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

TL;DR

This paper introduces a category-theoretic mathematical framework for the SBGN-PD visual language, enabling compositional analysis, abstraction, and integration of biochemical networks beyond visual representation.

Contribution

It develops a formal categorical structure for SBGN-PD, allowing systematic composition and abstraction of biochemical reaction networks.

Findings

Constructs a symmetric monoidal double category for SBGN-PD diagrams

Analyzes subnetwork influence on surrounding networks

Demonstrates framework applicability with standard examples

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.