New Algorithms for Modeling Flexibility in Proteins

拟议的研究通过发展和应用刚性理论（以前应用于工程和材料科学的数学分支）来表征蛋白质的刚性区域和灵活性，从而将数学和生物学联系起来。这些刚性区域可能具有受力部件，具有比稳定性所需的更多约束，以及仅仅是刚性的区域。刚性区域之间的柔性连接允许发生功能重要的运动。框架的刚性作为数学的一部分已经研究了一个多世纪。建议使用这一理论来开发蛋白质的准确和自然的表示，再加上现代计算技术来模拟和可视化蛋白质运动。这一由数学家、物理学家和生物化学家组成的新颖合作，将使基本的数学问题和重要的生物学应用得以解决。将有四个重点领域涉及数学，物理和生物。准确表征蛋白质内部的相互作用以预测其柔性和刚性区域，这涉及到将分子框架猜想应用于蛋白质三维键网络，以及发展张拉整体理论以更真实地表征蛋白质。这种约束网络方法将用于识别蛋白质内部和进化过程中的特定相互作用，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纳入模拟蛋白质折叠和展开的过程，并检查相变行为。这些动态方法也将用于探测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电位结合使用，以可视化蛋白质及其配体复合物的运动。这项研究的结果将以新的算法、蛋白质灵活性的三维地图和动画的形式发表，并在网上提供。因此，数学的发展将集中在两个方面。一个是推进刚性理论，通过严格证明和应用分子框架猜想，并通过张拉整体框架（其中包含不等式作为收缩和膨胀的约束）扩展分子框架的现实性。其次，三维多边形链展开，数学和计算几何的一个新兴领域，将在给定的一组约束下探索。在生物学中，三个具有挑战性的领域正在得到解决：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；建模对蛋白质与配体结合的灵活性具有重要的功能。因此，本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.