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

Technical Summary: A Mathematical Framework for Organising Principles in SBGN-PD

Problem Statement
Systems Biology Graphical Notation (SBGN) is a widely adopted standard for visualizing biochemical processes through three complementary languages: Process Description (SBGN-PD), Activity Flow (SBGN-ER), and Entity Relationship (SBGN-ER). While SBGN-PD effectively visualizes biochemical reaction networks at a detailed level, it remains purely a visual tool lacking an underlying mathematical framework. This absence limits the ability to perform compositional analysis, formal abstraction (zooming in/out), and integration with formal modeling approaches. Experimental studies often focus on subsystems due to the complexity of biological systems, yet there is a critical need for formal principles to compose these subsystems into larger networks and to abstract details without losing functional interfaces. Existing attempts to bridge SBGN visualizations with mathematical models (e.g., using asynchronous automata, Hybrid Functional Petri Nets, or textual representations) fail to support specific organizing principles: (a) formal composition and decomposition of networks, (b) formal zooming in/out of details, (c) compatibility between composition and abstraction, and (d) formal analysis of how a subnetwork depends on its environment and vice versa.

Methodology
The authors employ Applied Category Theory (ACT), specifically the theory of structured cospans within the framework of symmetric monoidal double categories (following Baez et al.). The methodology proceeds as follows:

1. Process Networks and Species: The authors define a process network as a mathematical structure modeling biochemical reactions. Unlike standard Petri nets which use source and target maps valued in natural numbers ($\mathbb{N}$), a process network consists of a set of entities $E$, a set of process species $B$, and $n$ "legs" (functions $l_i: B \to \mathcal{B}[E]$). These legs map process species to the free commutative monoid over $E$ with coefficients from the additive Boolean monoid $\mathcal{B} = (\{0,1\}, +)$. This choice reflects the SBGN-PD convention where precise stoichiometry is often absent; only the presence (1) or absence (0) of entities is tracked. The legs accommodate various arc types found in SBGN-PD (consumption, production, modulation, stimulation, catalysis, inhibition).
2. Category of Process Networks: The authors construct a category $\text{Process}_n$ where objects are process networks with $n$ legs and morphisms are pairs of functions preserving the network structure. They prove this category is finitely cocomplete (containing all finite colimits), enabling the composition of networks via pushouts.
3. Structured Cospans and Open Process Networks: To model networks with interfaces (allowing connection to other networks), the authors utilize structured cospans. They define an open process network as a cospan in $\text{Process}_n$ of the form $L(X) \xrightarrow{I} B \xleftarrow{O} L(Y)$, where $X$ and $Y$ are sets of interface entities, $B$ is the process network, and $L$ is a functor embedding sets into process networks.
4. Symmetric Monoidal Double Category: Using the theory of structured cospans, the authors construct a symmetric monoidal double category, $\text{OpenDouble}(\text{Process}_n)$.
   • Objects: Sets (interfaces).
   • Vertical 1-morphisms: Functions between sets.
   • Horizontal 1-morphisms: Open process networks.
   • 2-morphisms: Morphisms between open process networks that allow for the transformation of internal details (e.g., zooming out).
5. Macroscope: The authors introduce a combinatorial tool called a macroscope to analyze the influence between a process subnetwork and its environment. It quantifies how entities produced in the environment affect the subnetwork (inputs/modulators) and how entities produced in the subnetwork affect the environment (outputs).

Key Contributions and Results

• Formal Compositionality (Theorem 4.13): The paper establishes a symmetric monoidal double category where horizontal 1-morphisms correspond to SBGN-PD diagrams. This provides a rigorous mathematical mechanism to compose smaller biochemical networks into larger composite networks via pushouts and to combine them via monoidal products (disjoint union).
• Formal Abstraction (Zooming): The framework introduces morphisms of process networks that act as formal "zoom-out" or "zoom-in" operations. A morphism can map a detailed network to a coarser one by forgetting specific entities (e.g., omitting ATP/ADP in phosphorylation reactions) while preserving the structural relationships of the remaining components.
• Compatibility of Composition and Abstraction: A central result is the demonstration that the composition of networks and the process of zooming out are compatible. The horizontal composition of 2-morphisms ensures that if one zooms out two networks individually and then composes them, the result is equivalent to composing them first and then zooming out. This validates the consistency of the framework for multi-scale analysis.
• Macroscope for Subnetwork Analysis (Definition 4.27): The authors define the macroscope of a subnetwork $B'$ with respect to a larger network $B$. This tool formally captures the "interface" of influence: it identifies which entities produced by the environment modulate the subnetwork and which entities produced by the subnetwork modulate the environment. This allows for the study of dependencies between subsystems without simulating the entire network.
• Translation from SBGN-PD: The paper provides a sketch of how to translate SBGN-PD diagrams into process networks (Section 5), demonstrating the process on neuronal/muscle signaling and the MAPK cascade. This translation captures entity types, compartments, and connectivity structures, though it currently abstracts away auxiliary units and specific process node types.

Significance and Claims
The authors claim to be the first to analyze SBGN diagrams using category theory, building upon the foundational work of Baez and Master on Open Petri nets. The significance of this work lies in:

1. Bridging Visualization and Mathematics: It provides the first formal mathematical structure for SBGN-PD, moving beyond visual representation to enable compositional analysis.
2. Scalability and Modularity: By enabling the composition of networks and the formal abstraction of details, the framework addresses the "size issue" in systems biology, allowing researchers to study large-scale networks (like the Atlas of Inflammation Resolution) by composing smaller, experimentally verified subsystems.
3. Multi-Scale Analysis: The compatibility between composition and zooming allows for the construction of large networks from small ones and the subsequent abstraction of those networks to higher levels of organization, preserving functional interfaces.
4. Foundation for Future Tools: The framework is designed to be implemented in computational platforms based on category theory (e.g., AlgebraicJulia, CatColab), potentially leading to new software tools for the analysis and simulation of biochemical pathways.

Limitations and Future Directions
The authors acknowledge that the current work is a step toward a fully effective theory. Limitations include the lack of a complete, rigorous dictionary for translating all SBGN-PD features (such as auxiliary units, complex components, and logic/equivalence arcs) into process networks. Future directions include developing comprehensive compositional tools, implementing the framework in computational environments, creating functorial translations to existing formalisms (like ODEs or Petri nets), and extending the approach to the other SBGN languages (SBGN-AF and SBGN-ER).