Largescale analysis and computer modeling reveal hidden regularities behind variability of cell division patterns in Arabidopsis thaliana embryogenesis
Abstract
Noise plays a major role in cellular processes and in the development of tissues and organs. Several studies have examined the origin, the integration or the accommodation of noise in gene expression, cell growth and elaboration of organ shape. By contrast, much less is known about variability in cell division plane positioning, its origin and links with cell geometry, and its impact on tissue organization. Taking advantage of the firststereotypedthenvariable division patterns in the embryo of the model plant Arabidopsis thaliana, we combined 3D imaging and quantitative cell shape and cell lineage analysis together with mathematical and computer modeling to perform a large scale, systematic analysis of variability in division plane orientation. Our results reveal that, paradoxically, variability in cell division patterns of Arabidopsis embryos is accompanied by a progressive reduction of cell shape heterogeneity. The paradox is solved by showing that variability operates within a reduced repertoire of possible division plane orientations that is related to cell geometry. We show that in several domains of the embryo, a recently proposed geometrical division rule recapitulates observed variable patterns, thus suggesting that variable patterns emerge from deterministic principles operating in a variable geometrical context. Our work highlights the importance of emerging patterns in the plant embryo under iterated division principles, but also reveal domains where deviations between rule predictions and experimental observations point to additional regulatory mechanisms.
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Author Response
Reviewer #1 (Public Review):
Overall, it is an interesting work exploring stochastic and deterministic aspects of embryonic cell division in plants. The power of the authors' approach lies in the quantitative analysis of 3D cell geometries combined with quantitative computer modelling.
I am a bit confused about how authors relate stochasticity as an emergent property of a deterministic process. Typically, stochasticity is the lowlevel process resulting in variation of subcellular components those also related to the positioning of the cell division plane. Perhaps a more elaborated and clearer connection between stochasticity at the subcellular level and phenotypic variability should be provided.
Actually, our interpretation does not directly relate variability in division patterns to a deterministic selection of …
Author Response
Reviewer #1 (Public Review):
Overall, it is an interesting work exploring stochastic and deterministic aspects of embryonic cell division in plants. The power of the authors' approach lies in the quantitative analysis of 3D cell geometries combined with quantitative computer modelling.
I am a bit confused about how authors relate stochasticity as an emergent property of a deterministic process. Typically, stochasticity is the lowlevel process resulting in variation of subcellular components those also related to the positioning of the cell division plane. Perhaps a more elaborated and clearer connection between stochasticity at the subcellular level and phenotypic variability should be provided.
Actually, our interpretation does not directly relate variability in division patterns to a deterministic selection of division plane orientation. A key intermediate between these two scales is the variability in cell geometry. Based on our results, we propose that the selection of division plane orientation would obey a deterministic principle based on geometrical constraints. Variability in cell division patterns would ensue from the expression of this deterministic rule in the variable context of cell geometries. Variability in cell geometry would itself result from noise in the precise positioning of the division plane along the optimal, deterministic orientation. We have added a new summary Figure 10 to illustrate and clarify this interpretation.
I have a number of specific questions/concerns that I would like the authors to address as listed below:
 Major variability of cell shapes is observed in the apical domain as opposed to the basal domain. What would be an underlying principle to asymmetric shaping of the apicalbasal domain? The authors describe beautifully the observations but give relatively little discussion on this matter, leaving the reader guessing.
We added a new paragraph (before the last) in the Discussion on this point. The origin for a larger variability in the apical than in the basal domain can be found in part in the different cell shapes present in the two domains at stage 16C. Along the path tetrahedron > triangular prism > cuboid, the apical domain indeed appears farther from the final absorbing state represented by the cuboid shape, hence more time will be spent in the intermediate shapes in this domain. In addition, the tetrahedral and triangular prism shapes are closer to rotational symmetry and thus represent a larger source of variability in division plane orientation. Lastly, apical and basal cells have distinct environments, the basal cells being constrained between the suspensor, on one side, and apical cells, on the other. We can only speculate about the functional significance, if any, of the larger variability in the apical domain. For example, we can relate it to the future morphological transition that will characterize the apical domain with the emergence of the cotyledons at the heart stage. A variable pattern of cell walls could be required to establish a specific mechanical pattern at the tissue scale to favor this shape transition.
 Authors used graph theory to explain variability in cell division for the same topological feature. In light of quite a discrepancy between predictions and observations (i.e., Figure 4C) question arises of how this prediction could be affected by undergoing cell expansions as this element I believe is neglected in their graph theory approach?
It is indeed the case that cell edge lengths are ignored in the graphtheoretical approach described in Section 2.4. In this part of the paper, our objective was to objectively test if nontopological factors were implicated in the determination of division orientations. The rationale was thus to compare observed patterns to predictions obtained using topological information only. The strong discrepancy between observations and predictions (Figure 4C) confirmed that topology alone was not sufficient to predict observed patterns. The integration of cell geometry (including edge lengths) into the predictions is considered in the next sections (Sections 2.5 and 2.6).
We believe the modifications we made in Section 2.4 to answer Reviewer 3’s comment on this part (see below) should make this point clearer.
 Tetrahedron shape repeats in only 4% of embryos at the 16cell stage. What could be the criticality of this shape for the entire embryo patterning?
At the 16C stage, there are four domains (apical/basal x inner/outer) represented each by exactly one cell shape. The triangular prismatic shape is observed in two domains (outer apical and inner basal). The two other cell shapes, cuboid and tetrahedron, are specific to the two other domains, respectively outer basal and inner apical. Hence, the tetrahedron shape represents 25%, not 4% of cell shapes at the 16cell stage, a proportion large enough to potentially impinge on embryo patterning.
Our graph analysis shows that, under a topologically random regime of cell division, the tetrahedron shape should progressively vanish because, due to 4way junction avoidance, a tetrahedron cannot divide into two tetrahedra and because divisions of triangular prisms and cuboids generate a minor proportion of tetrahedra only in comparison with the other cell shapes (Figure 4C). In addition, our analysis also shows that the triangular prismatic shape is a necessary intermediate to transit from a tetrahedron to a cuboid shape.
Altogether, the presence of the tetrahedron shape in the inner apical domain at 16cell stage could be responsible for the large variability in cell shape subsequently observed in this domain. (See also our answer to Comment 1 of the Reviewer).
 It is not clear to me whether stochastic cell division modelling takes into account the mechanical influence of adjacent cells? In any case, authors should discuss how this could potentially affect their analysis.
Indeed, our stochastic cell division model only takes into account the geometry of the mother cell, ignoring the possible influence of the environment of the cell within the tissue (through mechanical signals or other, such as hormonal signals)  except of course for indirect effects that the environment could exert on cell shape. The possible mechanical influence of the cell environment was already discussed in the original version of our manuscript (last paragraph of the Discussion). In particular, we mentioned that the specific localization deep inside the embryo of the inner basal cells could mechanistically influence the positioning of the division plane, thus explaining the strong discrepancy between observations and predictions in this domain. This negative result illustrates how the cellautonomous model can be useful in pointing to possible environmental influence (or alternative geometrical rules yet to be identified) and in suggesting future directions of investigation.
To remove any ambiguity, we have made more explicit the cellautonomous nature of the model (Section 2.5 and Material and Methods).
 Authors should perform model parameter sensitivity analysis (i.e., position of surfaces) to confirm the convergence and robustness of their approach.
In the first version, we already reported in Supplementary Figures S8 to S15, for each domain and each orientation of division, the distribution plots (surface area x distance to cell center) of simulations performed in different cells. In each of these figures, the cells shared the same shape (cuboid, triangular prism or tetrahedron) but differed in their exact geometry. As can be seen from these graphs, similar point distributions were obtained for different cells and, more importantly, the simulations matching best with observed patterns shared the same relative localization within these distributions (except of course when the geometrical rule was not valid, as in the basal inner domain). Therefore, these results already provide a sensitivity analysis to shape fluctuations. To make this point more explicit, we now show in a new Supplementary Figure S12 the 3D shapes associated with the first set of graphs (basal outer domain; Figure S11) and added reference to this figure in Section 2.5.
In addition to biological variability, possible minor errors and uncertainties at the image processing and segmentation step may also affect mother cell geometry. To illustrate the robustness of our approach to this potential source of geometrical variability, we have added a new Supplementary Figure S10 showing the distribution plots of simulations performed within a raw mother cell mask and within its mask following filtering using a mathematical morphological opening with radius of 1, 2, or 3 voxels. Morphological opening is an image processing operation that smoothes binary objects, removing extrusions having a radius smaller than the prescribed radius. The obtained results show the robustness of our results to such alterations of mother cell geometry. In the four conditions (R=0,1,2,3), the simulated patterns matching best with the observed pattern are located at the same bottom left position of the plot, corresponding to the geometrical rule.
Lastly, we have also added a new Supplementary Figure S9 to illustrate the reproducibility of simulations results obtained within a given mother cell. In this figure, we show the distribution plots for two independent sets of 1000 simulations each. The graphs show similar distributions, with identical locations at the bottom left of the distribution of the simulations that best matched with the observed division plane.
Concerning the convergence of the 3D cell division computer model, we have added in Section 4.3 (Material and Methods: Computer modeling of cell divisions) the justification about the number of Monte Carlo cycles. We have added a new Supplementary Figure S7 illustrating the convergence of the algorithm over different independent runs. We also corrected a typo on the number of Monte Carlo cycles (which was 500 instead of 5000 as initially written).
Reviewer #2 (Public Review):
This is an interesting manuscript aiming at identifying minimal rules that account for cell divisions in early Arabidopsis embryos. This research has two main strengths. The authors consider cell division in 3 dimensions, whereas most other studies on the orientation of cell divisions are restricted to 2 dimensions. Based on their observations, the authors proposed that the previously proposed probabilistic rule for cell division can be replaced by a deterministic rule, with sources of stochasticity coming from irregularities/imperfections in cell geometry. The manuscript is overall wellwritten. I nevertheless have a few concerns.
 What is the effect of embryo fixation on cell geometry? Could the irregularities be an artefact due to fixation? How robust are the conclusions to numerical perturbations of the position of cell surfaces?
We used the fixation and staining protocol developed by one of us (JCP) (Truernit et al 2008). Yoshida et al. Developmental Cell (2014) used this same protocol, which they validated by comparison with live imaging data. The fixation and the following treatment could have an impact on cell geometry. For this reason, we have selected among a thousand embryo acquisitions, the embryos that are not or very few damaged with this treatment. The robustness of our results and conclusions to variability and alterations of cell geometries was also questioned by Reviewer #1. Please see above our indepth answer to this point.
 Section 2.7 on attractor patterns is essentially descriptive and the conclusions seem to be based on qualitative observation of a few cases. Can the authors support them with quantitative measures? Or with simulations?
We have completed this section with quantitative data when it was missing. In the apical outer domain, we had 135 observations at G6, which had been reached from G4 according to one or the other of the two main pathways shown in Figure 9A. These two possibilities accounted for 40% and 42% of observations, respectively.
The other attractors shown in Figure 9 are rare cases (Fig. 9B: 1 case over 173; Fig. 9C: less than 9 cases over 309). The case shown in Figure 9B was previously documented in Scheres et al 1995 (cited in the manuscript).
The lower frequency in the basal domain of alternative sequences leading to a same attractor pattern is consistent with the lower variability in this domain. However, the conclusion is the same as in the apical domain where the distribution between alternative sequences is more balanced: different sequences of division over several generations can lead to similar cell patterns.


Evaluation Summary:
The manuscript presents a highquality quantitative analysis of plant embryo cell division in 3D. Authors combine computer modeling with detailed microscopy imaging to reveal underlying patterns and biases in cell divisions. The manuscript will likely be of interest to cell and developmental biologists. The conclusion can be straightened following additional analysis and data interpretation.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)

Reviewer #1 (Public Review):
Overall, it is an interesting work exploring stochastic and deterministic aspects of embryonic cell division in plants. The power of the authors' approach lies in the quantitative analysis of 3D cell geometries combined with quantitative computer modelling.
I am a bit confused about how authors relate stochasticity as an emergent property of a deterministic process. Typically, stochasticity is the lowlevel process resulting in variation of subcellular components those also related to the positioning of the cell division plane. Perhaps a more elaborated and clearer connection between stochasticity at the subcellular level and phenotypic variability should be provided. I have a number of specific questions/concerns that I would like the authors to address as listed below:
Major variability of cell shapes is …
Reviewer #1 (Public Review):
Overall, it is an interesting work exploring stochastic and deterministic aspects of embryonic cell division in plants. The power of the authors' approach lies in the quantitative analysis of 3D cell geometries combined with quantitative computer modelling.
I am a bit confused about how authors relate stochasticity as an emergent property of a deterministic process. Typically, stochasticity is the lowlevel process resulting in variation of subcellular components those also related to the positioning of the cell division plane. Perhaps a more elaborated and clearer connection between stochasticity at the subcellular level and phenotypic variability should be provided. I have a number of specific questions/concerns that I would like the authors to address as listed below:
Major variability of cell shapes is observed in the apical domain as opposed to the basal domain. What would be an underlying principle to asymmetric shaping of the apicalbasal domain? The authors describe beautifully the observations but give relatively little discussion on this matter, leaving the reader guessing.
Authors used graph theory to explain variability in cell division for the same topological feature. In light of quite a discrepancy between predictions and observations (i.e., Figure 4C) question arises of how this prediction could be affected by undergoing cell expansions as this element I believe is neglected in their graph theory approach?
Tetrahedron shape repeats in only 4% of embryos at the 16cell stage. What could be the criticality of this shape for the entire embryo patterning?
It is not clear to me whether stochastic cell division modelling takes into account the mechanical influence of adjacent cells? In any case, authors should discuss how this could potentially affect their analysis.
Authors should perform model parameter sensitivity analysis (i.e., position of surfaces) to confirm the convergence and robustness of their approach.

Reviewer #2 (Public Review):
This is an interesting manuscript aiming at identifying minimal rules that account for cell divisions in early Arabidopsis embryos. This research has two main strengths. The authors consider cell division in 3 dimensions, whereas most other studies on the orientation of cell divisions are restricted to 2 dimensions. Based on their observations, the authors proposed that the previously proposed probabilistic rule for cell division can be replaced by a deterministic rule, with sources of stochasticity coming from irregularities/imperfections in cell geometry. The manuscript is overall wellwritten. I nevertheless have a few concerns.
What is the effect of embryo fixation on cell geometry? Could the irregularities be an artefact due to fixation? How robust are the conclusions to numerical perturbations of the …
Reviewer #2 (Public Review):
This is an interesting manuscript aiming at identifying minimal rules that account for cell divisions in early Arabidopsis embryos. This research has two main strengths. The authors consider cell division in 3 dimensions, whereas most other studies on the orientation of cell divisions are restricted to 2 dimensions. Based on their observations, the authors proposed that the previously proposed probabilistic rule for cell division can be replaced by a deterministic rule, with sources of stochasticity coming from irregularities/imperfections in cell geometry. The manuscript is overall wellwritten. I nevertheless have a few concerns.
What is the effect of embryo fixation on cell geometry? Could the irregularities be an artefact due to fixation? How robust are the conclusions to numerical perturbations of the position of cell surfaces?
Section 2.7 on attractor patterns is essentially descriptive and the conclusions seem to be based on qualitative observation of a few cases. Can the authors support them with quantitative measures? Or with simulations?

Reviewer #3 (Public Review):
Cell division pattern is a crucial factor to organize the tissue morphogenesis including the early embryo. However, it is still unclear how the robustness of cellular arrangement is ensured under the noise of cell division patterns. In this manuscript, the authors reconstructed the lineage of cell shape in Arabidopsis embryos by categorizing cellular topology in each stage. Through this reconstruction, the authors demonstrated that the diversity of cellular topology is reduced along embryo development, where most cells finally showed cuboid shape. Also, using graph theory, they showed that this transition of topology didn't follow a random division pattern. Next, using the modeling approach, the authors found that some domain in the embryo follows a cellular geometrical rule, in which the division plane …
Reviewer #3 (Public Review):
Cell division pattern is a crucial factor to organize the tissue morphogenesis including the early embryo. However, it is still unclear how the robustness of cellular arrangement is ensured under the noise of cell division patterns. In this manuscript, the authors reconstructed the lineage of cell shape in Arabidopsis embryos by categorizing cellular topology in each stage. Through this reconstruction, the authors demonstrated that the diversity of cellular topology is reduced along embryo development, where most cells finally showed cuboid shape. Also, using graph theory, they showed that this transition of topology didn't follow a random division pattern. Next, using the modeling approach, the authors found that some domain in the embryo follows a cellular geometrical rule, in which the division plane passes through the cell centroid with minimized area, while the other domain does not. In addition, they also showed the asymmetry of mother cell geometry is highly correlated with division patterns. Finally, the authors reported that the variability of division patterns is buffered along embryo development to enable robust cellular organization. These findings suggested that the variability of division patterns is restricted due to reduced diversity of cell topology and this limitation is highly important for the robustness of tissue organization. I agree this manuscript is attractive for people who are seeking fundamental mechanisms underlying proper cellular patterning in plant tissue morphogenesis. However, this manuscript lacks explanations to understand the novelty and superiority of their methods or results, especially for people of nontheoretical or mathematical backgrounds. For example, what "graph theory" is and what do Fig 4A and B indicate? How can we interpret Figure 57 (Which point corresponds to each division pattern in the right panels?) How can we conclude that variability is buffered from Figure 7C?

