Model-Based Analysis of Arabidopsis Leaf Epidermal Cells Reveals Distinct Division and Expansion Patterns for Pavement and Guard Cells 1[W]

To efficiently capture sunlight for photosynthesis, leaves typically develop into a flat and thin structure. This development is driven by cell division and expansion, but the individual contribution of these processes is currently unknown, mainly because of the experimental difficulties to disentangle them in a developing organ, due to their tight interconnection. To circumvent this problem, we built a mathematic model that describes the possible division patterns and expansion rates for individual epidermal cells. This model was used to fit experimental data on cell numbers and size obtained over time intervals of one day throughout the development of the first leaf pair of Arabidopsis ( Arabidopsis thaliana ). The parameters were obtained by a derivative-free optimization method that minimizes the differences between the predicted and experimentally observed cell size distributions. The model allowed us to calculate probabilities for a cell to divide into guard or pavement cells, the maximum size at which it can divide, and its average cell division and expansion rate at each point during the leaf developmental process. Surprisingly, average cell cycle duration remained constant throughout leaf development, whereas no evidence for a maximum cell size threshold for cell division of pavement cells was found. Furthermore, the model predicted that neighboring cells of different sizes within the epidermis expand at distinctly different relative rates, which could be verified by direct observations. We conclude that cell division seems to occur independently from the status of cell expansion, whereas the cell cycle might act as a timer rather than as a size-regulated machinery.

In most plant species, the above-ground plant body is dominated by leaves, the organs specialized in photosynthesis. This process converts carbon dioxide into organic components ultilizing energy from sunlight, making leaves the energy production site and the growth engine of plants. To maximize its light-capturing capacity, a leaf is typically flat and thin. This characteristic shape is established during the leaf developmental process. Leaves arise on the flanks of the shoot apical meristem (SAM) at auxin accumulation sites (Benková et al., 2003). At these positions, a number of cells start to bulge out from the meristem and eventually will form the basis of the leaf primordium when cell division proceeds (Reinhardt et al., 2000;Pien et al., 2001). Dorsiventrality is specified early during primordium development and defines the adaxial and abaxial sides of the leaf (Bowman, 2000). Divisions at the margin of the primordium drive leaf blade inception. Further expansion of the leaf blade is controlled by a strong preference for anticlinal divisions, leading primarily to lateral outgrowth of the different tissue layers, of which the epidermis is the main layer driving leaf growth (Donnelly et al., 1999;Savaldi-Goldstein et al., 2007).
During leaf development of dicotyledonous species, a cell proliferation phase, characterized by actively dividing cells, is followed by a cell expansion phase, characterized by cell growth and differentiation. After expansion, cells mature and the final leaf size is reached (Beemster et al., 2005). At the proliferation-to-expansion phase transition, cell division ceases along a longitudinal gradient from leaf tip to base (Donnelly et al., 1999). In the epidermis, the onset of differentiation coincides with the formation of stomata (De Veylder et al., 2001). A stomatal complex consists of two guard cells that control the aperture of the stomatal pore. Starting from a precursor meristemoid cell, a series of subsequent asymmetric divisions produce a number of guard mother and daughter cells. Subsequently, the guard mother cells divide symmetrically into two guard cells, ending the stomatal lineage. The daughter cells undergo cell fate specifications identical to those of the majority of the cells produced during the proliferation phase, resulting in puzzle-shaped pavement cells (Larkin et al., 1997;Geisler et al., 2000).
The final leaf size is determined by the total number of cells and the average cell size that result from cell division and cell expansion, respectively. Although the dynamics of these processes can be analyzed rigorously by the leaf growth kinematics (Fiorani et al., 2006), knowledge of cell cycle duration, cell expansion, and their interaction at the individual cell level is still poorly understood, not only because of technical obstacles to 6 study these phenomena, but also because a reduced cell proliferation is often compensated by an increase in cell size and vice versa (Tsukaya, 2002). Here, the individual cell sizes of pavement and guard cells were measured separately throughout leaf development. By fitting a mathematical model to these data, we could estimate the division and expansion parameters of pavement and guard cell populations within the growing leaf separately, allowing us to gain a better and more detailed insight into the processes that define leaf growth. Imaging of epidermal cells gave a good correlation between predicted and experimental cell growth data, supporting the model.

Following leaf growth during development
Recent developments in microscopic and imaging technologies suggest that cell tracking is the most suitable manner to disentangle cell division and expansion in plants, allowing cells to be followed for 3-4 days and applied successfully to the root, the SAM and sepals (Reddy et al., 2004;Campilho et al., 2006;Fernandez et al., 2010;Roeder et al., 2010). However, to cover the entire leaf development, a much longer time frame is needed. Furthermore, with these techniques only local observations can be made, which are not always easily correlated to global growth characteristics. Therefore, a general kinematic analysis of leaf growth in Arabidopsis (Arabidopsis thaliana) was used as a starting point for this research (De Veylder et al., 2001). In this approach, the first developing leaf pair (leaves 1 and 2) was harvested on a daily basis 5 to 25 days after sowing (DAS). Leaves 1 and 2 were selected because they are nearly indistinguishable and probably the best synchronized among replicate plants. Microscopic drawings of abaxial epidermal cells were made at 25% and 75% of the distance from the base to the tip of the leaf, giving a precise estimate of average cell area. To estimate the total cell number per leaf, these average cell areas were combined with the measured total leaf area. When analyzed on a daily basis, average cell division and relative expansion rates can be calculated. Plotting of the leaf size evolution on a logarithmic scale revealed a linear increase until day 11, indicating exponential growth (Fig. 1A). From day 12 onward, relative leaf expansion rates decreased and the mature leaf size was reached approximately at 20 DAS. A similar evolution could be observed for the total cell number 7 (Fig. 1B), with cell division rates being high until day 10 (Fig. 1C). Cell sizes remained relatively constant until day 10 (approximately 100 µm 2 ), whereas from day 10 onward, the average cell size increased approximately 10-fold by day 20, as the result of cell expansion in the absence of cell division (Fig. 1D). Coinciding with the decrease in cell division rate, the stomatal index (fraction of guard cells among all cells) increased linearly (Fig. 1E). The relative leaf expansion rates were the highest during the high division rate period (Fig. 1F). When pavement and stomatal cells are considered separately, the total number of pavement cells increased gradually from day 5 to day 14, while the number of guard cells continued to increase until day 17 (Supplemental Fig.   S1), indicating that divisions giving rise to guard cells continued approximately 3 days longer than those forming pavement cells.

Cell size distributions
Although the kinematic data give an indication of the general growth processes during leaf development, leaves are considered as homogenous cell populations, which is a simplification because the epidermis consists of multiple cell types, each with distinct size characteristics at different time points during development. Furthermore, pavement cells and guard cells are interdependent, because pavement cells are formed together with stomata (Geisler et al., 2000). Additionally, the size of pavement cells ranges from 50 up to 20,000 µm 2 , illustrating the heterogeneity of the population. A better insight into the cell area distribution was gained by extension of the image analysis algorithm used for the kinematic analysis to allow size measurements of individual guard and pavement cells.
To ascertain that the obtained data were representative for the complete leaf, we compared the data obtained from extrapolation of the measurements of cell sizes at two reference positions (25% and 75% between base and tip of the leaf blade) with those of microscopic drawings of five complete leaves at the transition from cell proliferation to cell expansion (namely in 9-day-old leaves). At this time point, the largest differences across the leaf would be expected because of the cell cycle arrest front that propagates along the leaf axis (Donnelly et al., 1999). Comparison of the data extracted from the complete leaves with those from the leaf reference sections revealed no significant differences for the average cell number and cell area. Furthermore, plotting of the cell size distributions also yielded a good overlay between the data resulting from the complete leaf and the leaf sections (Fig. S2), demonstrating that the data sampling at the two reference positions is a valid approach to estimate cellular parameters for the complete leaf.
According to the cell area measurements as described above, 97% of the pavement cells of 5-day-old leaves were smaller than 100 µm 2 , whereas at day 8, only 62% of the cells had a size below this threshold, indicating that pavement cell sizes increase already during the cell proliferation phase. Early in the expansion phase, at 10 DAS, the maximum cell size detected was approximately 1,600 µm 2 and 95% of the pavement cells were smaller than 500 µm 2 . From day 11 onward, the pavement cell area distribution broadened and the pavement cell population was distributed over a large range of cell sizes ( Fig. 2A). Guard cell sizes ranged from 25 to 150 µm 2 , with a mean area of approximately 75 µm 2 . During leaf development, cell sizes increased continuously, reaching a maximum size of approximately 300 µm 2 and an average area of 150 µm 2 at 25 DAS (Fig. 2B).
To obtain quantitative information about the changes in cell size distributions during leaf development, we used the frequency distribution of the cell areas of pavement and guard cells for the whole leaf on day i (for details, see Text S1). This absolute representation of the data revealed that from day 5 to 12, most pavement cells were very small (less than 300 µm 2 ) and that the number of these small cells increased significantly from day 6 to day 9 ( Fig. 2C), corresponding to the high cell division rate during these days. From day 9 until day 12, the peak of the distribution curves was less pronounced and was accompanied by a high proportion of large cells, of which the number increased until day 17, after which the distribution of pavement cells remained relatively stable. The guard cell distribution was different. The graph had a symmetrical bell shape with a peak at the mean, revealing a roughly normal size distribution of guard cell sizes (Fig. 2D).
Prior to day 9 the number of guard cells was low, but afterward increased significantly until day 17, indicating that most divisions of guard mother cells, leading to the formation of stomata, occurred relatively late during the epidermal development. After day 17, the complete guard cell size distribution continued to shift to the right, implying that cells had ceased dividing and continued to expand.

Mathematical model for leaf development
In order to study in more detail the crucial parameters for cell division and cell expansion during leaf development we build a general mathematical model. The model takes only the pavement and guard cells into account, because in our experimental system it is impossible to distinguish pavement from stomatal precursor cells. The model is based on the overall kinematics of leaf growth (Fig. 1), on the changes in size distribution of pavement and guard cells between successive days in function of cell expansion (Fig. 2), and on the changes in cell identity with each division event. Therefore, all possible transitions were considered that a cell can undergo from one day to the next: a precursor cell might either expand and divide into two pavement cells or two guard cells, or expand in the absence of division, whereas guard cells do not divide, but can expand. The model included a maximum guard cell size (GC max  Based on these assumptions, eight possible flows (F 1 to F 8 ) for pavement cells and one flow (F 9 ) related to guard cells were defined (Fig. 3B). Flow F 1 involves pavement cells that do not divide in one day, but only expanded. For dividing cells, possibilities differ based on whether the cell cycle duration time is shorter or longer than 24 h. For flows F 2 , F 3 , and F 6 , T c was ≤ 24 h and pavement cells divide within one day either once (F 2 ) or twice (F 3 ) into pavement cells or guard cells (F 6 ). In flow F 3 , for the sake of simplicity, both newborn pavement cells are assumed to be dividing into pavement cells.
Because T c > 18 h, not many cells take part in this flow and therefore the number of missed events must be small. When T c > 24 h, pavement cells can divide at most once in 1 day. In flow F 4 , pavement cells divide once, while in flow F 5 , they were dividing, but do not complete their division cycle within the one-day time step. Flow F 7 is related to the pavement cells that divide into guard cells and flow F 8 to those that are in the process of dividing into guard cells, but do not complete the division in the one-day time step.
Finally, flow F 9 represents guard cells that expanded only.
For these flows, we constructed functions for the transitions between the size distributions of pavement and guard cells from a given day i, to the next. Although all flows differed, they have a common basic structure. originates from a cell with area * a that follows the kth flow at day i . Therefore, for small distance ε , the radius around a , we have where 1 ε and 2 ε are both approximately equal to ε is the number of that originates from the k-th flow, then this corresponds at day i where k n is the number of cells at day 1 + i that originate from one cell at day i through the k -th flow. Hence the distribution of cells with area a at day . The mathematical description of each flow and explicit form of each k f 1, and k f 2, is given in the Model file (see Text S1). Using the functions we predicted the distribution of cells with area a at day 1 + i as follows, are the predicted distributions of pavement and guard cells, at day 1 + i respectively. The right hand sides of equations (6) and (7) are obtained by using (5)

Parameter estimation
The predicted distributions (6) and (7)  In the first phase of development, some a priori restrictions to the model are imposed to reduce the range in which parameters needed to be optimized. First, because no cell can divide into guard cell and pavement cell simultaneously, it is considered that Secondly, we assume that at the early stages of leaf development all cells divide (p 1 + p 2 = 1 at days 5, 6, and 7), which is supported by the uniform expression of cell cycle marker genes in young leaves (de Almeida Engler et al., 2009). Accordingly, the number of cells during early leaf development approximately doubled every 24 h (Table S1). Lastly, based on the size increase of guard cells over time (Fig. 2B), the maximum threshold for precursor cells to divide into guard cells (2GC max ) is based on GC max computed in the second phase, which is then applied to the first phase. With the model and the experimental distributions of the pavement and guard cells on a given day, we predicted the distributions for the next day. Although a best fit for the distribution of the guard cells at the earliest time points was difficult to find because of the low number of guard cells at the early stage of leaf development, experimental and computed data fitted well from 9 DAS onward (Figs. S3 and S4).

Constancy of average cell cycle duration
Cell division rates quantify the rate at which cells progress through the cell cycle. In

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the kinematic growth analyses, the cell division rate decreased progressively (Fig. 1C).
However, because division rates are calculated based on the total number of leaf cells, the observed decrease might be due to a reduction in the fraction of cells proliferating, an increase in the average cell cycle length, or a combination of both. To understand how the average division rate is controlled, it is essential to quantify the fraction of cells dividing into pavement (p 1 ) and guard (p 2 ) cells. As computed by the model, p 1 decreased gradually during leaf development, while p 2 increased, indicating a shift from basal proliferation to cell division in the stomatal lineage (Fig. 4A). The probability p 2 had the largest values between days 10 to 14, meaning that most stomata were produced during these days. After day 14, both probabilities dropped and decreased to 0, representing the exit of cell division during leaf development as indicated by an increase in probability p 3 , in which more than 80% of the cells did not divide after day 16.
Because the model allows us to split the total number of cells in a proliferative and an expanding population at the different stages of leaf development, a slow-down in the cell division rate can be discriminated from a reduction in the proliferative fraction.
Average cell cycle duration (T c ) can be derived from the cell division rate and was determined for each day during leaf development. As discussed above, the impact of T c in the model ended after day 18, when cell division stopped completely. Immediately prior to that, between days 16 and 18, the optimization results fluctuated a lot (Fig. S5), implying that the proliferative fraction was too small to obtain reliable results. For earlier time points, however, a stable output value was obtained. Remarkably, when different days are compared, the obtained T c value is nearly constant (Fig. 4B), indicating that the reduced cell division rate observed over the complete leaf during development by kinematic analysis was seemingly not caused by an increase in average cell cycle duration, but solely by a decrease in the proliferative fraction within the leaf.

Non-existence of size threshold for division
One of the parameters included in the model was the maximum guard cell size GC max , which was designated a maximum threshold because guard mother cells, identified as pavement cells in our analysis, must have an area smaller than 2GC max as a necessary condition for division into guard cells. As discussed above, the threshold for guard cells was calculated easily in the second stage of development (from days 18 to 25), when no cell divided. When we assumed the threshold to be constant, the parameter optimization procedure provided a value equal to 354 µm 2 , which was only a little above to the maximum guard cell size of 300 µm 2 found in the experimental data (Fig. 2), illustrating the accuracy of the optimization methods. The value of GC max indicated that cells with an area larger than approximately 700 µm 2 are unable to divide into guard cells.
In contrast to the robust value obtained for GC max in nearly all simulations, the optimized parameter value for the threshold above which no divisions into pavement cells occur (PC max ) yielded erratic values. In cases with an optimal solution, the experimental cell size distribution (used to build the model) and the computed cell size distribution profiles strongly deviated (Fig. 5A,B), implying that the found optimum was a spurious one. To strengthen this observation, simulations were done with fixed values for PC max = 300, 500, 1000, and 10,000 µm 2 at day 11 to 12, when the leaf consists of both dividing and expanding cells. Consistently with the behaviour of the model with fitted PC max , small fixed values of PC max resulted in a sharp deviation from the experimental data ( Fig.   5C). As the PC max value increased, the discrepancy between experimental and computed data gradually disappeared, which might be explained by the fact that most cells were smaller than the threshold and, thus that the PC max value became less relevant. These

Growth rates of epidermal cells
The difference in maximum cell sizes found for pavement and guard cells suggests that the two cell types might expand at different rates. Therefore, the model allows for different relative growth rates for pavement and guard cells. Cell division and cell expansion are considered independently and relative growth rates do not only apply to expanding, but also proliferation cells. growth rate from day 5 to day 13, with a profound increase between day 10 and 13 ( Fig.   6A). From day 13 onward, the relative growth rate declined steeply, suggesting that pavement cells grew faster in the young proliferating leaf than in the older expanding leaves. Eventually, growth stopped completely after day 18. These observations are consistent with the experimental data because the average cell area of pavement cells increased from 86 µm 2 in the young leaf to nearly 1,500 µm 2 between days 7 and 16, but did not change significantly after that (Fig. 1D). Similarly, guard cells displayed higher relative growth rates in the young than in the old leaves. Initially, the relative growth rate of pavement cells was higher than that of guard cells, but from day 18 onward, stomata grew faster than pavement cells (Fig. 6A).
After day 18 pavement cells did not divide; hence, the relative growth rate of small pavement cells (<300 µm 2 ) could be calculated by measuring the shift in the distribution peak in the experimental size distribution graphs (Fig. 2C). The average relative growth rate of small cells determined this way was 0.009/h, whereas the average relative growth rate from day 18 until day 25 for all pavement cells was calculated by the model to be 0.0009/h (Fig. 6A). These results suggest that during the last days of leaf development, small pavement cells grow 10-fold faster than the average population. To confirm the surprising finding that adjacent cells of different size grow at different rates, we performed confocal imaging to directly measure relative growth rates (RGRs) for individual pavement and guard cells over a 36 h time interval. Cell tracking experiments on leaves at 17 DAS confirmed that pavement cells smaller than 300 µm 2 grew faster than large ones (P-value = 3.65e-5, Student t-test) (Figs. 6C and 6D). The average RGRs of small pavement cells decreased steeply from ~ 0.015 to 0.005 in the size range between 30 and 300 µm 2 . In larger cells, a relatively constant RGR was measured. Small guard cells (<100 µm 2 ) also grew faster than larger ones (P-value = 0.0017, Student t-test) (Fig.   6C,E). In contrast to pavement cells, guard cells did not display a biphasic growth pattern, but a steady decrease in RGR with their cell size. Strikingly, at the transition of day 17 to 18 the average relative growth rate of pavement cells measured by live imaging (0.0065/h) was nearly identical to the value obtained by the model (0.0061/h), independently confirming the model predictions (Fig. 6A).

DISCUSSION
Leaf development of Arabidopsis is driven by two processes, cell division and cell expansion (Green and Bauer, 1977). As both processes are intimately intertwined, their individual contribution to leaf growth is not easily studied experimentally. Here, we developed a mathematical model based on the fates of the two epidermal cells types found in the abaxial epidermis of the first leaf pair of Arabidopsis to fit experimentally derived data obtained over the course of its development from a young dividing leaf primordium into an adult organ. This model allowed us to disentangle cell division and cell expansion parameters for the individual cell types.

Differential cell expansion within the leaf epidermis
Individual cell size measurements yielded cell area distributions during leaf development. When the relative growth rates of guard cells and pavement cells were compared, the growth of these two cell types followed distinctly different dynamics. The guard cells initially grow more slowly but steadily, whereas the large pavement cells initially grow faster, but stop around day 16. Differences in relative growth rates between adjacent cells are surprising and have, to our knowledge, not been investigated in leaves.
In the root, this problem has been studied by measuring cell sizes of different cell types (Beemster and Baskin, 1998). Root growth is linear and symplastic, meaning that neighbouring cells grow uniformly, without altering the positions of adjacent walls. As a consequence, at a given distance from the tip, all cells have by definition the same relative elongation rate (Green, 1976). This constraint implies that differences in mature cell sizes reflect differences in cell proliferation (Beemster and Baskin, 1998). In comparison, leaf growth is much more complex. As a leaf is a flat, exponentially growing structure with small and large cells dispersed over the epidermal surface, several parameters have to be taken into account in the equations. Time-lapse cell-tracking experiments on 17-day-old leaves indicate that small pavement and guard cells grow faster than large cells. Furthermore, modelling results indicated that relative cell growth rate of pavement cells is tightly regulated during development, peaking during the early leaf expansion phase, followed by a rapid reduction and no growth occurring at maturity.
In other words, cells exiting the proliferative status grow faster than those being in full expansion phase. A high increase in the relative growth rate was seen between days 10 and 14. Interestingly, this time frame coincides with the strongest increase in cellular DNA content through endoreduplication (Fig. S7). During the endoreduplication cycle, cells do not divide but continue to increase their DNA content. A positive correlation is often seen between cell size and the level of endoreduplication (Sugimoto-Shirasu and Roberts, 2003). The observed correlation between the increase in relative cell growth rate and elevation in DNA content supports the generally assumed hypothesis that endoreduplication supports cell growth.
Taken together, the model reveals substantial differences in relative growth rates within the epidermal layer. Because guard cells, and small and large pavement cells are dispersed throughout the leaf, due to the patterning in the stomatal lineage, adjacent guard and pavement cells of different sizes might expand at different relative rates within the developing epidermis. Differential expansion of tissue layers in cylindrical organs, such as stems and roots, results in tissue tension, often redirecting growth, as in the case of shoot phototropism and root gravitropic curvature (Liscum and Stowe-Evens, 2000; Swarup et al., 2005). Leaf epinasty is also the consequence of differential growth of the abaxial and adaxial sides (Kellet and Van Volkenburgh, 1997). Within one single tissue layer of a flat tissue structure, such as the abaxial leaf epidermis, tension is more difficult to translate into motion in order to release pressure. Disturbance of symplastic growth in the root by inducing differences in relative cell expansion rates between tissue layers causes cells to rupture and distort root growth (Ubeda-Tómas et al., 2009). However in the leaf, no cell ruptures are detected within the epidermis. Possibly, the lobes of epidermal pavement cells offer a way to dissipate the tissue tension. Remarkably, the characteristic jigsaw puzzle shape of pavement cells is established at the proliferation-toexpansion transition. Thus, the appearance of the puzzle shape fits with the moment at which differential relative cell expansion rates in adjacent cells would first appear.
Furthermore, the emergence of lobes leading to the puzzle shape is preceded by the reorganization of cortical microtubules (Panteris and Galatis, 2005;Kotzer et al., 2006) and application of external mechanical stresses to a tissue results in realignment of microtubules parallel to the maximal stress directions (Hamant et al., 2008). Based on these observations, it could be speculated that the occurrence of tissue tension within the epidermis layer, caused by differential cell expansion, could realign cortical microtubules and trigger the puzzle-shape formation. More experimentation will be required to validate this intriguing hypothesis.
Interestingly, at maturity, the guard cells in Arabidopsis leaves are elevated above the surrounding pavement cells, presumably to increase their evaporative capacity. Faster relative growth of the guard cells and surrounding smaller pavement cells compared to the larger pavement cells that make up the bulk of the tissue area could explain the development of this elevation.

Control of cell cycle duration
In the root, divisions in the meristematic zone ensure a constant cell production and Baskin, 2000). These data indicate that the constancy of cell cycle duration is widespread and that the basal cell division rate is strongly conserved at a cellular level (Baskin et al., 2000).

Interaction between cell division and cell growth
For decades, cell biologists have been interested in the process that links cell size and cell division (Neufeld and Edgar, 1998), but the interaction is currently unclear. Some reports have argued that epidermal pavement cells of Arabidopsis leaves divide only rarely at a cell size larger than 400 µm 2 (Donnelly et al., 1999;Geisler et al., 2000), suggesting a possible size threshold preventing cell division. By contrast, a study on the root meristem hinted at the lack of a maximum cell size threshold for cell division, because cells divide at very different sizes (Ivanov et al., 2002;Beemster et al., 2003).
Similarly, our mathematical model suggests no threshold for cell division, because such a threshold would result in a deviation from the observed cell size distributions.

Kinematic growth and image analysis
Leaf growth kinematics were analyzed as described (De Veylder et al., 2001). The leaf blade area of leaves 1 and 2 of five plants from 5 to 24 DAS was measured from dark-field binocular (days 8 to 24) or DIC light microscopy images (days 5 to 7). Growth rates of 53 pavement cells and 13 guard cells were tracked over time.

Flow cytometry analysis
Plant material was chopped in 200 µL of Cystain UV Precise P Nuclei extraction buffer (Partec), supplemented with 800 µL of staining buffer. The mix was filtered through a 50-µm green filter and read through the Cyflow MB flow cytometer (Partec).

Supplemental Data
The following materials are available in the online version of this article.
Supplemental Figure S1. Experimental and smoothed total number of pavement and guard cells during leaf development.

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Supplemental Figure S2. Validation of data extraction of two reference points for the whole-leaf cellular data analysis.
Supplemental Figure S3. Comparison between experimental and model-predicted cell size distributions of pavement cells.
Supplemental Figure S4. Comparison between experimental and model-predicted cell size distributions of guard cells.
Supplemental Figure S5. Result of 100 optimizations for the cell cycle duration.
An optimization program for 100 different initial guesses for the parameters was implemented.
Supplemental Figure S6. Putative recently divided pavement cells.
Supplemental Figure S8. Division parts related to the flows.
Supplemental Table S1. Initial estimation of the average cell cycle duration T c .
Supplemental Text S1. Mathematical description model.