An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich structure to account for a wide range of geometries that can be modelled after an energy landscape. Second, many standard data analysis tools initially developed for data in Euclidean space can also be generalised to data on a Riemannian manifold. In the context of protein dynamics, a conceptual challenge comes from the lack of a suitable smooth manifold and the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a novel local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold of point clouds modulo rigid body group actions and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. It performs exceptionally well on coarse-grained molecular dynamics simulated data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

翻译：一种日益普遍的观点认为，蛋白质动力学数据集存在于低构象能的非线性子空间中。因此，理想的数据分析工具应考虑这种非线性几何特征。黎曼几何框架适用于此有多种原因：首先，其丰富的结构能够模拟能量景观下的多种几何形态；其次，许多最初为欧氏空间数据开发的标准数据分析工具可推广到黎曼流形上的数据。在蛋白质动力学背景下，主要概念性挑战在于缺乏合适的平滑流形，以及缺少基于能量景观构建平滑黎曼结构的指导原则。此外，测地线及其相关映射的计算可行性也是一大难题。本研究针对这些挑战展开：论文第一部分开发了一种新型局部近似技术，能以计算可行方式计算黎曼流形上的测地线及相关映射；第二部分构建了基于刚体群作用模的点云平滑流形，以及基于蛋白质构象能量景观的黎曼结构。通过多项蛋白质动力学相关数据分析任务测试，该黎曼几何方法在粗粒化分子动力学模拟数据上表现优异。特别地，给定起止点的测地线能够近似重建中等变形有序转变过程中蛋白质的分子动力学轨迹。即使面对大尺度变形，该黎曼蛋白质几何方法也能在笔记本电脑上于数秒内给出物理上合理的汇总统计量并恢复潜在维度。