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

TL;DR

These findings identify the intracellular energetic state as a regulator of trigger-wave behavior, linking metabolic conditions to the spatial dynamics of wave propagation, and provides a general perspective on related energy-dependent cellular decision-making processes.

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.

Figures (5)

• Figure 1: Schematic illustration of bistable reaction kinetics and trigger-wave propagation. (a) Reaction term $f(u)$ versus $u$. Three real roots of $f(u)=0$ correspond to two stable steady states, $u_{\mathrm{lo}}$ and $u_{\mathrm{hi}}$, separated by an unstable state $u_{\mathrm{mid}}$. (b) Potential landscape defined by $F(u)=-\int^u f(u')\,\mathrm{d}u'$, with two wells for the stable states and a barrier for the unstable state. (c) Potential landscapes with identical local bistability but different global biases, corresponding to $\Delta F>0$, $\Delta F=0$, and $\Delta F<0$. The sign of $\Delta F$ sets the preferred invasion direction. (d) Traveling-front profile $u(z)$. The sign of $\Delta F$ determines the propagation direction, and the wave speed follows a Luther-type scaling, $c_0\sim\sqrt{Dk_{\mathrm{rxn}}}$. (e--f) Curvature correction and critical nucleus for spherical trigger waves in higher dimensions. The radial speed obeys $\mathrm{d}R/\mathrm{d}t=c_0-(d-1)D/R$, where $d$ is the spatial dimension. Fronts with $R<R_c$ collapse, whereas those with $R>R_c$ expand into sustained trigger waves.
• Figure 2: ATP and hydrolysis free energy regulate trigger waves in S-phase checkpoint Rad53 activation. (a) Rad53 activation circuit coupled to ATP hydrolysis. (b) Fixed points of the local reaction term $f(u)$ for varying non-equilibrium drive $\gamma$. (c--d) Bistable region and planar trigger-wave speed $c_0$ in the $\mathrm{ATP}$--$\gamma$ phase plane; representative parameter points (three cases) are selected and highlighted for the example kymographs in (d). (e--f) Trigger-wave speed versus $[\mathrm{ATP}]$ and $\gamma$; symbols are PDE simulations, and solid lines are theoretical predictions from the analytical wave-speed formula.
• Figure 3: Curvature correction and critical radius for spherical trigger waves. (a) Radial speed $\mathrm{d} R/\mathrm{d} t$ as a function of nucleus radius $R$. (b) Radius dynamics $R(t)$ for different initial radii. (c) Spatiotemporal evolution of the three-dimensional reaction--diffusion field for different initial radii. (d--f) ATP dependence of the planar wave speed $c_0$ and the curvature-induced correction $c_d$, showing three propagation regimes. (i) Region I: reverse-trigger regime, where $c_0<0$ and the high-activity region shrinks regardless of the initial radius. (ii) Region II: dilution-dominated regime, where $0<c_0<c_d$ and the initial high-activity nucleus is lost because curvature-induced dilution prevents expansion. (iii) Region III: forward-trigger regime, where $c_0>c_d$ and the nucleus expands into a sustained trigger wave. (g) Three-dimensional simulations at fixed initial radius with different ATP levels corresponding to the three regimes. In panels (c) and (g), the displayed $x$--$y$ sections pass through the equatorial plane of the spherical nucleus.
• Figure 4: ATP and hydrolysis free energy regulate trigger waves in CDK activation during the G2--M transition. (a) Schematic view of the CDK activation circuit. (b) $\gamma$-dependence of the Cdk activation reaction term. (c) Distribution of bistability and planar wave speed $c_0$. Labeled points $1$--$3$ denote representative parameter sets shown in (d--f). (d--f) Reaction term, potential function, and propagation profiles for representative cases (blue/purple/red points denote the reverse-wave regime, stationary-front boundary, and forward-wave regime, respectively). (g) Dependence of trigger-wave speed on $[\mathrm{Cdc13/Cdc2}^T]$ (fixed $\gamma$). (h) Dependence of trigger-wave speed on ATP (fixed $\gamma$). (i) Dependence of trigger-wave speed on ATP (fixed $[\mathrm{Cdc13/Cdc2}^T]$).
• Figure 5: Trigger waves in CDK activation with cyclin synthesis and APC-mediated degradation. (a) Schematic of the Cdk1--APC/C circuit. (b) Nullclines of the two-variable system: $\dot{A}=0$ (purple) and $\dot{T}=0$ (green). The enriched core activates after crossing the bistable boundary, with $T_{\rm core}\approx T_{\rm high}$. (c) Cytoplasmic wave speed is set by $T_{\rm cyto}$, with $T_{\rm core}=\eta T_{\rm cyto}$. (d) At $[\mathrm{ATP}]=3.9~\mathrm{mM}$ and $\gamma=10^{10}$, the $(\eta,R_{\rm core})$ plane is partitioned into a forward-wave region (colored by speed) and three nonpropagating regions: dilution-dominated, reverse-wave, and monostable. (e--h) ATP has competing effects on wave speed. By lowering the cyclin level required for core activation, increasing ATP reduces $T_{\rm cyto}$ at activation; the resulting ATP dependence of $c_0$ can therefore deviate from, or even reverse relative to, the monotonic increase observed at fixed $T$. Panel (h) summarizes the competition between the direct contribution $\left(\frac{\partial c_0}{\partial [\mathrm{ATP}]}\right)_{T_{\rm cyto}}$ and the indirect contribution $\left(\frac{\partial c_0}{\partial T_{\rm cyto}}\right)_{[\mathrm{ATP}]}\frac{\mathrm{d} T_{\rm cyto}}{\mathrm{d} [\mathrm{ATP}]}$. (i--k) One-dimensional simulations at different ATP levels with $\eta=1.5$ and $\gamma=10^{10}$ fixed. (l--n) One-dimensional simulations at different enrichment factors with $[\mathrm{ATP}]=4.0$ mM and $\gamma=10^{10}$ fixed. White lines in (i--n) indicate fitted wave fronts and inferred speeds.