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Fig. 1: Time evolution of mitotic waves.a Schematic view of the experimental preparation of cycling egg extracts in Teflon tubes. Bottom-Left: The regulatory network driving mitotic oscillations. b Top: Representative FRET ratio kymograph for extracts without sperm DNA. Color bar indicates FRET ratio. Two wavefronts are labeled in white (dashed, early time; solid, later time). The local wave speed is the inverse of the wavefront slope (dt/dx). Bottom: FRET ratio time course recorded at x = 10 mm. The period is defined as the interval between consecutive FRET ratio peaks. One early time region, T1, and one later time region, T2, are selected for (c). c Shifted mean-corrected FRET ratio for T1 and T2. Top: T1 showing in-phase activation with no clear wavefront. Bottom: T2 showing a traveling pulse, indicated by an arrow. Bidirectional arrows beside the colorbar indicate T1 and T2. d Time evolution of the period (top), slope (middle), and wave speed (bottom). Period and slope distributions (grayscale colormap) represent the time-normalized kernel density estimation. The solid curves show the locally weighted scatterplot smoothing (LOWESS) estimations. The speed here is the inverse of the slope’s LOWESS estimation. The onset of exponential decay (τ0, dashed blue line) is calculated via a moving horizon fitting (see Supplementary Fig. 1). The speed after τ0 is fitted by an exponential function (solid red line) to calculate the entrainment time (Δτ). The horizontal black line indicates the fitted terminal speed (vt) and the vertical dashed black line indicates τ0 + Δτ. e Speed-period relationship. Local measurements of the period and speed are represented by gray dots (2 % of all data points were shown). Combined speed-period relation (solid black line) is computed from the LOWESS estimations in (d). Markers (open circles with grayscale fillings) indicate multiples of 100 min (except for the first, shown for clarity). Color bar indicates time. The transition time points τ0 and τ0 + Δτ are indicated on a dashed guideline with the transition window highlighted in red. Data in (d, e) are pooled from two independent egg batches with three replicates each (n = 6).
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Fig. 2: Mathematical modeling explains the transition from phase to trigger waves.a Schematic representation of the mitotic dynamics influenced by scaling factors α (left), β (middle), and η (right), which scale for the rates of respective reactions indicated by black arrows. See Methods and Supplementary Fig. 3 for details. Unperturbed (scaling factor = 1, black lines) and perturbed (scaling factor < 1, red lines) dynamics are compared for phase-plane trajectories of total and active cyclin B-Cdk1 concentration (top) and active cyclin B-Cdk1 concentration time courses (bottom). b Spatiotemporal evolution of the cyclin B-Cdk1 activity showing the transition from fast to slow waves. Simulation has incorporated the experimental time-dependence of the period shown as 1/β(t) (top panel) and spatial variability in the synthesis term ks(x) (left panel, see also Methods). c Influence of spatial heterogeneity (Ak) on the wave speed entrainment. d Speed-period relation of the experiment (dashed line) and the numerical simulations (solid lines with respective Ak values labeled). The transition points τ0 and τ0 + Δτ are marked as in Fig. 1e. The time frames between τ0 and τ0 + Δτ are highlighted in red. e Dependence of transition time scales τ0 (blue) and Δτ (red) on spatial heterogeneity. Experimental measurements are given in dashed lines. Vertical lines correspond to simulation conditions illustrated in (d) with matching grayscale colors.
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Fig. 3: Nuclei entrain the system to the trigger wave regime.a Representative kymograph of the effects of adding sperm DNA (+XS). A magnified view of the region inside the red rectangle is provided in the lower panel to show nuclear growth before the entry into mitosis (red arrows). T1 and T2 mark the regions used in (b). b Comparison of shifted mean-corrected FRET ratio at early times with no clear wavefront (top) and distinctive traveling pulses at late times (bottom). c Comparison of the time dependence of the slope for the case of added sperm DNA (+XS, orange) and control (−XS, green). Kernel density estimations (normalized at each time) and LOWESS curves are given in the corresponding colors. d Time evolution of the period (top) and wave speed (bottom) for +XS. e Speed-period relations for both conditions are obtained from LOWESS curves in Fig. 1d (−XS) and Fig. 3d (+XS), respectively. Open circles (−XS) and open squares (+XS) along the line highlight multiples of 100 min. The transition points are marked on dashed guides for each, as done previously. Data are pooled from two independent egg batches with three replicates each (n = 6)
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Fig. 4: Nuclear localization of cyclin B-Cdk1 through import triggers mitotic waves.a Schematic representation of the nucleus (blue) and the surrounding microtubule structure (gray), which imports cell-cycle regulators such as cyclin B-Cdk1 via molecular motor transport, indicated with green arrows. The import strength is proportional to the concentration of cyclin B-Cdk1 (c) and is spatially described in the model by a Gaussian profile V(x, t). The spatial derivative of this profile gives the direction of the flux J, which becomes zero during mitosis when local activity crosses the NEB threshold. b Time evolution of active (red) and total cyclin B-Cdk1 (black) in space (right panel), showing depletion of cyclin B in the neighborhood and its accumulation at the nuclear position (xn) before activation. The accumulation of cyclin B triggers activation at the nuclear position before the surrounding areas, as indicated in the phase plane representation (left panel) at different positions. c Kymographs representing the spatiotemporal evolution of active (middle) and total (bottom) cyclin B-Cdk1 under the influence of nuclear import. The activation of import is shown in blue, relative to the activity at the position indicated in the kymograph (red, top). The strength of V(x, t) and positions of nuclei xi are represented at the boundaries according to the respective color scale.
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Fig. 5: CSF boundary-driven mitotic waves.a Schematic representation of the experimental setup to trigger boundary-driven mitotic waves by a 5 second dip of an extract-filled tube in CSF extract. CSF-induced arrest is highlighted in red. b Solitary pulse of high Cdk1 activity propagating with a speed of 40 μm/min (wavefront indicated by a white line) in an interphase-arrested extract triggered by CSF dipping. c Spatial profiles of the shifted FRET ratio from the kymograph in (b) showing the excitable pulse. The red vertical line corresponds to CSF-induced arrest. d Kymographs for boundary-driven traveling waves in cycling extracts without (left, − XS) and with nuclei (right, +XS) present. e Traveling waves from kymographs in d revealed by the shifted mean-corrected FRET ratio without (−XS) and with nuclei (+XS). f Wave speed as a function of time for the two conditions in (d) analyzed via later-time exponential fits (red lines). g Speed-period relationship combining both LOWESS estimations for conditions with (+XS) and without (−XS) nuclei. Transition points and guides are as previously defined. Data are pooled from two independent egg batches with at least eight replicates each (n = 24 for −XS and n = 21 for +XS).
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Fig. 6: Influence of the oscillator properties and spatial heterogeneity on the formation of phase and trigger waves.a Kymographs representing the spatiotemporal evolution of cyclin B-Cdk1 activity. (i) Simulation with a spatially homogeneous initial condition of activity and a spatially heterogeneous period dependence to introduce a pacemaker at x = 2.5 mm, exhibiting trigger waves for α = 1. (ii) Simulation with a spatially linear phase difference in the initial condition of activity and a spatially homogeneous period, exhibiting phase waves for α = 1. (iii) Same spatial heterogeneity as (i) for α = 0.1. (iv) Same spatial heterogeneity as (ii) for α = 0.1. b v and γ as functions of α resulting from fitting the long-term shapes of pacemaker-driven waves in (a) with the expression x = d + vtγ, showing a progressive transition to linearly propagating trigger waves (γ = 1) with a stable speed. c Temporal evolution with cycle number of the spatially averaged wave speed for phase and trigger waves in the shown kymographs (i) and (ii), (a). Speed is measured only at the wavefront segments indicated in red lines.
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Fig. 7: Spatial heterogeneity coordinates phase-to-trigger wave transitiona Time evolution of wave speed. Each vertical line with the corresponding arrow on the top indicates τ0 for each experimental condition (461, 115, 146, and 45 min for Control, +XS, +CSF, and +XS/+CSF, respectively). The horizontal line depicts the terminal speed for the excitable system (40 μm/min). b Exponential relaxation of late-time wave speed. Time is measured from respective τ0, and speed is offset by the terminal speed and then normalized against the speed at τ0. The negative reciprocal of the curve slope gives a visual estimation of Δτ, which are 526, 157, 396, and 81 min for each condition. Dotted guidelines correspond to 50 and 500-min time scale relaxations, respectively. c Transition time scales τ0 and Δτ (top) and terminal speeds (bottom). Terminal speeds are 44, 39, 27, and 31 μm/min for each condition, converging to a similar level and comparable to the traveling speed of the activation pulse in interphase extracts driven by CSF (40 μm/min).
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Fig. 7: Spatial heterogeneity coordinates phase-to-trigger wave transitiona Time evolution of wave speed. Each vertical line with the corresponding arrow on the top indicates τ0 for each experimental condition (461, 115, 146, and 45 min for Control, +XS, +CSF, and +XS/+CSF, respectively). The horizontal line depicts the terminal speed for the excitable system (40 μm/min). b Exponential relaxation of late-time wave speed. Time is measured from respective τ0, and speed is offset by the terminal speed and then normalized against the speed at τ0. The negative reciprocal of the curve slope gives a visual estimation of Δτ, which are 526, 157, 396, and 81 min for each condition. Dotted guidelines correspond to 50 and 500-min time scale relaxations, respectively. c Transition time scales τ0 and Δτ (top) and terminal speeds (bottom). Terminal speeds are 44, 39, 27, and 31 μm/min for each condition, converging to a similar level and comparable to the traveling speed of the activation pulse in interphase extracts driven by CSF (40 μm/min).
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Supplementary Fig. 1 Mean square residual (MSR) of the exponential fitting of the wave speed at [τ,∞) (top), and its τ-derivative (bottom). τ0 is defined as the largest τ when the derivative is below a threshold
(horizontal red line). τ0 for each experimental condition is indicated with a red arrow.
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Supplementary Fig. 4 a Spatiotemporal evolution of the activity of cyclin B-Cdk1 showing the transition
from phase to trigger waves with constant period with time using the parameter β(t) = 1 (bottom) and spatial
variability in the synthesis term. b Speed-period relation and temporal dependence of the period and wave
speed of the numerical simulation in panel a.
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Supplementary Fig. 5 Period, maximum activation rate, and slope for CSF boundary-driven mitotic waves.
a Period for no sperm DNA (−XS) and added sperm DNA (+XS) cases. Both columns feature the kernel
density estimation over time with solid lines representing the LOWESS estimation. b Maximum activation
rate. c Slope.
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Supplementary Fig. 6 a Time series of active Cdk1 concentrations (left, 10 oscillators) and the raster plot
of their peak times (right, 50 oscillators), obtained as realizations of the Gillespie’s SSA. The reaction
volume is assumed to be Vc = 1 pL as argued in the Methods section. The intrinsic noise is too weak to
make independent oscillators desynchronized within a number of cycles that we observe in our experiments,
suggesting its marginal role in establishing the wave dynamics in coupled, spatially extended systems. b
Time series of active Cdk1 concentrations (at 10 independent positions for each condition) at five different
noise levels (S = 0.10, 0.20, 1.00, 2.00, and 3.16 nM2
/min from top to bottom), obtained by solving Eqs. (10)
and (11) without diffusion (D = 0 µm2
/min). The smallest investigated level S = 0.10 nM2
/min produces
negligible randomness comparable to the one driven by the intrinsic noise (a, left). For more complete
characterization of the impact of stochasticity, we also explore the conditions where fluctuations immediately
desynchronize independent oscillators, for example, S = 3.16 nM2
/min. Simulations at these conditions
generates dynamics even noisier than what is observed in cell cycle circuits reconstituted in microemulsion
droplets1
. c Diffusion (D = 240 µm2
/min) greatly attenuates desynchronization, even for conditions where
any synchronization patterns are hard to be observed among independent oscillators. d A pacemaker introduced at x = 2.5 mm entrains the global oscillation in a way similar to the deterministic simulation
(Fig. 6a, (i)), regardless of the noise level.
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Supplementary Movie 1 Cdk1 wave dynamics in extracts without nuclei. Data shared with the kymograph
in Fig. 1. (Top) Pseudo-color movie of spatiotemporal dynamics of Cdk1 activity in bulk extracts. Color
scale as in Fig. 1. Early times exhibit phase waves which give way to trigger waves over time. (Bottom)
Detrended and smoothed FRET ratio (averaged over the width of the tube) plotted across the length of the
tube. This shows how the spatial profiles develop from diffuse phase waves to pulse-like trigger waves
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Supplementary Movie 2 Wave dynamics in extracts with reconstituted nuclei. Data shared with the
kymograph in Fig. 3. (Top) Pseudo-color movie of spatiotemporal dynamics of Cdk1 activity in bulk extracts.
Color scale as in Fig. 3. Early times exhibit phase waves which quickly give way to trigger waves emanating
from nuclei. Nuclei appear as hot-colored regions due to their import of active Cdk1. (Bottom) Detrended
and smoothed FRET ratio (averaged over the width of the tube) plotted across the length of the tube. The
curve represents the average over the width of the tube.
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Supplementary Movie 3 Time evolution of the active and total cyclin B-Cdk1 in space under the
influence of nuclear import. The video corresponds to the numerical simulation shown in Fig. 4. The active
part is indicated in red, and the total cyclin B-Cdk1 is indicated in black (top), with their respective color
scales shown (middle, bottom). Nuclear import strength as a function of time is represented by the color of
the circle, according to the respective color scale, and the size of the circle is scaled using an arbitrary factor
chosen for visual clarity.
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Supplementary Movie 4 Excitable pulse in interphase extract driven by CSF. Data shared with the
kymograph in Fig. 4b. (Top) Pseudo-color movie of excitable pulse of Cdk1 activity in interphase extract as
driven by CSF. (Bottom) Detrended and smoothed FRET ratio (averaged over the width of the tube) plotted
across the length of the tube. A singular pulse is driven by the source.
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Supplementary Movie 5 CSF-driven wave dynamics in extracts without nuclei. Data shared with the
kymograph in Fig. 4d, left. (Top) Pseudo-color movie of trigger wave pulses in cycling extracts as driven by
CSF. Phase wave dynamics are permanently abolished by driving. (Bottom) Detrended and smoothed FRET
ratio (averaged over the width of the tube) plotted across the length of the tube. The curve represents the
average over the width of the tube. The CSF source drives multiple trigger wave pulses.
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Supplementary Movie 6 CSF-driven wave dynamics in extracts with reconstituted nuclei. Data shared
with the kymograph in Fig. 4d, right. (Top) Pseudo-color movie of trigger wave pulses of Cdk1 activity in
cycling extracts with reconstituted nuclei as driven by CSF. Both nuclei and the source drive trigger waves,but the CSF source ultimately dominates. (Bottom) Detrended and smoothed FRET ratio (averaged over the
width of the tube) plotted across the length of the tube. The curve represents the average over the width of
the tube.
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Supplementary Fig. 2 a Definition of the maximum activation rate, dA/dt. The largest time derivative of
FRET ratio per cycle, indicated in red dots, is defined as dA/dt. The time course is taken at x = 10 mm of
the kymograph given in Fig. 1b. b Maximum activation rate in non-driven experiments without sperm DNA
(−XS), represented by the kernel density (gray colormap) and LOWESS (solid black line) estimations. c
Maximum activation rate in non-driven experiments with sperm DNA (+XS).
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Supplementary Fig. 3 Dependence of the period, da/dt, and wave speed for different parameter scalings.
Top: Phase space representation of the oscillator and the corresponding time series. Bottom: Period, da/dt,
and wave speed is represented using continuous, dashed, and dotted lines as function of timescale separation
α (left), decreasing synthesis and degradation rates with β (middle), and all rates with η (right). Vertical
lines indicate the parameter values used in the top panels with the respective colors.
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