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

Periodicity, mixed-mode oscillations, and multiple timescales in a phosphoinositide-Rho GTPase network

While rhythmic contractile behavior is commonly observed at the cellular cortex, the primary focus has been on excitable or periodic events described by simple activator-delayed inhibitor mechanisms. We show that Rho GTPase activation in nocodazole-treated mitotic cells exhibits both simple oscillations and complex mixed-mode oscillations. Rho oscillations with a 20- to 30-s period are regulated by phosphatidylinositol (3,4,5)-trisphosphate (PIP3) via an activator-delayed inhibitor mechanism, while a slow reaction with period of minutes is regulated by phosphatidylinositol 4-kinase via an activator-substrate depletion mechanism. Conversion from simple to complex oscillations can be induced by modulating PIP3 metabolism or altering membrane contact site protein E-Syt1. PTEN depletion results in a period-doubling intermediate, which, like mixed-mode oscillations, is an intermediate state toward chaos. In sum, this system operates at the edge of chaos. Small changes in phosphoinositide metabolism can confer cells with the flexibility to rapidly enter ordered states with different periodicities.

Nonlinear delay differential equations and their application to modeling biological network motifs

Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.

Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.

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

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.

Desynchronizing Embryonic Cell Division Waves Reveals the Robustness of Xenopus laevis Development

The early Xenopus laevis embryo is replete with dynamic spatial waves. One such wave, the cell division wave, emerges from the collective cell division timing of first tens and later hundreds of cells throughout the embryo. Here, we show that cell division waves do not propagate between neighboring cells and do not rely on cell-to-cell coupling to maintain their division timing. Instead, intrinsic variation in division period autonomously and gradually builds these striking patterns of cell division. Disrupting this pattern of division by placing embryos in a temperature gradient resulted in highly asynchronous entry to the midblastula transition and misexpression of the mesodermal marker Xbra. Remarkably, this gene expression defect is corrected during involution, resulting in delayed yet normal Xbra expression and viable embryos. This implies the existence of a previously unknown mechanism for normalizing mesodermal gene expression during involution.

How Does the Xenopus laevis Embryonic Cell Cycle Avoid Spatial Chaos?

Theoretical studies have shown that a deterministic biochemical oscillator can become chaotic when operating over a sufficiently large volume and have suggested that the Xenopus laevis cell cycle oscillator operates close to such a chaotic regime. To experimentally test this hypothesis, we decreased the speed of the post-fertilization calcium wave, which had been predicted to generate chaos. However, cell divisions were found to develop normally, and eggs developed into normal tadpoles. Motivated by these experiments, we carried out modeling studies to understand the prerequisites for the predicted spatial chaos. We showed that this type of spatial chaos requires oscillatory reaction dynamics with short pulse duration and postulated that the mitotic exit in Xenopus laevis is likely slow enough to avoid chaos. In systems with shorter pulses, chaos may be an important hazard, as in cardiac arrhythmias, or a useful feature, as in the pigmentation of certain mollusk shells.

Synchronization ability of coupled cell-cycle oscillators in changing environments

BACKGROUND

The biochemical oscillator that controls periodic events during the Xenopus embryonic cell cycle is centered on the activity of CDKs, and the cell cycle is driven by a protein circuit that is centered on the cyclin-dependent protein kinase CDK1 and the anaphase-promoting complex (APC). Many studies have been conducted to confirm that the interactions in the cell cycle can produce oscillations and predict behaviors such as synchronization, but much less is known about how the various elaborations and collective behavior of the basic oscillators can affect the robustness of the system. Therefore, in this study, we investigate and model a multi-cell system of the Xenopus embryonic cell cycle oscillators that are coupled through a common complex protein, and then analyze their synchronization ability under four different external stimuli, including a constant input signal, a square-wave periodic signal, a sinusoidal signal and a noise signal.

RESULTS

Through bifurcation analysis and numerical simulations, we obtain synchronization intervals of the sensitive parameters in the individual oscillator and the coupling parameters in the coupled oscillators. Then, we analyze the effects of these parameters on the synchronization period and amplitude, and find interesting phenomena, e.g., there are two synchronization intervals with activation coefficient in the Hill function of the activated CDK1 that activates the Plk1, and different synchronization intervals have distinct influences on the synchronization period and amplitude. To quantify the speediness and robustness of the synchronization, we use two quantities, the synchronization time and the robustness index, to evaluate the synchronization ability. More interestingly, we find that the coupled system has an optimal signal strength that maximizes the synchronization index under different external stimuli. Simulation results also show that the ability and robustness of the synchronization for the square-wave periodic signal of cyclin synthesis is strongest in comparison to the other three different signals.

CONCLUSIONS

These results suggest that the reaction process in which the activated cyclin-CDK1 activates the Plk1 has a very important influence on the synchronization ability of the coupled system, and the square-wave periodic signal of cyclin synthesis is more conducive to the synchronization and robustness of the coupled cell-cycle oscillators. Our study provides insight into the internal mechanisms of the cell cycle system and helps to generate hypotheses for further research.

Perturbation-based analysis and modeling of combinatorial regulation in the yeast sulfur assimilation pathway

Here we establish the utility of a recently described perturbative method to study complex regulatory circuits in vivo. By combining rapid modulation of single TFs under physiological conditions with genome-wide expression analysis, we elucidate several novel regulatory features within the pathways of sulfur assimilation and beyond.

In yeast, the pathways of sulfur assimilation are combinatorially controlled by five transcriptional regulators (three DNA-binding proteins [Met31p, Met32p, and Cbf1p], an activator [Met4p], and a cofactor [Met28p]) and a ubiquitin ligase subunit (Met30p). This regulatory system exerts combinatorial control not only over sulfur assimilation and methionine biosynthesis, but also on many other physiological functions in the cell. Recently we characterized a gene induction system that, upon the addition of an inducer, results in near-immediate transcription of a gene of interest under physiological conditions. We used this to perturb levels of single transcription factors during steady-state growth in chemostats, which facilitated distinction of direct from indirect effects of individual factors dynamically through quantification of the subsequent changes in genome-wide patterns of gene expression. We were able to show directly that Cbf1p acts sometimes as a repressor and sometimes as an activator. We also found circumstances in which Met31p/Met32p function as repressors, as well as those in which they function as activators. We elucidated and numerically modeled feedback relationships among the regulators, notably feedforward regulation of Met32p (but not Met31p) by Met4p that generates dynamic differences in abundance that can account for the differences in function of these two proteins despite their identical binding sites.