The authors have declared that no competing interests exist.
Conceived and designed the experiments: TI JOD MLM PCN AJL. Performed the experiments: JOD. Analyzed the data: TI LK. Contributed reagents/materials/analysis tools: TI JOD. Wrote the paper: TI AJL. Developed the model: TI LK MLM PCN TCL AJL.
Current address: Department of Bionanoscience, Delft University of Technology, Delft, The Netherlands
Mitosis in the early syncytial
The early embryos of many species, including
The nuclei of the
However, mitosis is also a mechanical phenomenon. In the syncytial embryo, nuclei are embedded in an elastic cytoskeleton, which contains both actin and microtubules
In this paper we report the results of both our image analysis of wavefronts in early
We treat the embryo theoretically as an excitable medium, consisting of nuclei that can be triggered into initiating metaphase or anaphase, thereby locally exciting the medium and thus signaling their neighbors. We not only consider the well-known case of nonlinear wavefront propagation in a chemically excitable medium
An example image of detected nuclei in a
a) Image of a
The onsets of metaphase and anaphase, as determined by the axes ratio (
The processes of metaphase and anaphase affect not only the shapes of the chromosomes, but also their positions. After each of the shape changes, the nuclei move
The displacement response to the onsets of metaphase and anaphase causes the nuclei to move to new equilibrium positions (
We quantify the wavefront speeds in
a)
From embryo to embryo there are large variations in wavefront speed (
The nuclei on the surface are separated by a well-defined distance
Data set 1 | ||||
cycle number | 10 | 11 | 12 | 13 |
nuclear spacing ( |
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wavefront speed ( |
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cycle duration (s) | ||||
mitosis duration (s) |
Data set 2 | |||
cycle number | 11 | 12 | 13 |
nuclear spacing ( |
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wavefront speed ( |
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cycle duration (s) | |||
mitosis duration (s) |
Data sets 1 and 2 correspond to two different sets of measurements, taken on different days. They correspond to respectively the closed and open symbols in
Our observation that the mitotic wavefronts propagate at constant speed across the embryo suggests that the embryo can be considered as an excitable medium that supports nonlinear front propagation. Alternatively, the nuclei could all have biological clocks that determine when mitosis starts, which operate independently; in that case the wavefront would be only a result of a lucky timing of those clocks. We discuss various timing models and show that they are inconsistent with our observations in the supplementary material. Here we concentrate on two distinct classes of models for front propagation in excitable media. In the first model the nuclei communicate by releasing a small chemical species, which then diffuses to neighboring nuclei, triggering them to initiate mitosis. In the second model we explore the novel idea that mitotic wavefronts in the early embryo can be described by wavefront propagation in a medium that is
At the end of a cycle, when all nuclei have completed the duplication of their DNA, we assume that they are in an excitable state, meaning that they can be triggered to initiate mitosis once they receive an appropriate signal. An obvious candidate for signaling between nuclei is a small protein (
To introduce chemical excitability, we assume that if the local concentration of
a) Color plot showing the chemical wavefront in two dimensions. The wave starts in the center (red dot) with a single Dirac delta peak release. The color coding indicates when a nucleus releases its chemical to the bulk, going from red through the different hues of the rainbow to violet. b) Plot showing the best fits (blue and purple lines) of the diffusion model with time delay to the to the two sets of experimental data (black and gray dots with error bars). Although the time delay manages to balance the trend that the wavefront speed increases in the region of interest (but not before), the model fails to describe the observed data. Here
In the case of zero delay time, the speed
Combining the parameters of our model, we define a nondimensional threshold and speed:
In a three-dimensional model the power of
We next consider the possibility of a delay
A key result of our analysis with a fixed time delay is that a physically unrealistic diffusion coefficient is required in order to reproduce our experimental observations. In order to obtain a strictly decreasing wavefront speed for the range of interest, a diffusion constant of more than
We also investigated the wavefront speed in the case where the delay time is allowed to vary from one cycle to the next. Naturally, given a value for the diffusion constant and the threshold, for each cycle we can find a delay time such that the speed predicted by the model matches the observed speed; these values are listed in Table S1 in
On the basis of these results, we conclude that it is very unlikely that a wavefront that propagates via diffusion of some chemical species would slow down with cycle number, as observed in our experiments. We also note that any model in which the biochemical signal is mediated by a method that is faster than diffusion (such as active transport) suffers from the same problem: the predicted wavespeed would go up with increasing cycle, because the spacing between the nuclei goes down.
The early embryo cannot support ordinary elastic waves because it is heavily damped by the viscosity of the cytosol. Consequently, displacements do not propagate ballistically as in a wave, but diffusively. However, just as diffusion of
In our model, a nucleus starts its program when the largest eigenvalue of the local stress tensor exceeds a threshold value
The term on the left represents the damping with damping factor
In order to introduce
Here
In order to compare the model results with the data, we need to estimate the values of the elastic constants and the damping parameter. The speed
Because of the mathematical similarity between the mechanical-signaling model (
These values for the diffusion constant are comparable to those found in microrheology experiments, which have measured the frequency-dependent complex shear modulus in a variety of living cells
The found value for the exponent
In addition, we note that the nuclear displacement profile provides a more discriminating test of the mechanical signaling model than the wavefront velocity. Although the velocity wavefront speed data alone can be fitted by either purely isotropic force dipoles or purely anisotropic force dipoles (and presumably anything in between), the displacement wavefront can only be fit with dipoles with a strong isotropic component. Moreover, although either the displacement or the velocity data can be fit with different combinations of the threshold and diffusion constant, the numbers given above are the only ones for which we can fit both quantities.
In summary, the mechanical signaling model agrees much better with the data than the biochemical signaling model in two important respects. First, it captures the dependence of the wavefront velocity on cell cycle number while the biochemical signaling model does not. From dimensional analysis, we have shown for both models that the wavefront velocity depends mainly on
Finally, we note that we have assumed that the elastic constants and damping coefficients vary from cycle to cycle but do not change much during the period that we are focusing on. However, the cytoskeleton reconstructs completely during the cell cycle. Our analysis will apply as long as the elastic constants and damping coefficient do not change appreciably from the time that the original triggering wavefront is generated to the time that the anaphasic wavefront occurs. Thus, the assumption is that cytoskeletal reconstruction occurs sometime during anaphase and is finished before the process of mitosis begins in the next cycle. In particular, this also means that our model should not be able to correctly predict the much larger displacements following anaphase (see
During the early cycles of
There are several markers of the mitotic process in each cycle, corresponding to the onsets of metaphase and anaphase, which are visible as wavefronts that travel across the embryo (
The speed of the mitotic wavefronts slows down in each successive cycle (
The onsets of metaphase and anaphase both trigger a mechanical response of the entire embryo in the form of displacements of the nuclei that also exhibit a wavefront pattern (
In addition to these observations, we add those of Parry et al.
There is a visible wavefront in calcium release that coincides with the onset of anaphase.
The speed of the calcium wavefront slows down in each successive cycle, presumably matching the speed of the mitotic wavefront.
We have considered two scenarios to assess whether they are consistent with these observations. In both cases, based on observations (1), (2) and (5), we take the observed metaphase, anaphase and calcium wavefronts to be different markers of the same mitotic process, and assume that the mitotic wavefront is triggered by a single event.
This prediction could be tested by looking for wavefronts in likely signaling species. If the wavefront propagates biochemically, then wavefronts should be observable in the appropriate signaling molecules (presumably CDKs or cyclins that are known to govern checkpoints in the cell cycle that precede the onset of metaphase
Scenario B is consistent as well with independent observations made in
We emphasize that Scenario B does not imply that the entire process of triggering mitosis is mechanical. Indeed, the mechanism by which additional stress is generated via a force dipole in our model must be biochemical. First, there must be some sensor components that are activated when the stress exceeds its threshold value. These components must then activate other biochemical species to eventually generate additional stress by creating a force dipole. If Scenario B is correct, there should be a way of incorporating our mechanically signaling model into models of the chemical networks that control the cell cycle, such as those of Tyson and Novak
In principle, the estimated elastic constants and damping coefficients could be obtained directly from experiments by measuring the storage and loss moduli of the embryo surface
Even though the process of mitosis is known to require chemical activation, the key assumption in Scenario B is that the initial wavefront also
Finally, we note that biochemical experiments could also test the mechanical-signaling model. The most straightforward test would be to to destroy or degrade the filaments that mechanically couple the nuclei. This should prevent the mechanical wavefronts from propagating and thus the nuclei from synchronizing their mitosis. This could be done by injecting colcemid or nocodazole to disrupt the microtubules or latrunculin which affects actin filaments, for example
The imaged flies were from a His-GFP stock with a P [w+ ubi-H2A-GFP] insertion on the third chromosome. All embryos were collected at 25°C and dechorionated in 100% bleach for 1 minute. They were picked using a
We visualized nuclear DNA/chromosomes by tagging their histones with GFP. To determine the positions, sizes, aspect ratios and orientations of the DNA/chromosomes from each video frame, we developed a new image analysis technique, explained in detail in
Because the images were taken at high frequency (typically 1 Hz), the nuclei move less than their own radius from one frame to the next, simplifying tracking. The obvious exception is when nuclei divide during anaphase, and the observed shape splits in two. Because we detect shapes as well as positions of the chromosomes in each nucleus, tracking divisions is easy as well: when a nucleus divides, the chromosomes become highly elongated just before they split, and produce two almost circular daughters close to the endpoints of the long axis of the mother immediately after it splits, which are easily identified.
Our image analysis results are for two different sets of experiments, which were carried out at ambient room temperature several months apart. The ambient temperature was higher for the second set, resulting in faster embryo development. We only used the data from those embryos which we could track from cycle 10–14 in the first set (Dataset 1, 3 embryos) and cycle 11–14 in the second set (Dataset 2, 4 embryos).
The average data from the two sets are given in
In addition to the data shown in
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We thank Thomas Gregor for providing resources for the experiments and for his careful reading of the manuscript, and Xiaoyang Long for assistance with acquiring the experimental data. We also thank Gareth Alexander and Michael Lampson for helpful discussions.