McIsaac RS et al. (2011), Does the potential for chaos constrain the embr...

XB-ART-43601

PLoS Comput Biol 2011 Jul 01;77:e1002109. doi: 10.1371/journal.pcbi.1002109.

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Does the potential for chaos constrain the embryonic cell-cycle oscillator?

McIsaac RS, Huang KC, Sengupta A, Wingreen NS.

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Although many of the core components of the embryonic cell-cycle network have been elucidated, the question of how embryos achieve robust, synchronous cellular divisions post-fertilization remains unexplored. What are the different schemes that could be implemented by the embryo to achieve synchronization? By extending a cell-cycle model previously developed for embryos of the frog Xenopus laevis to include the spatial dimensions of the embryo, we establish a novel role for the rapid, fertilization-initiated calcium wave that triggers cell-cycle oscillations. Specifically, in our simulations a fast calcium wave results in synchronized cell cycles, while a slow wave results in full-blown spatio-temporal chaos. We show that such chaos would ultimately lead to an unpredictable patchwork of cell divisions across the embryo. Given this potential for chaos, our results indicate a novel design principle whereby the fast calcium-wave trigger following embryo fertilization synchronizes cell divisions.

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Species referenced: Xenopus laevis
Genes referenced: ccnb1.2 cdk1

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	Figure 1. Diffusive spread of cell-cycle activity.(A) The cell-cycle is driven by a combination of positive and negative feedback loops, which modulate Cdk1 activity through phosphorylation/dephosphorylation. (B) 3.33-hour simulation where cell-cycle activity spreads across a mm embryo. In the spreading model, oscillations initially occur in only of the embryo (i.e., APC is initially assumed to be active only between and mm in a 1 mm long embryo). Through diffusion, oscillations are able to spread across the embryo via local activation of APC once a local critical threshold of overall Cdk1 activity is reached. Each species is initially at its pre-fertilization concentration, with the activity threshold chosen to be when Cyclin B-Cdk1-Tp reaches 4× its initial value. The species plotted in the top panel is Cyclin B-Cdk1 (active) and that plotted in the bottom panel is Cyclin B-Cdk1-YpTp (inactive). In the inactive complex, Cdk1 is phosphorylated at both Threonine-14 and Tyrosine-15.
	Figure 2. Modeling the cell-cycle in developing embryos.(A) Schematic of early development. Across species, fertilization results in a fast calcium wave that sweeps across the embryo. In X . laevis, the result is the destruction of CSF, the consequent restoration of APC-mediated negative feedback, and the subsequent initiation of synchronous nuclear divisions. (B,C) Cell-cycle simulations for spherical embryos with diameters of 1 mm. The fast and slow calcium waves have speeds of 0.36 and 0.125 mm/min, respectively. The number of mitotic divisions that occur in 8 hours along the black lines in the first panel are shown at right.
	Figure 3. Cell-cycle equations exhibit chaotic behavior with extreme sensitivity to initial conditions.(A) Simulations with two different calcium wave speeds in 1 mm embryos, with colormap showing the concentration of doubly phosphorylated Cyclin B - Cdk1 complex. Simulation cell is 1D with no flux boundary conditions. (B) Simulation in a large (6 mm) 1D embryo with periodic boundary conditions. (C) The space-time correlation function is computed using the mean-centered values in (B) between min. and min. (inset) The correlation length is computed to be mm at .
	Figure 4. Sensitivity analysis of cell-cycle equations.(A–D) The critical calcium wave speed, , and the period, , are computed for different values of the (A) embryo length, (B) protein diffusion constant, (C) rate of cyclin synthesis , and (D) rate of APC activity .
	Figure 5. Cell-cycle simulations with cellularization.The embryo is a 1 mm diameter sphere and the calcium wave speed is 1 mm/4 min4.17 m/s. Mitotic divisions are taken when Cyclin B-Cdk1-YpTp reaches half its concentration for uniform oscillations, and are modeled by a no-flux boundary dividing a cell into two equal volumes. By this metric, the first division occurs after 62 minutes, and each subsequent division occurs at 39.2-minute intervals. Solid diagonal lines indicate cellular divisions.
	Figure 6. Cellularization does not prevent chaos for a slow Ca waveSimulations as in Fig. 5, but for a calcium wave speed of 1 mm/10 min1.67 m/s. The solid diagonals indicate at least one cell division. Following the dashed line, even when the simulation continues for longer times, the displayed portions do not divide.

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Davidenko, Stationary and drifting spiral waves of excitation in isolated cardiac muscle. 1992, Pubmed