生物分子数据的多维持久性
Multidimensional persistence in biomolecular data
摘要：
持久同源性已经成为大数据（包括生物分子数据）拓扑简化的流行技术。多维持久性在弥合几何和拓扑之间的鸿沟方面有着相当大的希望。然而，其实用和坚固的结构一直是一个挑战。我们介绍了两类多维持久性，即伪多维持久性和多尺度多维持久性。前者是通过对高维数据（例如来自分子动力学或偏微分方程的结果）重复应用持久同源过滤而产生的。后者是通过各向同性和各向异性的尺度，创造新的simiplicial复合体和相关的拓扑空间。通过蛋白质折叠、蛋白质柔性分析、冷冻电子显微镜数据的拓扑去噪和纳米颗粒的尺度依赖性，证明了所提出的拓扑方法的实用性、鲁棒性和效率。在多维持久性中观察到部分折叠蛋白质与未折叠蛋白质之间的拓扑转变。通过Laplace-Beltrami流实现噪声拓扑特征和分子拓扑指纹的分离。多尺度多维持久同源性揭示了Betti-0不变量的相对局部特征和Betti-1和Betti-2不变量的相对全局特征。
英文摘要：
Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudo-multidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryo-electron microscopy data, and the scale dependence of nano particles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace-Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.
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