Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of topological data analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in data sets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features—connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex data sets. We first provide a TDA primer for biologists before exploring the use of TDA across biological sub‐disciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data are shape and shape is data, will be relevant as biology transitions into a data‐driven era where the meaningful interpretation of large data sets is a limiting factor.

中文翻译：

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未来事物的形态：拓扑数据分析和生物学，从分子到有机体。

形状就是数据，数据就是形状。生物学家习惯于思考生物分子、细胞、组织和生物体的形状是如何从遗传、发育和环境的影响中产生的。我们很少考虑数据本身具有形状和结构，或者可以测量数据的形状并对其进行分析。在这里，我们以生物学家和应用数学家都可以使用的方式回顾拓扑数据分析 (TDA) 在生物学中的应用。TDA 使用来自代数拓扑的原理来全面测量数据集中的形状。使用将数据点的相似性相互关联的函数，我们可以监控拓扑特征的演变——连接组件、回路和空隙。这种演变，一种拓扑特征，简洁地总结了大型、复杂的数据集。我们首先为生物学家提供 TDA 入门，然后再探索跨生物子学科、跨越结构生物学、分子生物学、进化和发展的 TDA 的使用。我们最后比较和对比不同的 TDA 方法及其在生物学中的应用潜力。TDA 的愿景，即数据就是形状，形状就是数据，随着生物学过渡到数据驱动时代，对大数据集的有意义的解释是一个限制因素，这一愿景将是相关的。