A mathematical framework to study organising principles in graphical representations of biochemical processes

TL;DR

This work develops a category-theoretic framework to formalize graphical biochemical representations in SBGN-PD. By employing structured cospans and constructing a symmetric monoidal double category, it provides a compositional, zoomable formalism for biochemical reaction networks, with process networks and morphisms capturing how networks are built, decomposed, and abstracted. A macroscope is introduced to formalize environment–subnetwork influences, and a translation pipeline converts SBGN-PD diagrams into process networks, enabling compositional analysis at multiple scales. The framework aims to unify visualization with mathematical structure, supporting large-scale analyses and integration with formal modelling approaches, while outlining limitations and directions for further development and computational implementation.

Abstract

The complexity of molecular and cellular processes forces experimental studies to focus on subsystems. To study the functioning of biological systems across levels of structural and functional organisation, we require tools to compose and organise networks with different levels of detail and abstraction. Systems Biology Graphical Notation (SBGN) is a standardised notational system that visualises biochemical processes as networks. Despite their widespread adoption, SBGN languages remain purely visual and lack an underlying mathematical framework, limiting their compositional analysis, abstraction, and integration with formal modelling approaches. SBGN comprises three complementary visual languages-Process Description (SBGN-PD), Activity Flow (SBGN-AF), and Entity Relationship (SBGN-ER)-each operating at a different level of abstraction. In this manuscript, we introduce a category-theoretic formalism for SBGN-PD, a visual language to describe biochemical processes as biochemical reaction networks. Using the theory of structured cospans, we construct a symmetric monoidal double category whose horizontal 1-morphisms correspond to SBGN-PD diagrams. We also analyse how a designated subnetwork influences the surrounding network and how external entities, in turn, affect the internal reactions of the subnetwork. Our work addresses a key gap between biological visualisation and mathematical structure. It provides precise organising principles for SBGN-PD, including compositionality, enabling the construction of large biochemical reaction networks from smaller ones, and zooming out, allowing the abstraction of detailed biochemical mechanisms while preserving their functional interfaces. Throughout the paper, the proposed framework is illustrated using standard SBGN-PD examples, demonstrating its applicability to large-scale biochemical reaction networks.

Figures (26)

• Figure 1: Example of a SBGN Process Description describing a biochemical reaction (top), and the visualization of its formal abstraction (bottom right) as done in our framework (Definition \ref{['ARDefinition: Biochemical process networks']}). In the bottom left, we illustrate a translation of the SBGN-PD visualization to its abstraction, whose detail is explained Example \ref{['Example:MAPK']}.
• Figure 2: Illustration of building an SBGN-PD visualisation of the MAPK cascade by composing three reaction networks (a), (b) and (c), and two reaction networks (d) and (e), using the Theorem \ref{['ARMain Theorem1']}. SBGN images are derived from the MAPK cascade example on Page 65 in rougny_systems_2019.
• Figure 3: An illustration of building an SBGN-PD visualisation of the Insulin-like Growth Factor (IGF) signalling by composing the MAPK cascade with two other biochemical molecular networks (marked in blue and grey) using the Theorem \ref{['ARMain Theorem1']}. SBGN images are derived from IGF signalling and MAPK cascade examples on Pages 64 and 65 in rougny_systems_2019.
• Figure 4: Some geometric shapes used in Figure \ref{['fig:construction-of-insulinsignalling']}. Here, (a), (b), (c), (d) and (e), respectively, denote association, complex node, stimulation arc, compartment node and phenotype. SBGN images are derived from the reference card on Page 75 in rougny_systems_2019.
• Figure 5: An illustration of the encapsulation node submap, the reference node tag and the equivalence arc in the SBGN-PD visualisation of the IGF signalling. The SBGN image is taken from the example of IGF signalling on Page 64, rougny_systems_2019.

Theorems & Definitions (50)

• Definition 3.1: Process network and process species
• Definition 3.2: Evaluation function of a process species
• Remark 3.3
• Definition 3.4: Morphism of process networks
• Lemma 3.5
• proof
• Example 3.6
• Proposition 4.1
• proof
• Lemma 4.2