^{1}

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^{2}

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Conceived and designed the experiments: ABP MCG RN. Performed the experiments: ABP WTG. Analyzed the data: ABP WTG MCG RN. Wrote the paper: ABP MCG RN.

The authors have declared that no competing interests exist.

The regulation of cleavage plane orientation is one of the key mechanisms driving epithelial morphogenesis. Still, many aspects of the relationship between local cleavage patterns and tissue-level properties remain poorly understood. Here we develop a topological model that simulates the dynamics of a 2D proliferating epithelium from generation to generation, enabling the exploration of a wide variety of biologically plausible cleavage patterns. We investigate a spectrum of models that incorporate the spatial impact of neighboring cells and the temporal influence of parent cells on the choice of cleavage plane. Our findings show that cleavage patterns generate “signature” equilibrium distributions of polygonal cell shapes. These signatures enable the inference of local cleavage parameters such as neighbor impact, maternal influence, and division symmetry from global observations of the distribution of cell shape. Applying these insights to the proliferating epithelia of five diverse organisms, we find that strong division symmetry and moderate neighbor/maternal influence are required to reproduce the predominance of hexagonal cells and low variability in cell shape seen empirically. Furthermore, we present two distinct cleavage pattern models, one stochastic and one deterministic, that can reproduce the empirical distribution of cell shapes. Although the proliferating epithelia of the five diverse organisms show a highly conserved cell shape distribution, there are multiple plausible cleavage patterns that can generate this distribution, and experimental evidence suggests that indeed plants and fruitflies use distinct division mechanisms.

Cell division is one of the key mechanisms driving organismal growth and morphogenesis. Yet many aspects of the relationship between local cell division (how a cell chooses an orientation to divide) and global tissue architecture (e.g., regular versus irregular cells) remain poorly understood. We present a computational framework for studying topological networks that are created by cell division; this framework reveals how certain tissue statistics can be used to infer properties of the cell division model. Recently it has been observed that five diverse organisms show almost identical cell shape distributions in their proliferating epithelial tissues, yet how this conservation arises is not understood. Using our model we show that the low variation observed in nature requires a strong correlation between how neighboring cells divide and that although the statistics of plants and fruitflies are almost identical, it is likely that they have evolved distinct cell division methods.

The spatial and temporal regulation of cell shape and cell proliferation are key
mechanisms that direct tissue morphogenesis during development. Much of our
knowledge of tissue morphogenesis comes from the study of simple epithelial
monolayers, 2D planar sheets of strongly adhering cells in which division occurs in
the plane of the epithelium. The strong structural constraints and developmental
importance of epithelia have inspired a multitude of theoretical and computational
models since the early 20^{th} century

The topology of an epithelium is defined as the network of connectivity between cells
(

(A) Polygonal lattice approximation of a larval stage wing disc epithelium
from _{n}_{n} = A_{avg}
(n−2)/4_{avg}

For these reasons, topological models have been useful both experimentally and
theoretically in understanding proliferating epithelia

The predicted distribution

Despite much experimental and theoretical progress, previous models have limitations
that make it difficult to address these questions. The major difficulty lies in
modeling the neighborhood and lineage dependence in cleavage plane choice. For
example, our previous model encodes a mean-field approximation that ignores the
variability in the number of neighbors gained via the division of neighboring cells

Here we present a computational model of cell division that enables us to explore a much larger class of biologically plausible division models by directly simulating the topology of a proliferating epithelium from generation to generation. This includes division schemes with spatial and temporal dependence between neighboring cells and mother-daughter cells. Given a division model, we can compute the equilibrium distribution of polygonal cell shapes, along with other tissue-level topological properties. Our findings show that the fraction of hexagons and the variability in cell shape are both important global indicators of local division parameters, and we propose that it may be possible to infer these parameters from empirical data. Furthermore, we describe several division schemes that can reproduce with high accuracy the cell shape distribution seen in five diverse organisms. We use this modeling framework to formulate and explore some of the central theoretical and empirical questions regarding the local-to-global regulation of cell shape in proliferating epithelia.

The topology of an epithelial cell sheet can be described mathematically as a planar
network of trivalent vertices, edges, and faces. The vertices represent tricellular
junctions, the edges represent cell sides, and the faces represent the cells
themselves (

The core of the topological model is the

(A) A cell's cleavage plane model (CPM) specifies the stochastic
rule by which a cell chooses its cleavage plane for the next division. In
this example, the hexagonal mother cell has equal chance of dividing into
two pentagons or one hexagonal and one quadrilateral cell. The choice of
cleavage orientation can also affect the neighbor cells in more than one
way, for example it may be biased towards smaller neighboring cells. After
division, daughter cells lose sides on average, while two neighboring cells
gain sides (orange). (B,C) A CPM is specified by the choice of first edge
(B) and second edge (C). The possible cleavage planes are shown as dashed
white lines. Probabilities of choosing a cleavage plane are shown adjacent
to the second edge. (D,E) Dynamics of the
O

The epithelial network is only modified by cell division. We do not consider any junctional rearrangements due to cell repacking, cell migration, or cell death.

Each cell divides exactly once per division cycle and the order in which cells divide is chosen uniformly at random from all possible orderings. All cells in an epithelium use the same algorithm, or CPM, for choosing their cleavage plane.

A parent cell divides into two daughter cells through the creation of two
trivalent vertices and one edge along the chosen cleavage plane. Thus
daughter cells always share an edge (

When a cell divides, its cleavage plane must consist of two non-adjacent edges of the original cell. This precludes the formation of tetravalent vertices and 3-sided cells, both of which are rarely observed empirically.

This model describes a generic proliferating epithelium with no/minimal cell
rearrangement. The assumptions are based on experimental evidence from the
larval stage wing disc of

The CPM is the core of the model and describes how mitotic cells select their
cleavage planes. The two main local parameters of a CPM that affect global
epithelial topology are: 1) The extent to which cleavage plane orientation
is directed by the local neighborhood surrounding the cell; and 2) The
symmetry with which a mitotic cell's neighbors are distributed to
the two daughter cells. Computationally, this is modeled as a two-stage
algorithm that first selects a cell side (Side1) based on local topology and
then selects a second side (Side2) based on topological symmetry (

The selection of Side1 models the influence of local neighborhood topology on
cleavage plane orientation. Biologically, the local topology surrounding the
cell could impact cleavage orientation if neighbors with fewer sides
influence physical properties such as tension in the mitotic cell cortex
_{1},
S_{1}, L_{1} and O_{1} (

_{1}.

_{1}.

_{1}._{1} model, here Side1 is chosen from
the neighboring cell with the largest number of neighbors. Though
biologically implausible, it will help us assess the impact of
division asymmetry on global tissue topology.

_{1}.

A second important factor in the determination of cleavage plane orientation
is the manner in which a mitotic cell's neighbors are segregated
between the two daughter cells. We refer to cell divisions that equally
segregate neighbors as symmetric, while divisions that segregate neighbors
unequally are asymmetric. The symmetry of a CPM is governed by the choice of
Side2, the second edge of the cleavage plane. We simulated four strategies
for the selection of Side2: R_{2},
E_{2}, B_{2},
and U_{2}, all of which are illustrated
in

_{2}

_{2}

_{2}

_{2}

Each pair of Side_{1} and Side_{2} algorithms constitutes a
distinct CPM, denoted by Side1|Side2. We simulated each of the 16 possible
CPMs for a total of 12 generations of cell division. In each generation,
every cell divides once and the order in which cells divide is random. We
simulated many different initial conditions (a single
^{12} = 4,096 cells. For each
simulation, we recorded the final topological shape distribution and the CPM
mean and standard deviation over 100 trials (

(A) Steady-state shape distributions for all simulated CPMs (color),
sorted from high to low cell shape variance. Also included are the
proliferating epithelia (grayscale) from _{0}_{0}

Previous work suggests that proliferating epithelia with a specific CPM will
converge to an equilibrium distribution of polygonal cell shapes _{1}|_{2}) CPM remains an
important question. We find that simulations of a completely random CPM
(R

Assuming negligible boundary effects, every CPM described herein should
converge at an exponential rate to a mean shape of 6 sides _{1}). _{2}). Our findings indicate that
highly symmetric and charitable CPMs suppress global cell shape
variability.

Our simulation results reveal a strong correlation between the degree of
division symmetry and the number of hexagons in the population. For every
Side1 strategy tested, the percentage of hexagons in the population
increased with increasingly symmetric Side2 CPMs (

The degree with which the Side1 CPM favors smaller neighbors also had a
noteworthy effect on the percentage of hexagons in the population. One can
order L_{1},
R_{1}, S_{1}
as explicitly increasing in charitability. For every Side2 algorithm tested,
increasingly charitable Side1 CPMs led to an increased percentage of
hexagons and a correspondingly lower variance (_{1}
appears to be implicitly charitable; the CPM favors the recently divided
sister cell which tends to have fewer sides due to its recent division. The
simulations suggest that this CPM lies between R_{1}
and S_{1} in its ability to reduce
shape variance. The simulations also reveal some complexities overlooked by
our earlier Markov chain model _{1}|B_{2}).
Also, a different CPM
(O_{1}|E_{2})
can reproduce the cell shape distribution observed in natural epithelia;
this CPM has lower charitability but higher symmetry. This illustrates that
the interplay between autonomous symmetry and non-autonomous charitability
critically determine the equilibrium shape distribution.

The CPM that minimized the variance in polygonal cell shape and produced the
largest percentage of hexagons was
S_{1}|E_{2},
which is both maximally charitable and maximally symmetric
(_{6}_{4}_{6}_{4}

The wide spectrum of shape distributions produced by different CPMs raises the
intriguing possibility of inferring the CPM based solely on empirical
observations of global epithelial topology. For example, a hypothesis for a cell
division strategy in a given epithelium might be rejected simply by comparing
the empirical cell shape distribution with the one predicted by the CPM. Here we
present the results of comparing our simulated CPMs to cell shape distribution
data from natural proliferating epithelia in a diverse array of organisms. We
use data, collected and published previously by our group

To compare simulated CPMs with natural epithelia, we sorted all distributions
(simulated and empirical) by variance. Compared with simulated topologies,
natural distributions exhibit a surprisingly low shape variance and a high
percentage of hexagons (_{n}_{n}

Of all division models tested, the
O^{4}) and has a nonzero fraction of
4-sided cells
(_{4}_{4}_{1}|B_{2}
CPM also matches the empirical data, with 43% hexagons and a
standard deviation of 0.72 sides and 5.3% 4-sided cells. Although
this is significantly different from the conserved empirical distribution,
it is possible that a similar CPM with higher symmetry than
B_{2} but lower symmetry than
E_{2} may generate the expected
distribution. We have derived such a CPM through simulation (

Previous results raise the possibility that the conserved distribution may
arise from distinct division strategies in different organisms. To test this
possibility, we compare our simulated distributions to those found in
related work on cell division in plants and in the larval wing disc of

Orthogonal regulation is a common mode of division in plant development _{1} rule. Thus, the
O

Although the

Given the evidence against O_{1} might translate into a
physical mechanism. One possibility is that for a given cell, the longest
edge is adjacent to the smallest neighbor and thus more likely to be cut by
a cleavage plane or exert the most tension

By comparing natural and simulated cell shape distributions, we can make several inferences about proliferating epithelia. First, the observed low variability in cell shape implies that division strategies are not only highly symmetric, but also moderately charitable: they directly or indirectly favor adding sides to smaller neighbors. Second, although the proliferating epithelia of five diverse organisms show a highly conserved shape distribution, there are multiple plausible CPMs that can generate this distribution, and experimental evidence suggests that indeed plants and fruitflies do have distinct division mechanisms. This raises the possibility that different organisms may have evolved distinct mechanisms to suppress shape variability during proliferation. Alternatively, the low shape variability may be an indirect outcome of other factors that favor symmetric and charitable divisions. Looking forward, as proliferation is better understood in other organisms, our topological framework can provide a background for hypothesis generation and testing as well as a basis for studying pattern formation in the presence of proliferation.

Hexagonal fraction p6* vs. mean shape. Most CPMs produce a mean shape
close to 6, even though the percentage of hexagons varies significantly. A
mean of 6 is expected for all CPMs, provided that the boundary effects are
minimal. Only a few CPMs, based on LargestNeighbor1 show a mean closer to 5,
as discussed in the

(3.46 MB TIF)

A SmallestNeighbor based CPM that matches empirical data. We interpolate the
symmetry value between SmallestNeighbor|Binomial and
SmallestNeighbor|EqualSplit by having each cell execute the first method
with probability a and the second method with probability
(1-

(5.22 MB TIF)

Empirical Cell Shape Distribution Data from Five Diverse Organisms. Shape
distribution data for proliferating epithelial in several organisms of
interest. The data for

(0.04 MB DOC)

Cell shape distribution data for all CPMs. Distribution data for simulated CPMs. Each data point is a result of 100 simulations, each with 12 generations of division and 4,096 cells. Modes of distributions are shown in red. This data supports the existence of an equilibrium distribution that depends on the CPM but is independent of initial conditions.

(0.07 MB DOC)

Cell shape distribution data for all CPMS (Sorted by percentage of hexagons). The same shape distribution data for simulated CPMs as shown in Table 2 but here sorted by the steady state fraction of hexagonal cells. As in Table 2, each data point is a result of 100 simulations, each with 12 generations of division and 4,096 cells. Modes of distributions are shown in red. Hexagonal frequencies are shown in bold. As division becomes more symmetric and charitable, the fraction of hexagons increases and eventually hexagons become the mode of the shape distribution.

(0.08 MB DOC)

Standard Deviation (%) of Equilibrium Fraction of
_{1}_{1} strategy) appear more likely to exhibit high
variability in equilibrium shape frequency. The large variance appears to be
a result of conflicting effects that increase and decrease shape variance
(e.g.

(0.07 MB DOC)

Comparison to other Relevant Models.

(0.03 MB DOC)

Includes relevant data, methodologies, and equations that supplement the main manuscript.

(0.05 MB DOC)

We thank M. Brenner and J. Dumais for insights and critical comments.