Proteins operate and interact with partners by dynamically exchanging between functional substates of a conformational ensemble on a rugged free energy landscape. Understanding how these substates are linked by coordinated, collective motions requires exploring a high-dimensional space, which remains a tremendous challenge. While molecular dynamics simulations can provide atomically detailed insight into the dynamics, computational demands to adequately sample conformational ensembles of large biomolecules and their complexes often require tremendous resources. Kinematic models can provide high-level insights into conformational ensembles and molecular rigidity beyond the reach of molecular dynamics by reducing the dimensionality of the search space. Here, we model a protein as a kinematic linkage and present a new geometric method to characterize molecular rigidity from the constraint manifold Q and its tangent space ${\mathcal{T}}_{\mathbf{q}}Q$ at the current configuration q. In contrast to methods based on combinatorial constraint counting, our method is valid for both generic and non-generic, e.g., singular configurations. Importantly, our geometric approach provides an explicit basis for collective motions along floppy modes, resulting in an efficient procedure to probe conformational space. An atomically detailed structural characterization of coordinated, collective motions would allow us to engineer or allosterically modulate biomolecules by selectively stabilizing conformations that enhance or inhibit function with broad implications for human health.

蛋白质通过在坚固的自由能格局上的构象集合的功能性亚状态之间动态交换，来运作并与伴侣互动。要了解这些子状态如何通过协调的集体运动联系起来，就需要探索一个高维空间，这仍然是一个巨大的挑战。尽管分子动力学模拟可以提供动力学方面的原子详细信息，但要充分采样大型生物分子及其复合物的构象集合的计算需求通常需要大量资源。运动学模型可以通过减小搜索空间的维数，为分子构象和分子刚度提供高级的洞察力，而这超出了分子动力学。这里，Q及其切线空间${\mathcal{Ť}}_{\mathbf{q}}问$在当前配置q。与基于组合约束计数的方法相比，我们的方法对通用和非通用（例如，单数配置）均有效。重要的是，我们的几何方法为沿着软盘模式的集体运动提供了明确的基础，从而提供了一种有效的探查构象空间的程序。原子序的，协调的集体运动的结构表征，将使我们能够通过选择性地稳定增强或抑制功能的构象，对生物分子进行工程改造或变构调节，从而对人体健康产生广泛影响。