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Thermodynamic Constraints Drive Hierarchical Preemption in Cellular Decision-Making: A Hybrid Petri Net Framework with Application to Bacillus subtilis Sporulation

Eugenio Simao

TL;DR

The problem addressed is how cells rapidly decide among competing pathways under energy scarcity. The authors develop SHYPN 2.0, a hybrid Petri net framework that couples stochastic regulatory logic with continuous energy flux and thermodynamic constraints to model Bacillus subtilis sporulation. They show that energy stress induces hierarchical preemption, enabling a rapid ATP-independent route that achieves a 16× improvement in energy efficiency and near-normal sporulation yield despite severe ATP depletion, driven by a thermodynamic free-energy landscape and energy buffering via ATP regeneration and GTP accumulation. This work provides a predictive, thermodynamically grounded account of energy-aware decision-making with potential applicability to other energy-limited cellular decisions and broader signal-hierarchy contexts.

Abstract

Cellular decision-making under stress involves rapid pathway selection despite energy scarcity. Here we demonstrate that thermodynamic constraints actively drive energy-efficient sporulation, where continuous metabolic sources enable system robustness through dynamic energy management. Using hybrid Petri nets (stochastic transitions with continuous sources) to model Bacillus subtilis sporulation, we show that stress conditions (ATP = 300 mM, 94% depletion) enable sporulation completion with extreme energy efficiency: 0.73 mM ATP per mature spore versus 11.6 mM ATP under normal conditions--a 16-fold efficiency gain. Despite ATP dropping to 1 mM (99.7% depletion) during the crisis, continuous ATP regeneration rescues the system, producing 67 mM mature spores (89% of normal yield) with only 49 mM total ATP consumption. This efficiency emerges from the interplay between stochastic regulatory transitions and continuous metabolic sources, where GTP accumulation (+4974 mM, 166% increase) provides an energy buffer while ATP regeneration (+240 mM) prevents complete depletion. The hybrid Petri net formalism--combining stochastic transitions for regulatory events with continuous sources for metabolic flux--extended with thermodynamic constraints through inhibitor arcs and energy-coupled rate functions, provides the mathematical foundation enabling this discovery by integrating discrete regulatory logic with continuous energy dynamics in a resource-aware concurrency model.

Thermodynamic Constraints Drive Hierarchical Preemption in Cellular Decision-Making: A Hybrid Petri Net Framework with Application to Bacillus subtilis Sporulation

TL;DR

The problem addressed is how cells rapidly decide among competing pathways under energy scarcity. The authors develop SHYPN 2.0, a hybrid Petri net framework that couples stochastic regulatory logic with continuous energy flux and thermodynamic constraints to model Bacillus subtilis sporulation. They show that energy stress induces hierarchical preemption, enabling a rapid ATP-independent route that achieves a 16× improvement in energy efficiency and near-normal sporulation yield despite severe ATP depletion, driven by a thermodynamic free-energy landscape and energy buffering via ATP regeneration and GTP accumulation. This work provides a predictive, thermodynamically grounded account of energy-aware decision-making with potential applicability to other energy-limited cellular decisions and broader signal-hierarchy contexts.

Abstract

Cellular decision-making under stress involves rapid pathway selection despite energy scarcity. Here we demonstrate that thermodynamic constraints actively drive energy-efficient sporulation, where continuous metabolic sources enable system robustness through dynamic energy management. Using hybrid Petri nets (stochastic transitions with continuous sources) to model Bacillus subtilis sporulation, we show that stress conditions (ATP = 300 mM, 94% depletion) enable sporulation completion with extreme energy efficiency: 0.73 mM ATP per mature spore versus 11.6 mM ATP under normal conditions--a 16-fold efficiency gain. Despite ATP dropping to 1 mM (99.7% depletion) during the crisis, continuous ATP regeneration rescues the system, producing 67 mM mature spores (89% of normal yield) with only 49 mM total ATP consumption. This efficiency emerges from the interplay between stochastic regulatory transitions and continuous metabolic sources, where GTP accumulation (+4974 mM, 166% increase) provides an energy buffer while ATP regeneration (+240 mM) prevents complete depletion. The hybrid Petri net formalism--combining stochastic transitions for regulatory events with continuous sources for metabolic flux--extended with thermodynamic constraints through inhibitor arcs and energy-coupled rate functions, provides the mathematical foundation enabling this discovery by integrating discrete regulatory logic with continuous energy dynamics in a resource-aware concurrency model.
Paper Structure (34 sections, 13 equations, 3 figures, 2 tables)

This paper contains 34 sections, 13 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Petri Nets as Foundation for Biological Systems
  3. Signal Hierarchy Theory and Extended BioPNs
  4. Bacillus Sporulation: A Model for Energy-Limited Decisions
  5. Contributions
  6. Methods
  7. Hybrid Petri Net Formalism
  8. Rate Functions and Enablement
  9. Inhibitor Arcs and Thermodynamic Constraints
  10. Continuous Dynamics
  11. Model Construction
  12. Bacillus Sporulation Network
  13. Experimental Conditions
  14. Thermodynamic Analysis
  15. Free Energy Landscape Reconstruction
  16. ...and 19 more sections

Figures (3)

  • Figure 1: Hybrid Petri net model of Bacillus subtilis sporulation.
  • Figure 2: Thermodynamic free energy landscape showing normal (dark blue) and stress (red) pathways overlaid on gray energy surface. Circles mark initial states, squares mark final states, and red star indicates ATP crisis minimum.
  • Figure 3: Phase space trajectories in SigmaF-Forespore commitment plane.