Cadherin-Dependent Cell Morphology in an Epithelium:
Constructing a Quantitative Dynamical Model
Ian M. Gemp1, Richard W. Carthew2, Sascha Hilgenfeldt3*
1Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois, United States of America, 2Department of Molecular Biosciences,
Northwestern University, Evanston, Illinois, United States of America, 3Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois,
United States of America
Cells in the Drosophila retina have well-defined morphologies that are attained during tissue morphogenesis. We present a
computer simulation of the epithelial tissue in which the global interfacial energy between cells is minimized. Experimental
data for both normal cells and mutant cells either lacking or misexpressing the adhesion protein N-cadherin can be
explained by a simple model incorporating salient features of morphogenesis that include the timing of N-cadherin
expression in cells and its temporal relationship to the remodeling of cell-cell contacts. The simulations reproduce the
geometries of wild-type and mutant cells, distinguish features of cadherin dynamics, and emphasize the importance of
adhesion protein biogenesis and its timing with respect to cell remodeling. The simulations also indicate that N-cadherin
protein is recycled from inactive interfaces to active interfaces, thereby modulating adhesion strengths between cells.
Citation: Gemp IM, Carthew RW, Hilgenfeldt S (2011) Cadherin-Dependent Cell Morphology in an Epithelium: Constructing a Quantitative Dynamical Model. PLoS
Comput Biol 7(7): e1002115. doi:10.1371/journal.pcbi.1002115
Editor: Douglas A. Lauffenburger, Massachusetts Institute of Technology, United States of America
Received December 24, 2010; Accepted May 23, 2011; Published July 21, 2011
Copyright: ? 2011 Gemp et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the NIH (GM077581, www.nih.gov). IMG acknowledges funding through an NSF-RTG (EMSW21, www.nsf.gov). The funders
had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: email@example.com
Tissues in multicellular organisms consist of a variety of cells
with specialized functions. The differences between cell types are
often manifest in the individual cells’ shapes, their relative
positions, and their neighbor relations . It has become
recognized that mechanical forces within cells control cell shapes
and their larger-scale organization within tissues [2,3,4]. These
forces are generated by specific molecules present within and upon
the surfaces of cells, including the actin-myosin cytoskeleton and
cell-cell adhesion molecules. A fundamental issue is whether tissue
morphology can be described as the sum of cell morphologies,
which are individually determined by autonomous force genera-
tors. An alternative description of tissue morphology assumes that
morphology is passively determined by equilibrium mechanics
once a small number of force parameters is established by force-
generating molecules within cells. This alternative is rationalized
by experimental observations finding that cell packing in the
Drosophila retina is a consequence of mechanical equilibrium
.Moreover, retinal tissue morphology can be quantitatively
modeled by assuming global minimization of interfacial energies
that are established by cellular force-generating molecules .
Models of other epithelial tissues using similar methods have also
successfully reproduced morphological properties [7,8,9], giving
credence to the approach.
The Drosophila retina is a pseudostratified epithelium containing
over 800 repeating units called ommatidia (Fig. 1). Each
ommatidium is on average D<9?m across its widest axis, and
consists of twenty cells, including eight photoreceptor neurons and
twelve accessory cells . Four of these accessory cells (called
cone cells) adhere together to form a transparent plate that acts as
both the floor of the simple lens and the roof of the underlying
pool of photoreceptors. Two primary pigment cells optically
insulate each cone-cell group; together with the cone cells, they
form the ‘‘core’’ structure of the ommatidium. Secondary and
tertiary pigment cells form the ommatidium ‘‘frame’’. Function-
ally, the cone cells form an ‘‘aperture’’ for focused light to be
transmitted from the lens to the photoreceptors, while the opaque
primary pigment cells delineate the aperture stop.
Cells in the retina express two kinds of cell-cell adhesion
molecules: E-cadherin and N-cadherin [5,11]. All cells contain E-
Figure 1. Drosophila eye geometry. (A) Adherens Junction (AJ) cross
section schematic, with the ‘‘core’’ of cone and primary pigment cells
and the ‘‘frame’’ of secondary and tertiary/bristle cells. (B) Side view of
an ommatidium with photoreceptor cells (R) below the AJ and the lens
(L) above it. (C) Double-stained confocal fluorescence image at the AJ
plane of a pupal retina (age 48 h post-pupation). Antibody staining
highlights E-cadherin (green) and N-cadherin (red); where the two
proteins are co-localized the color appears orange. Note the extreme
regularity of the structure.
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cadherin; however, only the four cone cells contain N-cadherin.
The cadherin protein molecules become localized in a thin band of
lateral cell membrane (b<50 nm wide) corresponding to the
adherens junction (AJ), which is the major site of adhesion between
cells (Fig. 1B). Cadherins in one membrane bind to cadherins
located in the membrane of a neighboring cell across the
intercellular gap, affecting the adherence of one cell with its
neighbor . The binding interactions appear to be homotypic; E-
and N-cadherin molecules do not bind to each other , even
though heterotypic binding has been found for Xenopus cadherins
. Intermolecular binding collectively generates adhesion
between facing membranes, thereby decreasing the interfacial
energy per AJ area and making expansion of the interfacial AJ
domain energetically favorable. Membrane that has established
such bonds is considered ‘‘active’’. However, the expansion of one
interfacial domain affects other cell-cell interfaces due to
constraints on the overall sizes of the AJ membrane domain and
cell volume,. Shape changes in one cell induce shape changes in
others, and alteration of the elastic energy of the membranes
around all cells. Ultimately, the mechanical energy of the entire
ommatidium needs to be minimized globally in order to find an
equilibrium configuration for the ommatidium [13,14]. This
configuration describes the entire retinal tissue since ommatidia
are arranged in the epithelium as identical tiles with six-fold axes
It is well known that cell-cell adhesion can play an important
role in cell sorting and morphogenesis of tissues [15,16], as
theoretically described and experimentally demonstrated [17,18].
More elaborate formalisms have been successfully used to simulate
a variety of tissues [19,20]. Our work indicates that in tissues like
the Drosophila retina, the role of cadherins goes beyond cell sorting
and in fact determines the details of their geometric shapes . In
order to understand how cadherin molecules control cell
geometry,, one must consider the distribution and dynamic
properties of cadherins. In the present work, we apply these
considerations to our mechanical model. We then test the fit of
various distribution and dynamic models with experimental data
for mutants in which certain cells produce altered levels of N-
cadherin. We demonstrate (i) that the model describes character-
istic shape changes in such mutants, (ii) that the simulations
distinguish between different mechanisms of how cadherin levels
are attained and controlled, and (iii) that the model, although
conceived as an equilibrium tool, incorporates important dynam-
ical features in morphogenesis, such as the temporal sequence of
cadherin expression and cell-cell contact remodeling.
The mechanical model computes cell shapes by the minimiza-
tion of a mechanical energy functional
with respect to global shape (deformations of all cells are taken into
account), as is described more fully in the Model section. Note that
(1) describes a two-dimensional energy functional defined in the AJ
plane. Since the AJ is only ,50 nm tall, the structure is two-
dimensional to a good approximation, and we can consider
mechanical forces and energies in this plane only. Any forces out
of plane have to fulfill separate, independent force balances that do
not enter into the model.
All cell membranes are assumed to share a uniform stretch
modulus. This modulus encompasses energies from actin cyto-
skeletal contractility and membrane curvature. Two parameters
(cE, cN) encode the adhesion strength of E- and N-cadherin,
respectively. These are the only adjustable variables when optimi-
zing lengths of edges between cells (Lij) and strains on cell
circumferences (Di). We find a unique combination of these
dimensionless parameters (cE<0.025, cN<0.032) provides the best
fit to experimentally described morphology within experimental
Figure 2 shows the modeled AJ structure of such an
ommatidium with labels corresponding to the names of each cell.
Tissues are intricate, heterogeneous systems, consisting of
individual cells whose shapes and relative positions are of
great importance to the tissue’s function, as well as to its
formation during morphogenesis. To make progress in our
understanding of the formation of organs, their malfunc-
tion, and their therapeutic replacement in regenerative
medicine, it is crucial to elucidate the connection between
shape and function. We have developed a quantitative
mechanical model of an epithelial tissue, the retina of
Drosophila, and compare the modeling results with
experimental data. The model successfully predicts shape
changes induced by different expression levels of cell-cell
adhesion molecules. Furthermore, the model gives new
insight into the changes a tissue undergoes during
morphogenesis. Comparing simulations and experiments,
we are able to accept or reject different hypotheses about
morphogenetic dynamics. In this way, we can identify the
time course of adhesion molecule synthesis and of cell-cell
contact, as well as gain new insight into the regulation of
adhesion strength. Given the prominent role of adhesion
in wound healing, cancer research, and many other fields,
our fundamental work introduces a novel modeling tool of
universal applicability and importance.
Figure 2. Nomenclature and geometry of the modeled
ommatidia. Indicated are the cell types (P,C1,C2), the ommatidial
scale D, and some of the quantities contributing to the quantification of
errors: Lcenand LPPare two examples of edge length quantities, in the
case of an asymmetrically deformed ommatidium (black dashed lines)
the asymmetry is quantified using Dx, while errors in angles between
edges are expressed in terms of tension ratios. The red circle enlarges
one example of a triple junction where the angles hj are used to
compute the ratios rjof the tensions tk, tlof the edges adjacent to each
angle. Note that E cadherin is active on all edges, while on the center
edge and the C1C2 edges (orange) N cadherin is also active.
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Anterior and posterior cone cells are treated equivalently and are
labeled C1; equatorial and polar cone cells are type C2; primary
pigment cells have index P. Interfaces are denoted by the names of
the cells on either side of a membrane interface, e.g. the interface
between a C1 and C2 cell is called a C1C2 edge. We refer to the
central C2C2 edge as the center edge. When necessary, we
explicitly denote angles by the sequence of cells surrounding them,
e.g. a C2/P/C1 angle is the angle between a C2P and a C1P edge.
While the simulation in Fig. 2 fits the geometric features of a
normal ommatidium, its two adhesion parameters are open to
interpretation. They are proportional to interfacial concentrations
of paired E- and N-cadherin molecules, respectively. Although our
model uses these concentrations to minimize AJ interfacial
energies, how the cells establish and maintain interfacial
concentrations of paired molecules is another matter. We have
tested a further application of the model by exploring how it is
affected by the way in which levels of cadherin protein pairing are
regulated. This investigation, as well as the remaining sections of
the present work, demonstrate that the energy functional model is
capable of addressing and answering questions about important
N-cadherin dynamics: Destruction vs. recycling
A cell synthesizes cadherin protein in the cytoplasm, and it is
transported to the AJ domain of a cell’s outer membrane [21,22].
If a molecule locally pairs with another molecule on a neighboring
membrane, then it is stabilized both spatially and temporally .
Cell biologists have long known that unpaired cadherin molecules
are internalized by endocytosis whereas paired molecules are not
[22,24,25,26,27]. Endocytosis is critical for maintenance of
epithelial adhesive integrity [28,29,30]. Once internalized, cad-
herins are either recycled back to the cell surface or trafficked to
lysosomes for destruction [31,32]. Trafficking to lysosomes entails
passage through Rab5- and Rab7-enriched compartments . In
contrast, recycling back to the AJ occurs by two routes: directly
from sorting endosomes or after transport to the recycling
endosome [34,35]. Faced with such disparate alternative path-
ways, what determines which route is taken after cadherins are
internalized? If unpaired cadherins follow a route to destruction,
then they do not have an opportunity to redistribute to other
membrane domains. If unpaired cadherins are recycled, they are
free to distribute on other domains of the membrane, and will
accumulate along AJ interfaces with neighboring cells that contain
the same cadherin.
We formulated two models that simulate these different
scenarios for N-cadherin. Both models assume a steady-state
concentration of N-cadherin proteins in a cell. The Destruction
Model assumes high rates of N-cadherin synthesis and turnover; if
unpaired, molecules are rapidly endocytosed and degraded
(Fig. 3A). Reaction equilibria are established locally, so that the
unpaired concentration determines the concentration of N-
cadherin pairs if the neighboring membrane contains the same
kind of cadherin. Such local reaction equilibria at interfaces have
been described in general terms [36,37,38]. The resulting
dimensionless binding energy per membrane length is indepen-
dent of the concentrations on other domains of the AJ in the same
The Recycling Model, by contrast, assumes a low rate of
synthesis and turnover of N-cadherin. The cadherin molecules are
long-lived and if unpaired, they recycle via endocytosis to and
from the AJ. They then redistribute along the AJ in different ways
depending on which interfaces are active, i.e. which neighbors
express N-cadherin as well (Fig. 3B). In the dimensionless energy
functional (1), an N-cadherin binding term along an AJ of length L
Figure 3. Destruction and Recycling models for N-cadherin distribution. Cone cells with unpaired N-cadherin (green), or paired N-cadherin
(red) along active interfaces (dashed orange). (A) Distribution and abundance of N-cadherin according to the Destruction Model. Green arrows in the
inset A9 illustrate that N-cadherin molecules are continually and rapidly synthesized and destroyed (Ø). Blue arrows illustrate that unpaired N-
cadherin traffics via endosomes (blue) to and from the cell surface. This results in a uniform coverage of active interfaces with an equilibrium
distribution of unpaired and paired molecules. (B) Distribution and abundance of N-cadherin according to the Recycling Model. The rates of synthesis
and destruction of N-cadherin are minor relative to trafficking of N-cadherin to and from the surface (B9). Transport of unpaired N-cadherin through
endosomes allows continual redistribution of N-cadherin along the cell surface, so that unpaired molecules have multiple opportunities to find
partners to pair with. The C2 cells, with the longest active edges, exhaust their unpaired N-cadherin supply. Unpaired molecules remain in the C1
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is written as 2cNL or alternatively 2NactBdim/ES, where Bdimis the
dimensional binding energy of a single N-cadherin pair bond, and
Nactis the number of such active bonds along the interface. Thus, if
a cell has Nactbonds along a total AJ length Lact, and these bonds
are distributed evenly, the effective dimensionless binding strength
becomes c=b Nact/Lact, where we define the constant b=Bdim/ES.
Note that Nactis not necessarily the total number of N-cadherin
molecules synthesized in a cell, since some molecules may not find
a binding partner. Cells contain a defined number N0of cadherin
molecules. We assume this number to be identical for C1 and C2
cells, both for simplicity and because genetic regulation of
cadherin expression is not known to differ between cone cells. It
is clear, however, that C2 cells have a longer active membrane
length than C1 cells (see Figs. 2 and 3). In fact, we can write
Only a C2 cell has all of its N-cadherin molecules bound and
active (NactC2=N0), whereas some unbound molecules remain
within the C1 cells (Fig. 3B). Quantitatively, the number of those
molecules is N0[122LC1C2/(2LC1C2+Lcen)]. The binding strength
established in the Recycling Model between any two cone cells is
thus, cN=bN0/(2LC1C2+Lcen). Instead of explicitly defining cNas a
parameter, the Recycling Model implicitly defines it via the
parameter bN0and the self-consistently determined interfacial AJ
lengths between cone cells. The C2 cells can be said to be limiting
for the process, because all of their N-cadherin is paired, whereas
the C1 cells retain some inactive cadherin. In practice, the
simulation protocol iteratively adjusts the cadherin strengths and
the interface lengths until convergence is reached. In all cases
presented here, accurate convergence takes very few iteration
The simulations of normal ommatidia are indistinguishable for
the two models, as both of them establish a uniform cNalong all
edges between cone cells. Hence, in order to determine if either
model is correct, we turned to situations in which certain cells
synthesize less N-cadherin than their neighbors. If asymmetries
arise in an ommatidium so that cone cells have different active
lengths of AJ with respect to N-cadherin, the two models predict
significantly different equilibrium morphologies.
Simulation of N-cadherin loss
Experiments can be performed in the Drosophila eye to create
ommatidia where some cells have a normal gene while other cells
are missing the gene . Although the cellular composition of such
mosaics is generated in a random manner, it is possible to screen
through hundreds of ommatidia and find examples where specific
cone cells are missing a gene. This was done for the N-cadherin
gene, and a number of mosaic ommatidia were found. The
advantage of working with these mutants is that no additional
parameter has to be introduced: normal cells retain normal levels
of N-cadherin synthesis, while for mutant cells there can be no
binding strength at all.
We first examined an ommatidium with only one mutant C1
cone cell (Fig. 4A). Three of the five cone/cone interfaces are
active for homophilic N-cadherin binding, while the two interfaces
juxtaposed to the mutant cell are seen to be shorter than normal.
An overall symmetry breaking of the ommatidium results.
Simulations using the Destruction Model resulted in a pattern
that reproduces the overall deformation of the ommatidium
(Fig. 4B). In the Recycling Model, a more elaborate distribution of
binding strength results. The normal C1 cell has a longer active
interface to populate than the C2 cells, and consequently becomes
limiting for cadherin distribution. However, the surplus N-
cadherin molecules in the C2 cells can form active bonds with
each other across the center edge. We thus obtain cN=0 on the
two inactive interfaces, cN<0.038 on the two active C1C2 edges
and cN<0.053 on the center edge, the latter number a much
higher binding strength than in the wildtype. The Surface Evolver
simulation according to the Recycling Model is shown in Fig. 4C,
and is also successful in describing the pattern seen in the
experimental image. The differences between Destruction and
Recycling simulations are subtle; we quantified the simulation
errors using length of the characteristic C1C2 edges on the left side
of the ommatidium, together with the angles around them (we use
the two tension ratios belonging to the angles P/C1/C2 and P/
C2/C1). The definitions of the quantitative individual errors feand
the total error function Feare found in the Model section. Table 1
shows that the Recycling Model does better in the total error
function Fe(it is smaller by about one third – see Table 1), but
there is not enough data to make this finding statistically
Therefore, we also analyzed an ommatidium that was missing
N-cadherin in the bottom C2 and right C1 cells (Fig. 5A). In the
Destruction Model, these two cone cells do not alter the
concentration of paired N-cadherin along the single active
interface, leading to a normal binding strength cN<0.032, while
all other interfaces between cone cells have cN=0. Model
simulation resulted in a characteristic asymmetry reflecting what
is observed in experiment (Fig. 5B). However, the details of the
configuration were poorly reproduced. In particular, the active
interface is not long enough and the angle under which it meets
with the cone/primary pigment cell edges does not fit the data.
In the Recycling Model, the entire N-cadherin molecule
population becomes distributed along the active interface between
the two normal cone cells. Not only does this redistribute the
cadherin molecules from other interfaces, but even the number of
paired N-cadherin molecules is higher because there are no
unpaired molecules left over in the C1 cell. As a result, the N-
cadherin binding strength along the active interface is enhanced to
bN0/LC1C2. While LC1C2adjusts itself during a Surface Evolver
simulation (and becomes longer because of the energetic
advantage of long interfaces carrying large cadherin strength), it
is clear that this binding strength is much larger than in the
Destruction Model. Indeed, the final value from the Surface
Evolver simulation is cN<0.081, almost three times the wild-type
binding strength (Fig. 5C). The Recycling Model not only
simulates the observed asymmetry, but the extreme length and
angle associated with the active interface.
The error measures in the simulations with the Recycling Model
are all significantly smaller than those with the Destruction Model
(Table 1); the total error Feis more than five times smaller. Thus, a
prediction of our mechanical energy model is that cone cells use
recycling to redistribute N-cadherin along distinct interfaces.
Further evidence supports a recycling/redistribution mechanism
in cone cells. In the mosaic experiment shown in Fig. 5, the AJs
were visualized using an antibody specific for the b-catenin protein
(Fig. 5D). This protein stoichiometrically associates with cadherin
protein on the cell membrane, where it helps anchor cadherin to
the AJ. Importantly, since b-catenin is produced in vast excess and
only bound molecules are stable, its abundance is directly
proportional to the abundance of cadherin . The fluorescence
intensity of b-catenin staining along C1C2 edges of normal
ommatidia is significantly lower than the fluorescence along C1C2
edges of mosaic ommatidia with two mutant cone cells, indicating
the presence of much more cadherin on the remaining active
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edges – a finding that is compatible with the Recycling Model
(Fig. 5D), but contradictory to the Destruction Model.
N-cadherin dynamics: timing and level of expression
The mechanical energy model simulations presented so far
implicitly assumed that N-cadherin protein expression is estab-
lished simultaneously in all four cone cells. Moreover, they
assumed equal protein synthesis in all cells. While descriptive
fluorescence microscopy supports these assumptions , it lacks
sufficient resolution and quantitativeness to rigorously demonstrate
their veracity. Moreover, the four cone cells do not behave
identically from a developmental standpoint. C1 cells begin
differentiation several hours before C2 cells, and C1 cells
specifically induce cells to a P cell fate whereas C2 cells do not
[39,40]. The mechanical energy model is, however, capable of
testing these assumptions in certain mutant ommatidia.
The Destruction Model for N-cadherin would be insensitive to
variability of N-cadherin expression since the eventual local
equilibria would be established no matter the sequence or level of
cadherin synthesis. However, the Recycling Model would be very
sensitive to N-cadherin expression. For example, if the C2 cells
were to synthesize N-cadherin earlier than the C1 cells and at a
stage where the C2-C2 center edge contact is established, all of the
N-cadherin from C2 cells would go onto the center edge, for a
strength cNcen=bN0/Lcen, much greater than observed. There
would be no N-cadherin left to bind along the C1C2 edges,
rendering the other binding energies cNC1C2=0. For similar
reasons, the Recycling Model would also be sensitive to changes in
the level, rather than the timing, of the expression – changes that
the Destruction Model, again, would fail to be affected by.
We therefore examined situations in which certain cells
synthesize N-cadherin at different times and levels from normal,
in order to (i) further test the Recycling versus Destruction Models,
and (ii) establish in what ways morphology is affected by these
Simulation of N-cadherin misexpression
We examined mosaic ommatidia in which some cells mis-
expressed N-cadherin protein. These ommatidia were made by a
Figure 4. Simulation of ommatidium with one N-cadherin mutant cone cell. (A) Experimental image of ommatidium with one C1 cell (left)
not expressing N-cadherin. N-cadherin-producing cone cells are marked in purple, while cone cells not marked purple do not produce N-cadherin.
Note that primary pigment cells, whether marked or not, do not normally synthesize N-cadherin. (B,C) Simulations using the Destruction (B) and the
Recycling (C) Models reflect the general asymmetry and deformation of the ommatidium.. Differences between the models are slight and manifest
largely in the center edge length. The width of red active edges is a measure of N-cadherin binding strength in the models. (D,E) Distribution of N-
cadherin according to the Destruction (D) and Recycling (E) Models. The Destruction Model predicts unaltered densities of N-cadherin molecules and
thus unchanged binding strength on the active edges. The Recycling model predicts rearrangement of the unpaired molecules in the C2 cells,
leading to increased binding strength across all remaining active edges.
Table 1. Error measures for N-cadherin knockout mutants.
Fig. 4, Experimental 0.08720.665 0.865 N/A
Fig. 4, Destruction 0.1210.607 0.8460.155
Fig. 4, Recycling 0.1170.6470.867 0.113
Fig. 5, Experimental0.209 1.861.74N/A
Fig. 5, Destruction0.183 0.921.05 0.425
Fig. 5, Recycling0.2171.401.52 0.0794
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random genetic switch that activated expression of a N-cadherin
transgene in certain cells. Since the switch can be triggered in cells
irrespective of their identity, pigment cells can potentially express
N-cadherin. Moreover, cone cells affected by the switch express N-
cadherin from two sources - the endogenous N-cadherin gene and the
transgene i.e., the total number of N-cadherin molecules is greater
than the normal number N0. Production of N-cadherin from the
transgene is defined as N+. Switching on of the N-cadherin transgene
does not occur synchronously with the onset of endogenous N-
cadherin expression but rather occurs 10 to 30 hours earlier in time
. Thus, the transgene not only perturbs the level but also the
timing of N-cadherin expression (Fig. 6A).
We modeled an ommatidium that had N-cadherin trangene
expression in two C2 cells and one P cell. With the Recycling
Model, the question of which cell is limiting for N-cadherin
depends on the ratio N+/N0. However, except for very extreme
ratios, the large active interface of the P cell makes it limiting. The
binding strength along the P cell is then cNP=bN+/(2LC2P+LC1P).
The next limiting cell is the right-hand C1 cell, which retains
N02LC1PcNPmolecules, distributed along the C1C2 edges, where
the binding strength is cNC1C2=b (N02LC1PcNP)/(2LC1C2). Thus,
the overexpressing C2 cells retain a number of N-cadherin
molecules given by
This number is to be distributed along the left-hand C1C2 edges
and the center edge. Depending on whether this number is greater
or smaller than N0/2, the left-hand C1 cell or the C2 cells will be
Because the transgene is expressed before the endogenous gene,
the P cell initially distributes its N-cadherin along the interfaces
contacting the two C2 cells also expressing the transgene, so that
cNP=bN+/(2LC2P). The remaining N+/2 molecules in each C2 cell
distribute to the center edge with a binding strength of
cNcen,1=bN+/(2Lcen). Thereafter, the endogenous N-cadherin gene
is expressed. We assumed that the transgenic N-cadherin would
not influence the binding behavior of the endogenous N-cadherin
molecules, so that they would distribute similarly (though not
identically) to a normal ommatidium. The result of a simulation
for N+/N0=0.9 is shown in Fig. 6B. The cadherin binding
strengths along different interfaces vary depending on the ratio of
N+/N0, with very strong binding along the center edge (Fig. 7A).
Qualitatively, the simulation matched an observed experimental
ommatidium with the same configuration of misexpression
(Fig. 6D). The right-hand C1 cell is bulged out with an elongated
C1P edge, while the misexpressing P cell has become more
compact and rounded, curving the PP edges at the top and bottom
of the ommatidium. We quantified the error in the simulation
compared to the observed shape to allow for an unbiased
comparison. All of the errors were evaluated as a function of the
ratio N+/N0. The overall penalty function showed a well-defined
minimum at N+/N0<1.25 (Fig. 7B, C). All in all, we evaluated 15
different error contributions (accounting for the left/right
asymmetry, there are 6 different tension ratios, 8 different edge
lengths, and the parameter Dx– see the Model section for exact
definitions of the error quantities). However, we found that only a
handful of these measures contribute significantly to the total error
(Fig. 7D). We also tested the shapes and penalty function values
Figure 5. Simulation of ommatidium with two N-cadherin mutant cone cells. (A) Experimental image of ommatidium with one C1 and one
C2 cone cell (right and bottom, respectively) not expressing N-cadherin (lack of purple marker). The orange circle indicates the characteristic obtuse
C2/P/C1 angle. (B) In the Destruction Model simulation, the length of the active edge (red) and C2/P/C1 angle are poorly reproduced. (C) The
Recycling Model simulation reproduces those quantities significantly better, particularly the C2/P/C1 angle. The thicker red line indicates the
predicted increased N-cadherin coverage on the active edge. (D) Anti-b-catenin antibody localization in cone cells as visualized by FITC fluorescence.
Asterisks mark cells with normal N-cadherin, and unmarked cells lack N-cadherin. The staining of the C1C2 edge from a normal ommatidium (left) is
lower than staining on the active C1C2 edges of mosaic ommatidia (center and right), as indicated with arrows. Fluorescence was quantitated on
each indicated active C1C2 edge using ImageJ software. The active C1C2 edge fluorescence in the mosaic ommatidia were 1.84- and 2.45-fold greater
than the active C1C1 edge in the normal ommatidium.
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assuming a Destruction Model for this example, but total error
values were very high in this case (data not shown). Thus, the
Recycling Model is strongly supported by simulations of N-
The importance of N-cadherin timing and cell-cell
We determined the sensitivity of the Recycling Model to timing
of N-cadherin expression, using the example already described.
We reran the simulation with the assumption that endogenous and
transgenic N-cadherin were expressed simultaneously. This result-
ed in a different pattern from observed (Fig. 6C). The penalty
function for the simulation was greater than the penalty for the
simulation with real timing (Fig. 7B). Thus, the Recycling Model
predicts that tissue morphogenesis is highly dependent on timing
of cadherin expression even if levels are unchanged.
Our Recycling Model illustrates how gene expression estab-
lished at different times in different cells can lead to dramatically
different morphologies. There is another implication to this; cells
frequently change neighbors in a prescribed sequence over time
during morphogenesis. Thus, even two cells that initiate N-
cadherin expression at the same time might not be able to pair
their molecules until such time that they become neighbors. This
could have profound consequences on the ultimate distribution of
N-cadherin and cell morphologies in a recycling scenario. To test
this hypothesis, we exploited the known sequence of neighbor
changes that P cells undergo (Fig. 8A). P cells first exclusively
contact C1 cells; then P cells form additional contact with C2 cells,
and only afterwards do they form contact with each other . In
normal ommatidia, this has no consequences for N-cadherin
binding, as P cells do not contain N-cadherin.
We modeled a mosaic ommatidium in which the N-cadherin
transgene is switched on in two neighboring P cells. Although this
switching occurs at the same time in the two cells, the N-cadherin
is unable to pair because the two P cells are not yet neighbors (cf.
Fig. 8A). By the time they do become neighbors, the endogenous
N-cadherin gene has been expressed in the cone cells for about
10 hours. Therefore, transgenic N-cadherin first pairs on the C1P
and C2P edges (Fig. 8A), increasing adhesion and elongating these
edges. The binding strength is cNP=bN+/(2LC2P+LC1P), assuming
that the P cells are limiting, which leaves N02N+ LC1P/
(2LC2P+LC1P) and N02N+ LC2P/(2LC2P+LC1P) molecules in the
C1 and C2 cells, respectively. The active interface lengths for the
two different cone cell species are different (the C2 cells have to
Figure 6. Timing and level of N-cadherin expression affects morphology. (A) Order of events (expression) in the misexpression experiment.
(B) Simulation using the Recycling Model where the N-cadherin transgene initiates expression in both C2 cells and one P cell before the endogenous
gene begins expression. The level of N-cadherin expression is 90% greater in the C2 cells than C1 cells (N+/N0=0.9). (C) Simulation using the
Recycling Model as in A but where the transgene and endogenous gene begin expression at the same time, and where the level of N-cadherin
expression is 60% greater in C2 cells than C1 cells (N+/N0=0.6). (D) Experimental image of ommatidium with the C2 cells and one P cell misexpressing
the N-cadherin transgene (marked by purple).
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cover the center edge as well), but so are the numbers of cadherins
to be distributed. The most important feature of the simulation is
that, up to N+/N0<1.2, the PP edges do not have paired N-
cadherin at all, and therefore have a relatively high tension,
leading to a characteristic acute P/C2/P angle seen in the
simulation (Fig. 8B). Strikingly, an experimental mosaic omma-
tidium with misexpression in two P neighbors shows a similar
morphology (Fig. 8D). A penalty function of the simulation
compared to experimental showed a well-defined minimum at N+/
N0<1.2 (Fig. 9A–C). While the best-fit simulation yielded the
cruciform shape with the acute P/C2/P angles seen in experiment,
any significant deviation from this optimal N+/N0led to large
shape changes. Note that this optimal ratio is nearly the same as
that obtained from the simulation for the misexpression mutant in
Fig. 7. The N-cadherin coverages of the various interfaces are
plotted in Fig. 9D; note that any abrupt change in slope of the
curves indicates a change in the sequence of limiting cells
To determine if the simulation worked because of the timing of
gene expression with respect to cell contact, we performed a
simulation that disregarded the ordered cell contacts and assumed
the P cells are in contact when N-cadherin expression begins. Since
transgenic N-cadherin is expressed first, it would only be distributed
along the PP edges, i.e., the binding strength there would be a
large value cNPP=bN+/(2LPP), and the effective tension on these
edges quite small. The effect of this change produces a simulation
that poorly fits the experiment (Fig. 8C). This failure was seen if N-
cadherin was expressed either simultaneously or sequentially from
the transgene and endogenous gene (Fig. 9A).
We have applied and extended the mechanical energy functional
modeling on the morphology of Drosophila eye tissue in several
crucial ways. This model relies entirely on passive global
minimization of a simple interfacial energy functional. However,
we have shown how to incorporate new features that, implicitly,
describe observed biological features of the tissue. These features
include: recycling and redistribution of unpaired N-cadherin
molecules from the surface of cells, the temporal dynamics of N-
cadherin gene expression, and the temporal dynamics of cell-cell
contacts. Note that we established these model features without any
Figure 7. Analysis of misexpression simulations. (A) Relative binding strengths on various interfaces as the N+/N0ratio varies. Binding strength
when N-cadherin misexpressed (OP) was normalized to the wildtype binding strength (WT). (B) Total error (values of the total error function Fe) as a
function of the N+/N0ratio. Red line describes error from simulation in which the transgene and endogenous gene begin expression sequentially.
Black line describes simulation in which the two genes begin expression simultaneously. (C) The effect of changing N+/N0on the shape of the
ommatidium in the sequential simulation. For small N+/N0(here shown for N+/N0=0.4 on the left), the asymmetry is weak; for large N+/N0=1.8 (right),
the differences are more subtle, and the quantitation through Feis necessary. (D) Total error is dominated by contributions fefrom the tension ratios
of cone/cone cell edges and C2/P cell edges, with significant contributions from the asymmetry parameter Dx.
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additional free parameters. Because the dynamics of morphogenesis
and gene expression are experimentally described phenomena, and
because the Surface Evolver simulations are capable of varying
these features, we incorporated these phenomena into a model that
describes the experimental data with a high degree of accuracy.
When we incorporate features into the model that are not observed
in the tissue, the model does not simulate the experiments to the
same degree. Two points can be made. First, it is essential for any
morphogenesis model to be flexible enough that critical biological
features can be modified. Second, our morphogenesis model not
only simulates normal morphology but also morphologies observed
in complex genetic mutants. In fact, it is the asymmetries in
ommatidialshapeand the cadherinexpressiondynamicsintroduced
through mutants that allowed us to successfully test these features
within the framework of the model.
The success of these simulations suggests that models employing
energy minimization can go beyond the quantitative computation
of local equilibrium morphologies, and become useful tools for
simulating morphogenetic changes that are quasistatic with respect
to the time scale of mechanical equilibration. In the case of
changes in gene expression or cell neighbor relations, this condi-
tion is fulfilled, as these changes can take hours [41,42,43]. In
contrast, mechanical equilibria are established in minutes at most,
depending on dissipative effects [44,45]. In the experiments we
have described, there is also sufficient separation between the
onset of cadherin expression and the changes in cell-cell contacts.
We simulated misexpression of a N-cadherin transgene to test
many of the biological features of the model. In all of these
simulations, the best-fit solutions predicted a level of transgenic N-
cadherin expression level to be 1.2-fold greater than that of the
endogenous N-cadherin gene. One would expect that independent
simulations predict the same level of transgene expression because
its expression is independent of which cell or combination of cells
express it. Also encouraging is that the level of transgene expre-
ssion is predicted by simulations to be comparable to the level of
endogenous N-cadherin gene expression. The transgene uses the
transcriptional promoter from the actin5c gene to indirectly drive
transcription of the N-cadherin coding region . Actin5c is one of
six actin isoforms and represents only 5–10% of the actin in pupal
head cells . Thus, we estimate that 50,000 to 100,000 actin5c
protein molecules would be present in these cells, and presumably
a similar number of heterologous proteins produced from an
Figure 8. Timing of N-cadherin expression with cell-cell remodeling is important for morphology. (A) Order of events (expression and
morphogenetic) in the P cell misexpression experiment. (B) Simulation in which N-cadherin expression in P cells begins before they contact one
another. All four cone cells only produce N-cadherin from its endogenous gene. Here N+/N0=1.2, close to the minimum of the total error function. (C)
Simulation in which N-cadherin expression in P cells begins at the same time as they contact one another. For N+/N0=1.0,, the shape compares poorly
with observed. This remains true for the entire range of N+/N0ratios. (D) Experimental image of ommatidium with both P cells misexpressing the N-
cadherin transgene (marked purple). In all figures, the orange circle marks the characteristic acute angle of the mutant P/C2/P junction.
Quantitative Dynamical Model of Cell Morphology
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actin5c transgene. About 50,000 endogenous cadherin protein
molecules are present in a typical animal cell . Therefore, it is
reasonable to find that transgenic and endogenous production of
N-cadherin are not different by orders of magnitude.
Most computer models of tissue morphogenesis simulate visual
changes in cell shape and neighbor relations that occur over time
. However, use of qualitative visual information to compare
normal and perturbed morphogenesis limits their abilities to find
optimum mechanical parameters. Our model introduces a novel
method for exploring morphogenesis: the effect of perturbations
during morphogenesis can be simulated by the final cell shapes
and neighbor relations that are achieved at the end of
morphogenesis. The simplicity of the final cell geometry allows
for unambiguous quantitative comparisons between different
morphogenetic conditions, particularly for highly ordered tissues
like the Drosophila retinal epithelium.
In previous work, we introduced a quantitative simulation
method for normal cell geometry in the two-dimensional plane of
the AJ based on two free parameters encoding for the binding
strengths of E- and N-cadherin. Quantitatively, we minimized the
mechanical energy functional (see equation (1))
using the Surface Evolver software . In (1), dimensional
are normalized by a scale LS=D/9 (making
LS<1 mm), so that L=Ldim/LS. Likewise, energies Edimare non-
dimensionalized as e=Edim/ES, where ES=KAb LS. Here, KAis a
uniform membrane stretch modulus (an energy per area), assumed
uniform for all cell membranes [13,50]. Note that all results we
report are independent of the numerical values of b, LSand ES, as
we will describe shapes through relative errors of the energy
Further, Di=(Li2L0 i)/L0 Iis the strain  on the membrane
of cell i, whose total circumference in the two-dimensional
projection of the AJ is Li. At unstressed equilibrium (measured
experimentally for cells detached from their neighbors), the
Figure 9. Analysis of P cell misexpression simulations. (A) Total error Fefor different simulations as a function of N+/N0. The red line describes
the simulation error when P cells express N-cadherin before cone cells, and P cells do not contact each other until after the onset of N-cadherin
expression in cone cells. The purple line describes simulation error when P cells express N-cadherin before cone cells but P cells are always in contact
with each other. The black line describes simulation error when P cells and cone cells simultaneously begin N-cadherin expression, and P cells always
are in contact with each other. (B) Error contributions to the best-fit model are dominated by the same feterms as in the other misexpression
simulation, except that the symmetric structure of this mutant has no Dxcontribution to the error. The minimum is located at N+/N0<1.2. (C) The
effect of N+/N0on the shape of the ommatidium, with only values near the error function minimum approximating the observed cruciform mutant
shape. (D) Binding strengths of N-cadherin on various edges of the structure as N+/N0varies. Note the absence of binding on PP edges for N+/N0,1.2.
Quantitative Dynamical Model of Cell Morphology
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circumference is L0 i. We scale the specific cadherin binding
energies CEand CNby KAto obtain the dimensionless quantities
cE=CE/KAand cN=CN/KA, which are the only parameters in the
model. If the same cadherin is active in two neighboring cells i and
j (note the Kronecker delta functions), the resulting binding energy
is proportional to Lij, the length of the membrane between the
cells. The minimization of (1) is performed under constraints of
constant cross-sectional areas of the central cells in the ommati-
dium, formalized through Lagrange multipliers in Surface Evolver
. Tests incorporating either bending energy terms or a finite
energy penalty for changes in cross sectional area did not yield
significantly different results (see  for more details).
We arrived at unique values for cEand cNby quantitatively
comparing experimental geometry data of normal ommatidia with
those determined from our computational simulations minimizing
(1), cf. . All lengths and angles characterizing the observed
ommatidium structure were reproduced within experimental
error. In order to quantify the deviations between experimental
and simulated structures, we summed up relative errors of both the
lengths of edges and the angles between edges in the structures.
Minimization of the total error function then yielded the best-fit
wild-type values cE=0.025?0.005 and cN=0.032?0.005, respec-
tively, where the errors reflect uncertainties in the geometrical data
Quantifying modeling errors
In order to have a more general measure for shape differences,
we generalized the penalty function of , taking into account
errors in (i) the interface lengths in the 2D structure, (ii) the angles
between interfaces, and (iii) in cases where the entire unit becomes
asymmetric, a measure of that global asymmetry. The latter
quantity, Dx, is the dimensionless distance of the PP edges to the
nearest vertex of the hexagonal ommatidial frame (see Fig. 2; note
that in a symmetric ommatidium with centered PP edges Dx=1/
4). The angles between edges (e.g. hj, hk, hlin Fig. 2) are a measure
of the differences in the mechanical tensions tj, tk, tlalong these
edges (if all tensions are equal, all angles are 120?). In fact, the ratio
of any two tensions is given by
and corresponding cyclic permutation of the indices (jkl). Even
subtle differences in angles can translate into strong differences in
tension ratios. Therefore, we adopt the quantities rlinstead of the
angles to obtain a more sensitive error measure.
Any quantity X contributes to the total error via a squared
Here, the superscript SE denotes the quantity obtained from
Surface Evolver simulations, while X without superscript is the
experimental value. Our total error function is the sum of all fe.
where the indices i,j denote cells and the indices l denote angles.
The smaller Fe, the better is the agreement between the measured
quantities and those obtained by simulations. In many cases, shape
differences are dominated by only one or a few fe, while in others
the complete set has to be included in Fe.
We thank Ken Brakke for his invaluable help with questions about the
Surface Evolver. Discussions with Steve Davis, Matthew Miklius, and
Taher Saif are greatly appreciated.
Conceived and designed the experiments: RWC. Analyzed the data: IMG
SH. Contributed reagents/materials/analysis tools: SH. Wrote the paper:
IMG SH. Performed Numerical Simulations: IMG. Developed theory:
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