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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

Abstract

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: sascha@illinois.edu

Introduction

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 [1]. 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

[5].Moreover, retinal tissue morphology can be quantitatively

modeled by assuming global minimization of interfacial energies

that are established by cellular force-generating molecules [6].

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 [10]. 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 [5]. The binding interactions appear to be homotypic; E-

and N-cadherin molecules do not bind to each other [5], even

though heterotypic binding has been found for Xenopus cadherins

[12]. 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

of symmetry.

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 [6]. 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.

Results

The mechanical model computes cell shapes by the minimiza-

tion of a mechanical energy functional

e~

X

i

1

2D2

iL0i{

X

i,j

LijcEdi,Edj,E{

X

i,j

LijcNdi,Ndj,N

ð1Þ

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

error [6].

Figure 2 shows the modeled AJ structure of such an

ommatidium with labels corresponding to the names of each cell.

Author Summary

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

morphogenetic mechanisms.

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 [23].

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 [33]. 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

cell.

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

cells.

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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

cC1

N~bNC1

act=2LC1C2, cC2

N~bNC2

act=(2LC1C2zLcen):

ð2Þ

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

steps.

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 [5]. 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

significant.

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 [22]. 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 [5], 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

changes.

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.

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Table 1. Error measures for N-cadherin knockout mutants.

LC1C2/D (active)

rP/C1/C2

rP/C2/C1

Fe

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

[5]. 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

NC2~N0zNz{2LC1C2cC1C2

N

=b

~2Nz(LC1PzLC2P)=(LC1Pz2LC2P):

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

limiting.

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-

cadherin misexpression.

The importance of N-cadherin timing and cell-cell

remodeling

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 [10]. 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

described above.

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).

Discussion

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 [5]. Actin5c is one of

six actin isoforms and represents only 5–10% of the actin in pupal

head cells [46]. 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.

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actin5c transgene. About 50,000 endogenous cadherin protein

molecules are present in a typical animal cell [47]. 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

[48]. 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.

Model

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))

e~

X

i

1

2D2

iL0i{

X

i,j

LijcEdi,Edj,E{

X

i,j

LijcNdi,Ndj,N

using the Surface Evolver software [49]. In (1), dimensional

lengths Ldim

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

functional only.

Further, Di=(Li2L0 i)/L0 Iis the strain [51] 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.

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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

[6]. Tests incorporating either bending energy terms or a finite

energy penalty for changes in cross sectional area did not yield

significantly different results (see [6] 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. [6]. 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

[6].

Quantifying modeling errors

In order to have a more general measure for shape differences,

we generalized the penalty function of [6], 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

rj:tl

tk~{coshj{sinhjcothk

ð3Þ

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

relative error,

fe(X):

XSE{X

X

??2

:

ð4Þ

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.

Explicitly,

Fe:

DSE

x{Dx

Dx

??2

z

X

i,j

LSE

ij{Lij

Lij

!2

z

X

l

rSE

l{rl

rl

??2

, ð5Þ

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.

Acknowledgments

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.

Author Contributions

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:

IMG SH.

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