Data Sheet 6_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）在生物标志物发现、精准肿瘤学与系统生物学领域应用广泛，但传统研究通常将其视为向量或复合评分，将本就具有多维属性的生物学组织压缩为一维总结。因此，其内部结构、上下文依赖性与功能一致性大多仍难以被解析。
方法
本研究提出一种几何框架，将组学特征重新定义为多维信息实体，其生物学意义源于结构组织而非仅分子组成。每个特征被嵌入整合了调控、表型、微环境、免疫与临床约束的共享隐空间，并以凸多面体（convex polytope）形式表征。该表征方式保留了内部组织特性，支持开展固有几何测量——包括重心距离、体积、各向异性与不对称性——以量化一致性、差异性与隐复杂度。我们将此框架应用于32种癌症基因组图谱（The Cancer Genome Atlas, TCGA）癌症类型中重构的24796条代谢调控环路，这些环路被编码为18维隐空间中的配对调控与代谢特征。
结果
几何分析结果显示，不一致性占主导：多数调控环路处于强或极端不一致状态，并呈现出高维且常为不对称的几何结构，而完全一致的环路极为罕见且结构受限。这些几何表型以可重复且非均匀的模式对代谢通路及超家族进行分层，这类模式无法通过传统的基于向量或网络的表征方式轻易捕捉。
讨论
通过将组学特征转化为可测量的几何对象，该框架为多组学生物标志物的对比与去冗余提供了严谨的方法，为分析癌症乃至其他领域的复杂调控系统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应用程序（https://sigpolytope.shinyapps.io/geometricatlas/）获取。