[Paper Review] ATP Level and Phosphorylation Free Energy Regulate Trigger-Wave Speed and Critical Nucleus Size in Cellular Biochemical Systems

Jianwei Li, Kai Meng+2 more|arXiv (Cornell University)|Mar 11, 2026

Microtubule and mitosis dynamics0 citations

TL;DR

The paper develops a thermodynamically consistent reaction–diffusion framework to show how ATP level and phosphorylation free energy (gamma) regulate trigger-wave speed and the critical nucleus size in ATP-driven bistable systems, with predictions for Rad53 and CDK activation circuits.

ABSTRACT

Trigger waves are self-regenerating propagating fronts that emerge from the coupling of nonlinear reaction kinetics and diffusion. In cells, trigger waves coordinate large-scale processes such as mitotic entry and stress responses. Although the roles of circuit topology and feedback architecture in generating bistability are well established, how nonequilibrium energetic driving shapes wave propagation is less well understood. Here, we employ a thermodynamically consistent reaction--diffusion framework to investigate trigger-wave dynamics in ATP-dependent phosphorylation--dephosphorylation systems. We first recapitulate general expressions for trigger-wave speed in the bistable regime and analyze curvature-induced corrections that determine the minimum critical nucleus required for sustained propagation in higher dimensions. We then apply this framework to two representative systems, treating ATP concentration and the nonequilibrium parameter $γ= [ATP]/(K_{\mathrm{eq}}[ADP][P_i])$ as independent control variables to examine how energetic driving regulates wave propagation. Our results show that ATP and $γ$ not only modulate wave speed, but can also reverse the direction of propagation and reshape the parameter regime supporting trigger waves. The critical excitation radius also depends on both ATP concentration and phosphorylation free energy. These findings identify the intracellular energetic state as a regulator of trigger-wave behavior, linking metabolic conditions to the spatial dynamics of wave propagation. More broadly, this framework connects classical reaction--diffusion theory with ATP-driven biochemical regulation and provides a general perspective on related energy-dependent cellular decision-making processes.

Motivation & Objective

• Motivate how nonequilibrium energy driving shapes wave propagation in cellular bistable circuits.
• Develop a thermodynamically consistent reaction–diffusion model incorporating ATP and gamma as control parameters.
• Derive analytical expressions for planar trigger-wave speed and curvature corrections.
• Apply the framework to Rad53 activation and CDK activation to predict how energetic state affects wave propagation.

Proposed method

• Formulate a bistable reaction–diffusion equation with energy-driven kinetics.
• Define a potential-like function F(u; theta) and relate wave speed to Delta F via c0 = DeltaF / integral (du/dz)^2.
• Derive curvature corrections for spherical fronts leading to dR/dt = c0 - D(d-1)/R and R_c = D(d-1)/c0.
• Introduce gamma = [ATP]/(K_eq[ADP][Pi]) as a nonequilibrium drive and show its effect on bistability and wave speed.
• Provide analytical expressions for c0 in terms of ATP and gamma for representative circuits.
• Perform 1D and 3D PDE simulations to validate front speed, direction, and curvature effects.

Experimental results

Research questions

• RQ1 How do intracellular ATP concentration and phosphorylation free energy gamma influence trigger-wave speed and direction in bistable reaction–diffusion systems?
• RQ2 What is the dependence of the critical nucleus size on ATP and gamma in higher dimensions?
• RQ3 How does curvature interact with metabolic driving to permit or suppress trigger-wave propagation in 3D geometries?
• RQ4 Can the framework predict qualitative and quantitative changes in Rad53-like and CDK-like trigger waves under energy variations?
• RQ5 What experimental tests can validate the predicted ATP- and gamma-dependent phase diagrams for trigger waves?

Key findings

• Trigger-wave speed depends on the thermodynamic driving DeltaF and the metabolic rate, with ATP and gamma modulating both speed and propagation direction.
• In three dimensions, curvature reduces front speed via a dilution term, yielding a critical nucleus radius Rc = D(d-1)/c0 for spherical fronts.
• Increasing ATP generally increases planar wave speed and reduces the critical nucleus size for sustained propagation in the bistable regime.
• The bistable region and wave propagation regimes (forward, stationary, reverse) shift in the ATP–gamma phase space, with a phase diagram separating reverse-wave, stationary, and forward-wave regions.
• For Rad53-like circuits, c0 scales with sqrt([ATP]) in certain ranges, and higher energy drive expands the forward-propagating regime while higher gamma enhances nonequilibrium driving.
• For CDK activation models, bistability exists only within a finite region of ATP–gamma space, and wave speed is sensitive to both total CDK complex concentration and energetic state.