Markovian Promoter Models: A Mechanistic Alternative to Hill Functions in Gene Regulatory Networks
Abstract
Gene regulatory networks are typically modeled using ordinary differential equations (ODEs) with phenomenological Hill functions to represent transcriptional regulation. While computationally efficient, Hill functions lack mechanistic grounding and cannot capture stochastic promoter dynamics. We present a hybrid Markovian-ODE framework that explicitly models discrete promoter states while maintaining computational tractability. Our approach tracks individual transcription factor binding events as a continuous-time Markov chain, coupled with deterministic ODEs for molecular concentrations. We validate this framework on seven gene regulatory systems spanning basic to advanced complexity: the GAL system, repressilator, Goodwin oscillator, toggle switch, incoherent feed-forward loop, p53-Mdm2 oscillator, and NF-$κ$B pathway. Comparison with stochastic simulation algorithm (SSA) ground truth demonstrates that Markovian promoter models achieve similar accuracy to full stochastic simulations while being 10-100$\times$ faster. Our framework provides a mechanistic foundation for gene regulation modeling and enables investigation of promoter-level stochasticity in complex regulatory networks.
Summary
This paper addresses the limitations of using Hill functions in modeling gene regulatory networks (GRNs). While Hill functions are computationally efficient, they lack mechanistic grounding and fail to capture stochastic promoter dynamics. The authors propose a hybrid Markovian-ODE framework to explicitly model discrete promoter states while maintaining computational tractability. This approach tracks transcription factor binding events as a continuous-time Markov chain, coupled with deterministic ODEs for molecular concentrations. The framework is validated on seven gene regulatory systems with varying complexity, including the GAL system, repressilator, and NF-κB pathway. The key finding is that Markovian promoter models achieve accuracy comparable to full stochastic simulations using the Stochastic Simulation Algorithm (SSA) while being significantly faster (10-100x). The authors demonstrate that their framework can capture ultrasensitivity, gene-specific responses, and stochastic fluctuations observed in GRNs. The model is parameterized using accessible transcriptomic and proteomic data, making it practical for real-world applications. This research matters to the field because it provides a more mechanistic and computationally efficient alternative to Hill functions for modeling GRNs, enabling the investigation of promoter-level stochasticity in complex regulatory networks and facilitating the integration of gene regulation with stochastic chemical kinetics frameworks.
Key Insights
- •The proposed Markovian-ODE framework models promoter state transitions as a continuous-time Markov chain, coupled with ODEs for molecular concentrations, providing a mechanistic alternative to Hill functions.
- •The framework achieves similar accuracy to full SSA simulations while providing a 10-100x speedup in computational time.
- •The framework is CME-compatible, ensuring that all reactions remain 0th, 1st, or 2nd order, allowing seamless integration with existing stochastic simulation frameworks.
- •The model parameters can be inferred from accessible experimental data, such as transcriptomic dose-response curves and time-series data, rather than requiring detailed binding constants.
- •The authors demonstrated that ultrasensitivity emerges naturally from discrete promoter states without requiring phenomenological Hill functions, as shown in the GAL system (n_eff = 3.2± 0.3)
- •The framework was validated on seven gene regulatory systems with different network architectures, dynamic behaviors, and biological functions, demonstrating its generalizability.
- •A detailed step-by-step conversion procedure for the GAL system is provided as a template for converting other gene regulatory systems from traditional ODE models to the Markovian promoter framework.
Practical Implications
- •The framework can be used to model complex GRNs with greater accuracy and computational efficiency, enabling the investigation of promoter-level stochasticity and its impact on cellular behavior.
- •Researchers in systems biology and synthetic biology can use this framework to design and analyze gene circuits with greater precision, leading to improved control and predictability of biological systems.
- •The framework can be integrated into whole-cell models to simulate the dynamics of cellular processes with greater fidelity, providing insights into the complex interactions between different cellular components.
- •The framework can be used to analyze experimental data from transcriptomic and proteomic studies, providing a more mechanistic interpretation of gene regulation and its response to environmental stimuli.
- •Future research directions include developing adaptive time-stepping methods to further improve computational efficiency and extending the framework to model more complex regulatory mechanisms, such as chromatin modifications and epigenetic regulation.