Wave propagation in a generic cell and corresponding representations of wave directionality.

A. Three-dimensional sketch of a generic cell showing the main features mentioned in the text. B. Schematic representations of active Cyclin B–Cdk1 propagation in traveling-wave–dominated system. C. Schematic representations of active Cyclin B–Cdk1 propagation in gradient-dominated system. D. Kymographs representing the dynamics sketched in B. Left: kymographs sketched along the cell diameter including the nucleus (the z-axis). Middle: kymographs sketched along the vertical angular coordinate ϕ introduced in A for the same dynamics. Right: kymographs adapted from biological systems exhibiting these dynamics. Top: mitosis in Xenopus laevis (adapted from Chang and Ferrell Jr (2013)). Bottom: meiosis II in starfish (adapted from Bischof et al. (2017)).

The Cyclin B-Cdk1 network and its dynamics.

A. Basic network structure regulating the Cyclin B–Cdk1 enzymatic complex. Through phosphorylation and dephosphorylation, Cyclin B–Cdk1 establishes positive feedback loops encompassing Cdc25 and Wee1. In addition, regulation of APC/C generates a delayed negative feedback loop. Together, these features form the essential ingredients for relaxation-like oscillations, characterized by rapid transitions between active and inactive states and robustness to perturbations. B. Experimental measurements of Cyclin B–Cdk1 activity in frog cell-free extracts, obtained using a FRET sensor. The time series displays the characteristic oscillations described in panel A. The spatiotemporal dynamics shown in the kymograph further illustrates how, in the presence of nuclei, activity becomes synchronized through traveling waves. Figure adapted from Puls et al. (2024) C. Computational models have successfully reconstructed the key mechanisms underlying the spatiotemporal dynamics of Cyclin B–Cdk1. In this work, to connect cellular geometry to emergent wave propagation regimes we use the model introduced by Yang and Ferrell Jr (2013) and extended to a spatial framework by Puls et al. (2024). Figure adapted from Puls et al. (2024).

Cyclin B–Cdk1 dynamics: characterization of regimes and wave velocities.

A. Dynamical regimes observed in the simulations. The wave front remains relatively stable, whereas the wave back behavior changes dramatically. Each regime arises from different initial conditions and diffusion coefficients; the corresponding parameter values are indicated by dots in panel C. B. Schematic representation of activity in a generic spatiotemporal system comparing control (γ = 1) and increased nuclear scaling factor (γ > 1) conditions. Under increaseded nuclear scaling factor, the velocity of the wave back can reverse direction. C. Phase diagram showing the distinct dynamical regimes identified in the cytoplasmic system: (i) propagation dominated by traveling waves; (ii) fully reversed wave backs; (iii) coexistence of traveling-wave-dominated fronts with reversed wave backs; and (iv) planar wave backs associated with homogeneous relaxation. Panel B is computed along the black dashed line. D. Quantification of wave front and wave back velocities as a function of the total active Cyclin B–Cdk1 level. Total activity is renormalized across two scenarios: variation of the nuclear scaling factor parameter γ, and increase of nuclear size under increased nuclear scaling factor conditions. Apparent divergences in vB arise from vanishing spatial domains over which the velocity is measured (see Supplementary Fig. S3C). Wave back velocities are fitted with a homographic function.

The RhoA network and its dynamics.

A. Basic network structure regulating RhoA activity. RhoA exists in active, GTP-bound state (Rho-GTP) and inactive, GDP-bound state (Rho-GDP). Activation depends on guanine nucleotide exchange factors such as Ect2, which is itself recruited and activated by Rho–GTP, forming a positive feedback loop. Inactivation occurs through GTP hydrolysis, a process catalysed by the GTPase-activating protein RGA-3/4. RGA-3/4 is recruited by F-actin, which is polymerised by a formin downstream of Rho–GTP activity, thereby closing a negative feedback loop. As in the Cyclin B-Cdk1 system, this combination of positive and negative feedback provides the essential prerequisites for establishing oscillations. B. Time series of Rho–GTP, Rho–GDP, and F-actin generated using the model introduced in Michaud et al. (2022), which is used throughout this paper to reproduce RhoA dynamics. C. Bifurcation diagram as a function of the parameter α, representing Ect2 concentration and used as a control parameter throughout this paper. Representative spatial patterns of Rho-GTP are shown for α = 0.8, 1.2, and 1.3. The model exhibits a rich variety of dynamical regimes, including spirals, wave trains and spriral core turbulence. D. Experimental dynamics of active RhoA in a starfish oocytes. Figure adapted from Bement et al. (2015).

Cortical dynamics and cytoplasmic inhibitory waves.

A. Frames of Rho-GTP dynamics coupled with a reduction in effective Ect2 caused by Cdk1 inhibition with heterogeneous F-actin disassembly. Scale bar: 50μm. B. Kymographs over the vertical dashed line in A of the Rho-GTP concentration and the effective Ect2. Scale bars: 25μm (y-axis) and 240s (x-axis). C. Averaged period over the horizontal dashed line in A using wavelet transformation and time series corresponding to (x, y) = (150, 150). D. Averaged period with homogeneous F-actin disassembly and Heaviside inhibition (Supplementary Video 6). E. Averaged period with heterogenous F-actin disassembly and Heaviside inhibition (Supplementary Video 7). F. Averaged period with homogeneous F-actin disassembly and Cdk1 inhibition (Supplementary Video 8).

Reconstitution of cortical dynamics during surface contraction waves in a three-dimensional cell cortex.

A. Cortical dynamic for each of the regimes introduced in Fig. 2. Upon release from inhibition, cortical activity reappears as a moving front whose direction is set by the underlying cytoplasmic Cdk1 waveform (see Supplementary Videos 9-12). B. Ect2 signal for different Cdk1 phosphorylation rates (inhibition strengths s). C. Spatiotemporal dynamics of the corresponding strong and weak inhibition cases given in B. The bottom shows the maximum activity of Rho GTP (green) and the percentage of the system that is active (blue) averaged over the equator of the system. D. Rho GTP signal after Cdk1 inhibition in starfish oocytes. Adapted from Bement et al. (2015). Full movie in Supplementary video 18.

Standard parameter values used in all simulations unless stated otherwise.

When multiple values are provided for a given parameter, they are listed in the order [Regime (i), Regime (ii), Regime (iii), Regime (iv)], corresponding to the four dynamical regimes defined in Fig. 2C (traveling-wave–dominated, counterpropagating back, coexistence, and homogeneous relaxation).

Initial conditions and increasing factor (γ).

A. Limit cycle of the Cyclin B-Cdk1 model with standard parameters and initial conditions in the nucleus and the cytoplasm. B. Profiles of the initial conditions for γ = 0 and γ = 1.5.

Fits of the wave fronts (A) and wave backs (B) for different increase factors (γ).

Quantification of wave velocities as a function of total Cyclin B–Cdk1.

Wave front and wave back velocities for: A. variable nuclear scaling factor and B. variable nuclear size. C. Renormalization to total Cyclin B–Cdk1 content. D. Fraction of the domain governed by traveling-wave–dominated and diffusion-driven wave-back dynamics.

Effect of system size on active Cyclin B–Cdk1 dynamics.

Kymographs and time series at the indicated times for: A. Diffusion D = 30 μm2/min. B. Diffusion D = 90 μm2/min. C. Diffusion D = 270 μm2/min. Small systems (high diffusion; panel C) exhibit a nearly homogeneous decay of Cyclin B–Cdk1 activity because diffusion rapidly smooths spatial gradients. In larger systems (low diffusion; panel A), gradients persist, leading to a combination of counterpropagating wave-back segments and forward-propagating traveling waves. D. Quantification of wave front and wave back velocities for different nuclear scaling factors. Increasing nuclear concentration leads to slower wave-back propagation because larger amounts of Cyclin B–Cdk1 must be degraded. E. Hyperbolic fits of wave-back velocities for different diffusion coefficients (left). The region of coexistence of oppositely propagating wave-back segments narrows as the diffusion coefficient increases (right).

Example of bubble-like patterns for α = 0.6 with heterogeneous F-actin degradation.

A. Kymographs (along the white dashed line) and corresponding frames (at the red markers indicated in the kymograph). Regions of higher activity act as pacemakers that emit target-like waves into less active or quiescent regions. B. Same simulation as in A but with diffusion set to zero. Local oscillation periods vary across the domain, and some regions do not oscillate, establishing the spatial heterogeneity that seeds bubble-like pattern formation. Scale bars: 25 μm and 180 s.

Effect of inhibition duration on the reactivation phase wave.

A. Short inhibition times preserve spatial inhomogeneities in the dynamical variables, leading to rapid pattern reactivation. Scale bar: 60° (y-axis). B. Long inhibition times drive the system closer to a homogeneous state; upon release, planar traveling waves appear before heterogeneities are re-amplified and the pattern reactivates. C. The previous behaviors occur only when F-actin degradation is homogeneous. When degradation is heterogeneous, parameter inhomogeneities are sufficient to reactivate the pattern immediately. Simulations were performed with D = 90/60 and γ = 1.

Effect of pacemaker position on the reactivation phase wave.

A. Schematic setup of the spherical system and definition of variables and pacemaker positions (rotational symmetry assumed). B. Time at which the inhibitory wave reaches each point on the sphere for different pacemaker positions. C. Kymograph for zpacemaker = 0.10r, with the Ect2 front indicated in red (semi-transparent curve shows the reference case zpacemaker = 0.40r). Scale bar: 30° (y-axis). D. Frames corresponding to the kymograph in C. Simulation parameters: D = 90/60 and γ = 1.