DEVELOPMENTAL DYNAMICS 244:1014–1021, 2015 DOI: 10.1002/DVDY.24298
RESEARCH ARTICLE
Evidence for Internuclear Signaling in Drosophila Embryogenesis a
Richard Buckalew,1* Kara Finley,2 Soichi Tanda,2 and Todd Young1 1
Mathematics, Ohio University, Athens, Ohio Biological Sciences, Ohio University, Athens, Ohio
Developmental Dynamics
2
Background: Syncytial nuclei in Drosophila embryos undergo their first 13 divisions nearly synchronously. In the last several cell cycles, these division events travel across the anterior–posterior axis of the syncytial blastoderm in a wave. The phenomenon is well documented but the underlying mechanisms are not yet understood. Results: We study timing and positional data obtained from in vivo imaging of Drosophila embryos. We determine the statistical properties of the distribution of division times within and across generations with the null hypothesis that timing of division events is an independent random variable for each nucleus. We also compare timing data with a model of Drosophila cell cycle regulation that does not include internuclear signaling, and to a universal model of phase-dependent signaling to determine the probable form of internuclear signaling in the syncytial embryo. Conclusions: The statistical variance of division times is lower than one would expect from uncoordinated activity. In fact, the variance decreases between the 10th and 11th divisions, which demonstrates a contribution of internuclear signaling to the observed synchrony and division waves. Our comparison with a coupled oscillator model leads us to conclude that internuclear signaling must be of Response/Signaling type with a positive impulse. Developmental C 2015 Wiley Periodicals, Inc. Dynamics 244:1014–1021, 2015. V Key words: Drosophila; mitosis; synchronization; synchrony; division wave Submitted 4 December 2014; First Decision 23 April 2015; Accepted 20 May 2015; Published online 29 May 2015
Introduction Cell division is one of the most essential biological processes for metazoans. In most multicellular organisms the first phase in development is the cleavage stage, characterized by rapid and synchronous cell divisions. Cell division in the cleavage stage of such organisms is controlled by maternally provided products independent of zygotic gene expression. This allows cells to increase their numbers quickly. It is thus of special interest to understand the mechanisms that control synchronous and rapid cell divisions. Drosophila melanogaster is a model organism that has been studied for many years, helping to provide insights into many biological processes, including genetics, epigenetics, metabolism, development, and so forth (Singh and Irvine, 2012). Because of its fast embryonic development, which takes approximately 1 day from egg deposition to hatching, D. melanogaster is particularly well suited to investigations of genetic, cellular, and molecular mechanisms of embryogenesis. Several phenomenological observations have been made of the early embryo, including the comprehensive work of Foe and Alberts (1983). Foe observed a remarkable synchrony between the embryonic nuclei as they proGrant sponsor: NIH; Grant number: NIH-NIGMS R01GM090207. *Correspondence to: Richard Buckalew, Mathematics, Ohio University, Athens, Ohio 45701. E-mail: [email protected] Dr. Buckalew’s present address is Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio, 43210
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gress through the cell cycle and undergo mitosis. The exact mechanism for this synchronization has not been identified. The first 13 mitotic divisions are synchronous and proceed without cytokinesis to form a single syncytium of around 6,000 nuclei. During the first seven nuclear division cycles in D. melanogaster, all dividing nuclei are contained in the interior of the syncytium. Around cycle 7, most nuclei and their surrounding protoplasmic islands begin migrating toward the surface of the embryo, while the “yolk nuclei” are formed from the nuclei remaining in the interior. The migrating nuclei arrive at the surface during the 10th cycle, forming the blastoderm monolayer, then undergo meta-synchronous division during cycles 10–13 (Foe and Alberts, 1983). These meta-synchronous divisions occur in the form of waves, starting at one or both ends of the embryo. The cell cycle lengths increase during cycles 10–13, and at cycle 14 the cell cycle length increases extensively and the embryo undergoes cellularization, becoming the cellular blastoderm. During cycles 1–13, the nuclei exhibit cell cycles consisting only of S and M phases, with no G1 or G2 phase involved. The growth phases appear to be skipped because the embryo is able to use maternally provided gene products to drive the embryonic cell cycle, with very limited zygotic gene transcription occurring (Lee and Orr-Weaver, 2003). Cycle 14 marks the beginning of the Article is online at: http://onlinelibrary.wiley.com/doi/10.1002/dvdy. 24298/abstract C 2015 Wiley Periodicals, Inc. V
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regular cell cycle with G1 and G2 phases, contributing to the longer duration of this cell cycle. Foe and her colleagues observed division waves extending across the syncytium during cycles 10–13 (Foe and Alberts, 1983), beginning at both the posterior and anterior portions of the embryo (such waves may also travel in only one direction, as in the data set we use in this study). The mechanisms responsible for the duration and regulation of the early cell cycles have been studied extensively (Foe and Alberts, 1983; Edgar et al., 1994; Stiffler et al., 1999; Jun-Yuan et al., 2004; Calzone et al., 2007; McCleland et al., 2009; Qing and Pomerening, 2012; Xuemin et al., 2012). The Cyclin B/Cdk1 (Cyclin-dependent kinase 1) complex regulates mitotic entry, including chromosome condensation and nuclear envelope breakdown, and is an obvious candidate for the main driver of the observed synchrony. Cyclin B/Cdk1 complexes only form once a threshold level of Cyclin B has been reached (Qing and Pomerening, 2012), and exit from mitosis requires the dissociation of this complex at the beginning of anaphase, presumably by APC (anaphase-promoting complex) (Stiffler et al., 1999). Although binding to Cyclin B is required for its activity, the regulation of Cdk1 activity is complex, including two feedback loops. The first one is a double negative feedback loop between Cdk1 and Wee1, which keeps the level of Cdk1 activity low during interphase. The second one is a positive feedback loop between Cdk1 and String tyrosine phosphatase. The removal of the phosphate at Y15 by String activates Cdk1 by counteracting the effect of Wee1. Cdk1 in turn activates String by phosphorylating it. Thus, at the entry to M phase, high Cdk1 activity reached by accumulation of Cyclin B and dephosphorylation at Y15 by String tyrosine phosphatase leads to a sharp decrease in Wee1 activity. As Wee1 activity is decreased, the positive feedback loop between Cdk1 and String dominates, further raising the level of Cdk1 activity. Studies show that cytoplasmic Cdk1 remains active and at a relatively constant level until cycle 8, when local fluctuations in Cdk1 activity begin to occur as the complex dissociates (Edgar et al., 1994). The start of oscillation seems to coincide with declining Cyclin B levels. Cyclin B measured in the early embryo at a global level does not appear to oscillate until cycle 7, although it shows a steady decrease (Edgar et al., 1994). However, Cyclin B levels exhibit localized oscillations corresponding to the cell cycle. During the end of metaphase, spindle-associated Cyclin B is degraded, contributing to the localized oscillations and allowing for the progression into anaphase (Lee and Orr-Weaver, 2003). As Edgar et al. proposes, the continuing degradation of Cyclin B may contribute to the lengthening of cell cycles 9–14, because the embryo needs more time to accumulate the levels of Cyclin B required for mitotic entry due to the higher nucleo:cytoplasmic ratio (Edgar et al., 1994). The time required for translation of the maternal Cyclin B mRNA to meet the Cyclin B threshold causes a longer S phase in these later cycles, while mitosis length remains the same. As Cyclin B levels start to oscillate, the aforementioned positive and double negative feedback loops operate to induce robust activation of the Cyclin B/Cdk1 complex. Calzone et al. (2007) modeled some of the factors regulating the mitotic cycle in the syncytial embryo. Their work accurately accounts for the slowdown of the cycles during later divisions, due largely to depletion of maternal Cyclin B mRNA. In that work individual nuclei were not modeled, instead the total “nuclear matter” was treated as a single compartment, and so internuclear
coupling was not considered. Chang and Ferrell (2013) implicate the Cyclin B/Cdk1 complex and its interactions with Wee1, Cdc25 (String in Drosophila), and APC, as the driver of synchronous division in Xenopus embryos. This coupling is by means of the mechanism of trigger waves, which propagate to neighboring nuclei as a high activity level of the Cyclin B/Cdk1 complex maintained by a positive feedback loop between Cdc25 and the Cyclin B/Cdk1 complex and a double negative feedback loop between Wee1 and the Cyclin B/Cdk1 complex. We speculate that a similar mechanism is at work in the Drosophila embryo. For multiple periodic processes to be synchronized, they must (A) have nearly identical clocks, (B) have coupling mechanisms, or have a combination of (A) and (B). Situation (A) can produce synchrony for long periods of time, but not indefinitely because even very small differences will eventually lead to out-of-phase clocks. Situation (B) requires strong coupling if the processes possess very different clocks. It has long been known that between two or more mechanical or electronic oscillators, a combination of (A) and (B) will cause them to become synchronized indefinitely. The strength of coupling needed to maintain synchrony is small and grows with the difference between the two frequencies. Because the syncytial nuclei of a single early Drosophila embryo are nearly identical in every way, it is reasonable that their mitotic cycles are almost identical. Because the synchrony only lasts for a relatively short time, 13 divisions, it is possible that the observed synchrony is entirely due to good clocks (A). The exact mechanism has not been identified, but we have shown that feedback processes between cells, i.e., both (A) and (B), can cause similar synchronizing behavior in yeast and many of the properties of such a system were analyzed in Young et al. (2012). In the present study, we give evidence from recently obtained data that the synchrony of mitotic cycles observed in the syncytial nuclei is not solely due to nearly identical clocks, but in fact must also have an element of coupling, i.e., both (A) and (B).
Results Automated Tracking Data Tomer et al. (2012), demonstrated a novel method of video microscopy and used it to create compelling images and large data sets relating to Drosophila embryogenesis, available at www.janelia.org/digitalembryo. They imaged Drosophila at many stages of development, including mitotic divisions 10 through 13 in the syncytial blastoderm. In correspondence, the authors provided us with automated tracking data for the 12th and 13th divisions. The authors imaged a developing embryo 128 times, at intervals of 25 sec. Their imaging technique allows for resolution in three spatial dimensions, so that the initial data set consists of volumetric luminescence data (obtained from histone-eGFP tagging) at sub-micron resolutions. Using an image processing algorithm, they identified individual nuclei in the data for each time point. They then compared neighboring time points algorithmically to track individual nuclei from image to image, and to identify when a mitotic division has occurred. The resulting data set consists of 51,482 lines of data, where each line contains the position of a single nucleus at a single time point, plus lineage data indicating whether it is the continuation of a nucleus at the previous time point, or the result of a division.
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Fig. 1. Division waves and linear fit lines for the 12th and 13th mitotic divisions in the automated tracking data. Vertical units are pixels, with the head of the embryo near x ¼ 200, and horizontal units are video frames, recorded once every 25 sec.
By reconstructing this lineage data, we identified many lines of descent; branching trees starting at a single nucleus that follow its descendants and each of theirs, and so on. The data were very noisy, and included a large number of lines of descent that we classified as erroneous for various reasons. We thus trimmed our data set to include only those lines of descent which satisfied the following criteria: (1) the line must start with a single nucleus that is present at the first time step; (2) that nucleus must divide at approximately the time when the division wave reaches it; (3) each of its two daughters must persist until dividing approximately when the second wave reaches them, and (4) their four total daughter nuclei must persist without dividing until the last time point present in the data. Any line of descent satisfying all four criteria can reasonably be considered to correspond to an actual nuclear lineage. We identified 457 such lines consisting of 3,199 nuclei which we can track with high confidence. In Figure 1, we plot the positions of mitotic divisions recovered from the lineage data. The horizontal axis is time, corresponding to the 128 separate images taken at 25-sec intervals. The vertical axis represents the distance from the anterior end of the embryo. Two division waves are apparent in the data as lines with positive slope originating from the anterior end and propagating toward the posterior.
Manual Tracking Data For this data set, automated tracking data are only available for the 12th and 13th mitoses, but reconstructed image data are available for the 10th through the 13th mitotic divisions. Similar to the automated data, the data set includes 201 images taken at 25-sec intervals. The images are two-dimensional projections from the lateral side of the embryo, and do not include depth information. To perform more robust statistical analysis than possible with the automated data, we obtained positional information for a portion of the nuclei by tracking them by hand in the image data. In all, we followed 654 nuclei in 117 lines of descent. These manual data cover four division waves, including the three mitotic
Fig. 2. Division waves and linear fit lines for the 10th, 11th, 12th, and 13th divisions in the manual tracking data.
cycles between them, plus two partial cycles at the beginning and end of the data. We recorded n ¼ 63; 94; 136, and 180 divisions in each wave. Figure 2 shows the positions of divisions in the manual data (which were measured from the anterior of the embryo). Four division waves of decreasing speed are evident.
Statistical Methods We will treat the mitotic divisions as events in a stochastic process. Let ti be a random variable denoting the time of the ith division of a particular line of nuclei. It is clear that ti is actually a sum of random variables: ti ¼ c1 þ c2 þ ::: þ ci ; where cj ; j ¼ 1; 2; :::; i, represents the duration (period) of the j th cycle. We may consider the variance of the variable ti among different nuclei within an embryo to be a measure of the synchronization at the ith division. It is a well-known fact that if c1 ; :::; cn is a collection of independent random variables distributed Pwith variances s21 ; . . . ; s2n , then the random variable ti ¼ ij¼1 cj has variance varðti Þ ¼
i X
s2j ;
(1)
j¼1
in other words, the variance of the sum is the sum of the variances. If the cell cycles of different nuclei are not coupled, then ci and ti would be independent of the nucleus and the distributions of ci and ti within the embryo should have the characteristics of independent random variables. Particularly, they should satisfy the variance condition (Eq. 1) and the variables ci ¼ ti ti1 and ciþ1 ¼ tiþ1 ti should not be correlated. Because the mitotic divisions occur in waves, a nucleus’s position affects its division time. In the data set we analyze, nuclei near the anterior of the embryo divide before those near the posterior. The speed of the waves is variable as well; from the manual tracking data, we see that the division wave for the 10th mitosis travels at approximately 136:3 microns per minute, the 11th at 119:3 microns per minute, the 12th at 79:1 and the 13th
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TABLE 1. Linear Fits for Division Waves in the Automated and Manual Tracking Data Automated tracking i
mi
bi
r
TABLE 2. Distributional Parameters for Divisions From the Automated Tracking Data
Manual tracking i
mi
bi
12 78.4 648.2 0.919 10 156.7 2278.9 13 73.6 4973.8 0.863 11 137.2 5993.0 12 91.0 7309.8 13 94.1 13145.7
i 12 13
r 0.946 0.979 0.977 0.950
n 457 914
li 16
4 10 3 1015
r2
skewness
2.137 4.108
3.097 2.210
p (normality) 8 1079 3 10112
Suppose a nucleus undergoes its ith mitotic division at time ti and position xi . Its wave-adjusted deviation ~t i is given by: ~t i ¼ ti t i ðxÞ: Our subsequent analysis will center on ~t i .
Developmental Dynamics
Analysis
Fig. 3. Distributions of t~12 (left) and t~13 (right) from the automated tracking data.
at 81:8 microns per minute. These speeds are relatively constant through the duration of the wave, indicating that simple diffusion of a background chemical cannot drive the progression of the wave. Instead, we suspect “trigger wave”-like mechanisms propagating from nucleus to nucleus as described in Chang and Ferrell (2013). Because the waves travel at distinct speeds, absolute division times ti are not suitable for measuring synchrony. We want to measure the degree to which nearby cells are synchronized, and ignore the variations caused by location in the embryo. For this reason we introduce the wave-adjusted deviations ~t i which are defined as follows. From the division time and location data in Figures 1 and 2, we obtain a linear fit xi ¼ mi t þ bi for each division wave. The results are summarized in Table 1. We want to define a function t i ðx Þ that represents the expected time of division for a nucleus at position x during the ith division wave. This can be obtained from the fit line xi ¼ mi t þ bi by solving for t, giving: t i ðxÞ ¼ x bi : mi
(2)
The amount of synchronization in the ith division wave can be measured by looking at the difference between the actual time of division ti and the expected time of division t i given by Equation 2. This is what we call the wave adjusted deviation ~t i , representing the “de-trended” deviations. Formally we have:
The distribution of ~t i for i ¼ 12; 13 from the automated data is shown in Figure 3, and the parameters for the distributions are given in Table 2. We note that the distributions are not normal, and that the variance increases dramatically from the 12th to the 13th division, which indicates the beginning of the breakdown in synchrony described by Foe and Alberts (1983), as well as Calzone et al. (2007). The underlying cause of this loss of synchrony is not completely understood, but it is likely caused in part by the switch to zygotic expression rather than the embryo’s earlier use of maternally provided gene products at the mid-blastula transition at the 13th division Budirahardja and Gonczy (2009). Figure 4 and Table 3 show the distributions and distribution parameters for waves 10 through 13 in the manual tracking data. Figure 5 shows the v-squared 95% confidence intervals for the population variance based on each sample. There is a clear decrease in variance between the 10th and 11th divisions, which strongly indicates that division times between these two division cycles are not independent. Also shown in Figure 5 are data obtained from simulations of an SDE version of the Calzone model. (See the Appendix for a description of the Calzone model and our implementation of it.) For each of 10 values of stochastic noise, the model was run 100 times and the division times were recorded. The gray curves are the variances of the division times for each of divisions 10 through 13, in s2. It is clear that there are no mechanisms in the model that give rise to a decrease in variance at the 11th, or any other, division. We suspect this is because it treats all the nuclei as a single compartment, and thus does not include the potential for internuclear signaling. Now consider the lengths ci of the division cycles. Recall that ci ¼ tiþ1 ti , where ti is the time of the ith division. There is again an effect of position on ci due to the division waves: because the waves are decreasing in speed (see Table 3) a nucleus at the posterior end of the embryo has a longer expected cycle length ci than one near the anterior. Thus we define the expected ith cycle length at position x to be c ðx Þ ¼ t iþ1 ðx Þ t i ðxÞ. We present a statistical analysis of the wave-adjusted cycle length deviations ~c i defined as: ~c i ¼ ci c : Our null hypothesis is that there is no communication between nuclei. In this case, the deviation from a nucleus’ expected division time will be a random variable, independent from that of its mother’s. From the manual tracking data, we have two sets of
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Fig. 4.
Distributions of t~10 (top left), t~11 (top right), t~12 (bottom left), and t~13 (bottom right) from the manual tracking data.
TABLE 3. Distributional Parameters for Divisions From the Manual Tracking Data i 10 11 12 13
n 63 94 136 180
li 15
1 10 4 1016 2 1015 4 1015
mother-daughter pairs where we can make such comparisons, between the 11th and 12th divisions, and between the 12th and 13th. Figure 6 shows the plot of ~c 12 vs. ~c 11 and ~c 13 vs. ~c 12 . The results indicate a negative correlation between the lengths of the 12th and 13th division cycles, with correlation coefficient 0:25 and P-value 0:00086. Thus, although the effect is not very strong, we consider it to be a real effect. As described above, such a correlation would not exist if the values of ~c i were independent random variables. Even the case of the 11th and 12th cycles, where the correlation is essentially zero, provides an affirmative result, for if there were no internuclear communication, then we would expect a positive correlation: genetic or epigenetic effects causing a nucleus to divide late should cause daughter nuclei to divide later than expected as well. These results indicate that some form of internuclear signaling related to cell cycle regulation must be occurring.
Comparison With a Universal Coupled Oscillator Model In Young et al. (2012), the authors introduced a simple, but universal model of cell cycle signaling in a community of cells based on earlier work by Bozcko et al. (2010). The conclusions were confirmed in the
r2
skewness
p (normality)
0.290 0.154 0.335 0.861
0.173 0.0256 0.146 0.587
0.723 0.293 0:755 0:005
presence of dynamic delay by Gong et al. (2014). We specify two stages of the mitotic cycle as signaling (S) and responsive (R), and cells currently in the signaling stage influence cells in the responsive stage to either speed up or slow down their cell cycle progression. In the context of Drosophila embryogenesis, we conjecture that the responsive stage R is associated with the end of S-phase (or G2 in later cycles) and the signaling stage S with the beginning of M-phase. This is because the trigger wave phenomenon occurs around the checkpoint regulating mitotic entry, and the positive feedback loop between Cyclin B/Cdk1 and Cdc25 (String) occurs at this point in the cell cycle (Chang and Ferrell, 2013). In this model, the progression of cell i in the cell cycle is represented by the variable yi , which takes values in the interval ½0; 1Þ. This value represents a proxy for the cell’s state in the cell cycle; as originally proposed for budding yeast, yi represented the log of the cell’s volume, but it should be interpreted more broadly as simply an indication of progression through the cell cycle. Thus the value of yi can only increase in the model, and when it reaches 1 (division) it begins again at 0. It is therefore natural to think of the state of a population of n cells in this model as a point on the n-torus Sn .
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Fig. 5. Sample variances from the manual tracking data for divisions 10, 11, 12, and 13, and the 95% confidence intervals for the corresponding populations, as well as the calculated variances from 10 simulations of the stochastic Calzone model (see Appendix 1).
Fig. 6. Correlation between mother and daughter deviations from expected division times.
To model the cell cycle signaling described above, we proposed the differential equations: ( = R 1; when yi 2 dyi ; (3) ¼ dt 1 þ f ðIÞ; when yi 2 R where I ¼ fj j yj 2 Sg=n is the fraction of cells which are in the signaling region. The response function f ðIÞ must satisfy f ð0Þ ¼ 0 and it must be monotone (either increasing or decreasing) but it can be nonlinear. For instance, from biological considerations, we expect f to be sigmoidal (S-shaped), and to satisfy f ðI Þ 2 ð1; 1Þ for any I. Thus a cell yi will move forward with speed 1 when it is not in the responsive stage, and it will move with speed 1 þ f ðIÞ when it is in the responsive stage. While the actual processes involved in cell cycle communication among cells could in fact be extremely complex (and beyond our current knowledge), it is easy to see that all such communication, regardless of its details, must act by cells in some stage of
Fig. 7. RS-positive vs. SR-negative feedback processes. The cell cycle positions of two nuclei, nearby both in space and in the cell cycle, are represented by the solid disks, where cell cycle progression is represented by movement from left to right (the illustrations do not show any spatial information). The boundaries of the responsive and signaling regions R and S are indicated by the small vertical lines. In each illustration, three “snapshots” of the system are shown in order from top to bottom. In an RS-positive system (left), nuclei which are ahead in the cell cycle (i.e., closer to or already undergoing mitosis) accelerate the progression of those behind (illustrated by arched arrows), so the responsive nucleus progresses faster than the signaling nucleus. In SR-negative feedback (right), nuclei which are behind in the cell cycle slow down those ahead (arched inhibitory arrows), so that the responsive nucleus progresses more slowly than the signaling nucleus. In both cases, the distance between their positions in the cell cycle decreases, leading to synchronization.
the cycle (signaling) producing chemicals that effect the cell cycle progression of cells in another stage. If there is in fact internuclear signaling in the Drosophila embryo, we therefore expect such a model to capture its effects to a first order. Among the conclusions of Young et al. (2012), the following is most relevant to the current work: there are essentially two ways to produce synchrony in such a model, which we call RS-positive and SR-negative feedback processes. In an RS-positive process, when two nuclei are nearly synchronized, the one that is slightly ahead in the cell cycle (i.e., the one which will enter mitosis sooner) “reaches back” to accelerate the progression of the one that is slightly behind. This corresponds to a response function f that is increasing in I. RS refers to the fact that the nucleus ahead in the cell cycle signals to the one behind, and positive refers to the speeding-up effect. In contrast, in an SR-negative process the nucleus which is behind in the cell cycle retards the progression of the nucleus ahead of it, effectively allowing the one that is behind to “catch up,” corresponding to a decreasing f . These mechanisms are illustrated in Figure 7. Although both processes lead to synchrony, the mechanisms are distinct and produce different distributions of cells within the cell cycle under random perturbations. Figure 8 shows data obtained from numerical simulations of both processes (see Appendix 2). In both cases, a system beginning with a single cell progressed through the cell cycle through 13 divisions in a stochastic version of Equation 3. The times of all divisions at the end of the 13th cycle were recorded for 50 trials, each adjusted to have mean 0. We then calculated density functions (histograms) using 21 bins. To make the simulation data comparable to the experimental data, we discarded data points greater than 4 in the automatic tracking data (which we believe to be false positives from the tracking software), and scaled the simulation time units to make the variances equal. The SR-negative distribution has positive skew (skew ¼ 0:7632) and the RS-positive has negative skew (skew ¼ 0:6286). When data points greater than 4 are
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Fig. 8. Distributions of the division time deviations for the 13th division from the automated tracking data (blue) and a stochastic simulation of (5 for RS-positive feedback (red) and SR-negative feedback (green). Each distribution is plotted as a histogram with 21 bins.
discarded, the automated tracking data have a skew value of 0:4956. We also calculated the Kullback-Leibler divergence measure for the SR-negative and RS-positive distributions with respect to the automatic distribution, getting values of 3:52 and 0:42, respectively (Fig. 8). The conclusion is that the synchronization mechanism in the early Drosophila embryo is likely to be RS-positive signaling.
the loss of maternal Cyclin B mRNA is shown to be a key factor in the lengthening of the cell cycles, and the relative concentration of Cyclin B as the limiting factor for mitotic entry. This provides a natural mechanism for RS-positive signaling in the form of trigger waves, as the positive feedback loop between Cyclin B/ Cdk1 and Cdc25 drives local diffusion of the active Cyclin B/ Cdk1 complex (overcoming the double negative feedback between Cyclin B/Cdk1 and Wee1) and influences the cell cycles of nearby nuclei. Our hypothesis provides an explanation for the striking increase in synchronization at the 11th division: it is due to a decrease in internuclear distance due to the formation of the syncytial blastoderm. As the nuclei migrate from the yolk to form a two-dimensional monolayer, the relative density of nuclei increases. The 11th division is the first division after the syncytial blastoderm is fully formed, and the switch from a distribution in the volume of the yolk to the two dimensional surface of the syncytial blastoderm increases local density, allowing for faster and more efficient propagation of the synchronizing signals between nearby nuclei Daniels et al. (2012). Our results provide more evidence for intercellular signaling as a mechanism for cell cycle regulation and control in embryogenesis. The implication is that a “systems-level” approach is important in understanding even relatively simple cell cycle processes like regulation of mitosis in early embryogensis. These results suggest further directions of study for understanding synchronization mechanisms in early embryos.
Appendix 1
Discussion
The Calzone Model
We have presented three statistical arguments for the hypothesis that synchronization between nuclear divisions is affected by communication among nuclei. We see that: (a) the variance of the division time distribution decreases at the 11th mitosis, (b) this decrease in variance is unaccounted for in a model of the embryo that does not include internuclear communication, and (c) the deviations from expected division times when compared mother to daughter are uncorrelated or negatively correlated. We compared the distributions of division times with a general coupled oscillator feedback model to conclude that nuclei that are at nearby points in the cell cycle should be drawn toward synchronization through internuclear signaling, and the signaling must be of RS-positive type, where nuclei that are “ahead” of others in the cell cycle (but not necessarily in space) act to speed up the progression of those that are “behind.” We can summarize the mathematical results as follows. The observed synchronization of division cycles in the nuclei of the early Drosophila embryo is much tighter than one would expect if the nuclei did not influence the timing of each other’s divisions. The observed effect, summarized in Figure 6, is that when a nucleus divides “late” as compared with those around it, its daughters will have shorter mitotic cycles than their neighbors so that they can “catch up” to those around them. We are not proposing a causative effect between mother and daughter nuclei; the presence of internuclear signaling mechanisms alone is sufficient to explain this observation. A synchronizing effect has been noted in Xenopus embryos (Chang and Ferrell, 2013) and we expect a similar biochemical mechanism to be at work in Drosophila. In Calzone et al. (2007),
In the Analysis section of this manuscript, we refer to a model of early Drosophila embryogenesis due to Calzone et al. (2007). In this Appendix we provide the model, which is a differential
TABLE A1. Dynamical Variables in the Calzone Model variable ½MPF n ½preMPF n ½MPF c ½preMPF c ½IE ½FZY ½StgP n Stg n ½StgP c ½Stg c ½Wee1P n ½Wee1n ½Wee1c ½Stgm ½Xm ½Xp
description Nuclear Mitosis Promoting Factor (Cyclin B/Cdk1) Phosphorylated nuclear MPF Cytosolic MPF Phosphorylated cytosolic MPF Unknown Intermediate Enzyme between Wee1 and APC Fizzy Nuclear phosphorylated String Nuclear String Cytosolic phosphorylated String Cytosolic String Nuclear phosphorylated Wee1 Nuclear Wee1 Cytosolic Wee1 String mRNA Unknown String-promoting enzyme mRNA Unknown String-promoting enzyme protein
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equations model, as well as our implementation of it as a stochastic differential equations model. The Calzone model includes 16 dynamical variables, listed in Table A1, representing the concentrations of various proteins in two compartments representing the nuclear envelope and the cytosol. The variables satisfy a system of coupled differential equations, as well as several balancing equations which are used to define four auxiliary variables. Transport between compartments is linear, with individual rate constants for each protein species. Interactions between proteins follow Michaelis-Menten kinetics, with rate constants specific to each interaction. A nuclear division in the model occurs when the variable [FZY] increases through a threshold level. When this occurs, the volume of the nuclear compartment is doubled and the volume of the cytoplasmic compartment is decreased, while all concentrations are scaled appropriately to reflect these new volumes. Note that the Calzone model explicitly assumes that all nuclei are identical, treating them collectively as a single “nuclear mass”. Being a system of first order differential equations, the Calzone model can be written as dx ¼ FðxÞ dt
(4)
where x 2 R16 represents a vector of protein concentrations. For the simulations pictured in Figure 5, we modified Equation 2 to form the It^ o stochastic differential equation: dx ¼ Fðx Þdt þ s dW ; where W is a vector of Wiener processes with mean 0 and variance 1, and s varies in the range ð0; 0:1Þ. For the simulations presented in Figure 5, we used the Euler-Maruyama method. To handle the case where stochastic noise might cause a double count of divisions as the level of [FZY] varied randomly near the threshold, we used a switching method that triggered on the first upward crossing of the threshold, and then reset once the value of [FZY] was sufficiently low.
Appendix 2 The Stochastic Response/Signaling Model We adapted Equation 3 in two ways to generate the results shown in Figure 8. First, we added dynamic population size: Initially there is a single cell y1 , traveling with speed 1. When y1 reaches 1 0 it divides, which we represented by increasing the cell count by one and introducing y2 ¼ 0. In the absence of noise, the now two cells will travel together with speed 1 until they both divide upon reaching 1. Next we added stochasticity by adapting Equation 3 to an It^ o SDE in the usual way and integrated the system with noise s ¼ 0:01 using the Euler-Maruyama method.
Acknowledgments The authors thank Fernando Amat and Philipp Keller for their generous contributions of data and members of the Ferrell lab for personal communications about trigger waves. We also thank Dr. Janet Duerr and Dr. Mark Berryman for their helpful comments on the manuscript.
References Boczko E, Stowers C, Gedeon T, Young T. 2010. ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast. J Biolog Dyn 4:328–345. Budirahardja Y, Gonczy P. 2009. Coupling the cell cycle to development. Development 136:2861–2872. Calzone L, Thieffry D, Tyson J, Novak B. 2007. Dynamical modeling of syncytial mitotic cycles in Drosophila embryos. Mol Syst Biol 3:131. Chang JB, Ferrell J. 2013. Mitotic trigger waves and the spatial coordination of the Xenopus cell cycle. Nature 7464;500:603–607. Daniels BR, Rikhy R, Renz M, Debrowsky TM, LippencottSchwartz J. 2012. Multiscale diffusion in the mitotic Drosphila melanogaster syncytial blastoderm. Proc Natl Acad Sci U S A 109:8588–8593. Edgar BA, Sprenger F, Duronio RJ, Leopold P, O’Farrell PH. 1994. Distinct molecular mechanisms regulate cell cycle timing at successive stages of Drosophila embryogenesis. Genes Dev 8: 440–452. Foe VE, Alberts BM. 1983. Studies of nuclear and cytoplasmic behaviour during the five mitotic cycles that precede gastrulation in Drosophila embryogenesis. J Cell Sci 61:31–70. Gong X, Buckalew R, Young T, Boczko E. 2014. Cell cycle feedback model with a gap. J Biol Dyn 8:79–98. Jun-Yuan J, Squirrel JM, Schubiger G. 2004. Both Cyclin B levels and DNA-replication checkpoint control the early embryonic mitoses in Drosophila. Development 131:401–411. Lee LA, Orr-Weaver TL. 2003. Regulation of cell cycles in Drosophila development: intrinsic and extrinsic cues. Annu Rev Genet 37:545–578. McCleland ML, Farrell JA, O’Farrell PH. 2009. Influence of cyclin type and dose on mitotic entry and progression in the early Drosophila embryo. J Cell Biol 184:639–646. Qing K, Pomerening JR. 2012. Punctuated cyclin synthesis drives early embryonic cell cycle oscillations. Mol Biol Cell 23:284–296. Singh A, Irvine KD. 2012. Drosophila as a model for understanding development and disease. Dev Dyn 241:1–2. Stiffler LA, Ji J, Trautmann S, Trusty C, Schubiger G. 1999. Cyclin A and B functions in the early Drosophila embryo. Development 1265505–5513. Tomer R, Khairy K, Amat F, Keller PJ. 2012. Quantitative highspeed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy. Nat Methods 9:755–763. Xuemin L, Drocco J, Wieschaus EF. 2010. Cell cycle regulation via inter-nuclear communication during the early embryonic development of Drosophila melanogaster. Cell Cycle 9:2908–2910. Young T, Fernandez B, Buckalew R, Moses G, Boczko E. 2012. Clustering in cell cycle dynamics with general responsive/ signaling feedback. J Theor Biol 292:103–115.
RESEARCH ARTICLE
Evidence for Internuclear Signaling in Drosophila Embryogenesis a
Richard Buckalew,1* Kara Finley,2 Soichi Tanda,2 and Todd Young1 1
Mathematics, Ohio University, Athens, Ohio Biological Sciences, Ohio University, Athens, Ohio
Developmental Dynamics
2
Background: Syncytial nuclei in Drosophila embryos undergo their first 13 divisions nearly synchronously. In the last several cell cycles, these division events travel across the anterior–posterior axis of the syncytial blastoderm in a wave. The phenomenon is well documented but the underlying mechanisms are not yet understood. Results: We study timing and positional data obtained from in vivo imaging of Drosophila embryos. We determine the statistical properties of the distribution of division times within and across generations with the null hypothesis that timing of division events is an independent random variable for each nucleus. We also compare timing data with a model of Drosophila cell cycle regulation that does not include internuclear signaling, and to a universal model of phase-dependent signaling to determine the probable form of internuclear signaling in the syncytial embryo. Conclusions: The statistical variance of division times is lower than one would expect from uncoordinated activity. In fact, the variance decreases between the 10th and 11th divisions, which demonstrates a contribution of internuclear signaling to the observed synchrony and division waves. Our comparison with a coupled oscillator model leads us to conclude that internuclear signaling must be of Response/Signaling type with a positive impulse. Developmental C 2015 Wiley Periodicals, Inc. Dynamics 244:1014–1021, 2015. V Key words: Drosophila; mitosis; synchronization; synchrony; division wave Submitted 4 December 2014; First Decision 23 April 2015; Accepted 20 May 2015; Published online 29 May 2015
Introduction Cell division is one of the most essential biological processes for metazoans. In most multicellular organisms the first phase in development is the cleavage stage, characterized by rapid and synchronous cell divisions. Cell division in the cleavage stage of such organisms is controlled by maternally provided products independent of zygotic gene expression. This allows cells to increase their numbers quickly. It is thus of special interest to understand the mechanisms that control synchronous and rapid cell divisions. Drosophila melanogaster is a model organism that has been studied for many years, helping to provide insights into many biological processes, including genetics, epigenetics, metabolism, development, and so forth (Singh and Irvine, 2012). Because of its fast embryonic development, which takes approximately 1 day from egg deposition to hatching, D. melanogaster is particularly well suited to investigations of genetic, cellular, and molecular mechanisms of embryogenesis. Several phenomenological observations have been made of the early embryo, including the comprehensive work of Foe and Alberts (1983). Foe observed a remarkable synchrony between the embryonic nuclei as they proGrant sponsor: NIH; Grant number: NIH-NIGMS R01GM090207. *Correspondence to: Richard Buckalew, Mathematics, Ohio University, Athens, Ohio 45701. E-mail: [email protected] Dr. Buckalew’s present address is Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio, 43210
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gress through the cell cycle and undergo mitosis. The exact mechanism for this synchronization has not been identified. The first 13 mitotic divisions are synchronous and proceed without cytokinesis to form a single syncytium of around 6,000 nuclei. During the first seven nuclear division cycles in D. melanogaster, all dividing nuclei are contained in the interior of the syncytium. Around cycle 7, most nuclei and their surrounding protoplasmic islands begin migrating toward the surface of the embryo, while the “yolk nuclei” are formed from the nuclei remaining in the interior. The migrating nuclei arrive at the surface during the 10th cycle, forming the blastoderm monolayer, then undergo meta-synchronous division during cycles 10–13 (Foe and Alberts, 1983). These meta-synchronous divisions occur in the form of waves, starting at one or both ends of the embryo. The cell cycle lengths increase during cycles 10–13, and at cycle 14 the cell cycle length increases extensively and the embryo undergoes cellularization, becoming the cellular blastoderm. During cycles 1–13, the nuclei exhibit cell cycles consisting only of S and M phases, with no G1 or G2 phase involved. The growth phases appear to be skipped because the embryo is able to use maternally provided gene products to drive the embryonic cell cycle, with very limited zygotic gene transcription occurring (Lee and Orr-Weaver, 2003). Cycle 14 marks the beginning of the Article is online at: http://onlinelibrary.wiley.com/doi/10.1002/dvdy. 24298/abstract C 2015 Wiley Periodicals, Inc. V
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regular cell cycle with G1 and G2 phases, contributing to the longer duration of this cell cycle. Foe and her colleagues observed division waves extending across the syncytium during cycles 10–13 (Foe and Alberts, 1983), beginning at both the posterior and anterior portions of the embryo (such waves may also travel in only one direction, as in the data set we use in this study). The mechanisms responsible for the duration and regulation of the early cell cycles have been studied extensively (Foe and Alberts, 1983; Edgar et al., 1994; Stiffler et al., 1999; Jun-Yuan et al., 2004; Calzone et al., 2007; McCleland et al., 2009; Qing and Pomerening, 2012; Xuemin et al., 2012). The Cyclin B/Cdk1 (Cyclin-dependent kinase 1) complex regulates mitotic entry, including chromosome condensation and nuclear envelope breakdown, and is an obvious candidate for the main driver of the observed synchrony. Cyclin B/Cdk1 complexes only form once a threshold level of Cyclin B has been reached (Qing and Pomerening, 2012), and exit from mitosis requires the dissociation of this complex at the beginning of anaphase, presumably by APC (anaphase-promoting complex) (Stiffler et al., 1999). Although binding to Cyclin B is required for its activity, the regulation of Cdk1 activity is complex, including two feedback loops. The first one is a double negative feedback loop between Cdk1 and Wee1, which keeps the level of Cdk1 activity low during interphase. The second one is a positive feedback loop between Cdk1 and String tyrosine phosphatase. The removal of the phosphate at Y15 by String activates Cdk1 by counteracting the effect of Wee1. Cdk1 in turn activates String by phosphorylating it. Thus, at the entry to M phase, high Cdk1 activity reached by accumulation of Cyclin B and dephosphorylation at Y15 by String tyrosine phosphatase leads to a sharp decrease in Wee1 activity. As Wee1 activity is decreased, the positive feedback loop between Cdk1 and String dominates, further raising the level of Cdk1 activity. Studies show that cytoplasmic Cdk1 remains active and at a relatively constant level until cycle 8, when local fluctuations in Cdk1 activity begin to occur as the complex dissociates (Edgar et al., 1994). The start of oscillation seems to coincide with declining Cyclin B levels. Cyclin B measured in the early embryo at a global level does not appear to oscillate until cycle 7, although it shows a steady decrease (Edgar et al., 1994). However, Cyclin B levels exhibit localized oscillations corresponding to the cell cycle. During the end of metaphase, spindle-associated Cyclin B is degraded, contributing to the localized oscillations and allowing for the progression into anaphase (Lee and Orr-Weaver, 2003). As Edgar et al. proposes, the continuing degradation of Cyclin B may contribute to the lengthening of cell cycles 9–14, because the embryo needs more time to accumulate the levels of Cyclin B required for mitotic entry due to the higher nucleo:cytoplasmic ratio (Edgar et al., 1994). The time required for translation of the maternal Cyclin B mRNA to meet the Cyclin B threshold causes a longer S phase in these later cycles, while mitosis length remains the same. As Cyclin B levels start to oscillate, the aforementioned positive and double negative feedback loops operate to induce robust activation of the Cyclin B/Cdk1 complex. Calzone et al. (2007) modeled some of the factors regulating the mitotic cycle in the syncytial embryo. Their work accurately accounts for the slowdown of the cycles during later divisions, due largely to depletion of maternal Cyclin B mRNA. In that work individual nuclei were not modeled, instead the total “nuclear matter” was treated as a single compartment, and so internuclear
coupling was not considered. Chang and Ferrell (2013) implicate the Cyclin B/Cdk1 complex and its interactions with Wee1, Cdc25 (String in Drosophila), and APC, as the driver of synchronous division in Xenopus embryos. This coupling is by means of the mechanism of trigger waves, which propagate to neighboring nuclei as a high activity level of the Cyclin B/Cdk1 complex maintained by a positive feedback loop between Cdc25 and the Cyclin B/Cdk1 complex and a double negative feedback loop between Wee1 and the Cyclin B/Cdk1 complex. We speculate that a similar mechanism is at work in the Drosophila embryo. For multiple periodic processes to be synchronized, they must (A) have nearly identical clocks, (B) have coupling mechanisms, or have a combination of (A) and (B). Situation (A) can produce synchrony for long periods of time, but not indefinitely because even very small differences will eventually lead to out-of-phase clocks. Situation (B) requires strong coupling if the processes possess very different clocks. It has long been known that between two or more mechanical or electronic oscillators, a combination of (A) and (B) will cause them to become synchronized indefinitely. The strength of coupling needed to maintain synchrony is small and grows with the difference between the two frequencies. Because the syncytial nuclei of a single early Drosophila embryo are nearly identical in every way, it is reasonable that their mitotic cycles are almost identical. Because the synchrony only lasts for a relatively short time, 13 divisions, it is possible that the observed synchrony is entirely due to good clocks (A). The exact mechanism has not been identified, but we have shown that feedback processes between cells, i.e., both (A) and (B), can cause similar synchronizing behavior in yeast and many of the properties of such a system were analyzed in Young et al. (2012). In the present study, we give evidence from recently obtained data that the synchrony of mitotic cycles observed in the syncytial nuclei is not solely due to nearly identical clocks, but in fact must also have an element of coupling, i.e., both (A) and (B).
Results Automated Tracking Data Tomer et al. (2012), demonstrated a novel method of video microscopy and used it to create compelling images and large data sets relating to Drosophila embryogenesis, available at www.janelia.org/digitalembryo. They imaged Drosophila at many stages of development, including mitotic divisions 10 through 13 in the syncytial blastoderm. In correspondence, the authors provided us with automated tracking data for the 12th and 13th divisions. The authors imaged a developing embryo 128 times, at intervals of 25 sec. Their imaging technique allows for resolution in three spatial dimensions, so that the initial data set consists of volumetric luminescence data (obtained from histone-eGFP tagging) at sub-micron resolutions. Using an image processing algorithm, they identified individual nuclei in the data for each time point. They then compared neighboring time points algorithmically to track individual nuclei from image to image, and to identify when a mitotic division has occurred. The resulting data set consists of 51,482 lines of data, where each line contains the position of a single nucleus at a single time point, plus lineage data indicating whether it is the continuation of a nucleus at the previous time point, or the result of a division.
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Fig. 1. Division waves and linear fit lines for the 12th and 13th mitotic divisions in the automated tracking data. Vertical units are pixels, with the head of the embryo near x ¼ 200, and horizontal units are video frames, recorded once every 25 sec.
By reconstructing this lineage data, we identified many lines of descent; branching trees starting at a single nucleus that follow its descendants and each of theirs, and so on. The data were very noisy, and included a large number of lines of descent that we classified as erroneous for various reasons. We thus trimmed our data set to include only those lines of descent which satisfied the following criteria: (1) the line must start with a single nucleus that is present at the first time step; (2) that nucleus must divide at approximately the time when the division wave reaches it; (3) each of its two daughters must persist until dividing approximately when the second wave reaches them, and (4) their four total daughter nuclei must persist without dividing until the last time point present in the data. Any line of descent satisfying all four criteria can reasonably be considered to correspond to an actual nuclear lineage. We identified 457 such lines consisting of 3,199 nuclei which we can track with high confidence. In Figure 1, we plot the positions of mitotic divisions recovered from the lineage data. The horizontal axis is time, corresponding to the 128 separate images taken at 25-sec intervals. The vertical axis represents the distance from the anterior end of the embryo. Two division waves are apparent in the data as lines with positive slope originating from the anterior end and propagating toward the posterior.
Manual Tracking Data For this data set, automated tracking data are only available for the 12th and 13th mitoses, but reconstructed image data are available for the 10th through the 13th mitotic divisions. Similar to the automated data, the data set includes 201 images taken at 25-sec intervals. The images are two-dimensional projections from the lateral side of the embryo, and do not include depth information. To perform more robust statistical analysis than possible with the automated data, we obtained positional information for a portion of the nuclei by tracking them by hand in the image data. In all, we followed 654 nuclei in 117 lines of descent. These manual data cover four division waves, including the three mitotic
Fig. 2. Division waves and linear fit lines for the 10th, 11th, 12th, and 13th divisions in the manual tracking data.
cycles between them, plus two partial cycles at the beginning and end of the data. We recorded n ¼ 63; 94; 136, and 180 divisions in each wave. Figure 2 shows the positions of divisions in the manual data (which were measured from the anterior of the embryo). Four division waves of decreasing speed are evident.
Statistical Methods We will treat the mitotic divisions as events in a stochastic process. Let ti be a random variable denoting the time of the ith division of a particular line of nuclei. It is clear that ti is actually a sum of random variables: ti ¼ c1 þ c2 þ ::: þ ci ; where cj ; j ¼ 1; 2; :::; i, represents the duration (period) of the j th cycle. We may consider the variance of the variable ti among different nuclei within an embryo to be a measure of the synchronization at the ith division. It is a well-known fact that if c1 ; :::; cn is a collection of independent random variables distributed Pwith variances s21 ; . . . ; s2n , then the random variable ti ¼ ij¼1 cj has variance varðti Þ ¼
i X
s2j ;
(1)
j¼1
in other words, the variance of the sum is the sum of the variances. If the cell cycles of different nuclei are not coupled, then ci and ti would be independent of the nucleus and the distributions of ci and ti within the embryo should have the characteristics of independent random variables. Particularly, they should satisfy the variance condition (Eq. 1) and the variables ci ¼ ti ti1 and ciþ1 ¼ tiþ1 ti should not be correlated. Because the mitotic divisions occur in waves, a nucleus’s position affects its division time. In the data set we analyze, nuclei near the anterior of the embryo divide before those near the posterior. The speed of the waves is variable as well; from the manual tracking data, we see that the division wave for the 10th mitosis travels at approximately 136:3 microns per minute, the 11th at 119:3 microns per minute, the 12th at 79:1 and the 13th
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TABLE 1. Linear Fits for Division Waves in the Automated and Manual Tracking Data Automated tracking i
mi
bi
r
TABLE 2. Distributional Parameters for Divisions From the Automated Tracking Data
Manual tracking i
mi
bi
12 78.4 648.2 0.919 10 156.7 2278.9 13 73.6 4973.8 0.863 11 137.2 5993.0 12 91.0 7309.8 13 94.1 13145.7
i 12 13
r 0.946 0.979 0.977 0.950
n 457 914
li 16
4 10 3 1015
r2
skewness
2.137 4.108
3.097 2.210
p (normality) 8 1079 3 10112
Suppose a nucleus undergoes its ith mitotic division at time ti and position xi . Its wave-adjusted deviation ~t i is given by: ~t i ¼ ti t i ðxÞ: Our subsequent analysis will center on ~t i .
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Analysis
Fig. 3. Distributions of t~12 (left) and t~13 (right) from the automated tracking data.
at 81:8 microns per minute. These speeds are relatively constant through the duration of the wave, indicating that simple diffusion of a background chemical cannot drive the progression of the wave. Instead, we suspect “trigger wave”-like mechanisms propagating from nucleus to nucleus as described in Chang and Ferrell (2013). Because the waves travel at distinct speeds, absolute division times ti are not suitable for measuring synchrony. We want to measure the degree to which nearby cells are synchronized, and ignore the variations caused by location in the embryo. For this reason we introduce the wave-adjusted deviations ~t i which are defined as follows. From the division time and location data in Figures 1 and 2, we obtain a linear fit xi ¼ mi t þ bi for each division wave. The results are summarized in Table 1. We want to define a function t i ðx Þ that represents the expected time of division for a nucleus at position x during the ith division wave. This can be obtained from the fit line xi ¼ mi t þ bi by solving for t, giving: t i ðxÞ ¼ x bi : mi
(2)
The amount of synchronization in the ith division wave can be measured by looking at the difference between the actual time of division ti and the expected time of division t i given by Equation 2. This is what we call the wave adjusted deviation ~t i , representing the “de-trended” deviations. Formally we have:
The distribution of ~t i for i ¼ 12; 13 from the automated data is shown in Figure 3, and the parameters for the distributions are given in Table 2. We note that the distributions are not normal, and that the variance increases dramatically from the 12th to the 13th division, which indicates the beginning of the breakdown in synchrony described by Foe and Alberts (1983), as well as Calzone et al. (2007). The underlying cause of this loss of synchrony is not completely understood, but it is likely caused in part by the switch to zygotic expression rather than the embryo’s earlier use of maternally provided gene products at the mid-blastula transition at the 13th division Budirahardja and Gonczy (2009). Figure 4 and Table 3 show the distributions and distribution parameters for waves 10 through 13 in the manual tracking data. Figure 5 shows the v-squared 95% confidence intervals for the population variance based on each sample. There is a clear decrease in variance between the 10th and 11th divisions, which strongly indicates that division times between these two division cycles are not independent. Also shown in Figure 5 are data obtained from simulations of an SDE version of the Calzone model. (See the Appendix for a description of the Calzone model and our implementation of it.) For each of 10 values of stochastic noise, the model was run 100 times and the division times were recorded. The gray curves are the variances of the division times for each of divisions 10 through 13, in s2. It is clear that there are no mechanisms in the model that give rise to a decrease in variance at the 11th, or any other, division. We suspect this is because it treats all the nuclei as a single compartment, and thus does not include the potential for internuclear signaling. Now consider the lengths ci of the division cycles. Recall that ci ¼ tiþ1 ti , where ti is the time of the ith division. There is again an effect of position on ci due to the division waves: because the waves are decreasing in speed (see Table 3) a nucleus at the posterior end of the embryo has a longer expected cycle length ci than one near the anterior. Thus we define the expected ith cycle length at position x to be c ðx Þ ¼ t iþ1 ðx Þ t i ðxÞ. We present a statistical analysis of the wave-adjusted cycle length deviations ~c i defined as: ~c i ¼ ci c : Our null hypothesis is that there is no communication between nuclei. In this case, the deviation from a nucleus’ expected division time will be a random variable, independent from that of its mother’s. From the manual tracking data, we have two sets of
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Fig. 4.
Distributions of t~10 (top left), t~11 (top right), t~12 (bottom left), and t~13 (bottom right) from the manual tracking data.
TABLE 3. Distributional Parameters for Divisions From the Manual Tracking Data i 10 11 12 13
n 63 94 136 180
li 15
1 10 4 1016 2 1015 4 1015
mother-daughter pairs where we can make such comparisons, between the 11th and 12th divisions, and between the 12th and 13th. Figure 6 shows the plot of ~c 12 vs. ~c 11 and ~c 13 vs. ~c 12 . The results indicate a negative correlation between the lengths of the 12th and 13th division cycles, with correlation coefficient 0:25 and P-value 0:00086. Thus, although the effect is not very strong, we consider it to be a real effect. As described above, such a correlation would not exist if the values of ~c i were independent random variables. Even the case of the 11th and 12th cycles, where the correlation is essentially zero, provides an affirmative result, for if there were no internuclear communication, then we would expect a positive correlation: genetic or epigenetic effects causing a nucleus to divide late should cause daughter nuclei to divide later than expected as well. These results indicate that some form of internuclear signaling related to cell cycle regulation must be occurring.
Comparison With a Universal Coupled Oscillator Model In Young et al. (2012), the authors introduced a simple, but universal model of cell cycle signaling in a community of cells based on earlier work by Bozcko et al. (2010). The conclusions were confirmed in the
r2
skewness
p (normality)
0.290 0.154 0.335 0.861
0.173 0.0256 0.146 0.587
0.723 0.293 0:755 0:005
presence of dynamic delay by Gong et al. (2014). We specify two stages of the mitotic cycle as signaling (S) and responsive (R), and cells currently in the signaling stage influence cells in the responsive stage to either speed up or slow down their cell cycle progression. In the context of Drosophila embryogenesis, we conjecture that the responsive stage R is associated with the end of S-phase (or G2 in later cycles) and the signaling stage S with the beginning of M-phase. This is because the trigger wave phenomenon occurs around the checkpoint regulating mitotic entry, and the positive feedback loop between Cyclin B/Cdk1 and Cdc25 (String) occurs at this point in the cell cycle (Chang and Ferrell, 2013). In this model, the progression of cell i in the cell cycle is represented by the variable yi , which takes values in the interval ½0; 1Þ. This value represents a proxy for the cell’s state in the cell cycle; as originally proposed for budding yeast, yi represented the log of the cell’s volume, but it should be interpreted more broadly as simply an indication of progression through the cell cycle. Thus the value of yi can only increase in the model, and when it reaches 1 (division) it begins again at 0. It is therefore natural to think of the state of a population of n cells in this model as a point on the n-torus Sn .
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Fig. 5. Sample variances from the manual tracking data for divisions 10, 11, 12, and 13, and the 95% confidence intervals for the corresponding populations, as well as the calculated variances from 10 simulations of the stochastic Calzone model (see Appendix 1).
Fig. 6. Correlation between mother and daughter deviations from expected division times.
To model the cell cycle signaling described above, we proposed the differential equations: ( = R 1; when yi 2 dyi ; (3) ¼ dt 1 þ f ðIÞ; when yi 2 R where I ¼ fj j yj 2 Sg=n is the fraction of cells which are in the signaling region. The response function f ðIÞ must satisfy f ð0Þ ¼ 0 and it must be monotone (either increasing or decreasing) but it can be nonlinear. For instance, from biological considerations, we expect f to be sigmoidal (S-shaped), and to satisfy f ðI Þ 2 ð1; 1Þ for any I. Thus a cell yi will move forward with speed 1 when it is not in the responsive stage, and it will move with speed 1 þ f ðIÞ when it is in the responsive stage. While the actual processes involved in cell cycle communication among cells could in fact be extremely complex (and beyond our current knowledge), it is easy to see that all such communication, regardless of its details, must act by cells in some stage of
Fig. 7. RS-positive vs. SR-negative feedback processes. The cell cycle positions of two nuclei, nearby both in space and in the cell cycle, are represented by the solid disks, where cell cycle progression is represented by movement from left to right (the illustrations do not show any spatial information). The boundaries of the responsive and signaling regions R and S are indicated by the small vertical lines. In each illustration, three “snapshots” of the system are shown in order from top to bottom. In an RS-positive system (left), nuclei which are ahead in the cell cycle (i.e., closer to or already undergoing mitosis) accelerate the progression of those behind (illustrated by arched arrows), so the responsive nucleus progresses faster than the signaling nucleus. In SR-negative feedback (right), nuclei which are behind in the cell cycle slow down those ahead (arched inhibitory arrows), so that the responsive nucleus progresses more slowly than the signaling nucleus. In both cases, the distance between their positions in the cell cycle decreases, leading to synchronization.
the cycle (signaling) producing chemicals that effect the cell cycle progression of cells in another stage. If there is in fact internuclear signaling in the Drosophila embryo, we therefore expect such a model to capture its effects to a first order. Among the conclusions of Young et al. (2012), the following is most relevant to the current work: there are essentially two ways to produce synchrony in such a model, which we call RS-positive and SR-negative feedback processes. In an RS-positive process, when two nuclei are nearly synchronized, the one that is slightly ahead in the cell cycle (i.e., the one which will enter mitosis sooner) “reaches back” to accelerate the progression of the one that is slightly behind. This corresponds to a response function f that is increasing in I. RS refers to the fact that the nucleus ahead in the cell cycle signals to the one behind, and positive refers to the speeding-up effect. In contrast, in an SR-negative process the nucleus which is behind in the cell cycle retards the progression of the nucleus ahead of it, effectively allowing the one that is behind to “catch up,” corresponding to a decreasing f . These mechanisms are illustrated in Figure 7. Although both processes lead to synchrony, the mechanisms are distinct and produce different distributions of cells within the cell cycle under random perturbations. Figure 8 shows data obtained from numerical simulations of both processes (see Appendix 2). In both cases, a system beginning with a single cell progressed through the cell cycle through 13 divisions in a stochastic version of Equation 3. The times of all divisions at the end of the 13th cycle were recorded for 50 trials, each adjusted to have mean 0. We then calculated density functions (histograms) using 21 bins. To make the simulation data comparable to the experimental data, we discarded data points greater than 4 in the automatic tracking data (which we believe to be false positives from the tracking software), and scaled the simulation time units to make the variances equal. The SR-negative distribution has positive skew (skew ¼ 0:7632) and the RS-positive has negative skew (skew ¼ 0:6286). When data points greater than 4 are
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Fig. 8. Distributions of the division time deviations for the 13th division from the automated tracking data (blue) and a stochastic simulation of (5 for RS-positive feedback (red) and SR-negative feedback (green). Each distribution is plotted as a histogram with 21 bins.
discarded, the automated tracking data have a skew value of 0:4956. We also calculated the Kullback-Leibler divergence measure for the SR-negative and RS-positive distributions with respect to the automatic distribution, getting values of 3:52 and 0:42, respectively (Fig. 8). The conclusion is that the synchronization mechanism in the early Drosophila embryo is likely to be RS-positive signaling.
the loss of maternal Cyclin B mRNA is shown to be a key factor in the lengthening of the cell cycles, and the relative concentration of Cyclin B as the limiting factor for mitotic entry. This provides a natural mechanism for RS-positive signaling in the form of trigger waves, as the positive feedback loop between Cyclin B/ Cdk1 and Cdc25 drives local diffusion of the active Cyclin B/ Cdk1 complex (overcoming the double negative feedback between Cyclin B/Cdk1 and Wee1) and influences the cell cycles of nearby nuclei. Our hypothesis provides an explanation for the striking increase in synchronization at the 11th division: it is due to a decrease in internuclear distance due to the formation of the syncytial blastoderm. As the nuclei migrate from the yolk to form a two-dimensional monolayer, the relative density of nuclei increases. The 11th division is the first division after the syncytial blastoderm is fully formed, and the switch from a distribution in the volume of the yolk to the two dimensional surface of the syncytial blastoderm increases local density, allowing for faster and more efficient propagation of the synchronizing signals between nearby nuclei Daniels et al. (2012). Our results provide more evidence for intercellular signaling as a mechanism for cell cycle regulation and control in embryogenesis. The implication is that a “systems-level” approach is important in understanding even relatively simple cell cycle processes like regulation of mitosis in early embryogensis. These results suggest further directions of study for understanding synchronization mechanisms in early embryos.
Appendix 1
Discussion
The Calzone Model
We have presented three statistical arguments for the hypothesis that synchronization between nuclear divisions is affected by communication among nuclei. We see that: (a) the variance of the division time distribution decreases at the 11th mitosis, (b) this decrease in variance is unaccounted for in a model of the embryo that does not include internuclear communication, and (c) the deviations from expected division times when compared mother to daughter are uncorrelated or negatively correlated. We compared the distributions of division times with a general coupled oscillator feedback model to conclude that nuclei that are at nearby points in the cell cycle should be drawn toward synchronization through internuclear signaling, and the signaling must be of RS-positive type, where nuclei that are “ahead” of others in the cell cycle (but not necessarily in space) act to speed up the progression of those that are “behind.” We can summarize the mathematical results as follows. The observed synchronization of division cycles in the nuclei of the early Drosophila embryo is much tighter than one would expect if the nuclei did not influence the timing of each other’s divisions. The observed effect, summarized in Figure 6, is that when a nucleus divides “late” as compared with those around it, its daughters will have shorter mitotic cycles than their neighbors so that they can “catch up” to those around them. We are not proposing a causative effect between mother and daughter nuclei; the presence of internuclear signaling mechanisms alone is sufficient to explain this observation. A synchronizing effect has been noted in Xenopus embryos (Chang and Ferrell, 2013) and we expect a similar biochemical mechanism to be at work in Drosophila. In Calzone et al. (2007),
In the Analysis section of this manuscript, we refer to a model of early Drosophila embryogenesis due to Calzone et al. (2007). In this Appendix we provide the model, which is a differential
TABLE A1. Dynamical Variables in the Calzone Model variable ½MPF n ½preMPF n ½MPF c ½preMPF c ½IE ½FZY ½StgP n Stg n ½StgP c ½Stg c ½Wee1P n ½Wee1n ½Wee1c ½Stgm ½Xm ½Xp
description Nuclear Mitosis Promoting Factor (Cyclin B/Cdk1) Phosphorylated nuclear MPF Cytosolic MPF Phosphorylated cytosolic MPF Unknown Intermediate Enzyme between Wee1 and APC Fizzy Nuclear phosphorylated String Nuclear String Cytosolic phosphorylated String Cytosolic String Nuclear phosphorylated Wee1 Nuclear Wee1 Cytosolic Wee1 String mRNA Unknown String-promoting enzyme mRNA Unknown String-promoting enzyme protein
INTERNUCLEAR SIGNALING IN DROSOPHILA 1021
Developmental Dynamics
equations model, as well as our implementation of it as a stochastic differential equations model. The Calzone model includes 16 dynamical variables, listed in Table A1, representing the concentrations of various proteins in two compartments representing the nuclear envelope and the cytosol. The variables satisfy a system of coupled differential equations, as well as several balancing equations which are used to define four auxiliary variables. Transport between compartments is linear, with individual rate constants for each protein species. Interactions between proteins follow Michaelis-Menten kinetics, with rate constants specific to each interaction. A nuclear division in the model occurs when the variable [FZY] increases through a threshold level. When this occurs, the volume of the nuclear compartment is doubled and the volume of the cytoplasmic compartment is decreased, while all concentrations are scaled appropriately to reflect these new volumes. Note that the Calzone model explicitly assumes that all nuclei are identical, treating them collectively as a single “nuclear mass”. Being a system of first order differential equations, the Calzone model can be written as dx ¼ FðxÞ dt
(4)
where x 2 R16 represents a vector of protein concentrations. For the simulations pictured in Figure 5, we modified Equation 2 to form the It^ o stochastic differential equation: dx ¼ Fðx Þdt þ s dW ; where W is a vector of Wiener processes with mean 0 and variance 1, and s varies in the range ð0; 0:1Þ. For the simulations presented in Figure 5, we used the Euler-Maruyama method. To handle the case where stochastic noise might cause a double count of divisions as the level of [FZY] varied randomly near the threshold, we used a switching method that triggered on the first upward crossing of the threshold, and then reset once the value of [FZY] was sufficiently low.
Appendix 2 The Stochastic Response/Signaling Model We adapted Equation 3 in two ways to generate the results shown in Figure 8. First, we added dynamic population size: Initially there is a single cell y1 , traveling with speed 1. When y1 reaches 1 0 it divides, which we represented by increasing the cell count by one and introducing y2 ¼ 0. In the absence of noise, the now two cells will travel together with speed 1 until they both divide upon reaching 1. Next we added stochasticity by adapting Equation 3 to an It^ o SDE in the usual way and integrated the system with noise s ¼ 0:01 using the Euler-Maruyama method.
Acknowledgments The authors thank Fernando Amat and Philipp Keller for their generous contributions of data and members of the Ferrell lab for personal communications about trigger waves. We also thank Dr. Janet Duerr and Dr. Mark Berryman for their helpful comments on the manuscript.
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