Moduli Spaces of Polygons with Applications to Protein-Macrocycle Docking

合成大环化合物（MC）因其作为治疗剂的潜力而受到越来越多的关注。除了作为治疗方法的前景之外，由于其复杂的拓扑结构，MC还具有巨大的数学兴趣。然而，由于几个原因，缺乏用于建模MC及其蛋白质靶标的3D复合物的有效方法。MC对于用于对接小的非环状药物分子的当前方法来说太大且太复杂。它们对于蛋白质-蛋白质对接的方法来说也太灵活了。这个项目的目标是开发几何方法来有效地模拟大分子的柔顺相互作用，也被称为柔性对接问题。目前的方法是基于暴力直接计算机模拟。分子的大尺寸和待探索空间的受限性质使得这些不切实际。用于大环结构建模的软件BRIKARD和分子对接套件CLUSPRO中使用的高效精确分析方法将通过数学结构的发展结合到一个新的对接协议中，能够有效地处理灵活的配体蛋白质对接。由此产生的软件将立即用于制药工业中的大环药物设计领域。此外，分子柔性和机器人技术之间的联系为创建简单的模型提供了丰富的基础，这些模型将用于向高中生介绍机器人技术，柔性和分子建模的基本思想的外展活动。蛋白质-大环对接的核心在于全局搜索能量最小值，这是一个具有挑战性的问题，因为搜索空间的高维性，崎岖的能量景观，能量函数评价成本高。大环形状空间的非平凡结构有效地排除了使用当前方法成功建模蛋白质-大环相互作用。 通过适当的流形描述对应于不同程度的分子柔性的构象空间，将克服这一障碍。搜索将通过引入黎曼度量来引导，以适当地权衡配体-蛋白质旋转-平移和环状配体的灵活性。还将并入能量项以解释配体和结合口袋的空间相互作用。这将允许低能量区域的有效采样，避免接合构象空间的高能量区域。 由此产生的混合方法将开始与粗粒度的全球采样，以确定有前途的领域的蛋白质配体shapespace，其次是中期范围的调整和优化，使用更详细的评分功能，最后，有针对性的精细resscribe，以获得高分辨率的结构。所创建的对接框架将适用于广泛的任务，包括无约束配体对接和蛋白质环建模。智力方面的优点包括：（i）理解具有固定边长和角度的欧几里德空间R3中的多边形的模空间，（ii）导出用于在蛋白质-配体复合物中的相对放置和顺应形状变化中搜索的适当权重，以及（iii）改进一般对接协议的有效性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Synthetic Macrocycles (MCs) are receiving increased attention for their potential as therapeutic agents. Beside their promise as therapeutics, MCs also have great mathematical interest due to their complex topological structure. However, for several reasons effective methods for modeling 3D complexes of MCs and their protein targets are lacking. MCs are too large and complex for current methods used for docking small non-cyclic drug molecules. They are also too flexible for methods of protein-protein docking. The goal of this project is to develop geometrical methods to efficiently model the compliant interaction of macromolecules, also known as the Flexible Docking Problem. Current methods are based on brute force direct computer simulations. The large size of the molecules and the confined nature of the space to be explored renders these impractical. The efficient exact analytical methodologies used in the software BRIKARD for macrocycle structure modeling and in the molecular docking suite CLUSPRO will be combined through the development of mathematical structures into a new docking protocol, capable of efficiently handling flexible ligand protein docking. The resulting software will find immediate use in the pharmaceutical industry in the area of macrocyclic drug design. Moreover, the connection between molecular flexibility and robotics offers a rich ground for creating simple models that will be used for outreach activities introducing high school students to basic ideas of robotics, flexibility and molecular modeling.At the heart of protein-macrocycle docking lies a global search for an energy minimum, a challenging problem due to high dimensionality of the search space, rugged energy landscape, and high cost of energy function evaluation. The non-trivial structure of macrocycle shapespace effectively precludes successful modeling of protein-macrocycle interactions using current methods. This obstacle will be overcome by describing conformational spaces corresponding to different degrees of molecular flexibility by appropriate manifolds. The search will be guided by introducing a Riemannian metric to properly weigh ligand-protein rotation-translations and cyclic ligand flexibility. Energetic terms will also be incorporated to account for steric interactions of the ligand and binding pocket. This will allow efficient sampling of low energy regions, avoiding high energy areas of the joint conformational space. The resulting hybrid method will begin with coarse grained global sampling to identify promising areas of protein-ligand shapespace, followed by medium-range adjustments and optimization using more detailed scoring functions, and, finally, targeted fine resampling to obtain high resolution structures. The docking framework created will be applicable to a broad spectrum of tasks including unconstrained ligand docking and protein loop modeling. Intellectual merits include: (i) understanding moduli spaces of polygons in Euclidean space R3 with fixed side lengths and angles, (ii) deriving proper weights for searches in the relative placement and compliant shape changes in a protein-ligand complex, and (iii) improvement of effectiveness of general docking protocols. The resulting software will be released as open source and made freely available through the widely used CLUSPRO server.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.