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# Control of the mitotic cleavage plane by local epithelial topology

^{1,}

^{2,}

^{4,}

^{5}James H. Veldhuis,

^{3}Boris Rubinstein,

^{2}Heather N. Cartwright,

^{2}Norbert Perrimon,

^{4}G. Wayne Brodland,

^{3}Radhika Nagpal,

^{5}and Matthew C. Gibson

^{2,}

^{6,}

^{*}

^{1}Program in Biophysics, Harvard University (Cambridge, MA 02138, USA)

^{2}Stowers Institute for Medical Research (Kansas City, MO 64110, USA)

^{3}Department of Civil and Environmental Engineering, University of Waterloo (Waterloo, ON N2L 3G1, Canada)

^{4}Department of Genetics, Howard Hughes Medical Institute, Harvard Medical School (Boston, MA 02115, USA)

^{5}School of Engineering & Applied Sciences, Harvard University (Cambridge, MA 02138, USA)

^{6}Department of Anatomy and Cell Biology, Kansas University Medical Center (Kansas City 64110, KS)

## SUMMARY

For nearly 150 years, it has been recognized that cell shape strongly influences the orientation of the mitotic cleavage plane (e.g. Hofmeister, 1863). However, we still understand little about the complex interplay between cell shape and cleavage plane orientation in epithelia, where polygonal cell geometries emerge from multiple factors, including cell packing, cell growth, and cell division itself. Here, using mechanical simulations, we show that the polygonal shapes of individual cells can systematically bias the long axis orientations of their adjacent mitotic neighbors. Strikingly, analysis of both animal epithelia and plant epidermis confirm a robust and nearly identical correlation between local cell topology and cleavage plane orientation *in vivo*. Using simple mathematics, we show that this effect derives from fundamental packing constraints. Our results suggest that local epithelial topology is a key determinant of cleavage plane orientation, and that cleavage plane bias may be a widespread property of polygonal cell sheets in plants and animals.

## INTRODUCTION

The active control of the mitotic cleavage plane is crucial to numerous processes, and the consequences of cleavage plane mis-orientation can be catastrophic, ranging from polycystic kidney disease and organ malformation to tumorogenesis (Baena-Lopez et al., 2005; Fischer et al., 2006; Gong et al., 2004; Quyn et al., 2010; Saburi et al., 2008). Although the control of cleavage plane orientation is usually understood from a molecular viewpoint (Buschmann et al., 2006; Fernandez-Minan et al., 2007; Johnston et al., 2009; Siller and Doe, 2009; Speicher et al., 2008; Thery et al., 2005; Traas et al., 1995; Vanstraelen et al., 2006; Walker et al., 2007; Wright et al., 2009), more than a century of observations show that mitotic cells in both plants and animals tend to divide orthogonal to their geometric long axis as a default mechanism (Gray et al., 2004; Hofmeister, 1863; O'Connell and Wang, 2000; Strauss et al., 2006). In plants, the geometric location of the division plane can be predicted by cytoskeletal structures (Kost and Chua, 2002; Palevitz, 1987; Pickett-Heaps and Northcote, 1966; Sinnott and Bloch, 1940), and biophysical experiments suggest that the cytoskeleton may be involved in the process of orienting the division plane as dictated by cell geometry (Flanders et al., 1990; Goodbody et al., 1991; Katsuta et al., 1990; Lloyd, 1991). Further, in animal cells, recent studies implicate the geometry of cell-matrix adhesions as a key determinant of cleavage plane orientation (Thery et al., 2007; Thery et al., 2005). Cell geometry is thus a critical determinant of cleavage plane orientation, at both the molecular and biophysical level.

While the regulation of mitotic cleavage plane orientation by geometric cues has been extensively probed in unicellular systems, far less attention has been given to adherent epithelial and epidermal layers. In this context, cell geometry does not exist in isolation, because cell shapes emerge from the combined effects of growth, mitosis, and cellular packing. *A priori*, this complex interplay of biological processes, and the diverse genetic programs that have evolved to control them in plants and animals, would appear to suggest a staggering range of possible cell geometries within an epithelium. However, spatial considerations impose powerful constraints on the shapes of cells in monolayer sheets, from the distribution of polygonal cell types (Rivier et al., 1995) to their neighbor correlations (Peshkin et al., 1991) and relative sizes (Rivier and Lissowski, 1982). Indeed, empirical investigation confirms that many monolayer cell sheets across the plant and animal kingdoms converge on a default equilibrium distribution of cellular shapes, with approximately 45% hexagons, 25% pentagons, and 20% heptagons (Gibson et al., 2006; Korn and Spalding, 1973; Lewis, 1928). Such clear conservation of cellular network architecture raises the question as to whether conserved cellular division mechanisms are responsible for generating such similar packing arrangements of cells, as numerous studies have proposed (Dubertret et al., 1998; Gibson et al., 2006; Korn and Spalding, 1973; Miri and Rivier, 2006; Patel et al., 2009). The strongest evidence to date that common mechanisms are used among plants and animals to generate conserved packing relationships can be found in the *mitotic shift*, wherein the distribution of mitotic cell shapes is shifted by a single polygon class to have a heptagonal mean, as opposed to a hexagonal mean as seen in interphase cells.

Here, we use computational modeling, experimental observation, and mathematical analysis to report that, as a default property, neighbor cell shape can strongly bias cleavage plane orientation in the monolayer cell sheets of both plants and animals. Intriguingly, we show that this bias increases the structural regularity of an epithelium by increasing the frequency of hexagons. Our analysis indicates that simultaneously, cleavage plane bias is also involved in specifying the mitotic shift. The mechanism through which cleavage plane bias accomplishes these effects is differential side-gaining, whereby dividing cells preferentially cleave their common interfaces with sub-hexagonal cells such as quadrilaterals, and avoid cleaving their common interfaces with super-hexagonal cells such as octagons. Together, our results suggest a common emergent mechanism in plants and animals for the control of tissue-level architecture by packing-mediated control of the mitotic cleavage plane.

## RESULTS

### The shape of a cell is predicted to be influenced by local topology

In epithelia, the tissue-level architecture at the apical junctions is a contiguous tiling of polygonal cell shapes (Figures 1A and B). This pattern can be described as a simple planar network wherein a cell’s number of neighbors (topology) is equivalent to its polygon class (Figure 1B’). To investigate the effect of polygonal cell packing on mitotic cell shape, and by extension, cleavage plane orientation, we tested whether a cell’s long axis is systematically influenced by the polygon class of neighboring cells.

To address this, we numerically solved for the minimal energy configuration of a local cellular neighborhood (Prusinkiewicz and Lindenmayer, 1990), defined to be a central mitotic cell (*M*) and its first-order polygonal neighbors. Geometrically, cells were idealized as polygonal prisms with constant height (Figure 1A). For relaxation, cell mechanics were modeled in terms a balance between edge-length tensions, described using ideal springs, and internal pressure, modeled as an ideal gas (Figure 1C). The central mitotic cell, *M*, was a heptagon, consistent with the fact that the average mitotic cell is seven-sided *in vivo* (Aegerter-Wilmsen et al., 2010; Gibson et al., 2006). Parameters were chosen to be uniform for every cell, and initial conditions were arbitrary (Figures 1D–F). Given these choices, the effect of local topology on the shape of the central cell was an emergent property of the relaxed mechanical network at equilibrium (Figures 1D–F; Figure S1; Extended Experimental Procedures).

To analyze the impact of local topology on the long axis of *M*, we replaced one neighbor hexagon with a single N-sided cell, *N*. Strikingly, inserting any non-hexagonal neighbor induced a clear long axis in *M*, with opposite orientation of the long axis for N<6 versus N>6 (Figures 1D–F; ;2A).2A). Specifically, the presence of quadrilateral or pentagonal neighbors induced a long axis parallel to the *NM* interface, while heptagons and octagons induced a long axis orthogonal to interface *NM*. These results suggest that in cell sheets, the orientation of a mitotic cell’s longest axis can be strongly influenced by the polygon class of a single neighboring cell. As a consequence of this effect, neighbor cells with fewer sides (such as quadrilaterals and pentagons) tend to lie along the shortest axis of M, which is the location of the presumed cleavage plane.

**The orientation of a cell’s short axis is predicted to correlate with its quadrilateral and pentagonal neighbors, and to anti-correlate with heptagonal and octagonal neighbors**

To test whether this effect was robust under more realistic conditions, we numerically relaxed heterogeneous local neighborhoods that were stochastically generated from the known polygonal cell shape distribution of the *Drosophila melanogaster* wing epithelium (Figure 2B) (Aegerter-Wilmsen et al., 2010; Gibson et al., 2006). Even under these conditions, more than 70% of quadrilateral neighbors were positioned on the central cell’s short axis, double the percentage expected by chance (Figure 2C). To quantify this relationship, we defined an acute angle, θ, with respect to the presumed cleavage plane along the central cell’s short axis (see Figure 2D). On average, as a function of increasing θ, the neighbor polygon class in direction θ increased monotonically (Figure 2E). Therefore, even in a heterogeneous context, the topology of a cellular neighborhood robustly and systematically influenced the orientation of the long axis in a central cell.

### Cleavage plane bias in the *Drosophila* wing disc

In both plants and animals, cells are thought to divide their long axis by forming a cleavage plane along the short axis of the cell (Hofmeister, 1863; Strauss et al., 2006). If a cell’s short axis consistently bisects its cellular neighbors having the fewest sides (Figure 2), then mitotic division planes should be disproportionately biased towards quadrilaterals and pentagons *in vivo*. To test this, we measured the correlation between neighbor cell polygon class and cleavage plane orientation in the *Drosophila* wing imaginal disc (Figure 3A). Here, cell division proceeds through a stereotyped process of cell rounding at the apical epithelial surface (Figures 3B–D;(Gibson et al., 2006)). To define the frequency with which different classes of polygonal neighbors were bisected by the cleavage plane, we examined 420 cells at the cytokinetic stage, which is the most stable and easily scored phase of mitosis (Figure 3E). For each case, we recorded the position of all primary neighboring polygons and computed the frequency with which each polygon class occupied the cleavage plane position (Figures 3F,G).

**In both plants and animals, a dividing cell’s cleavage plane correlates with its quadrilateral and pentagonal neighbors, and anti-correlates with heptagonal and octagonal neighbors**

If the orientation of cell division were random with respect to local topology, approximately 28.6% of any given polygon class would be expected to correlate with the cleavage plane (two randomly-chosen cells out of seven neighbors). However, in the wing disc, we found that more than 50% of quadrilaterals in the primary neighborhood occupied the division plane position (Figure 3H; *n*=46/83). Further, octagons were anti-correlated with the division plane, and occupied that position with less than 10% probability (*n*=6/77). As predicted by the mechanical model, this cleavage plane bias was monotone decreasing across all polygon types. We conclude that in the *Drosophila* wing disc, the polygonal topology of local neighborhoods strongly influences cleavage plane orientation in mitotic cells.

In order to test the assumption that *Drosophila* wing disc cells actually divide their longest axis, we next performed time-lapse analysis of proliferating *Drosophila* wing discs in *ex vivo* culture (see Movie S1; Experimental Procedures). For each of 198 mitotic cells (Figure 4A), we measured the geometric long axis orientation during both interphase (Figure 4A’, far left), and cytokinesis (Figure 4A’, far right). We found a strong tendency for cells to follow a long-axis division mechanism, although with moderate noise in the orientation (Figure 4B). This tendency to divide the longest axis correlated with the interphase geometry (Figure 4B), and increased with the cell’s interphase elongation ratio (the ratio of the long axis to the short axis; Extended Experimental Procedures). For example, for the 99 cells having an elongation ratio below the median value of 1.68, the average deviation from a long axis-division mechanism was about 33°; by contrast, for the 99 cells having an elongation ratio above the median value, the average deviation was about 21° (data not shown). This dependence on the relative axis lengths suggests that these cells might be able to adjust their spindle orientations to their newly acquired shapes following mechanical strain, as has been previously reported in cell culture and in vertebrate embryonic cells (Black and Vincent, 1988; Gray et al., 2004; O'Connell and Wang, 2000; Strauss et al., 2006).

To test whether deviation from the long axis division mechanism could explain the discrepancy between our cleavage plane bias predictions and the empirical measurements, we incorporated the measured deviation into our original model (Figure 4C; Extended Experimental Procedures). Interestingly, when the measured deviation was incorporated, the mechanical predictions were significantly improved (compare the red and black curves in Figure 4C), closely matching the empirically measured bias (Figure 4C, blue curve). Therefore, cleavage plane bias is likely to be robust to noise in the cleavage plane mechanism, and may be present even when cell divisions do not perfectly obey a long axis division scheme.

### Cleavage plane bias in plant epidermis

Because our original predictions were mechanically motivated (Figures 1 and and2),2), and are expected to persist even when there is moderate noise in the cleavage plane (Figure 4), we reasoned that cleavage plane bias should be equally likely to appear in plant tissues. To test this, we used data from FT Lewis’s classical study of cucumber epidermal cell topology (*Cucumis sativus*) to compute the probability with which an N-sided polygonal cell occupies the division plane of a mitotic neighbor (Extended Experimental Procedures; Lewis, 1928). Remarkably, in *Cucumis*, the cleavage plane bias was almost indistinguishable from that measured in the *Drosophila* wing disc (Figure 3H). We once again observed strong enrichment for 4-sided cells along the cleavage planes of mitotic cells, while 8-sided cells were underrepresented. In order to verify our inferences from Lewis’s data (1928), we also directly examined the relationship between local topology and cellular long axis orientation in the epidermis of *Cucumis* (Figure S3A). From fixed samples of cucumber epidermis, we studied a population of 501 epidermal cells having the same polygonal distribution as the original population of 500 mitotic cells studied by Lewis (1928). Cells were selected in a spatially constrained, impartial manner based solely on polygon class (Extended Experimental Procedures). We next tested whether a naïve long-axis division rule was sufficient to generate cleavage plane bias in *Cucumis*. Based on an ellipse of best fit to each cell’s geometry (Figure S3A; Extended Experimental Procedures), we were able to reproduce not only the cleavage plane enrichment observed in Lewis’s original data (Figure S3C), but also the inferred cleavage plane bias (Figure S3D). Taken together, our results suggest that cleavage plane bias occurs in polygonal cell sheets as an emergent effect of cell packing, independent of species-specific considerations.

### Cleavage plane bias and the topological constraints on cell geometry

The quantitative similarity of cleavage plane bias in plants, animals, and *in silico* suggests the underlying mechanism is geometric, rather than molecular. In fact, fundamental geometric constraints imposed by the internal angles of neighboring polygons are sufficient to explain this phenomenon. For illustration, consider the comparison between a tiling of three hexagons versus two hexagons and a square (Figures 5A and B). From elementary geometry, a square (N=4) has internal angles of 90°, while the internal angles of a hexagon (N=6) average 120° (for an N-sided polygon, average internal angles are 180° (N-2)/N). In the context of a contiguous layer, the presence of 90° internal angles within the square forces the internal angles of the adjacent hexagon to increase to 135° (Figure 5B). Intuitively, this deformation results in elongation of these hexagons parallel to the interface with the square, thus generating a cellular long axis.

The constraints imposed by the internal angles of one cell upon the long axis of its neighbor can be formalized for the arbitrary case of an N-sided cell, surrounded by N symmetric hexagonal neighbors (Figure 5C). Assume that a mitotic cell, *M*, is situated vertically above cell *N*, resulting in a horizontal interface *NM* of length L. In the simplest case, all side lengths, including L, are equal, and without loss of generality can be set to one. Further, the internal angles α_{N} and β_{M} can be computed as a function of N. Using simple trigonometry and exploiting the symmetric configuration of neighbors, we can solve for the ratio of the horizontal axis, *d _{m}*, to the vertical axis,

*h*, for the ellipse of best fit to cell

_{m}*M*(Figure 5C; Extended Experimental Procedures):

In this framework, the direction of *M*’s short axis (presumed cleavage plane) is described by the ratio *d _{m}:h_{m}*, which the above equation shows is determined by the N value (Figure 5D). Geometrically, the ratio

*d*varies with N because the length

_{m}:h_{m}*d*decreases for N>6 and increases for N<6 (Figure 5E). Consequently, when N>6 (

_{m}*d*<1),

_{m}:h_{m}*d*forms the short axis parallel to interface

_{m}*NM*. Conversely, if N<6 (

*d*>1), then

_{m}:h_{m}*h*forms the short axis, or presumed cleavage plane, in the direction of

_{m}*N*, perpendicular to the interface

*NM*.

### Cleavage plane bias is predicted to be robust to side length and cell size differences

Intuitively, differential side lengths of N-sided neighbors would also affect the short axis orientation of *M* (Figure 5F). To analyze the relative contributions of angular constraints versus side lengths, consider the more realistic case when the edge lengths are non-uniform (L≠1). Here, *d _{m}:h_{m}* depends on both N and L (Figure 5D and Extended Experimental Procedures):

For the simplified case when L=1, the direction of the short axis undergoes a 90° rotation (between horizontal and vertical) when *d _{m}:h_{m}* passes through the value 1, which corresponds to N=6 (

*red line*, Figure 5D). Changing the value of L changes the length

*d*(Figure 5F), and thus alters the N value at which this transition occurs (

_{m}*black lines*, Figure 5D). The long axis orientation of

*M*is thus determined by the interplay between the polygon class and apposed side length of each neighbor,

*N*. In the

*Drosophila*wing disc, the value of L fluctuates by about 40% on average (Table 1). Equation (2) predicts that a 40% deviation in L value would change the point of rotation by only a single N value, suggesting that cleavage plane bias should be noisy yet reproducible.

Supporting this analysis, cell size has a surprisingly weak influence compared to polygon class in our mechanical simulations (Figure S4). Consistent with our simulations, based on live imaging analysis of local neighborhoods surrounding dividing cells in the *Drosophila* wing disc epithelium, there was no discernable difference in average area for cells occupying the cleavage plane position (Figure S4D). We conclude that internal angle constraints linked to the polygon class of neighboring cells are likely the dominant cause of cleavage plane bias, with a lesser contribution from the effects of differential side lengths.

### Cleavage plane bias is predicted to alter global tissue topology

Numerous recent studies have used mathematical or computational approaches to understand the equilibrium topology of proliferating epithelia (Aegerter-Wilmsen et al., 2010; Cowan and Morris, 1988; Dubertret et al., 1998; Dubertret and Rivier, 1997; Gibson et al., 2006; Korn and Spalding, 1973; Miri and Rivier, 2006; Patel et al., 2009). Intuitively, cleavage plane bias must alter the topology of a cell sheet because it modulates the rates at which specific polygon classes gain sides due to neighbor cell mitoses. We therefore investigated the implications of cleavage plane bias for the distribution of polygonal cell shapes. We used two distinct computational simulations informed by the empirical division parameters (Figure S2A–C) to model global topology with and without cleavage plane bias (Figures 6 and S5–S6). For both simulation types, the cleavage plane bias values approximated those measured empirically (Figures S5F and S6H). Both an abstract, topological simulator using a Monte-Carlo framework based on topological weights (Figure 6A) (Patel et al., 2009) and a mechanical model of tissue growth based on long-axis divisions (Figure 6D) (Brodland and Veldhuis, 2002) confirmed that cleavage plane bias affects the frequency of hexagonal cells (Figures 6B,E). Moreover, the distribution of mitotic polygonal cells was severely disrupted in the absence of bias, resulting in decreased frequencies of heptagons and increased frequencies of octagons and nonagons (Figures 6C,F). Taken together, these results suggest that cleavage plane bias is required to achieve the normal equilibrium distribution of cell shapes.

## DISCUSSION

The results presented here raise several important questions. First, while our analysis provides a geometrical rationale for cleavage plane bias based on interphase polygon topology (Figure 5), we still cannot rule out the simultaneous action of molecular cues at the cell cortex. In metazoans, epithelial cells often undergo mitosis-induced deformation prior to cleavage (Figures 3C,C’; Figures 4A,A’) (Gibson et al., 2006; Thery and Bornens, 2008), and our live imaging results from *Drosophila* strongly suggest that a cellular long axis present in interphase can inform spindle orientation during mitosis (Figure 4). One intriguing possibility is that the interphase distribution of cell-cell contacts determines the localization of cortical cues important for spindle alignment, as has been previously reported (Thery et al., 2007; Thery et al., 2005).

For plant cells, by contrast, our results indicate that local cell packing influences, either directly or indirectly, the placement of the phragmosome and/or pre-prophase band (Pickett-Heaps and Northcote, 1966; Sinnott and Bloch, 1940). There are multiple ways in which this might be accomplished, potentially including stress or strain sensing mechanisms (Hamant et al., 2008; Lintilhac and Vesecky, 1984; Lynch and Lintilhac, 1997), or more simply, based on cytoskeletal mechanisms that are able to sense cell shape (Flanders et al., 1990; Goodbody et al., 1991; Katsuta et al., 1990). To conclude, in addition to our purely geometrical interpretation, our results are also consistent with a hypothesis that in both animals and in plants, local epithelial topology may coordinately specify both the cellular long axis and the distribution of cortical determinants of the eventual cleavage plane.

A second important question concerns the broader implications of cleavage plane bias for the emergence of cell shape. Previous studies of proliferating cell sheets in *Drosophila* and in *Cucumis* have shown that the distribution of mitotic cell shapes is shifted to have a heptagonal mean, as opposed to the hexagonal mean observed in the population of cells overall (Aegerter-Wilmsen et al., 2010; Gibson et al., 2006; Lewis, 1928). Our simulations (Figures 6A,D) suggest that the mitotic cell distribution is disrupted in the absence of cleavage plane bias (Figures 6C,F), which is consistent with the view that in both *Drosophila* and *Cucumis*, interphase cells passively gain additional sides as a consequence of neighbor cell divisions. This interpretation contrasts with the idea that the mitotic shift solely reflects modulation of the cell cycle by topology-dependent mechanical stress (Aegerter-Wilmsen et al., 2010). Moreover, cleavage plane bias is actually expected to synergize with the mitotic shift. By enriching for super-hexagonal cells in the mitotic cell population, which are entropically favored to neighbor sub-hexagonal cells (Peshkin et al., 1991), the mitotic shift intuitively must amplify the effects of cleavage plane bias.

In summary, by varying the orientation of cell division based on neighbor cell geometry, cells and tissues are able to achieve increased geometric regularity via a dynamic, topology-mediated feedback and control system. Precisely how the default geometric forces that bias cleavage plane orientation interact with other mechanisms of division plane control (Baena-Lopez et al., 2005; Gong et al., 2004; Li et al., 2009; Segalen et al., 2010; Siller et al., 2006; Willemsen et al., 2008) should be an important avenue for future research.

## Experimental Procedures

### Fly strains

To visualize the septate junctions, we used a *neuroglian-GFP* exon trap line, which was described in a previous study (Morin et al., 2001). To visualize the chromosomes in parallel, we generated a stock also carrying a Histone-2 RFP marker ((Schuh et al., 2007); Bloomington stock 23650).

### Wing disc sample preparation & imaging

Wing discs were dissected from wandering 3^{rd} instar larvae in Ringers’ solution, fixed in 4% paraformaldehyde in PBS, and then mounted in 70% glycerol/PBS. For live imaging, discs were carefully dissected and placed in a 50:50 mixture of Ringer’s solution (130mM NaCl, 5 mM KCl, 1.5 mM MgCl_{2}), and a second solution (adapted from (Aldaz et al., 2010)), consisting of 2% FBS (Gibco) and 0.5% Pen/Strep (Gibco; 5,000 units/mL penicillin; 5,000 µg/mL streptomycin) in Shields and Sang M3 Insect media (Sigma). Live discs were mounted between two pieces Scotch doublesided tape (3M). Air bubbles were added to the medium using an insulin syringe (BD Ultra-fine with a 30-gauge needle) to potentiate gas exchange. Wing discs were maintained in culture for up to four hours, and imaged at intervals of 15–30 seconds. All samples, live and fixed, were imaged on a Leica SP5 or Leica SP2 confocal microscope with a 63X glycerol or oil objective.

*Cucumis* sample preparation and imaging

Epidermis was collected from freshly gathered cucumbers approximately 10 cm in length and 2 cm in diameter (Red Ridge Farm, Odessa, MO). Epidermis was peeled in thin layers and fixed in 4% paraformaldehyde in 50 mM KPO_{4}, 5.0 mM EDTA and 0.2% Tween20 (pH 7) for at least 2 hours at room temperature (adapted from Gallagher and Smith (1999)). Tissue pieces were then washed 2–5 times in dH_{2}O, and incubated in 5 mg/mL Calcofluor White (Sigma) in PBS for at least 10 minutes before imaging. Images were collected using a Zeiss LSM 510 Meta with a 20x Plan-Apochromat objective, NA 0.8.

### Error bars

Unless otherwise specified, error bars refer to a single standard deviation. For the case of ratio distributions, we have reported an average value of the standard deviation. This was computed as follows: the data were randomized and broken into three sub-samples of equal size in order to compute an average value for the standard deviation, based on 1000 random shuffles of the data.

### Annotation of *Drosophila* wing disc cytokinetic figures in fixed preparations

A total of 420 cytokinetic figures and their 2946 cellular neighbors were scored by hand, in multiple focal planes to ensure accuracy of topological counts. Out of the 2946 neighbors, 840, or exactly two per cytokinetic figure, were designated as being in the division plane position. Cells were interpreted to be in the division plane position when they occupied the majority of the cytokinetic furrow. Due to the ambiguity of division ordering, cytokinetic figures adjacent to other cytokinetic figures were not considered for analysis.

### Annotation of fixed *Drosophila* wing disc epithelial cell sheets

Images of contiguous epithelial regions from *Drosophila* wing disc epithelia were annotated by hand using Microsoft Powerpoint. We used custom-built software to digitize the annotations for analysis in MATLAB. A total of three such cell sheets, containing 254, 195, and 233 cells, respectively, were analyzed to compute the effective L value (Figure 5C; Table 1), which is described in the text. See Extended Experimental Procedures for additional details.

### Live imaging analysis of mitosis in the *Drosophila* wing disc

From live movies, a total of 198 mitotic cells in the *Drosophila* wing disc epithelium were analyzed by hand using ImageJ. With the exception of cells located on compartment boundaries, every scoreable cell on the epithelium was used. To control for possible mechanical influences due to neighboring divisions, we did not consider dividing cells neighboring each other to be scoreable if they rounded up at the same time. Interphase geometry measurements were based on the earliest available time point (the first movie frame), except in rare cases when epithelial morphology obscured the cell in question, in which case a slightly later time point was used. The long axis orientation of each cell was computed using ImageJ, including curvature, based on manual input from the Polygon Selections tool. The identical procedure was used for each cell at later stages, including the eventual cytokinetic figure (see Figure 4A’ for an illustration). See the Extended Experimental Procedures for additional details.

### Analysis of *Cucumis* epidermal cell sheets

Images of contiguous regions of *Cucumis* epidermis were annotated by hand using ImageJ. Cell geometry was outlined using the Polygon Selections tool, with one node placed per tri-cellular junction, except in cases of very curved cellular edges, in which additional nodes were used to increase annotation accuracy. To visualize the ellipse of best fit to cell geometry, we used a custommade ImageJ macro. See the Extended Experimental Procedures for additional information.

### Algorithm for computing the minimal energy configuration for local cellular neighborhoods

We used a mechanical relaxation algorithm for cellular networks that has been previously described (Prusinkiewicz and Lindenmayer, 1990). For relaxation (Figure 1), cellular networks were modeled in terms of a balance between edge length tensions (described using ideal springs) and internal pressure (Figure S1). Relaxation was implemented in terms of a system of ordinary differential equations that were solved numerically using the ODE45 solver in MATLAB (Mathworks). See the Extended Experimental Procedures for additional information.

### Topological simulations of proliferation

Proliferation was simulated in terms of a network containing exclusively tri-cellular nodes, with wrapping boundary conditions. All division parameters, including division likelihoods of polygonal cells, the statistical partitioning of mother cell nodes, and the likelihoods of orienting the division plane in the direction of specific polygonal neighbor cell types, are matched to the empirically measured statistics for the *Drosophila* wing disc (see Figure S2A–C). The algorithmic details are described in the Extended Experimental Procedures.

### Finite element models of proliferating cell sheets

The FEM simulations (Brodland and Veldhuis, 2002; Chen and Brodland, 2000) model apical contractility, cell-cell adhesion, and all other forces along the cellular edge lengths in terms of a net, interfacial tension, γ, which is generated by rod-like finite elements. Proliferation is modeled in terms of long-axis divisions. Cell-cell rearrangements (T1 transitions) are permitted when cellular edge lengths shrink below a threshold value. See Figure S6 for a comparison between simulations in which T1 transitions are active, versus those for which they are inactive. Additional details are described in the Extended Experimental Procedures.

- Neighbor cell topology biases cleavage plane orientation in monolayer cell sheets.
- This “cleavage plane bias” is observed in both plants and animals.
- This effect can be explained by fundamental packing constraints.
- Cleavage plane bias influences global epithelial topology.

## ACKNOWLEDGEMENTS

The authors gratefully acknowledge support for this research from the Stowers Institute for Medical Research and the Burroughs Wellcome Fund (to M.C.G.), from the National Science Foundation (to R.N.), from the Natural Sciences and Engineering Research Council of Canada (to G.W.B) and the Howard Hughes Medical Institute (to N.P.). W.T.G. was supported in part by NIH/NIGMS Molecular Biophysics Training Grant #T32 GM008313 and CTC grant 1029 to MCG.

## Footnotes

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- Modeling and inferring cleavage patterns in proliferating epithelia.[PLoS Comput Biol. 2009]
*Patel AB, Gibson WT, Gibson MC, Nagpal R.**PLoS Comput Biol. 2009 Jun; 5(6):e1000412. Epub 2009 Jun 12.* - Mechanisms of regulating cell topology in proliferating epithelia: impact of division plane, mechanical forces, and cell memory.[PLoS One. 2012]
*Li Y, Naveed H, Kachalo S, Xu LX, Liang J.**PLoS One. 2012; 7(8):e43108. Epub 2012 Aug 17.* - Drosophila aPKC is required for mitotic spindle orientation during symmetric division of epithelial cells.[Development. 2012]
*Guilgur LG, Prudêncio P, Ferreira T, Pimenta-Marques AR, Martinho RG.**Development. 2012 Feb; 139(3):503-13.* - Cell topology, geometry, and morphogenesis in proliferating epithelia.[Curr Top Dev Biol. 2009]
*Gibson WT, Gibson MC.**Curr Top Dev Biol. 2009; 89:87-114.* - Spindle orientation in animal cell mitosis: roles of integrin in the control of spindle axis.[J Cell Physiol. 2007]
*Toyoshima F, Nishida E.**J Cell Physiol. 2007 Nov; 213(2):407-11.*

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- Control of the mitotic cleavage plane by local epithelial topologyControl of the mitotic cleavage plane by local epithelial topologyNIHPA Author Manuscripts. Feb 4, 2011; 144(3)427PMC

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