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Published in final edited form as: Cell Rep. 2015 Jul 23;12(5):892–900. doi: 10.1016/j.celrep.2015.06.070

How does the Xenopus laevis embryonic cell cycle avoid spatial chaos?

Lendert Gelens 1,2, Kerwyn Casey Huang 3,4, James E Ferrell Jr 1,5,*
PMCID: PMC4531097  NIHMSID: NIHMS710536  PMID: 26212326

Summary

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.

Introduction

In the amphibian Xenopus laevis, embryogenesis begins when the sperm penetrates the cell cycle-arrested egg, initiating a wave of elevated cytoplasmic calcium that brings about the completion of meiosis II, and allows the mitotic cell cycles to begin (Gerhart, 1980; Hausen and Riebesell, 1991). The first mitotic cleavage occurs approximately 85 min after fertilization, and is followed by 11 rapid, 25-min cell cycles that lead up to the mid-blastula transition. These rapid early embryonic cell cycles are remarkably regular, with the cell cycle period varying little from cycle to cycle and from cell to cell.

These cell cycle oscillations are driven by a biochemical oscillator circuit centered on the cyclin B-cyclin-dependent kinase 1 (Cdk1) complex (Minshull et al., 1989; Murray and Kirschner, 1989). The oscillator circuit possesses a bistable trigger (Goldbeter, 1993; Novak and Tyson, 1993b; Pomerening et al., 2003; Sha et al., 2003), which is built from interlinked positive (Cdk1 activates Cdc25, which activates Cdk1 (Hoffmann et al., 1993; Solomon et al., 1990)) and double-negative (Cdk1 inhibits Wee1 and Myt1, which inhibit Cdk1 (McGowan and Russell, 1993; Mueller et al., 1995a; Mueller et al., 1995b; Parker and Piwnica-Worms, 1992; Tang et al., 1993)) feedback loops. As levels of cyclin B rise during S phase, the switch eventually flips, driving the cell irreversibly into mitosis. Mitotic exit is then initiated through a negative-feedback loop involving the anaphase-promoting complex/cyclosome (APC/CCdc20), which polyubiquitinates cyclin B, tagging it for degradation by the proteosome (Cdk1 activates APC/CCdc20, which inhibits Cdk1 (King et al., 1996)). The strengths and response functions of the different feedback loops have been experimentally measured using Xenopus extracts (Kim and Ferrell, 2007; Pomerening et al., 2005; Pomerening et al., 2003; Trunnell et al., 2011; Tsai et al., 2014; Yang and Ferrell, 2013), providing a detailed quantitative accounting of the biochemical reactions of the cell cycle oscillator.

Much can be learned by considering a biochemical oscillator circuit to be a system of spatially coupled homogeneous biochemical reactions that can be modeled by ordinary differential equations (ODEs). This assumption is well justified for very small or well-stirred systems. However, the Xenopus egg is neither. This underscores the need to examine the spatial dynamics of cell cycle oscillations, as well as the temporal dynamics. The first model to include spatial diffusion of all proteins involved in the early cell cycle control system was recently presented by McIsaac et al. (McIsaac et al., 2011). Surprisingly, the authors noticed that the system had the potential to generate spatially chaotic dynamics. Although it is well-known that ODE cell cycle models can generate temporal chaotic oscillations (Romond et al., 1999), in this case the reactions of the ODE model were not chaotic, and chaos only emerged once diffusive spatial coupling was added to the model. In particular, the authors used simulations to predict that if the post-fertilization calcium wave were slowed down even by a factor of two compared to its physiological speed, chaotic cell cycle oscillations would emerge, giving rise to an unpredictable patchwork of cell divisions across the embryo.

Using the same biochemical model as in (McIsaac et al., 2011), Figure 1 illustrates these two different types of spatial dynamics in a periodic system of diffusively coupled cell cycle oscillators. This model (Experimental Procedures) was numerically evaluated in a system with periodic boundary conditions. The kinetic parameters were taken to be uniform over the whole space, but the oscillations were initiated at different times at different positions as determined by when the calcium wave passed that location. This leads to spatial heterogeneity in the initiation time of cell cycle oscillations.

Fig 1. A slower calcium wave leads to chaotic dynamics in a cell cycle model.

Fig 1

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Simulated time series of Cdk1 activity using cell cycle model 1 (SI: Models) where the cell cycle oscillations are initiated upon passing of (A) a fast calcium wave with speed vel = 50 μm/s, and (B) a slow wave with vel = 10 μm/s. Shown is the activity of Cdk1 as a function of position and time, in arbitrary units. (C,D) The spatio-temporal correlation for (A,B), respectively. Periodic boundary conditions were assumed; only half of the domain is shown.

For a fast calcium wave, the oscillations are nearly synchronous (Figure 1A). However, when the calcium wave is slowed down, synchrony is lost. Instead, different regions in space exhibit transient oscillatory dynamics with constantly evolving periods and amplitudes (Figure 1B). This loss of synchrony is confirmed by the spatio-temporal correlation function (Figure 1C,D). For a fast calcium wave, there is strong correlation over all space at zero time difference and at time differences corresponding to integer multiples of the period of the cell cycle oscillations (Figure 1C). For the slowed calcium wave, such correlations no longer exist, although correlation signatures can still be found, mainly around half of the cell cycle period (Figure 1D). Thus, it appears that this shift from spatially synchronized to complex, poorly coordinated spatial dynamics represents a transition into chaos. In this work, we will use the term chaos to describe any irregular dynamics that are poorly correlated in space and time, that have a deterministic chaotic attractor (see Figure 4), and that have different responses with respect to slight changes in initial conditions (see Figure S1).

Fig 4. Both chaotic and periodic dynamics show a clear attractor in phase space.

Fig 4

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Time series illustrating the long-term behavior of u in the spatially coupled FHN model with γ = 1 (A) and γ = 0.01 (B) at two locations: x = 0 (magenta) and x = 250 (green). (C,D) Trajectories for the systems shown in (A, B) plotted in phase space, beginning after a transient of 15,000 time units. The u nullcline is shown in solid black, and the v nullcline in dashed black.

Intrigued that chaos might be lurking so close to the normal physiological operating range of the cell cycle oscillator, we set out to test this hypothesis experimentally. In this study, we demonstrate that cell divisions proceed with their normal near synchrony even when the calcium wave is slowed down by a factor of ~3. We then investigated why the in vivo system displays less tendency for chaos than the model system. We found that the model becomes most susceptible to transitions into chaos when the cell cycle oscillations have a very short pulse duration of Cdk1 activation, probably shorter than in the physiological case (Pomerening et al., 2003; Tsai et al., 2014; Yang and Ferrell, 2013). Finally, to see whether there is a general link between the pulse duration of the oscillatory reactions and the propensity of a spatially distributed system to exhibit chaos, we made use of the FitzHugh-Nagumo model, a well-explored model in physics and theoretical biology inspired by the action potential. In general, it appears that any heterogeneity, be it a slow calcium wave or a heterogeneous distribution of biochemical parameters in space, can trigger a chaotic response if coupled to oscillatory reactions with sufficiently short pulses. Thus, the observed spatial dynamics put constraints on the temporal dynamics of the biochemical reactions.

Results

Xenopus embryos with slow calcium waves show no evidence of chaotic cell division

We hypothesized that one way to slow the initial calcium wave would be to transiently cool the fertilized egg. To this end, we designed a setup that allows precise and accurate control of the temperature of eggs and embryos. This setup consists of a chamber with a capacity of several hundred embryos (Figure 2A). The temperature of the aluminum chamber is electrically controlled to ±0.01°C using Peltier elements. We fertilized eggs in this device at various temperatures and measured the velocity of the calcium wave by monitoring the pigment rippling that accompanies the wave. As has been previously noted, the calcium wave propagated approximately linearly from the sperm entry point to the opposite side of the embryo (Busa and Nuccitelli, 1985; Fontanilla and Nuccitelli, 1998). Therefore, we could estimate the speed of the calcium wave by constructing kymographs and fitting a straight line to the observed wave of pigment changes (Figure 2B, Movies S1,S2). The calcium wave speed increased approximately linearly with temperature, ranging from 3.6 μm/s at 11°C to about 9.3 μm/s at 23°C (Figure 2C). Thus, by decreasing the temperature from 23°C to 13°C, the calcium wave was slowed down by a factor of ~3.

Fig 2. Slowing the calcium wave speed by lowering temperature does not affect synchronization or embryonic development.

Fig 2

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(A) Device to control the temperature of eggs and embryos. (B) Kymographs showing the calcium wave after fertilization at T = 23°C and T = 13°C. Image contrast and brightness has been adjusted to accentuate the calcium waves. Wave speeds were estimated assuming that each embryo is 1.2 mm in diameter. (C) The calcium wave speed increases with increasing temperature, as shown by data pooled from 12 independent experiments. (D) The timing of cell division in one embryo fertilized at 13°C and then returned to 23°C after the calcium wave had passed (blue) and one embryo fertilized and maintained at 23°C (red). Each data point represents the division of one cell within the embryos. Note that as embryogenesis proceeds, cell division becomes metachronous, with cell division sweeping from one side of the embryo to the other. The pattern and timing of this metachronous cell division wave was indistinguishable in the control vs. experimental embryos. Average periods are expressed as means ± S.D. (E) Video frames of the embryos analyzed in (D) (see also Movie S3).

We next compared subsequent cell cycle progression in embryos that were fertilized at 13°C and then returned to 23°C after the calcium wave had passed, versus embryos fertilized and maintained at 23°C. Data from one control 23°C/23°C and one 13°C/23°C embryo are shown in Figure 2D,E. The embryo fertilized in the cold underwent its mitotic divisions about 14 min after the control embryos. However, the embryonic cell cycles proceeded with a normal period, and there was no evidence of desynchronization across the embryo (Figure 2D,E, and Movie S3). Moreover, both the control and 13°C/23°C embryos developed into healthy tadpoles (Figure 2E, Movie S3). Thus, there was no evidence for chaos in embryos whose cell cycles have been initiated by a slow calcium wave. The results shown in Figure 2D,E and Movie S3 are representative of the ~20 control and ~10 experimental embryos examined in each of eight independent experiments.

Spatially coupling relaxation oscillators of short pulse duration leads to chaos

This experimental finding prompted the question of why slower calcium waves do not seem to affect the correct development of the Xenopus embryo. To address this question, we examined the underlying dynamical principles that are of importance in generating the chaotic waves observed in previous modeling studies (McIsaac et al., 2011). The model used in (McIsaac et al., 2011) (and before that in (Pomerening et al., 2005; Pomerening et al., 2003); hereafter referred to as cell cycle model 1; see supplemental information) consists of 9 ODEs, including the various different phosphorylation states of the cyclin B-Cdk1 complex and polo-like kinase 1 (Plx1) activity (Figure 3A). In a model with this high degree of complexity, it is relatively difficult to understand exactly why the spatial chaos emerged. More recently, a simpler (but still realistic) cell cycle model for the early Xenopus embryo was proposed containing only 2 ODEs, where the various parameters have been empirically determined (Yang and Ferrell, 2013) (Figure 3B; hereafter referred to as cell cycle model 2; see supplemental information). Using the latter model, no sign of chaos was observed, not even when dramatically slowing down the calcium wave (Figure 3H, top).

Fig 3. Duration of the reaction response dictates the transition to chaotic dynamics in three different relaxation-oscillator models.

Fig 3

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(A,B) Two different cell cycle models, and (C) the FHN model. In each case, changing one parameter resulted in a transition from long (top) to short (bottom) pulses as shown in (D-F) for the models in (A-C), respectively. (G-I) The spatial dynamics corresponding to (D-F) as a result of introducing diffusive coupling. Oscillations were initiated after the passing of a calcium wave with speed vel = 10 μm/s in (G), vel = 10 μm/min in (H), and vel = 1 in (I).

Why did one model produce chaos and the other did not? Comparison of the two models showed that in cell cycle model 1 the relaxation oscillations had a very short pulse duration (McIsaac et al., 2011) (Figure 3D, bottom), whereas in cell cycle model 2 they were longer (Yang and Ferrell, 2013), with a much slower time constant for the switch from high to low cyclin B-Cdk1 levels (Figure 3E, top). To ascertain whether the profiles of these oscillations (short vs. longer pulse duration) were important determinants of the generation of chaotic dynamics, we first decreased the rate of degradation due to APC/C by a factor of 100 in cell cycle model 1. This change prolonged the time spent in M-phase and indeed restored the synchrony of mitosis (Figure 3D,G). Conversely, we increased the degradation rate due to APC/C by a factor of 10 in cell cycle model 2, which decreased the pulse duration of the oscillation reactions (Figure 3E) and triggered chaotic dynamics (Figure 3H). Finally, we also verified these results in a third cell cycle model introduced by Novak and Tyson (Novak and Tyson, 1993a). The cell cycle oscillations in this model had a more sustained M-phase and therefore maintained their synchrony when diffusively coupled.

Although this observation suggested that chaotic dynamics might be generally triggered by spatially coupling short pulsatile relaxation oscillators in the presence of a slow calcium wave, it remained possible that this is a phenomenon specific to these particular cell cycle models. To address this, we examined a more generic model that admits relaxation oscillations, the FitzHugh-Nagumo (FHN) model (Fitzhugh, 1961; Nagumo J., 1964), and added a parameter γ to alter the pulse duration of the oscillations (following the lead of Hall and Glass (Hall and Glass, 1999); see supplemental information). We found oscillations of long pulse duration for γ = 0.01 and oscillations with short pulses for γ = 1 (Figure 3F). Introducing spatial diffusive coupling into the FHN model produced the same transition from synchronous oscillations to chaotic dynamics as seen in the cell cycle models (Figure 3F,I), suggesting that the transition to chaos mediated by short pulses is a general phenomenon.

Dynamics in the FHN model can be interpreted in the two-dimensional phase space spanned by two variables: u, the rapidly changing variable, and v, the slowly changing variable. We examined the structure of the different long-term attractors that exist in the FHN model and how those attractors are altered upon changes in the pulse duration of the oscillation reactions. We found that the FHN system can relax into two different attracting states depending on the reaction parameters of the system. First, the system can show a complex series of oscillations that are uncorrelated at points sufficiently separated in space and time (Figure 4A). The phase space plot in this case shows that the system does not have a simple periodic structure (Figure 4C). This is typical for a chaotic attractor and argues that the observed dynamics are in fact chaotic. Nonetheless, this chaotic attractor still has clear structure, whereby the trajectories visit the cubic Z-shaped nullcline deterministically and head toward the bottom leg at random times. Second, the system can instead relax into perfect synchronous oscillations when the pulses are longer, jumping back and forth between the two legs of the u-nullcline (Figure 4B,D).

Spatial heterogeneities can generally trigger a transition to chaotic dynamics

These findings argue that chaotic dynamics can generally arise when spatially coupled relaxation oscillators of short pulse duration are initiated at different times (Figures 3,4). Using the FHN model, we now turn to the question of whether other spatial heterogeneities can trigger a transition from synchrony to chaos. We explored three different ways of introducing a heterogeneous spatial distribution, either in the initial conditions (Figure 5A) or in the reaction parameters (Figure 5B,C).

Fig 5. Chaotic dynamics are triggered in the FHN model by spatial heterogeneities, either in the initial conditions (A) or in the reaction parameters (B,C).

Fig 5

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In (C), the logarithm of ε, which defines the time scale of the slow variable v, was chosen at random for each discretized point in space from a Gaussian distribution with mean 3 and standard deviation 0.1. Two sets of simulations are shown in the presence of these heterogeneities: first using oscillators with longer pulse duration (D-F), and second using oscillators with short pulse duration (G-I).

First, we considered a system in which all reaction parameters are homogeneous in space, but the oscillations are initiated at times that vary linearly with their distance from a fixed point, corresponding to a spreading calcium wave. As above (Figure 3F,I), for small values of γ (long pulses) the waves remained correlated and slowly became more synchronous in time (Figure 5D). In contrast, for large values of γ (short pulses) the waves quickly broke up, lost correlation, and evolved chaotically (Figure 5G).

As a comparison, we next considered a system in which oscillations were initiated at the same time everywhere in space, and the heterogeneity was induced by varying the reaction parameter ε, which defines the slow time scale of the variable v, and as such also controls the period of the oscillations. A small region at the center was assigned a slightly larger value of ε (Figure 5B), such that this region oscillates faster than its surroundings when spatially uncoupled. Upon inclusion of spatial diffusive coupling, for oscillations with longer pulses, a trigger wave originated from this central region and spread out with a fixed velocity until it controlled the entire space (Figure 5E). Trigger waves initiated in this fashion by pacemakers appear to be a common phenomenon in biology (Gelens et al., 2014). In the context of the cell cycle, trigger waves have been recently observed in cell-free Xenopus egg extracts. It has been suggested that the centrosome serves as a pacemaker that controls mitotic entry and exit in the early Xenopus embryo (Chang and Ferrell, 2013; Gelens et al., 2014), perhaps by concentrating pro-mitotic factors such as Cdc25 and cyclin B (Bonnet et al., 2008; Jackman et al., 2003).

At higher values of γ (short pulse duration), chaotic dynamics spread out with a constant speed in much the same manner as the trigger wave observed for oscillations with longer pulses (Figure 5H). These results suggest that perhaps any type of heterogeneity is able to trigger chaotic oscillations. To test this idea further, we varied one of the parameters in the FHN model, ε, in a Gaussian noisy way (Figure 5C). For large values of γ (short pulses), the system indeed quickly developed chaos over the whole region (Figure 5I). In contrast, for small values of γ (long pulses), the spatially coupled oscillations emitted trigger waves that self-organized until eventually one location dominated the whole space (Figure 5F), closely resembling the mitotic waves observed in Xenopus egg extracts (Chang and Ferrell, 2013; Gelens et al., 2014). Diffusive processes that mix their immediate neighborhood are likely to orchestrate this self-organization process. Diffusion increasingly mixes large spatial regions as time increases according to the diffusion length given by 2DT. In these simulations, the diffusion constant was D = 1, such that over a time interval of t = 1000, ~60 length units become mixed. We applied a low-pass filter to the parameter distribution of ε, using a cut-off frequency based on this diffusion length (Figure 5C, dark blue). Strikingly, the locations of the primary local minima of this filtered response corresponded precisely to the points of origin of the trigger waves, and the global minimum ended up being the location of the dominant trigger wave (Figure 5F).

Thus, various types of heterogeneities can trigger chaos in the FHN system, provided that the reactions have a sufficiently short pulse duration. Such heterogeneities can arise in the initial conditions as demonstrated in (McIsaac et al., 2011), or in reaction parameter values as shown here. We note that the influence of heterogeneities has been studied before in FHN systems, especially in the context of cardiac electrophysiology (Bub et al., 2002; Ermentrout and Rinzel, 1996; Panfilov et al., 2005; Prat and Li, 2003; Sridhar et al., 2010; Steinberg et al., 2006). However, in these works, the influence of pulse duration in the presence of heterogeneities had not been studied, which we show to be crucial to trigger a transition into chaos. Given how much biological heterogeneity would be expected in, for example, the Xenopus egg, it seems likely that the pulsatile oscillatory reactions are simply not short enough to support chaos. In other words, our measurement of spatial synchrony in the face of heterogeneities provides us with key information about the shape of the underlying local reaction processes, which are difficult to directly assess experimentally. It helps to explain the biological need to maintain M-phase active for a sufficient amount of time. Various labs have noticed the presence of a time delay in the activation of the APC/C complex that starts mitotic exit by degrading cyclins (Vinod et al., 2013; Yang and Ferrell, 2013). Such a time delay can help to ensure a sufficient time in M-phase to avoid spatial chaos.

Characteristics of the transition into chaos

Given that the pulse duration of the reactions plays an essential role in the generation of chaos, we asked whether diffusion strength could also promote or inhibit chaotic dynamics. Using the same central pacemaker region as in Figure 5E,H, we explored the dynamics for varying values of the pulse duration (γ) and the diffusion strength (D). We found that, at constant γ, decreasing the diffusion strength promoted the generation of chaotic dynamics (Figure 6A-C). When D is decreased, the pacemaker in the center of the region loses its ability to dominate the whole space (Figure 6B). Such traveling wave instabilities have been widely studied in the context of cardiac and neural dynamics, where it has been analyzed how they can be blocked and reflected (also referred to as back-propagation or echo waves) (Booth and Erneux, 1995; Deng, 1991; Ermentrout and Rinzel, 1996; Keldermann et al., 2007; Li, 2003; Rabinovitch et al., 1999; Zhou and Bell, 1994). In the current work, however, the region captured by the initial trigger wave shrinks as D decreases and multiple defects appear that eventually no longer vary periodically with time and chaos sets in (Figure 6C). As shown in Figure 6D, there is a trade-off between the pulse duration of the reactions and the diffusion strength in the generation of chaos.

Fig 6. Decreasing pulse duration, decreasing diffusion strength, and increasing domain widths promote defects and chaotic dynamics.

Fig 6

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(A-C) show the transition into chaos when varying the diffusion strength D (with pulse duration γ = 0.5) in the FHN reaction. The simulations show the dynamics in the presence of a central pacemaker region (as in Figure 5E,F). (D) Two characteristic dynamical regimes are found as a function of γ and D: continuous trigger waves, and defects or chaotic dynamics. (E) The pulse duty cycle (percentage of the time u > 0) initially varies linearly with the control parameter γ (in log10 scale), but then deviates as γ increases further and the pulse no longer reaches the upper branch of the u-nullcline. (F-H) show the stabilization when decreasing the domain width L (with D = γ = 1). (I) The boundary between continuous trigger waves and defects or chaos shifts with the domain width L. (J) The critical diffusion coefficient at the boundary shown in (I) initially scales linearly with L (for γ = 1), but becomes constant for large values of L.

In Figure 6E, we show how the pulse duty cycle changes with γ, where we define the duty cycle as the area under the curve during a pulse (when u > 0) vs. the area under the curve when u < 0. The duty cycle scales linearly with γ on a log-log scale up to when γ is approximately 0.2, after which the curve deviates from the linear fit (red line). This deviation coincides with the moment that the system does not completely reach the upper branch of the v-nullcline (see insets). It is also at this moment that the system becomes susceptible to chaos and defect formation, which motivates the need to postpone APC/C activation and cyclin degradation such that the M-phase state can be fully reached.

Finally, we examined the influence of the domain size L on the transition to spatial chaos. Figure 6F-H shows that for fixed values of the diffusion strength and pulse duration (D = 1, γ = 1), spatial chaos is lost when the domain size decreases below a critical value. Figure 6I depicts in further detail how the region of defects and chaos shrinks with decreasing domain sizes. We analyze the critical diffusion strength D that defines the border between chaos/defects (smaller values of D) and continuous trigger waves (larger values of D). Figure 6J shows that the square of this critical diffusion strength (D2) initially scales linearly with the domain width L and then levels out for large values of L. Thus, larger domains are increasingly susceptible to defect and chaos creation.

To relate these criteria for transitioning into chaos to real physiological values, we review how we can relate the results of cell cycle model 2 (with experimentally measured parameters) presented in Figure 3E,H to our generic dimensionless analysis in the FHN model. The diffusion strength D of the relevant proteins involved in the cell cycle is taken to be 10 μm2/s, and the period of the cell cycle oscillations in extracts is around 40 min (Yang and Ferrell, 2013). The diameter of a typical Xenopus laevis egg is about 1 mm. We assume that the pacemaker region driving the cell cycle and triggering the waves is either the pericentriolar material, with a width of ~1 μm (Chang and Ferrell, 2013; Jackman et al., 2003), or the pronuclei, with diameters ~10 μm. For these physiological parameters, the cell cycle was spatially well coordinated and quasi-synchronous in our simulations (Figure 3H), while an increase in APC/C-dependent degradation by a factor of 10 led to defects and irregular dynamics (Figure 3H). In order to reproduce oscillations of similar period in the FHN model, we can rescale the dimensionless time such that one time unit corresponds to two seconds (while keeping all other parameters in the FHN model the same). The pulse duration can then be adjusted to match the oscillations shown in Figure 3H by setting γ ~ 0.002 (physiological case) or γ ~ 0.2 (tenfold increase in APC/C-dependent degradation) (Figure S2).

As one time unit in the FHN model corresponds to two seconds, and we choose one unit of space in the FHN model to be 1 μm, the diffusion coefficient D = 10 μm2/s corresponds to D = 20 in the FHN model. Examining Figure 6D, we see that the physiological case (D = 20 and γ ~ 0.002) is very stable and far away from any irregular dynamics, while the case of a tenfold increase in APC/C-dependent degradation (D = 20 and γ ~ 0.2) lies very close to the boundary between continuous trigger waves and irregular dynamics, considering a domain width L = 500. Indeed, in cell cycle model 2, decreasing the domain width twofold abolishes the defects and chaos. Similarly, increasing the APC/C-dependent degradation only fivefold instead of tenfold (Figure S2G), or increasing the diffusion strength twofold (Figure S2H) also prevent defects and chaos from developing. Although it is clear that the FHN model does not capture the finer details of the cell cycle model (such as the exact shape of the nullclines and pulse shape), this simple back-of-the-envelope calculation shows that many results can still be gleaned from a generic analysis using the FHN model. Thus for the FHN model, as was the case for the cell cycle model 2, spatial chaos becomes possible at parameter values that are not far from physiological values.

Discussion

We began by testing the hypothesis of McIsaac et al. that slowing down the fertilization-initiated calcium wave would lead to spatio-temporal chaos (Figure 1) (McIsaac et al., 2011). We showed experimentally that varying temperature can slow down the calcium wave and used this approach to test whether normal development depends on a rapid initial calcium wave. Cell cycle oscillations proceeded normally and healthy tadpoles developed even when the speed of the calcium wave was reduced by a factor of 3 (Figure 2).

To understand where the discrepancy lay between the model and the experimental system, we carried out simulations of various relaxation oscillator models and found that chaos only emerged when the oscillatory reactions were sufficiently short in pulse duration (Figure 3,4). A variety of heterogeneities, including differences in the initial conditions or in the reaction parameters, were found to trigger such chaos (Figure 5), and the transition into chaos sometimes spread in a trigger wave-like fashion through a homogeneous medium (Figure 5). Similar results were found in simulations with two spatial dimensions (Figure S3).

Since our experiments showed no sign of chaos when the calcium wave was slowed, and moreover frog cytoplasm is likely to contain the sorts of heterogeneities that can trigger chaos, we propose that the reactions of the Xenopus cell cycle oscillator do not generate pulses in short enough duration to admit chaos. Although it is difficult to experimentally determine exactly how short the duration of M-phase is, because experiments are done on embryos and extracts whose volumes are sufficiently large that they are far from the well-stirred limit, these conclusions are consistent with past experimental work (Pomerening et al., 2005; Solomon et al., 1990; Tsai et al., 2014; Yang and Ferrell, 2013).

In other biological systems, reaction dynamics with short pulse durations may in fact be present. One example is cardiac dynamics, where abrupt depolarization of the myocyte is important for normal force/velocity relationships and cardiac contractility. In a system like this, chaos might be an important hazard. The FHN equations have in fact been widely used to model heart conduction, and some arrhythmias have been interpreted as chaotic modes of the underlying spatio-temporal dynamics (Babloyantz, 1994; Chialvo and Jalife, 1987; Duckett and Barkley, 2000; Rappel et al., 1999; Starmer et al., 1993; Tyson and Keener, 1988; van der Pol and van der Mark, 1928; Winfree, 1989). Another related example is the complex chaotic (tent) patterns of pigment that are found on the shells of some mollusks (Meinhardt, 2003). Such mollusk patterns have been modeled using cellular automata (Wolfram, 1984) and reaction-diffusion equations (Ermentrout et al., 1986; Meinhardt and Klingler, 1987). In this case, the complex beautiful dynamics of the spatially chaotic system may be a feature of the system that has been selected for rather than a hazard to be avoided.

Experimental procedures

Xenopus experiments

All animal work was conducted according to relevant national and international guidelines. Animal protocols were approved by the Stanford University Administrative Panel on Laboratory Animal Care. Female Xenopus laevis frogs were induced by human chorionic gonadotropin injection and pelvic massage was performed to collect eggs 12 to 20 h after induction. In vitro fertilization was performed in the device shown in Figure 2A by mixing eggs with smashed testes for 1 min in several drops of 0.1× Marc’s Modified Ringer’s (MMR) buffer. The MMR buffer was first brought to the temperature of interest before fertilization of the eggs. The embryonic divisions were imaged with a stereoscope Nikon SMZ 1500 with a Leica DFC425 camera. Frame rate was 1 frame/30 s to image the long-term development and 1 frame/10 s to image the calcium wave.

Mathematical models

Here, we briefly discuss the origin and variables involved in the different models used throughout this work. The full set of equations, the parameters used, and the details of each numerical simulation can be found in the Supplementary Information.

Cell cycle model 1

This cell cycle model, used in (McIsaac et al., 2011) (and introduced before in (Pomerening et al., 2005; Pomerening et al., 2003)), consists of 9 ODEs, representing the different phosphorylation states of the cyclin B-Cdk1 complex and the activities of cyclin-B, Cdc25, Wee1, Plx1, and APC/C.

Cell cycle model 2

This cell cycle model, introduced in (Yang and Ferrell, 2013), consists of 2 ODEs. The first ODE describes the activity of Cdk1, and the second ODE describes the synthesis and destruction of the mitotic cyclins.

Asymmetric FitzHugh-Nagumo model

The general FitzHugh-Nagumo model (FHN) (Fitzhugh, 1961; Nagumo J., 1964) was adjusted to allow for tuning of the sharpness of the relaxation oscillations. An additional parameter γ was introduced that changes the time scale of the evolution of the v variable.

Supplementary Material

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Highlights.

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    Spatially coupled relaxation oscillators of short pulse duration can generate chaos

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    Spatial heterogeneities can trigger a transition to chaotic dynamics

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    Mitotic exit in the Xenopus laevis cell cycle is likely slow enough to avoid chaos

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    Spatial chaos can be a feature or a hazard in biology

Acknowledgments

We thank Lauren Chircus, who carried out preliminary experiments for this study, and Thomas Erneux, Graham Anderson and the Ferrell lab for helpful comments and discussions. This work was supported by National Institutes of Health grants GM046383 and GM107615 (to J.E.F.) and DP2-OD006466 (to K.C.H.). L.G. acknowledges support by the Research Foundation-Flanders, the Belgian American Educational Foundation, and the Research Council of the Vrije Universiteit Brussel.

Footnotes

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Supplemental information

Supplemental Information includes a full description of the models for biochemical reaction dynamics, description of the numerical methods used, three figures, and three movies and can be found with this article online at (link).

Author contributions

L.G. and J.F. designed the study; L.G. and K.C.H. performed the computational analyses; L.G. performed the Xenopus experiments; L.G., K.C.H., and J.E.F. wrote the manuscript and contributed to the ideas through discussions.

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