Modeling GRNs with a Probabilistic Categorical Framework

Abstract: Understanding the complex and stochastic nature of Gene Regulatory Networks (GRNs) remains a central challenge in systems biology. Existing modeling paradigms often struggle to effectively capture the intricate, multi-factor regulatory logic and to rigorously manage the dual uncertainties of network structure and kinetic parameters. In response, this work introduces the Probabilistic Categorical GRN(PC-GRN) framework. It is a novel theoretical approach founded on the synergistic integration of three core methodologies. Firstly, category theory provides a formal language for the modularity and composition of regulatory pathways. Secondly, Bayesian Typed Petri Nets (BTPNs) serve as an interpretable,mechanistic substrate for modeling stochastic cellular processes, with kinetic parameters themselves represented as probability distributions. The central innovation of PC-GRN is its end-to-end generative Bayesian inference engine, which learns a full posterior distribution over BTPN models (P (G, {\Theta}|D)) directly from data. This is achieved by the novel interplay of a GFlowNet, which learns a policy to sample network topologies, and a HyperNetwork, which performs amortized inference to predict their corresponding parameter distributions. The resulting framework provides a mathematically rigorous, biologically interpretable, and uncertainty-aware representation of GRNs, advancing predictive modeling and systems-level analysis.