Data Sheet 3_Geometric multidimensional representation of omic signatures.zip

IntroductionMulti-omic signatures are widely used in biomarker discovery, precision oncology, and systems biology, yet they are typically treated as vectors or composite scores that collapse intrinsically multidimensional biological organization into one-dimensional summaries. As a result, their internal structure, contextual dependencies, and functional coherence remain largely inaccessible.
MethodsHere, we introduce a geometric framework that reconceptualizes omic signatures as multidimensional informational entities whose biological meaning arises from structural organization rather than molecular membership alone. Each signature is embedded in a shared latent space integrating regulatory, phenotypic, microenvironmental, immune, and clinical constraints, and represented as a convex polytope. This representation preserves internal organization and enables intrinsic geometric measurements—including barycenter distance, volume, anisotropy, and asymmetry—that quantify concordance, divergence, and latent complexity. We applied this framework to 24,796 metabolic regulatory circuitries reconstructed across 32 TCGA cancer types, encoded as paired regulatory and metabolic signatures in an 18-dimensional latent space.
ResultsGeometric analysis shows that discordance predominates: most circuitries occupy strong or extreme discordance regimes and display high-dimensional, frequently asymmetric geometries, whereas fully concordant circuitries are rare and structurally constrained. These geometric phenotypes stratify metabolic pathways and superfamilies in reproducible, non-uniform patterns that are not readily captured by conventional vector- or network-based representations.
DiscussionBy transforming omic signatures into measurable geometric objects, this framework provides a principled approach for the comparison and de-redundancy of multi-omic biomarkers, providing a scalable method for analyzing complex regulatory systems across cancer and beyond. All geometric representations and derived descriptors are available through the SigPolytope Shiny application (https://sigpolytope.shinyapps.io/geometricatlas/).

引言：多组学特征（multi-omic signatures）目前已广泛应用于生物标志物发现、精准肿瘤学与系统生物学研究中，但现有研究通常将其视为向量或复合得分，将本征多维的生物组织压缩为一维总结。因此，其内部结构、上下文依赖关系与功能一致性在很大程度上仍难以获取。
方法：本研究提出一种几何框架，将组学特征重新概念化为多维信息实体，其生物学意义源于结构组织而非仅由分子成员决定。每个特征被嵌入一个整合了调控、表型、微环境、免疫与临床约束的共享潜空间，并以凸多面体（convex polytope）形式表征。该表征保留了内部结构，支持本征几何测量——包括重心距离、体积、各向异性与不对称性——以量化一致性、差异性与潜复杂度。我们将此框架应用于32种TCGA癌症类型中重构的24,796个代谢调控回路，这些回路以18维潜空间中的配对调控特征与代谢特征形式进行编码。
结果：几何分析显示，不一致性占据主导地位：大多数回路处于强或极端不一致状态，呈现高维且常为不对称的几何结构，而完全一致的回路极为罕见且受结构约束。这些几何表型以可重复的非均匀模式对代谢通路与超家族进行分层，这一特性无法通过传统的向量或基于网络的表征轻易捕捉。
讨论：通过将组学特征转化为可测量的几何对象，本框架为多组学生物标志物的比较与去冗余提供了一套严谨的方法，为跨癌症及其他领域的复杂调控系统分析提供了可扩展的手段。所有几何表征与衍生描述符均可通过SigPolytope Shiny应用（https://sigpolytope.shinyapps.io/geometricatlas/）获取。