# Spiral phyllotaxis in the moss Physcomitrium patens emerges from simple division rules of the apical cell

Joseph Cammarata, Sören Strauss, Brendan Lane, Robert Kelly-Bellow, Laure Mancini, Teva Vernoux, Yoan Coudert, Adrienne Roeder, Richard Smith

► To cite this version:

Joseph Cammarata, Sören Strauss, Brendan Lane, Robert Kelly-Bellow, Laure Mancini, et al.. Spiral phyllotaxis in the moss Physcomitrium patens emerges from simple division rules of the apical cell. Current Biology, In press, 10.2139/ssrn.5374572 . hal-05487452v2

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<figure><img src="image_1.png" /></figure>
# Current Biology

## Spiral phyllotaxis in the moss *Physcomitrium patens* emerges from simple division rules of the apical cell

### Graphical abstract

How does *P. patens* control phyllotaxis with a single apical cell?

<figure><img src="image_2.png" /></figure>

Apical cell time-lapse
3D segmentation & quantification

3D Phyllotaxis Model
(shortest wall divisions)

<figure><img src="image_3.png" /></figure>

- tetrahedral shape

- asymmetric cell geometry

- isotropic growth

<figure><img src="image_4.png" /></figure>

fails
to produce
phyllotaxis

<figure><img src="image_5.png" /></figure>

- divisions always avoid
oldest neighbor wall

<figure><img src="image_6.png" /></figure>

symmetric
phyllotaxis
(120°)

<figure><img src="image_7.png" /></figure>

- division occurs through
displaced cell centroid
(pushed from oldest walls)

<figure><img src="image_8.png" /></figure>

spiral
phyllotaxis

### Authors

Joseph Cammarata, Sören Strauss,
Brendan Lane, ..., Yoan Coudert,
Adrienne H.K. Roeder,
Richard S. Smith

### Correspondence

ahr75@cornell.edu (A.H.K.R.),
richard.smith@jic.ac.uk (R.S.S.)

### In brief

Cammarata et al. explore how cell division orientation generates phyllotaxis in moss with a unicellular meristem. They show that successive rotational divisions in the apical cell are precisely placed through a modified shortest wall rule that avoids older neighboring cell walls. This spacing mechanism mirrors that of plants with multicellular meristems.

### Highlights

* Time-lapse analysis reveals the dynamics of cell division in the unicellular apex

* New walls form through a shortest wall rule with a displaced division point

* Rotational divisions are patterned by new walls displaced from old walls

* These division rules create spiral phyllotaxis in *P. patens*
<figure><img src="image_9.png" /></figure>

## Article

# Spiral phyllotaxis in the moss *Physcomitrium patens* emerges from simple division rules of the apical cell

Joseph Cammarata,<sup>1,2,6,7</sup> Sören Strauss,<sup>3,7</sup> Brendan Lane,<sup>4,7</sup> Robert Kelly-Bellow,<sup>4,7</sup> Laure Mancini,<sup>5</sup> Teva Vernoux,<sup>5</sup> Yoan Coudert,<sup>5</sup> Adrienne H.K. Roeder,<sup>1,2,*</sup> and Richard S. Smith<sup>4,7,8,*</sup>

## SUMMARY

In many plants, leaves are arranged around the stem in a pattern called Fibonacci spiral phyllotaxis. These patterns have been well studied in flowering plants and are thought to arise from a spacing mechanism based on the cell-to-cell transport of the plant hormone auxin. This causes new primordia to emerge as far as possible from previous ones in the available space on a multicellular meristem. However, it is not clear how a spacing mechanism can create spirals in plants with a unicellular meristem. Through time-lapse imaging, quantification, and computer modeling, we study the single tetrahedral apical stem cell of the moss *Physcomitrium patens* and the emergence of a spiral pattern of leaf-like structures. We find that the apical cell divides asymmetrically in a spiral pattern, giving rise to a leaf progenitor daughter cell and another apical cell; thus, phyllotaxis in *P. patens* is controlled by cell division orientation. Apical cell divisions are asymmetric both in fate and geometry, the latter being explained through displacement of the new wall from the centroid of the apical cell. Modeling shows that incorporation of displacement from the centroid with the default division plane selection by the minimal wall area ("shortest wall" rule) is sufficient to explain the spiraling division planes leading to phyllotaxis. Thus, the whole architecture of the shoot is defined by the orientation of cell division. Some cell types in flowering plants undergo a similar spiraling division plane pattern, suggesting that this may be a common mechanism across phyla.

## INTRODUCTION

Phyllotaxis, the pattern of leaves or flowers around the stem, is one of the most striking examples of patterning in nature. The conspicuous rows of intersecting spirals in alternate directions on a sunflower head are often consecutive numbers in the Fibonacci sequence. To generate these patterns, the divergence angle between successive primordia must be close to the golden angle of 137.5°. Although phyllotaxis appears mathematically and geometrically complex, these patterns can emerge from a simple rule: new organ primordia emerge as far as possible from previous ones.<fnref n="1" /> Recent studies suggest a molecular spacing mechanism at the meristem where older primordia inhibit the emergence of new ones.<fnref n="2" />,<fnref n="3" /> The current theory is that this spacing mechanism relies on the active transport of the plant hormone auxin. In the model angiosperm *Arabidopsis*, PIN-FORMED (PIN) proteins transport auxin cell-to-cell, with their polarity determined by auxin concentration in neighboring cells. This feedback loop organizes PIN polarity in space and time, generating maxima that trigger primordium formation.<fnref n="5" />⁻⁹

In flowering plants with multicellular shoot apices, this tissue mechanism explains spiral phyllotaxis. In contrast, the apical meristem in certain bryophyte shoots, such as the moss *Physcomitrium patens* gametophore (upright shoot with leaf-like phyllids), consists of a single tetrahedral apical stem cell<sup>6,7</sup> (Figure S1; Video S1). Intriguingly, *P. patens* also produces spiral phyllotaxy from its apical cell, which undergoes repeated asymmetric divisions with one daughter initiating a phyllid and the other remaining meristematic (Figures 1B and 1C). Successive divisions rotate, producing phyllids in Fibonacci spiral phyllotaxis<sup>7,8</sup> (Figure 1A). How the division plane orients to rotate consistently and generate phyllotactic patterning in *P. patens* remained unclear.

Plant cell divisions are often oriented to produce the minimum-area wall passing through the cell's geometric center (centroid). In two dimensions, this is known as the "shortest wall" rule, a simplification of Errera's minimal energy rule inspired by soap bubbles.⁹ The default division plane can be predicted from cell shape according to this simple rule,¹⁰ which holds for cases like early *Arabidopsis* embryo divisions up to the 8-cell stage.¹¹

<footnote n="1"><sup>1</sup>School of Integrative Plant Science, Section of Plant Biology, Cornell University, Ithaca, NY 14853, USA</footnote>

<footnote n="2"><sup>2</sup>Weill Institute for Cell and Molecular Biology, Cornell University, Ithaca, NY 14853, USA</footnote>

<footnote n="3"><sup>3</sup>Department of Comparative Development and Genetics, Max Planck Institute for Plant Breeding Research, 50829 Cologne, Germany</footnote>

<footnote n="4"><sup>4</sup>Department of Computational and Systems Biology, John Innes Centre, Norwich Research Park, Norwich NR4 7UH, UK</footnote>

<footnote n="5"><sup>5</sup>Laboratoire Reproduction et Développement des Plantes, Université de Lyon, ENS de Lyon, CNRS, INRAE, 69342 Lyon, France</footnote>

<footnote n="6"><sup>6</sup>Present address: Syngenta Seeds Research and Development, 9 Davis Drive, Research Triangle Park, Durham, NC 27709, USA</footnote>

<footnote n="7"><sup>7</sup>These authors contributed equally</footnote>

<footnote n="8"><sup>8</sup>Lead contact</footnote>

<footnote>*Correspondence: ahr75@cornell.edu (A.H.K.R.), richard.smith@jic.ac.uk (R.S.S.)
https://doi.org/10.1016/j.cub.2026.01.059</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 1.** Phyllotaxis in *Physcomitrium patens* arises from repeated rotating divisions of a single apical cell</figcaption></figure>

(A) Scanning electron micrograph of a gametophore showing spiral phyllid arrangement.

(B) Diagram of the apical cell and its three most recent daughter cells (neighbors): magenta = most recent (1), green = second (2), and blue = oldest (3). Older daughters may have divided. Numbers indicate phyllid maturity (1 = youngest).

(C) Time-lapse confocal micrographs of a gametophore grown on 5 µM NAA to suppress phyllid outgrowth (propidium iodide cell wall stain). Inset: apical cell and its three neighboring cells (numbering corresponds to future phyllid order). The apex underwent two divisions over 48 h (red squares at 24 and 48 h).

(D and E) Cell surface segmentations with tracked lineages from (C) at 12 h (before apical division) and 24 h (after division), showing a new apical cell (An) and a new daughter cell (dn).

(F and G) Phyllotactic angle quantification.

(F) Measurement method: angle between new walls of two consecutive daughter cells (black and red dashed lines). Average angle is 126.9° (SD 9.0°, *n* = 19).

(G) Violin and box plot of phyllotactic angle distribution.

Scale bars, 100 µm (A) and 20 µm (C–E).

See also **Figures S1–S3** and **Video S1**.

It has been proposed as a universal rule for symmetric division in plants.<sup>12,13</sup>

In *P. patens*, the apical cell has a tetrahedral shape.<sup>14</sup> A perfect tetrahedron would allow four equivalent shortest-wall planes, making reliable selection of a specific plane for spiral patterning unclear. Couturier et al.<sup>15</sup> examined apical cell shapes that maintain cell shape under a minimal energy division rule. In a surface

model, they proposed spirals emerge from a triangular shape if the new wall always intersects the most recent daughter cell. Alternatively, division may not follow the shortest-wall rule, implying another mechanism. Various hypotheses exist for division orientation. Hofmeister<sup>16</sup> proposed divisions occur perpendicular to the primary growth direction. This idea was elaborated by Hejnowicz<sup>17</sup> to include the other two orthogonal principal
<figure><img src="image_11.png" /></figure>

directions, a rule that appears to encompass almost all divisions in the root. Lintilhac and Vesecky¹⁸ suggested stresses orient division planes, with Louveaux et al.¹⁹ proposing stress can override the shortest-wall rule in areas of highly anisotropic stress in the shoot apex (see Rasmussen and Bellinger²⁰ for a review on division plane orientation in plants). In some cases, the minimal-area plane through the centroid is not followed; for example, the formative divisions in the *Arabidopsis* embryo from the 8- to 16-cell stage that establishes the pre-epidermal layer.¹¹,²¹,²² Here, we use 3D time-lapse live imaging combined with quantitative cell shape analysis and simulation modeling to test possible mechanisms and understand how the correct division plane in the apical cell of *P. patens* is reliably chosen to produce spiral phyllotaxis.

## RESULTS

### Live imaging of the moss apex reveals a spiral pattern of division of the apical cell

Observing the *P. patens* apical cell is difficult because the apex is covered by growing phyllids (Figure 1A) and its small size makes dissection challenging. When grown on the synthetic auxin 1-naphthaleneacetic acid (NAA), phyllid outgrowth is inhibited,²³,²⁴ yet the shoot apex and apical cell continue to grow and divide. By culturing moss shoots on medium with 5 µM NAA, we could perform live confocal imaging of exposed apical cells and their neighbors during growth and division (Figure 1C). In total, 12 apices were imaged every 12 h over 48 h (5 time points), capturing 19 apical cell divisions (1–2 per apex). Each apex contained a single apical cell that divided in a spiral pattern. The central daughter cell retained stem cell identity²³ and tetrahedral shape, becoming the new apical cell (Figure S1), while the outer daughter cell (normally a phyllid precursor) divided only once or twice before arresting due to exogenous auxin. All divisions in the dataset maintained consistent phyllotaxy direction, with no reversals or repeated divisions on the same sides. Of the 12 apices, 8 showed counterclockwise spirals and 4 were clockwise (Figure S2), consistent with prior observations.¹⁴ To test whether NAA's effect on phyllid outgrowth is permanent, we grew gametophytes on NAA for 3 weeks, then removed them, and monitored recovery (Figure S3). In 18 of 21 samples, apices resumed phyllid production, indicating reversible inhibition and likely preserved daughter cell identity specification. Thus, NAA treatment preserves apical cell division properties, including spiral rotation of the division plane, confirming that spiral phyllotaxy arises from these rotating divisions.

To quantitatively analyze apical cell growth, shape, and division, we used MorphoGraphX (https://morphographx.org/)²⁵,²⁶ to extract curved 2.5D surfaces from confocal images, segment them into cells, and track lineages across time points (Figures 1D and 1E). The new division wall consistently avoided the wall shared with the oldest daughter cell. We measured the angle between the new division plane and the previous division wall (Figure 1F), finding an average divergence angle of 126.9° (SD 9.0°, *n* = 19) (Figure 1G). This is lower than the 135° previously reported for *P. patens* from mature phyllids.⁸ Modeling the apical cell surface in 2D as an equilateral triangle predicts a 120° divergence angle, so the observed value reflects an average ~6.9° tilt

in the phyllotactic spiral direction. This suggests post-division asymmetry in the apical cell, deviating from a perfect equilateral triangle.

### The shortest wall rule alone is not sufficient to orient the division plane

We tested whether the shortest wall through the cell centroid (the default division plane) can predict the observed apical cell division plane. As the shortest wall depends on cell geometry, we quantified the surface geometry of apical cells at each time point. In our dataset, apical cells showed variable shapes (Figure S2) and were generally smaller in surface area than neighboring daughter cells (Figures 2A and 2B). This size difference was already present immediately after division, indicating geometrically asymmetric division (Figure 2C). To assess surface shape anisotropy, we measured individual cell wall lengths. On average, the wall adjacent to the second youngest daughter (wall 2) was 14.5% longer (SD 16.5%, *n* = 60) than the other two walls (wall 1 and wall 3), which had similar lengths (Figure 2D). We then calculated cell length and width relative to the oldest wall (Figure 2E; STAR Methods). The average length was 17.0 µm (SD 3.7 µm, *n* = 60), compared with a mean width of 14.1 µm (SD 2.7 µm, *n* = 60), yielding an average length/width ratio of 1.22 (SD 0.23, *n* = 60) (Figures 2F–2H). Thus, the apical cell surface is not a perfect equilateral triangle but is typically elongated perpendicular to the oldest wall.

These findings suggest that apical cell asymmetry might orient the division plane via the shortest wall rule. If so, all cells with rotating division planes should show clear elongation perpendicular to the oldest wall. However, the distributions of length/width ratios (Figure 2F) and wall length (Figure 2D) ratios revealed substantial shape variation. Approximately 25% of cells had near-isotropic shapes (13/60 with a length/width ratio of 1.0–1.1), and ~15% were elongated parallel to the oldest wall (10/60 with a ratio of <1.0). Despite this variation, all apical cells in the dataset exhibited consistent rotating division patterns. We therefore conclude that apical cell shape asymmetry and the shortest wall rule alone cannot reliably explain the orientation of the rotating division plane.

### The principal direction of growth is not sufficient to orient the division plane

We hypothesized that the rotating division plane might result from anisotropic growth and align perpendicular to the primary growth direction, as proposed by Hofmeister.¹⁶ Using cell lineages from time-lapse data, we analyzed growth and growth directions of the apical cell. Overall growth (Figures 3A and 3C) was similar in the apical cell and its neighbors, though areal growth decreased as daughter cells aged, possibly due to NAA treatment inhibiting phyllid outgrowth. Growth was generally isotropic, with low growth anisotropy (mean 1.11, SD 0.09, *n* = 48; Figures 3B and 3D), indicating uniform expansion in all surface directions. Direct measurement of individual cell wall growth in the apical cell also revealed no consistent anisotropy (Figure 3E). Since we found no consistent anisotropic growth or a well-defined maximal growth direction, a rule positioning the new division plane perpendicular to the maximal growth direction cannot reliably generate spiral phyllotaxis. Moreover, the apical cell's asymmetric shape (Figures 2E–2H, S1, and S2)
<figure><img src="image_12.png" /><figcaption>**Figure 2.** The apical cell is elongated in one axis and grows isotropically on the surface</figcaption></figure>

(A) Representative surface segmentation of a *P. patens* shoot apex with a cell-area heatmap.

(B) Violin and box plots of surface area for the apical cell (A), youngest daughter (D1, magenta), second youngest (D2, green), and oldest daughter (D3, cyan).
Mean areas (µm²): apical = 261.1 µm² (SD 87.4), youngest = 334.4 µm² (SD 109.7), second youngest = 539.3 µm² (SD 210.0), and oldest = 733.1 µm² (SD 321.9).

(C) Violin and box plot of surface area ratio (daughter/apical cell) immediately after division.

(D) Violin and box plots of wall lengths for the apical cell's youngest (1, magenta), second youngest (2, green), and oldest wall (3, cyan). Mean lengths: wall 1 = 21.2 µm (SD 3.8), wall 2 = 24.0 µm (SD 4.7), and wall 3 = 21.0 µm (SD 4.5). Linear mixed-effects model showed a significant effect of wall type on length (*p* < 0.001); Tukey's HSD post-hoc tests confirmed that wall 2 was significantly longer than wall 1 and wall 3 (both *p* < 0.001). Data from 12 plants, 5 time points each (n = 60).

(E) Schematic of shape quantification: a regular grid (gray dashed lines) is aligned with the wall between the apical cell (A) and the oldest daughter (3) to measure length and width.

(F) Violin and box plot of apical cell length/width ratio. Mean = 1.22 (SD 0.23, n = 60).

(G and H) Violin and box plots of apical cell measurements of (G) length and (H) width.

Scale bar, 10 µm (A).

See also Figure S2.

combined with isotropic growth (Figures 3B and 3D) suggests that this asymmetry is not caused by differential growth during expansion but is instead established at division through placement of a new, asymmetric division plane.

## Analysis of apices in 3D confirms the shortest wall rule alone is not sufficient to orient the division plane

Since we could not identify the apical cell characteristics causing rotating division from surface quantification, we examined its 3D shape. Using a convolutional neural network (CNN) to enhance images for 3D processing²⁷,²⁸ (STAR Methods), we generated accurate 3D meshes of the apical cell and its neighbors. Although the apical cell approximates a tetrahedron in 3D, it is deflected outward where neighboring cell division walls attach (Figure S1; Video S1). Quantification of 3D dimensions (Figure 4; similar to surface measurements in Figure 2E) showed that the apical cell is typically larger in depth (mean 21.8 µm, SD 2.8 µm) than in width (mean 15.5 µm, SD 3.2 µm, *p* < 0.001) or

length (mean 19.0 µm, SD 3.8 µm, *p* < 0.001) (Figures 4A-4E). Unlike surface data, 3D volumetric growth of the apical cell exceeded that of neighbors (*p* < 0.001; Figure 4F), indicating stronger depth increase.

We then tested whether the 3D version of the shortest wall through the centroid rule could predict the division plane. Using MorphoGraphX division analysis tools, we generated minimal-area planes (local minima) through the centroid of 3D apical cell meshes (STAR Methods). Before division (Figures 5A and 5B), the smallest-area plane was frequently periclinal (parallel to the surface; dark blue in Figure 5B), a configuration never observed experimentally, while the actual division was often close to another local minimum plane. To quantify accuracy, we extracted the real division plane from the post-division time point (Figure 5C), approximated the pre-division centroid from the combined daughter and new apical cell mesh, and calculated the least-area plane through that centroid (Figure 5D). This predicted plane deviated on average by 27° (SD 22.8°, *n* =
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Figure 3. Areal growth and anisotropy do not differ between the apical cell and its daughter cells</figcaption></figure>

(A and B) Heatmaps of areal growth (A) and growth anisotropy (B) in representative apices over 24 h.

(C and D) Violin and box plots of areal growth (C) and growth anisotropy (D) over 12 h for the apical cell (A, black), youngest daughter (D1, magenta), second youngest daughter (D2, green), and oldest daughter (D3, cyan). Data from 12 plants, each measured at 4-time windows (total 48 observations).

(C) Mean areal extension of apical cell = 1.40 (SD 0.2). Linear mixed-effects model: all *p* > 0.39.

(D) Mean growth anisotropy of apical cell = 1.11 ± 0.09 (mean ± SD). Linear mixed-effects model with Tukey-adjusted pairwise comparisons: no significant differences (all *p* > 0.26).

(E) Violin and box plots of individual cell wall growth rates over 12 h. Linear mixed-effects model showed no significant difference in growth rate among the three walls (all Tukey-adjusted pairwise *p* > 0.3).

Scale bar, 10 µm (A and B).

See also Figure S1.

19) from the actual division wall (Figure 5F), indicating that the shortest wall rule alone does not sufficiently explain division plane orientation.

## Principal stress directions do not appear to orient the division plane

It has been proposed that cell division can follow the principal direction of stress¹⁸ and that stress can override the shortest wall rule in the meristem¹⁹ when it is strongly anisotropic. To test whether principal stress directions orient division in the apical cell, we analyzed 3D segmented apices from a time point just before division. Using a finite element method (FEM) simulation of 3D cells,²⁹ we pressurized the tissue under uniform turgor to reveal the stress pattern (Figure S4) (https://morphomechanx.org/). If stress directed the division plane, we would expect anisotropic stress aligned with the observed division plane. However, stress directions were mostly isotropic across the apex, and no consistent pattern emerged when compared with the actual division plane position in the subsequent time point (Figure S4). This indicates that stress alone is unlikely to be the primary determinant of division plane orientation in the apical cell.

## The shortest wall rule through a division point displaced from the centroid can explain division plane orientation

We observed that the apical cell consistently had a smaller volume than the daughter cell after division (Figure 5G), despite occasional cases where apical cell area was larger on the surface (Figure 2C). This asymmetry was reflected in an average volume ratio of daughter cell to new apical cell of 1.67 (SD 0.31, *n* = 19). The volume difference suggested that the division wall does not pass through the cell centroid. Measurements confirmed that the centroid of the actual division plane was displaced from the

centroid of the combined pre-division cell by an average of 2.21 µm (SD 1.13 µm, *n* = 19; Figure 5H). We therefore tested whether a shortest wall plane passing through this displaced division point could predict the division plane. Using the plane of least area that passes through the centroid of the actual division plane (an estimate of displacement; Figure 5E), we found it to be a strong predictor, deviating from the actual division plane by only 7° on average (SD 3.4°, *n* = 19; Figure 5F). In summary, analysis of 3D segmented meshes of the apical cell and its neighbors indicates that the apical cell divides according to a plane of minimal area whose center is displaced from the cell centroid.

## A 3D simulation model of moss phyllotaxis identifies rules defining division orientation in the apical cell

To identify division rules that generate spiraling division planes, we developed a 3D simulation of the growing shoot apex using uniform isotropic growth to match experimental observations in the apical cell and immediate neighbors (Figures 3A–3D). Simulations started from a tetrahedral apical cell with three daughter cells created by spiral divisions at prescribed angles (STAR Methods), featuring a curved top surface approximating the apex (Figures 6A and 6B). The apical cell divided upon reaching a threshold volume, producing a phyllid primordium and a new apical cell. Only division plane placement rules varied between runs. We measured average divergence angle and apical/daughter volume ratio over 20 divisions, targeting biological values of 126.9° for the angle (Figure 1G) and 1.67 for the volume ratio (Figure 5G).

We tested division hypotheses as rules. Rule 1 (minimal-area plane through centroid) produced periclinal divisions due to depth-elongated cell shape (Figure 6C; cf. Figures 4C and 4D), but these were never observed experimentally. Adding rule 2 (division plane must be anticlinal, passing through the outer face),
<figure><img src="image_15.png" /><figcaption>Figure 4. The apical cell is elongated in depth more than in length or width</figcaption></figure>

(A–C) Violin and boxplots of apical cell geometry from 3D segmentation: length (A), width (B), and depth (C).

(D) Depth/length ratio of the apical cell in 3D.

(E) Length/width ratio of the apical cell in 3D, comparable to 2D results (Figure 2F). n = 60.

(F) Violin and box plots of volumetric growth (volume ratio) for the apical cell (A, black), youngest daughter (D1, magenta), second youngest (D2, green), and oldest (D3, cyan). Using a linear mixed-effects model, D1, D2, and D3 showed significantly reduced volume growth compared with the apical cell (*p* < 0.0001 for all), with no significant differences among D1, D2, and D3 (*p* > 0.51). n = 48.

See also Figure S1.

equivalent to the shortest wall through the surface, still yielded unstable phyllotaxis with direction switches (Figure 6D; Video S2). Experimental data showed that new walls were roughly parallel to, and never intersected, the oldest wall. We therefore introduced rule 3 (new plane cannot intersect the oldest wall), which prevented periclinal divisions and rendered rule 2 redundant. Combining rule 1 and rule 3 generated stable spiral divisions (Figure 6E; Video S3), similar to the Couturier et al.¹⁵ model with minimum energy plus intersection with most recent daughter cell. However, it produced a divergence angle of 119.9° and volume ratio of 1.31, insufficient to match experimental data.

## Modeling displacement of the division wall away from older walls generates spiral phyllotaxis

Live imaging revealed new wall centroids were displaced ~2 µm from the cell centroid away from the oldest wall (Figure 5H). We thus added rule 4 (division point displaced from centroid). Displacing away from the oldest wall (making rule 3 redundant), along with rules 1, 2, and 4, produced stable spirals (120°) across distances up to 4 µm (Data S1A). A 1.25 µm displacement best fit the volume ratio data (119.9°, ratio 1.69; Figure 6F; Videos S4 and S5; Data S1A) and tolerated substantial noise (≤120% of displacement; Data S1B) and initial shape variations (Data S1C) but failed to generate the characteristic tilt required to yield angles >120°. Displacing away from both the oldest and second-oldest walls (63 combinations, total displacements 0.26–3.37 µm) gave >120° angles in 92% of cases (max. 124.3°), with 8% unstable (Data S1D). Equal displacements were stable, although more sensitive to noise. Among 10 values with an angle >123° and a ratio <2.1, the most robust was 2.75 µm (oldest) + 2.5 µm (second oldest): divergence 123.5°, ratio 1.87, average total displacement 2.58 µm, noise tolerance up to 166% (Figure 6G; Videos S6 and S7; Data S1E), and robust to initial shape scaling (0.6–1.5; Data S1F). In summary, simulations indicate that apical division plane positioning is governed by three rules: minimal-area plane (rule 1), anticlinal orientation passing through the outer face (rule 2), and division point

displaced away from the two oldest walls (rule 4). These rules are sufficient to generate the spiral phyllotaxis observed in *P. patens*.

## A 2D model shows the division rules converge to a fixed shape

In order to understand how division rules interact with growth and geometry to create consistent apical cell shapes, we developed a 2D triangular model of the apical cell. Analogous to the 3D simulation, the cell divides upon reaching a threshold area. Note that rule 2 (must pass through the outer face) does not apply in 2D. Using only rule 1 (shortest wall through the centroid) with different input shapes, the model initially divides opposite the vertex with the smallest angle and rapidly converges from any starting shape to an isosceles triangle, with successive divisions adopting the same orientation and becoming parallel (Figure S5A). When rule 1 is combined with rule 3 (not intersecting the oldest wall), or when rule 1 is used with displacement of the division point away from the oldest wall (rule 4), the model makes spiral divisions and converges to an equilateral triangle from any initial shape, with division angles approaching 120° (Figure S5B). Applying rule 1 while displacing the division point away from both older walls introduces a tilt into the division angle (Figure S5C). Unlike the 3D model, the 2D model can generate division angles that reach or exceed the 126.9° observed in the dataset, although this occurs at the expense of producing unrealistic area ratios between daughter and apical cells. Results mirroring the parameter exploration and robustness analysis of the 3D model are provided in Data S1G–S1L. Note that displacement distances are not directly comparable, as displacements in three dimensions include movement along the vertical axis.

## DISCUSSION

*Physcomitrium patens* exhibits spiral phyllotaxis in its phyllid arrangement, with a 135° divergence angle,⁸ close to the golden angle (137.5°) and similar to that in *Arabidopsis*.³⁰ Live imaging supports the view that phyllid positioning results
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Figure 5. The apical cell divides along the shortest wall, displaced from the cell centroid</figcaption></figure>

(A and B) 3D division analysis revealing local minima in division wall area through the cell centroid. Planes colored by area; note the perpendicular division plane is the global minimum.

(C-E) Analysis of 3D-segmented, combined apical plus daughter cell post-division. Asterisk (*) = apical cell, gray = cell outlines, red sphere = geometric centroid of combined apical and daughter cells, cyan = observed division plane, and yellow = predicted division plane.

(C) Best-fit plane to the observed division wall (note that it does not pass through the cell centroid).

(D) Shortest predicted plane through the geometric centroid.

(E) Shortest predicted plane through the centroid of the actual division plane.

(F) Violin/box plots showing angle difference between actual division plane and minimal-area plane through (left) cell centroid and (right) centroid of actual division plane.

(G) Violin/box plot of volume ratio (daughter cell: new apical cell) in apical cell divisions.

(H) Violin/box plot of division plane displacement from centroid (distance from division plane center to centroid of combined apical plus daughter cells).

(F-H) *n* = 19. Scale bars, 5 µm. See also Figure S4 and Data S1.

from successive spiral-oriented divisions of the apical cell, with each daughter cell forming a phyllid.⁷,³¹ This places new phyllids maximally distant from previous ones. In NAA-treated apices, the average divergence angle between successive daughter cells was 126.9° (Figure 1G), lower than the 135° seen in mature gametophores, suggesting post-initiation changes due to stem growth or twisting. By contrast, *Arabidopsis* shows minimal post-initiation shifts in organ divergence angle.³²

Geometric and developmental quantification of the apical cell and neighbors (surface: Figures 2 and 3; 3-dimensional: Figures 4 and 5) and FEM simulation (Figure S4) revealed that neither growth directions nor stress patterns direct division. Instead, cell geometry and a displaced division center can explain division orientation. Modeling demonstrated that displacement away from the oldest wall generates spiral divisions (Figure 6F), while an additional smaller displacement

from the second-oldest wall shifts the angle closer to the observed value (Figure 6G). Thus, spiral phyllotaxis in *P. patens* likely arises from the shortest-wall rule combined with displacement from the two oldest walls, a mechanism allowing single-apical-cell plants to form phyllotactic patterns without the spacing mechanisms used in multicellular apices. 3D modeling confirmed that these rules reliably produce spiral phyllotaxis and are robust to parameter variation, nuclear position noise, and initial cell shape. In our simpler 2D model, the pure shortest-wall rule converges to an isosceles triangle (a 1D subset of triangle shape space), allowing drift under perturbation (Figure S5D). Adding displacement from the oldest wall or from both older walls (rule 4) yields single attracting fixed points (Figures S5E-S5F), so cell shape returns to the fixed point after small perturbations, explaining the observed rarity of phyllotaxis direction changes.

Plant cell division orientation often follows minimal-area division through the centroid, producing equal-sized daughters, a rule observed in early *Arabidopsis* embryos until the 8-cell stage.¹¹,²² Later divisions (8 to 16 cells) deviate, showing geometric asymmetry with outer daughters over two times inner volume.¹³ Similarly, *P. patens* apical divisions are asymmetric in fate and geometry, with daughters averaging 1.67-fold the apical cell volume (Figure 5G). Because the division plane passes
<figure><img src="image_18.png" /><figcaption>Figure 6. 3D growth model of moss phyllostaxis</figcaption></figure>

(A) Whole and exploded views of the initial template: apical cell (gray), youngest daughter (magenta), second youngest (green), and oldest (cyan).

(B) Side view at 10 cells showing apical curvature. Simulation rules as in (G).

(C–G) Simulations of apical cell (gray) division with daughter cells colored on a gradient of red to blue from youngest to oldest. Misplaced division walls marked with a white asterisk (*).

(C) Rule 1 (minimal area division plane) gives periclinal division (*). Bottom row: side view with oldest daughter removed.

(D) Rule 1 (shortest wall) plus rule 2 (plane must intersect outer surface) gives spiral direction switch (*).

(E) Rules 1 plus 3 (division plane avoids oldest wall) give ~120° angles.

(F) Rules 1 + 2 + 4 (centroid of division plane displaced 1.25 µm from oldest wall) give ~120° angles and improved daughter/apical cell volume ratio.

(G) Rules 1 + 2 + 4 (division point displaced 2.75 µm from oldest wall and 2.5 µm from second-oldest wall) gives tilted division, angle ~123.5°. Scale bars, 20 µm (A and B).

See also Figure S5, Data S1, and Videos S2, S3, S4, S5, S6, and S7.

through the pre-division nucleus, this implies significant nuclear displacement from the centroid.

Single-celled apical meristems occur across bryophytes, yet some lack spiral phyllotaxis.¹⁵ Simulations with single displacement reliably yield ~120° (Figure 6F; Data S1A–S1C), recapitulating tristichous patterns in *Fontinalis antipyretica*, which has a similar apical cell shape to *P. patens*.⁸ Distichous patterns in *Fontinalis taxifolius* may arise from its lenticular apical cell.³¹,³³ These systems also maximize primordium spacing; however, it is an open question whether they rely on geometry plus division-plane displacement, as in *P. patens*. An additional question is how shoot tip shape is coordinated to maintain the apical cell centrally. Our model uses small global shifts (<0.05 µm) per step to prevent drift. One hypothesis is that the apical cell also organizes the growth field to preserve tip shape to counteract noise and displacement from the center due to division.

In flowering plants, auxin initiates primordia via polar transport, which creates spaced concentration peaks.³,⁵,³²,³⁴ Such transport-feedback spacing is hard to envision in a single-cell apex. In *P. patens*, phyllotaxis emerges directly from the spiral division sequence of the apical cell, which acts as a self-spacing mechanism. The new division angle places primordia maximally distant from existing ones, paralleling, yet mechanistically differing from, auxin-based spacing in higher plants. Although auxin regulates shoot apex function in *P. patens*,³¹ its role in phyllotactic patterning is unknown.

Spiral division patterns of stomatal divisions in some eudicots have remarkable resemblance to the *P. patens* apex (see Figure 2G in Rudall³⁵). In the *Arabidopsis* stomatal lineage, spiral divisions are thought to be controlled by nuclear displacement via BREAKING OF ASYMMETRY IN THE STOMATAL LINEAGE (BASL) that localizes to cell corners and displaces the nucleus/division wall.³⁶³⁷⁻³⁹ Note that the role of BASL in the stomatal lineage varies among eudicots.⁴⁰ Models for how cells might locate the shortest wall have been proposed, hypothesizing that cytoplasmic strands of actin and/or microtubules connect the nucleus to the cortical microtubule network. Based on a minimal energy principle, this would center the nucleus and find the shortest wall.⁴¹ BASL may disrupt these connections. Our finding that displacement from two walls gives the best fit to experimental data (Figure 6G) suggests that such a localization of a protein in *P. patens* might straddle the cell corner. Although no BASL homolog exists in *P. patens*, two BREVIS RADIX (BRX)-like proteins are present. In *Arabidopsis*, BRX co-localizes with BASL and is required for correct division plane orientation.⁴² Nuclear displacement via a polarity protein such as BRX could generate asymmetric apical divisions in *P. patens*. Possible localization mechanisms include cell-autonomous age-dependent wall signals that deplete over time or repelling factors from differentiating daughter cells diffusing through walls.

In conclusion, we propose that division plane displacement away from older walls, combined with a shortest-wall division rule, suffices to explain the spiral division sequence in the
# Current Biology
Article

<figure><img src="image_19.png" /></figure>

P. patens apical cell, producing spiral phyllotaxis. Nuclear displacement away from older daughters resembles inhibition models for multicellular meristems, a developmental principle in P. patens that parallels, yet diverges from, the mechanism in flowering plants.

## RESOURCE AVAILABILITY

### Lead contact

Requests for further information and resources should be directed to the lead contact, Richard Smith (richard.smith@jic.ac.uk).

### Materials availability

This study did not generate new materials.

### Data and code availability

* Confocal images and MorphoGraphX meshes are available at http://www.doi.org/10.5281/zenodo.18197802

* Code and simulation models are available at http://www.doi.org/10.5281/zenodo.18197802

* Any additional information required to run the simulations or analyze the data is available from the lead contact upon request.

## ACKNOWLEDGMENTS

We thank Miltos Tsiantis, Jill Harrison, and Michael Scanlon for useful discussions and Rhea Stinger for cryo-SEM expertise. Research reported in this publication was supported by the National Institute of General Medical Sciences of the National Institutes of Health under award number R01GM134037 and National Science Foundation award number MCB-2203275 to A.H.K.R.; the Human Frontiers Science Program grant RPG0067/2021 to Y.C., T.V., and R.S.S.; and a Biological Sciences Research Council Institute Strategic Program grant BB/X01102X/1 to the John Innes Centre. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health or other funders.

## AUTHOR CONTRIBUTIONS

Conceptualization, J.C., A.H.K.R., and R.S.S.; investigation, J.C., S.S., B.L., R.K.-B., and L.M.; data analysis, J.C., S.S., B.L., and R.K.-B.; supervision, T.V., Y.C., A.H.K.R., and R.S.S.; funding acquisition, T.C., Y.C., A.H.K.R., and R.S.S.; writing and revision, all authors.

## DECLARATION OF INTERESTS

The authors declare no competing interests.

## DECLARATION OF GENERATIVE AI AND AI-ASSISTED TECHNOLOGIES IN THE WRITING PROCESS

During the final preparation of this work, Grok was used to reduce the word count to comply with journal limits. Grok was instructed to reduce the word count without removing any references or scientific content. During the use of this tool, all changes were subsequently verified for accuracy and edited accordingly. The authors take full responsibility for the content of the published article.

## STAR★METHODS

Detailed methods are provided in the online version of this paper and include the following:

* KEY RESOURCES TABLE

* EXPERIMENTAL MODEL AND STUDY PARTICIPANT DETAILS

* Moss culture and hormone treatment

* METHOD DETAILS

* Confocal Microscopy

* QUANTIFICATION AND STATISTICAL ANALYSIS

* Statistics

* ADDITIONAL RESOURCES

## SUPPLEMENTAL INFORMATION

Supplemental information can be found online at https://doi.org/10.1016/j.cub.2026.01.059.

Received: July 29, 2025

Revised: November 17, 2025

Accepted: January 26, 2026

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<figure><img src="image_20.png" /></figure>

## STAR★METHODS

### KEY RESOURCES TABLE

<table><thead><tr><th>REAGENT or RESOURCE</th><th>SOURCE</th><th>IDENTIFIER</th></tr></thead><tbody><tr><td colspan="3">Chemicals, peptides, and recombinant proteins</td></tr><tr><td>Propidium Iodide</td><td>Sigma-Aldrich</td><td>P4170</td></tr><tr><td>1-Naphthylacetic acid</td><td>Sigma-Aldrich</td><td>N1641</td></tr><tr><td colspan="3">Deposited data</td></tr><tr><td>Confocal stacks (Cammarata 2025 Stacks)</td><td>This paper</td><td>www.doi.org/10.5281/zenodo.18197802</td></tr><tr><td>Apex meshes (Cammarata 2025 Meshes)</td><td>This paper</td><td>www.doi.org/10.5281/zenodo.18197802</td></tr><tr><td>Simulation software (MorphoMechanX) and models</td><td>This paper</td><td>www.doi.org/10.5281/zenodo.18197802</td></tr><tr><td colspan="3">Experimental models: Organisms/strains</td></tr><tr><td>Physcomitrium patens</td><td>Laboratory of Magdalena Bezanilla</td><td>GRANSDEN</td></tr><tr><td colspan="3">Software and algorithms</td></tr><tr><td>MorphoGraphX</td><td>Barbier de Reuille et al.<sup>25</sup>; Strauss et al.<sup>26</sup></td><td>www.MorphoGraphX.org</td></tr><tr><td>MorphoMechanX</td><td>N/A</td><td>https://MorphoMechanX.org</td></tr><tr><td>R</td><td>N/A</td><td>https://www.r-project.org/</td></tr><tr><td>Pressure model (Cammarata2025PhyscoPressure)</td><td>This paper</td><td>http://www.doi.org/10.5281/zenodo.18197802</td></tr><tr><td>3D division model (Camarata2025PhyscoApex)</td><td>This paper</td><td>http://www.doi.org/10.5281/zenodo.18197802</td></tr><tr><td>2D division model (Camarata2025 TwoDimensionalModel)</td><td>This paper</td><td>http://www.doi.org/10.5281/zenodo.18197802</td></tr><tr><td colspan="3">Other</td></tr><tr><td>Zeiss LSM-710 confocal microscope</td><td>Zeiss</td><td>N/A</td></tr><tr><td>FEI Nova NanoSEM 450 equipped with Gatan cryo-system</td><td>FEI</td><td>N/A</td></tr></tbody></table>

## EXPERIMENTAL MODEL AND STUDY PARTICIPANT DETAILS

### Moss culture and hormone treatment

All experiments and lines were of the Gransden strain of Physcomitrium patens (previously Physcomitrella patens). Small tufts of growing moss filaments were placed on agar BCD plates and grown for three to five weeks. All moss was grown under continuous light at 25 °C. NAA stocks were diluted to 5 µM in BCD media and plants were grown continuously in the presence of the hormone.

## METHOD DETAILS

### Confocal Microscopy

Moss shoots for imaging were dissected from three-to-five week old colonies grown on minimal media (BCD) with 5µM NAA. Colonies were flooded with water prior to dissection and tapped vigorously to liberate air bubbles. Dissected shoots were placed on a slide with a well made of vacuum grease and filled with 5 µg/ml propidium iodide (PI). Slides were then sealed with a coverslip and incubated for at least fifteen minutes before imaging. All confocal imaging was conducted using a Zeiss LSM-710 laser scanning microscope. PI-stained tissues were imaged using a 514 nm laser for excitation. Emission wavelengths between 566 and 650 nm were collected. Samples were imaged every 12 hours for 48 hours.

### Image analysis in 2.5D

Surface segmentations (Figures 1D and 1E) were created from the confocal images using MorphoGraphX following the standard pipeline (see MorphoGraphX user guide for details). The apex surface was extracted from the blurred raw data, and a curved surface mesh was generated with the cell wall signal projected onto the mesh. For samples where the apical cell was obscured by a growing hair cell, these were manually removed via voxel editing before extracting the surface mesh. The mesh was then segmented into cells. Consecutive time points of the same sample were manually lineage tracked. Cell type labels for the apical cell and its progeny were assigned to facilitate the downstream data analysis. For the surface mesh analysis, we computed cell sizes, area extension and principal directions of growth (PDGs) (Figures 2A, 3A, and 3B). Following the MorphoGraphX analysis we exported the quantification to CSV files to generate plots and perform statistics using R and the packages ggplot2.
For the surface wall length analysis (Figure 2D), we smoothed the wall vertices to remove noise, and the previously assigned cell type labels were used to merge the daughter cells of the apical cells to obtain a single wall on each side of the apical cell. Finally, we exported the wall length and growth rates to CSV files for analysis in R.

For the surface shape analysis (Figures 2E-2H), the opposite wall direction was determined by manually placing a Bezier grid parallel to the apical cell surface aligned with the junctions between the apical cell and its oldest neighboring daughter cell. Although this was manually placed, bias was minimal as the junction points are well defined. The grid's directions were used to compute custom directions for the apical cell so that the X direction was orthogonal to the next division wall. To obtain the directional cell sizes, we computed a principal component analysis (PCA) on the apical cell's mesh triangles to obtain the cell's main axis. Next, we computed the size components towards and parallel to the opposite wall using the custom directions from the first step to compute the shape anisotropy with respect to the next expected division to obtain the cell length and width.

### 3D segmentation

For 3D segmentations we used the CNN pipeline in MorphoGraphX for some samples to improve segmentation quality. The 3D shape analysis (Figures 4A-4E) was done as described above for the surface segmentations. Additionally, the cell depth was quantified.

### 3D division plane predictions

For the 3D division analysis (Figure 5) we used the division analysis tools in MorphoGraphX.²⁶ Dividing planes through the cell centroid (or centroid of the actual division plane) were sampled over a covering range of angles, and the plane of minimal area calculated. The actual division plane was calculated after division with the best planar approximation to the shared wall between the cells. Analysis was performed on the single apical cell before division, and on the pair of daughter cells after division considering the merged pair as a single cell. In the latter case, the angular difference between the minimal-area plane and the actual plane was calculated (Figures 5D-5F). Local minima before division (Figures 5A and 5B) were calculated with the cell division analysis tools in MorphoDynamX (https://morphographx.org/).

### Stress visualization model

As stresses cannot be directly measured in vivo, we used a finite element method (FEM) model to infer the stress induced by turgor pressure on their 3D shape. Three of the apices used for quantification were loaded into a MorphoMechanX²⁹ model adapted from Sapala et al.⁴³ An isotropic material of 100 MPa with a Poisson's ratio of 0.3 was used, and a turgor pressure of 0.5 MPa. Total stress (trace) was visualized in a heat map with the principal directions visualized as lines that vary in length depending on the magnitude of the stress in that direction (Figure S4).

### 3D Model of growth and apical cell division

The model was developed in the MorphoDynamX modelling framework (https://morphographx.org/). A 3D growth field for the *P. patens* shoot apex was built by extending the 2.5D surface shoot apex model of Smith et al.⁵ to full 3D by including interior points. This model uses a surface of revolution of a predefined contour shape, which in this case was chosen to match the shape of the *P. patens* shoot apex. Cells were represented as polyhedra with triangular faces that lay in the surface of revolution (on the surface of the apex) or beneath it (inside the apex). Points defining the vertices of the triangles have 3D positions in Euclidean space and additionally represented in intrinsic coordinates in the model. In the previous 2.5D model³ the longitudinal coordinate represented the distance along the surface from the tip, and the angle of the vertex with respect to the central axis. In our 3D extension, an additional coordinate was added to represent the relative distance of the point from the surface to the central axis. As with the 2D apex, this allowed growth to be implemented by changing the longitudinal intrinsic coordinate and recalculating the 3D position. Points were grown uniformly in the longitudinal coordinate, that is the rate of expansion of any longitudinal section is relative to its length; this means that points farther from the tip of the apex move away faster. This surface growth is interpolated to the interior of the apex by making planar sections perpendicular to the axis of rotation remain planar under growth. For the cells to maintain a polyhedral approximation of the apex surface even under growth, the triangular faces were subdivided when they become too large, with new points moved to the surface. Over time with successive divisions the apical cell will drift away from the center of the apex. In order to enable the simulation to continue for many divisions, we implemented a slight global shift in position of all points at each time step ($\langle= 0.05 \ \mu m\rangle$) to recenter the apical cell.

Cell division was performed when the apical cell reaches a threshold size. Each cell division cuts the polyhedron representing the cell in two through a planar division wall. All of the triangular facets of the cell are tested for intersection with the division plane. Those that intersect are divided in two along the plane and any non-triangular part is again divided into triangles. The new edges thus created become the boundary of the new dividing wall, which is triangulated using the Triangle library.⁴⁴ These interior vertices are transported and subdivided under growth and model the 3D change in the cells' shape over time.

The initial template (Figure 6A) was generated by starting with a single large cell and repeatedly dividing it along prescribed planes. These planes passed through a point 8 microns below the tip of the apex and had a slope of 4:1. Successive planes were rotated about the axis of rotation by a fixed phyllotactic angle of 120 degrees. To test our division rules, we divided the apical cell by the wall of least area passing through a central point, possibly with the additional constraints. The possible constraints were that the division wall must avoid the cell face shared with the oldest daughter cell, or that it must divide the surface face of the apical cell. In some scenarios the central point was the centroid of the apical cell, and in other scenarios it was displaced from the centroid away from specified daughter cell shared walls (oldest, 2^nd^ oldest). For each division, 500 candidate walls were tested with approximate uniform coverage. The cell was then divided along the wall of minimal area that also satisfied any required constraints.
## 2D Model of apical cell division

In two dimensions, an apical cell is modelled as a planar triangle, with the edges distinguished as the oldest, middle, and youngest. Division walls are straight lines; each division splits the apex into a triangle (the new apex) and a quadrilateral (the leaf initial). Division walls are chosen by finding the shortest line through some point; either the triangle centroid, or a division point shifted by some amount perpendicular to one or more walls. To find the division wall, we find the division point then calculate the lengths of 3143 division walls through that point (at an angular separation of 0.001 radian), choosing the shortest. In the new apex, the new division wall is tagged as the youngest, with the other two ageing up appropriately. To create a series of divisions, growth is simulated by scaling the new triangle so its area matches that of its parent at division; in this way, each division is applied to an apical cell of a constant area.

A triangle shape is placed in the phase space visualized in Figure S5 by translating the vertex shared by the oldest and youngest walls to the origin, then scaling and rotating so the vertex shared by the second-oldest and oldest walls is at the point (1,0). Then the position of the third vertex is the phase-space coordinate of the triangle. Triangles with counterclockwise phyllotaxis lie in the upper half of the coordinate plane, while triangles with clockwise phyllotaxis lie below the x-axis.

## QUANTIFICATION AND STATISTICAL ANALYSIS

### Statistics

For the analyses in Figures 2D, 3C, 3D, and 4F linear mixed-effects models were used to assess the effect of wall type or cell type on wall length, areal growth, growth anisotropy and volume growth (R-packages: lme4, ImerTest, and emmeans). Wall type or cell type was included as a fixed effect, and replicate (plant) as a random intercept to account for repeated measurements across five time points per plant (12 plants). Post-hoc pairwise comparisons between groups were performed using Tukey's HSD. The full dataset included 60 observations, with 48 observations used for time-window-specific variables such as areal growth and growth anisotropy. For Figure 2D the analysis revealed a significant main effect of wall number on length (*p* < 0.001). Post-hoc pairwise comparisons using Tukey's HSD indicated significant differences between the lengths of wall 1 vs wall 2 and wall 2 vs wall 3 (*p* < 0.001), but no significant difference between the lengths of wall 1 vs wall 3 (*p* = 0.77). For Figures 3C and 3D, the analysis revealed no significant main effect of cell type on area extension (all *p* > 0.39) and growth anisotropy (all *p* > 0.26).

To assess differences between Length, Width and Depth of the apical cell (Figures 4A–4C) first a Shapiro-Wilk test was conducted to assess the normality of the three variables. For all three measures the test indicated a significant deviation from normality (*p* = 0.00696, *p* = 0.02071, *p* = 0.0028), suggesting that the distribution of the values is not consistent with a normal distribution. Further, a Wilcoxon signed-rank test was conducted to perform a pairwise comparisons. This non-parametric test was chosen due to the paired nature of the data and violation of the normality assumption. The test revealed statistically significant differences between all three variables: Width and Depth (*p* = 1.67 × 10⁻¹¹), Length and Depth (*p* = 3.16 × 10⁻⁸) and Length and Width (*p* = 5.54 × 10⁻¹¹).

For Figure 4F the analysis revealed a significant main effect of cell type on volume growth (*p* < 0.001). Post-hoc pairwise comparisons using Tukey's HSD indicated that apical cells had significantly higher volume growth rates than neighboring cells (*p* < 0.001). When separating neighboring cells into their age (D1 the youngest, D2 the middle and D3 the oldest) pairwise statistical differences were found between A and all neighboring cells (D1: *p* = 0.02; D2: *p* = 0.01; D3: *p* = 0.037). Differences between the neighboring cells (D1, D2 and D3) were not significant.

## ADDITIONAL RESOURCES

Data, code and software available at: 10.5281/zenodo.18197802