Biological macromolecules have intricate structures that underpin their biological functions. Understanding their structure-function relationships remains a challenge due to their structural complexity and functional variability. Although de Rham-Hodge theory, a landmark of twentieth-century mathematics, has had a tremendous impact on mathematics and physics, it has not been devised for macromolecular modeling and analysis. In this work, we introduce de Rham-Hodge theory as a unified paradigm for analyzing the geometry, topology, flexibility, and Hodge mode analysis of biological macromolecules. Geometric characteristics and topological invariants are obtained either from the Helmholtz-Hodge decomposition of the scalar, vector, and/or tensor fields of a macromolecule or from the spectral analysis of various Laplace-de Rham operators defined on the molecular manifolds. We propose Laplace-de Rham spectral-based models for predicting macromolecular flexibility. We further construct a Laplace-de Rham-Helfrich operator for revealing cryo-EM natural frequencies. Extensive experiments are carried out to demonstrate that the proposed de Rham-Hodge paradigm is one of the most versatile tools for the multiscale modeling and analysis of biological macromolecules and subcellular organelles. Accurate, reliable, and topological structure-preserving algorithms for implementing discrete exterior calculus (DEC) have been developed to facilitate the aforementioned modeling and analysis of biological macromolecules. The proposed de Rham-Hodge paradigm has potential applications to subcellular organelles and the structure construction from medium- or low-resolution cryo-EM maps, and functional predictions from massive biomolecular datasets.

中文翻译：

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生物分子的 de Rham-Hodge 分析和建模

生物大分子具有支撑其生物功能的复杂结构。由于其结构复杂性和功能可变性，理解它们的结构-功能关系仍然是一个挑战。尽管德拉姆-霍奇理论作为二十世纪数学的里程碑，对数学和物理学产生了巨大影响，但它并不是为高分子建模和分析而设计的。在这项工作中，我们引入 de Rham-Hodge 理论作为分析生物大分子的几何、拓扑、柔性和 Hodge 模式分析的统一范式。几何特征和拓扑不变量可以从大分子的标量场、矢量场和/或张量场的亥姆霍兹-霍奇分解中获得，也可以从分子流形上定义的各种拉普拉斯-德拉姆算子的谱分析中获得。我们提出了基于拉普拉斯-德拉姆光谱的模型来预测大分子的灵活性。我们进一步构造了一个 Laplace-de Rham-Helfrich 算子来揭示冷冻电镜固有频率。进行了大量的实验来证明所提出的 de Rham-Hodge 范式是生物大分子和亚细胞细胞器的多尺度建模和分析的最通用的工具之一。已经开发出用于实现离散外微积分（DEC）的准确、可靠和拓扑结构保持的算法，以促进上述生物大分子的建模和分析。所提出的 de Rham-Hodge 范式对于亚细胞器、中低分辨率冷冻电镜图的结构构建以及大量生物分子数据集的功能预测具有潜在的应用。