In recent years, an abundance of new molecular structures have been elucidated using cryo-electron microscopy (cryo-EM), largely due to advances in hardware technology and data processing techniques. Owing to these exciting new developments, cryo-EM was selected by Nature Methods as the "Method of the Year 2015," and the Nobel Prize in Chemistry 2017 was awarded to three pioneers in the cryo-EM field: Jacques Dubochet, Joachim Frank, and Richard Henderson "for developing cryoelectron microscopy for the high-resolution structure determination of biomolecules in solution" [93]. The main goal of this article is to introduce the challenging and exciting computational tasks involved in reconstructing 3D molecular structures by cryo-EM. Determining molecular structures requires a wide range of computational tools in a variety of fields, including signal processing, estimation and detection theory, high-dimensional statistics, convex and nonconvex optimization, spectral algorithms, dimensionality reduction, and machine learning. The tools from these fi elds must be adapted to work under exceptionally challenging conditions, including extreme noise levels, the presence of missing data, and massive data sets as large as several terabytes. In addition, we present two statistical models, multireference alignment (MRA) and multitarget detection (MTD), that abstract away much of the intricacy of cryo-EM while retaining some of its essential features. Based on these abstractions, we discuss some recent intriguing results in the mathematical theory of cryo-EM and delineate relations with group, invariant, and information theories.

中文翻译：

---

单粒子冷冻电子显微镜：数学理论、计算挑战和机遇


近年来，由于硬件技术和数据处理技术的进步，人们利用冷冻电子显微镜（cryo-EM）阐明了大量新的分子结构。由于这些令人兴奋的新进展，冷冻电镜被《自然方法》评选为“2015 年度方法”，2017 年诺贝尔化学奖授予了冷冻电镜领域的三位先驱：雅克·杜博切特 (Jacques Dubochet)、约阿希姆·弗兰克 (Joachim Frank)、和 Richard Henderson“开发冷冻电子显微镜，用于溶液中生物分子的高分辨率结构测定”[93]。本文的主要目标是介绍通过冷冻电镜重建 3D 分子结构所涉及的具有挑战性和令人兴奋的计算任务。确定分子结构需要各个领域的广泛计算工具，包括信号处理、估计和检测理论、高维统计、凸和非凸优化、谱算法、降维和机器学习。这些领域的工具必须适应在异常具有挑战性的条件下工作，包括极端的噪声水平、丢失数据的存在以及高达数 TB 的海量数据集。此外，我们提出了两种统计模型，多参考对齐（MRA）和多目标检测（MTD），它们抽象了冷冻电镜的大部分复杂性，同时保留了其一些基本特征。基于这些抽象，我们讨论了冷冻电镜数学理论中最近一些有趣的结果，并描述了与群论、不变量论和信息论的关系。