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

Abstract: Systems Biology Graphical Notation (SBGN) is a standardised notational system that visualises biochemical processes as networks. These visualizations lack a formal framework, so that the analysis of such networks through modelling and simulation is an entirely separate task, determined by a chosen modelling framework (e.g. differential equations, Petri nets, stochastic processes, graphs). A second research gap is the lack of a mathematical framework to compose network representations. 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. We address these challenges by introducing a category-theoretic formalism for biochemical processes visualised using SBGN Process Description (SBGN-PD) language. Using the theory of structured cospans, we construct a symmetric monoidal double category and demonstrate its horizontal 1-morphisms as SBGN Process Descriptions. We obtain organisational principles such as 'compositionality' (building a large SBGN-PD from smaller ones) and 'zooming-out' (abstracting away details in biochemical processes) defined in category-theoretic terms. We also formally investigate how a particular portion of a biochemical network influences the remaining portion of the network and vice versa. Throughout the paper, we illustrate our findings using standard SBGN-PD examples.