions that are just rigid. Flexible linkages between the rigid regions allow functionally important motions to occur. The rigidity of frameworks has been studied as part of mathematics for over a century. It is proposed to use this theory to develop an accurate and natural representation of proteins, coupled with modern computational techniques for simulating and visualizing protein motion. This novel collaboration, involving a mathematician, a physicist, and a biochemist, will allow fundamental mathematical questions and important biological applications to be addressed. There will be four focus areas involving mathematics, physics and biology. Representing the interactions in proteins accurately to predict their flexible and rigid regions, which involves applying the molecular frameworks conjecture to three-dimensional protein bond networks, and developing tensegrity theory to represent proteins even more realistically. This constraint-network approach will be used to identify the specific interactions within proteins, and across evolution, that especially stabilize or destabilize the structure. Flexibility predictions, Monte Carlo methods, and polygonal chain unfolding theory will be incorporated to simulate the processes of protein folding and unfolding and to examine the phase transition behavior. These dynamic approaches will also be used to probe how proteins and their molecular partners (ligands) flex in order to optimize binding together. These applications of rigidity theory will be used in conjunction with realistic potentials to visualize the movements accessible to proteins and their complexes with ligands. The results of this research, in the form of new algorithms and three-dimensional maps and animations of protein flexibility, will be published and made available on the web. Thus, the mathematical developments will focus in two areas. One is advancing rigidity theory, through rigorously proving and applying the molecular framework conjecture and extending the realism of molecular frameworks through tensegrity frameworks (which incorporates inequalities as constraints for contraction and expansion). Secondly, three-dimensional polygonal chain unfolding, an emerging area of mathematics and computational geometry, will be explored under a given set of constraints. In biology, three challenging areas are being addressed: simulating protein unfolding pathways, including the unfolding phase transition; identifying key stabilizing interactions within protein frameworks; and modeling functionally important flexibility upon protein-ligand binding. Therefore, this research tackles several major challenges in mathematics and biology, while coupling the mathematical advances to biological applications.