Mitotic waves in the early embryogenesis of Drosophila: Bistability traded for speed

Abstract

Early embryogenesis of most metazoans is characterized by rapid and synchronous cleavage divisions. Chemical waves of Cdk1 activity were previously shown to spread across Drosophila embryos, and the underlying molecular processes were dissected. Here, we present the theory of the physical mechanisms that control Cdk1 waves in Drosophila The in vivo dynamics of Cdk1 are captured by a transiently bistable reaction-diffusion model, where time-dependent reaction terms account for the growing level of cyclins and Cdk1 activation across the cell cycle. We identify two distinct regimes. The first one is observed in mutants of the mitotic switch. There, waves are triggered by the classical mechanism of a stable state invading a metastable one. Conversely, waves in wild type reflect a transient phase that preserves the Cdk1 spatial gradients while the overall level of Cdk1 activity is swept upward by the time-dependent reaction terms. This unique mechanism generates a wave-like spreading that differs from bistable waves for its dependence on dynamic parameters and its faster speed. Namely, the speed of "sweep" waves strikingly decreases as the strength of the reaction terms increases and scales as the powers 3/4, -1/2, and 7/12 of Cdk1 molecular diffusivity, noise amplitude, and rate of increase of Cdk1 activity in the cell-cycle S phase, respectively. Theoretical predictions are supported by numerical simulations and experiments that couple quantitative measurements of Cdk1 activity and genetic perturbations of the accumulation rate of cyclins. Finally, our analysis bears upon the inhibition required to suppress Cdk1 waves at the cell-cycle pause for the maternal-to-zygotic transition.

Keywords: Drosophila; bistability; cell cycle; noise; waves.

Fig. 1.

Spatiotemporal patterns of Cdk1 activity differ in wild type and mutants of the mitotic switch. (A) Confocal images of embryos expressing His2Av-mRFP allow us to visualize the wave-like spreading of mitotic events across the embryo. (Scale bar: 10 $\mu \mathrm{m}$.) B and C show the spatial profile of Cdk1 activity measured experimentally by using a FRET-based biosensor (curves are separated by 60 s and 90 s, respectively). (B) In the wild type, the Cdk1 activity grows synchronously, with minor distortions of the gradients. (C) In the mutant lacking the mitotic switch feedbacks (see Fig. S1 for details), a sharp front propagates across the embryo. (D) In the wild type, the Cdk1 activity crosses a given threshold level of activity with delays that vary piecewise linearly in space, as for traveling waves. (E) Wild-type waves are faster than in the mutant in C, as shown by the time delays in threshold crossings. (F and G) The reaction–diffusion model Eq. 1 of Cdk1 activity recapitulates experimental observations.

Fig. 2.

Reaction–diffusion models of Cdk1 activity recapitulate experimental observations. (A) Scheme of the molecular interactions used in our reaction–diffusion model Eq. 1. (B) The force field in the model for an initial Chk1 level $h_0=0.55$, which roughly corresponds to the cell cycle 13. The lowest curve corresponds to $t=0$ and successive curves are separated by $2$ min. (C and D) Heat map of the Cdk1 activity from simulations of our reaction–diffusion model ($h_0=0.55$ as in B) in wild type (C) and the mutant lacking the mitotic switch feedback (D). The dashed curves indicate the times at different spatial points for the Cdk1 activity to cross a threshold level roughly corresponding to metaphase. (E and F) Experimental heat maps of the Cdk1 activity for the cell cycle 13 of a wild-type (E) and a mutant (F) embryo. (G) Heat map of Cdk1 activity for a metastable potential frozen at the time $t=6.5$ min ($h_0=0.55$). The bistable wave was obtained by poising the middle region close to the stable point of the force $G$ in B and the rest of the space at its metastable point (with a sharp transition in between). (H) Solid lines show the speed of deterministic waves computed using the mechanical analogy in Time-independent case; circles show the speed of bistable waves measured in simulations of Eq. 1 with $G$ frozen at different times and in the presence of different noise levels; diamonds show the speed in full simulations of Eq. 1 for different noise levels at a given $h_0$. The four groups of data are for $h_0=0.4,0.45,0.5,0.55$.

Fig. 3.

The dynamics of Cdk1 over a single cell cycle display three distinct phases. (A) Force field for the Cdk1 model at early times. Inset shows the timescale of relaxation to the low steady state is the inverse of the negative slope of the force near the fixed point. (B) Temporal evolution of Cdk1 activity (from experimental data) as a function of space demonstrates the formation of gradients of increasing length. (C) Force field for the Cdk1 model at times around the loss of bistability. (D) Temporal evolution of Cdk1 activity (from experimental data) as a function of space demonstrates that gradients are swept up largely undeformed during this phase. (E) Force field for the Cdk1 model at times when the system is evolving rapidly toward the only remaining high stable state. (F) Temporal evolution of Cdk1 activity (from experimental data) as a function of space demonstrates that gradients change, yet the time delays among different spatial points to reach a given Cdk1 threshold of activity are conserved (Inset).

Fig. 4.

A time-dependent theoretical model for chemical waves. (A and B) The force field (A) and the potential (B) in Eq. 3, which display transient metastability (curves are shown for values of $\zeta /F_0=\mathrm{0,0.02},0.04$, and $0.06$). (C) The values of the metastable ${\varphi }_0$, unstable ${\varphi }_1$, and stable ${\varphi }_2$ points as a function of $\zeta /F_0$. (D) Dependency of the speed of bistable waves on the dynamic parameter $\zeta /F_0$.

Fig. 5.

Two different regimes are observed for the time-dependent reaction–diffusion Eq. 2. (A) The wave speed $u$ vs. the drive $\beta$ in Eq. 3. The yellow line is the fit with the prediction ${(-\log \beta )}^{2/5}$ in Eq. 13 for small $\beta$. The blue line is the fit with the prediction ${\beta }^{7/12}$ in Eq. 11 for the regime of fast drive. A, Inset shows a zoom-in of the ${(-\log \beta )}^{2/5}$ fit. (B and C) Maximum (blue) and minimum (yellow) of the field $\varphi$ as a function of $\zeta =\beta t$ for $\beta ={10}^{-4}$ and $\beta =2$, respectively. The former illustrates bistable waves: While the wave spreads, the maximum is close to the upper stable point, while the rest of the field is still near the lower metastable point. Conversely, in the fast regime of large $\beta$, the field $\varphi$ grows more uniformly across space, indicating the different nature of the dynamics. (D) The growth of the spatial average $⟨\varphi ⟩$ vs. $\zeta =\beta t$ for $\beta =5$ (yellow circles, the stable fixed points in Fig. 4C; light blue squares, $⟨\varphi ⟩$; blue triangles, the numerical solution of Eq. 2 without noise). The dashed black linear fit defines the rate $k_S$ of early growth (which corresponds to the S phase of the cell cycle, whence the notation).

Fig. 6.

Verification of theoretical predictions in the Cdk1 model Eq. 1. A and B show the speed $u$ of Cdk1 waves vs. the rate $k_S$ of Cdk1 activation growth in the S phase, which is defined in Fig. 5D. Different values of the amplitude $\nu$ of the noise are shown in A and collapsed by the rescaling $u{\nu }^{1/2}$ in B, which supports the prediction Eq. 11. (C) Plot of $u$ vs. the Cdk1 diffusion coefficient $D$ (dashed blue line, best fit; green line, the $D^{3/4}$ scaling predicted for sweep waves; yellow line, the $D^{1/2}$ scaling expected for bistable waves (Time-independent case). (D) Plot of $u$ vs. $G_0$ defined in Eq. 1 (dashed black line, best fit; yellow line, the predicted scaling $G_0^{-1/3}$). (E) A scheme of the interactions in the mutant lacking the mitotic switch feedbacks. (F) Force field for the reaction–diffusion model of the mutant in E. The lowest curve corresponds to $t=0$, and successive curves are separated by $3$ min.

Fig. 7.

Experimental data show that Cdk1 waves are sweep waves. (A) Emission ratio of the Cdk1 FRET biosensor as a function of time for a wild-type embryo (blue line) and a cycA cycB double-heterozygous mutant embryo (yellow line) for cell cycle 13. Linear fits (light blue lines) define the S-phase rate $k_S$ of Cdk1 activity growth. (B) Emission ratio of the Cdk1 biosensor as in A. Green curves, quadratic fits to estimate the parameter $\beta$; dashed lines, the time frame used to estimate the gradients of Cdk1 activity. (C) The Cdk1 activity across the anterior–posterior axis, whence the gradient $g$ is measured by fitting the data with two lines of the same slope. (D and E) Heat maps of the spatiotemporal Cdk1 activity for the wild type (D) and the cycA cycB double-heterozygous mutant (E). Black solid lines, the time frame used to estimate the gradients in Cdk1 activity; dashed lines, the times of mitotic entry and exit. (F) The Cdk1 waves’ speed $u$ vs. the prediction $u\propto {\beta }^{2/3}/g$ in Eq. 10. The black line is the identity. (G) Scaling $u$ vs. $\beta$ (dashed line, best fit; solid line, theoretical prediction $u\sim {\beta }^{7/12}$). (H) The S-phase rate of Cdk1 growth $k_S$ strongly correlates with the values of $\beta$ fitted in B. The combination of results in G and H explains (noise makes $1/2$ indistinguishable from $7/12$) the empirical scaling $u\sim k_S^{1/2}$ previously reported in ref. .

Fig. 8.

Inhibition of Cdk1 waves at the maternal-to-zygotic transition. (A–D) Nuclear density for wild-type embryo (A) and three embryos with reduced DNA content (B–D), obtained from a cross between wild-type females and $C(2)EN$ males. These embryos show three different phenotypes: normal nuclear density, uniformly increased nuclear density due to an extra cell division, and nonuniform nuclear density due to a patchy extra division. The zoom-in in D highlights the sharp boundary between the region that underwent the extra division and the one that did not. (E–H) Heat maps of Cdk1 activity from cell cycle 11 to the maternal-to-zygotic transition for the wild-type embryo (E) and the three embryos with reduced DNA content (F–H). In cell cycle 14, the patchy embryo shows a wave of Cdk1 activity, which is unable to propagate throughout the entire embryo (H), resulting in the differences in nuclear density observed in D. (I–L) Plots of Cdk1 activity at cell cycle 14 for different regions of the embryos for the wild type (I) and for the three embryos with reduced DNA content (J–L). In cell cycle 14 of the patchy embryo, Cdk1 activity dampens as it propagates (L) until it rapidly drops and becomes unable to initiate mitosis across the entire embryo. A small dampening is also observed in embryos undergoing an extra division (K), but the effect is small and mitosis is initiated and completed across the entire embryo. For further details, see Fig. S8.