Tissue dynamics

Adherens junctions between epithelial cells

Green  :  ac)n;  Magenta  :  E-­‐cadherin  

Takeichi,  Nat  Rev  Mol  Cell  Biol  (2014)  

Differen'al  adhesion  hypothesis  

Po3s  model  and  Monte-­‐Carlo  simula'on  

J  Glazier  and  F  Graner  PRL  1992,  PRE  1993.  

-­‐Each  cell  characterized  by  a  spin  state  (as  many  spin  states  as  cells  (~1000)).  -­‐cells  of  different  types  (~2-­‐3)  

Interac)on  between  neighboring  spins:  -­‐same  state/cell  :  lowest  energy  -­‐states  corresponding  to  adhesive  cells    of  the  same  type  lower  energy  (favored)    as  compared  to  cells  of  different  types.    -­‐one  intercellular  medium  state  (no  area  constraint,  High  interface  energy  with  the  cell  states).  

SIMULATION OF THE DIFFERENTIAL ADHESION DRIUEN. . . 2129

duce biologically observed effects would give strong evi-

dence that cell rearrangement requires at least the

cooperation of a mechanism besides differential adhesion.

The visual similarity between the pattern of bubbles in

a soap froth and that of cells in an epithelial tissue was

noticed very early [26]. More recently, the analogy be-

tween surface tension driven boundary length minimiza-

tion in a soap froth and biological cells has providedquantitative models for observed cell arrangements [27].These models are static, explaining the statistics of stablepatterns rather than the dynamics of pattern formation.However, the analogy between froth and tissue is a fruit-

ful one, and (since we now understand the basic dynamicsof soap froth fairly well) [28] suggests that we could also

understand the dynamics of biological tissues, providedthat we recognize the intrinsic short-wavelength cutoffimplied by each cell s finite size: indeed, each cell has avolume-dependent (or area-dependent, in two-

dimensional models) energy term or constitutive relation.Thus we can write a generalized continuous Hamiltonian:

G ~(= f j(r(»(ze(s), r,„„;z,(s) )ds

cell surface

+ g f, (a, ), (1)cells, i

where j(r;„„z,(s), r,„„;z,(s)) is the energy of a unit of cellmembrane as a function of the membrane type, r;„„~,(s),and surface (or medium) with which it is in contact,

r,„„;z,(s), ds is a unit of cell membrane at location s, and

f, describes the constitutive relation of the cells as a

function of their volume a;. We assume that f; encodesall information concerning bulk cell effects, e.g., mem-

brane elasticity, cytoskeletal properties, etc. In this form,our energy resembles that of magnetic bubbles even more

closely than that of soap froth [29]. Thus we explicitly

assume that the cell bulk is isotropic, though we have al-

lowed for membrane surface-energy fluctuations within a

cell by making the membrane type position dependent.

There are many possible ways to implement such an

area-constrained surface-tension Hamiltonian. Since the

true system is dissipative rather than conservative, the re-

sults of the model depend sensitively on the choice of thedynamics. Possible choices include vertex dynamic,

center dynamic, and boundary dynamic models. In this

article, we use the extended two-dimensional Potts modelwhich we have presented earlier [30] to simulate various

observed cases of cell rearrangement and show how

differential adhesion can cause rearrangement without

cell motility. While we are currently extending our simu-

lations to three dimensions, the two-dimensional simula-

tion is sufficient to reproduce the typical types of cell-sorting behavior, including mixing (also known as the

checkerboard), complete and partial cell sorting, position

reversal and dispersal. We also discuss the convergence

of the model, and look in more detail at the quantitative

effects of temperature on the simulation to attempt to dis-

tinguish thermally activated from spontaneous processes.

II. THE MODEL

A. An extended Potts mode1

1. The Hamiltonian

We use a simple extension of the standard large-Q

Potts model [31] to include area constraints and type-

dependent boundary energies. We have described the

differences in detail elsewhere [30]. Beginning with Eq.(1), we make the additional simplifying assumptions that

the surface of each cell is isotropic, i.e., the energy of aunit of cell membrane depends only on the types of thecells on either side [r(s)=const], and that the constitu-

tive relation is a simple quadratic elastic term, with all

cells of a given type having the same natural volume orarea.The extended Potts model discretizes the continuous

cellular pattern onto a lattice, with a spin (T(i,j) definedat each lattice site (i,j) In th.e large-Q limit, we assign a

separate spin, o. H [1, . . . , N], to each of the N cells in

the pattern, with all lattice sites with a given o. compos-

ing a cell o. . Thus each cell extends over many latticesites and need not be simply connected. Typically, eachcell in our simulations covers approximately 40 latticesites, whose spins share the same o., and we treat approxi-mately N = 1000 cells, so o. can assume about 1000different values.

Each cell also has an associated cell type r(cr ), for ex-

ample, endodermal or ectodermal epithelium. Bonds be-

tween like spins have energy 0, that is, the energy inside acell is zero. Between unlike spins (i.e., at cell boundaries)there is a cell-type-dependent surface energy J. In addi-tion to their surface energy, biological cells have general-

ly a fixed range of sizes, which we include in the form ofan elastic term with elastic constant A, , and a fixed targetarea, which may depend on cell type. Therefore, ourHamiltonian is

Potts{i,j){i',j')neighbors

J(~(cr(i,j)),r(o(i',j')))(1—5 (; ) (, '))+X g (a(o)—A~ )) e(A,( )),splns o

(2)

where r(o ) is the cell type associated with the cell (T and

J(r, r') is the surface energy between spins of type r and~', A, is a Lagrange multiplier specifying the strength ofthe area constraint, a (o ) is the area of a cell o, A, is thetarget area for cells of type r, and B(x)= [O, x (0; 1,x 0] . This Hamiltonian is nearly identical to the

lowest-order expansion for the magnetic bubble Hamil-

tonian [29].

2. Procedure

All our simulations employ a second-nearest-neighborsquare lattice. The simulated cells are of two types, low

Negative interface energy : checkerboard patterns

Glazier & Graner, Phys Rev E (1993) Maturation of quail oviduct epithelium, Honda et al (1986)

Positive interface energy : cell sorting

Glazier & Graner, Phys Rev E (1993)

Chick retinal cell aggregate, P Armstrong (1989)

T Lecuit, PF Lenne, Nat Rev Mol Cel Biol (2007)

Actin cortex

Although our 2D description does not explicitly take intoaccount cell height, area elasticity does so indirectly.Assuming that cell volume stays constant, altering thecross-sectional area of a cell at the junctional level in-volves a deformation of the cell in three dimensions.Whereas the model describes single cells as elastic ob-jects, the cellular network is plastic because the modeltakes into account junction remodeling (see below).Line tensionLij (Box 1) describes forces resulting from

cell-cell interactions along the junctional regions of spe-cific cell boundaries. Multiple mechanisms might influ-ence line tension, which could vary from edge to edge.For example, adhesive interactions between cells couldfavor cell-boundary expansion, whereas the subcorticalactin cytoskeleton might oppose it. As the length of thecell boundary [ij between two vertices i and j increases,this term in the energy function decreases if the linetension Lij is negative and increases if Lij is positive. Inaddition to the subcortical actin cytoskeleton, manyepithelial cells assemble an actin-myosin belt that un-derlies the cortex at the level of apical junctions (Fig-ure Ab). Because actin-myosin contractility would tendto reduce the perimeter of each cell, it not only contrib-utes to the line tension but is also expected to generatethe coefficient Ga, describing the dependence of con-tractile tension on cell perimeter La (Box 1). This contrac-tility term involves the whole cell perimeter and is moti-vated by the fact that the actin-myosin ring appears tospan the entire cell.Before studying tissue morphologies, we first discuss

some general properties of the model. An important fea-ture is the ground state, or the most relaxed networkconfiguration. As discussed below, these ground statesdo not correspond to realistic tissue morphologies;however, ground states are important reference states.For a situation when all cells are identical, A!0"

a = A!0",Ka = K, Ga = G for all cells, and Lij = L for all edges. Fig-ure 1 shows that two main types of ground states existas a function of the model parameters. The geometryof the ground state is determined by the two normalizedparameters G = G/K A!0" and L=L=K!A!0""3=2. Here, G isa normalized contractility; when small, it implies thatcontractile forces are small compared to those fromarea elasticity. Similarly, L is a line tension, normalizedto area elastic tensions. When it is negative, cell

boundaries tend to expand; when it is positive, theytend to shrink.Two regions exist in the ground-state diagram shown

in Figure 1. In the gray region, regular hexagonal packinggeometry is the single ground state. This network con-figuration has both a bulk modulus and a shear modu-lus—i.e., work is required to compress or expand the tis-sue and also to shear it. In the blue region, the groundstate is degenerate, i.e., there existmany packing geom-etries all with the same minimal energy. They share thecommon feature that the area of all cells is equal to thepreferred area A!0", and their perimeters are equal toL0 = 2L/2G. As a consequence of this degeneracy, theconfiguration can be shearedwithout anywork required.The system is soft with vanishing shear modulus andbehaves more like a liquid in which cells can movepast one another easily.We call this state a soft network.Whereas the hexagonal ground state in the gray regionis regular, the soft network ground states are typicallyirregular. Examples of configurations corresponding topoints III and IV are shown. Analysis of the ground statesreveals that the morphologies andmaterial properties ofjunctional-network configurations depend strongly onthemodel parameters and that there exists a phase tran-sition between a solid and a soft state.

Packing Irregularity Induced by Cell DivisionIn addition to theglobal energyminimumorgroundstate,there exists for any choice of parameters an even largernumber of local energyminima,which all could representstable cell packing geometries. We think of a developingepithelium as assuming a sequence of stable networkconfigurations, which undergo rearrangements in re-sponse to local perturbations that affect the stable con-figuration. Such perturbations include cell division andapoptosis but might also correspond to slow changesin cellular properties. This quasistatic approximationallows us to define a history of stable configurations byslowly and locally modifying model parameters. Thus,a particular packing geometry is the consequence ofthe history of such perturbations.By using our model, we can numerically simulate the

evolution of cell packing geometry during tissue growth.We randomly select one cell and divide it by the follow-ing algorithm: We doubled the preferred area of the cell

Figure 1. Ground-State Diagram of theVertex Model

Diagram of the ground states of the energyfunction E(Ri) as a function of the normalizedline tension L and contractility G. In the grayregion, the ground state is a hexagonal net-work as indicated. In the blue region, theground states are irregular soft networks (anexample corresponding to case III is shown),and many configurations coexist. In thestriped region to the right, where L is large,cell areas vanish and the model breaksdown. Green dots indicate parameter valuesof five different cases (case I: L = 0.12, G =0.04; case II: L = 0, G = 0.1; case III: L = 20.85,G = 0.1; case IV: L = 20.32, G = 0.04; and caseV: L = 0, G = 0.04).

Cell Division and Epithelial Cell Packing Geometry2097

epithelial cells, we develop a 2D network model thatdescribes forces that act to displace vertices. Cellpackings correspond to stable and stationary networkconfigurations obtained by minimization of a potentialfunction. We use this approach to study the role of cellmechanics and cell division in determining networkpacking geometry. We compare our results to the prolif-erating larval wing epithelium of Drosophila and esti-mate the parameters characterizing the effects of con-tractility and adhesion in this tissue. We independentlyestimate parameter values by analyzing movements ofthe junctional network after laser ablation of individualcell boundaries and comparing them to the correspond-ing behaviors in our model.

Results

Physical Description of Cell Packing: A Vertex ModelApical junctions can be considered as a 2D network thatdefines the cell packing geometry. By using a vertexmodel, we describe the packing geometry of the junc-tional network (Box 1). Cells are represented aspolygons with cell edges defined as straight lines

connecting vertices—a good approximation for thewing disc epithelium. Junctional-network configurationsthat are stable on timescales shorter than those of celldivision correspond to those network configurations inour model for which the packing geometry is stable andstationary. These configurations obey a force balance,which implies that the net force Fi (Box 1) acting oneach vertex vanishes for all vertices. In general, forcesacting on the junctional network need not be forces de-rived from an energy. However, the forces we considerhere can in our simple description be represented bythe energy function E(Ri) presented in Box 1. Any stableand stationary configuration of the network then corre-sponds to a local minimum of the energy function. Thisprovidesa framework forcalculatingstable cellularpack-ing geometries. Similar energy functions have been usedinpreviousworks [14–16].Our vertexmodel is different inthat we represent only the network of apical junctions,and we introduce a quadratic perimeter energy.

We consider three contributions to the potential en-ergy E(Ri) of a particular configuration of the epithelialjunctional network given in Box 1: area elasticity, linetension along apical junctions, and contractility.

Box 1. Physical Description of Cell Packing in Epithelia

Epithelia are composed of a sheet of cells of similarheight that are connected via cell-cell adhesion. Theadhesion molecule Cadherin and components ofthe actin cytoskeleton are enriched apicolaterally(Figure Aa). Cell packing geometry can be definedby the network of adherens junctions (Figure Ab).This network is described by a vertex model withNC polygonal cells numbered by a = 1 . NC and NV

vertices, numbered i = 1 . NV at which cell edgesmeet. Stationary and stable network configurationssatisfy a mechanical force balance; this implies thatat each vertex, the total force Fi vanishes. We de-scribe these force balances as local minima of anenergy function

E!Ri"=X

a

Ka

2

!Aa 2A!0"

a

"2

+X

<i; j>

Lij[ij +X

a

Ga

2L2a

for which Fi = 2vE

vRi.

The energy function describes forces due tocell elasticity, actin-myosin bundles, and adhesionmolecules. The first term describes an area elasticitywith elastic coefficients Ka, for which Aa is the area ofcell a and A!0"

a is the preferred area, which is deter-mined by cell height and cell volume. The secondterm describes line tensionsLij at junctions betweenindividual cells. Here, [ij denotes the length of thejunction linking vertices i and j and the sum over <ij>is over all bonds. Line tensions can be reducedby increasing cell-cell adhesion or reducing actin-myosin contractility. The third term describes thecontractility of the cell perimeter La by a coefficientGa, which could reflect, for example, the mechanicsand contractility of the actin-myosin ring (Figure Aa).

Figure A. Adhesion and Contractility at Apical Junctions

Current Biology Vol 17 No 242096

Fahradifar  et  al,  Curr.  Biol  (2007);  Staple  et  al,  EPJE  (2010).  

   1__  4(3)1/2  

Fahradifar  et  al,    Curr.  Biol  (2007)  

Figure 2. Cell Division, Topology, and Morphology

(A) Cell division in the vertexmodel. The preferred area of a randomly chosen cell is increased, and the network is relaxed. A new cell boundary isintroducedwith a randomorientation. Both new cells are assigned the initial preferred area, and the resulting network is again relaxed. The yellowdot indicates the average vertex position of the original cell through which the new boundary is initially formed.(B) Normalized energy per cell of a growing network as a function of the number NC of cells for parameter values corresponding to case I(see Figure 1 and Box 1). Generation number is also indicated. The energy approaches a value that is greater than the ground-state value ofthe hexagonal network. The standard deviation, averaged over 250 individual divisions, is indicated as a function of cell number in the inset.(C) Fraction Pn of cells with n neighbors as a function of generation number in a growing network for case I.(D) Logarithmic plot of standard deviation of Pn as a function of generation number for the simulation of case I. The slope of the lines representsthe characteristic generation number, at which the standard deviation decreases 10-fold.

Current Biology Vol 17 No 242098

Tissue  growth  in  the  vertex  model  

Fahradifar  et  al,  Curr.  Biol  (2007)  Figure 2. Cell Division, Topology, and Morphology

(A) Cell division in the vertexmodel. The preferred area of a randomly chosen cell is increased, and the network is relaxed. A new cell boundary isintroducedwith a randomorientation. Both new cells are assigned the initial preferred area, and the resulting network is again relaxed. The yellowdot indicates the average vertex position of the original cell through which the new boundary is initially formed.(B) Normalized energy per cell of a growing network as a function of the number NC of cells for parameter values corresponding to case I(see Figure 1 and Box 1). Generation number is also indicated. The energy approaches a value that is greater than the ground-state value ofthe hexagonal network. The standard deviation, averaged over 250 individual divisions, is indicated as a function of cell number in the inset.(C) Fraction Pn of cells with n neighbors as a function of generation number in a growing network for case I.(D) Logarithmic plot of standard deviation of Pn as a function of generation number for the simulation of case I. The slope of the lines representsthe characteristic generation number, at which the standard deviation decreases 10-fold.

Current Biology Vol 17 No 242098

Figure 2. Cell Division, Topology, and Morphology

(A) Cell division in the vertexmodel. The preferred area of a randomly chosen cell is increased, and the network is relaxed. A new cell boundary isintroducedwith a randomorientation. Both new cells are assigned the initial preferred area, and the resulting network is again relaxed. The yellowdot indicates the average vertex position of the original cell through which the new boundary is initially formed.(B) Normalized energy per cell of a growing network as a function of the number NC of cells for parameter values corresponding to case I(see Figure 1 and Box 1). Generation number is also indicated. The energy approaches a value that is greater than the ground-state value ofthe hexagonal network. The standard deviation, averaged over 250 individual divisions, is indicated as a function of cell number in the inset.(C) Fraction Pn of cells with n neighbors as a function of generation number in a growing network for case I.(D) Logarithmic plot of standard deviation of Pn as a function of generation number for the simulation of case I. The slope of the lines representsthe characteristic generation number, at which the standard deviation decreases 10-fold.

Current Biology Vol 17 No 242098 Tissue  growth  in  the  vertex  model  

to the cut boundary can be compared to those obtainedin simulations for estimating parameter values of thevertex model. In the vertex model, we can study the dis-placements that follow the removal of individual bondsin a stable and stationary network configuration. Wecan thus mimic bond-cutting experiments startingfrom a stable network morphology generated by thegrowth algorithm described in the last section. To simu-late laser ablation, we randomly select bonds for re-moval. We set to zero the normalized line tension Lij ofthe cut bond as well as the normalized contractility Ga

of the cells sharing the cut bond. Removal of both theseterms is necessary for achieving quantitative agreementbetween simulations and experiments (see Supplemen-tal Experimental Procedures). We then numerically de-termine the new relaxed network configuration afterthese local parameter changes by using the conjugategradient method described previously. For a given pairof parameters G and L for all other cells, we determinethe resulting distributions of area and perimeterchanges of the pair of cells sharing the removed bonds.The gray points in Figures 5C and 5D show normalizedarea and perimeter changes, with respect to the normal-ized change in bond length, for many different removedbonds in the case that L = 0.12 and G = 0.04 for cellssurrounding the cut cells (case I). Here, experiment andtheory match quantitatively. As we move away from theparameter values of case I, the calculations and experi-ment begin to differ (for example Figures 5E and 5F).From these calculations, we conclude that there isa small parameter region near case I (outlined in greenin Figure 3) where theorymatches the experiments. Sim-ulations with these parameters also accurately describethe displacement of vertices in the field of cells sur-rounding the cut bond (Supplemental Data and FiguresS6 and S7).By performing simulations of both proliferation and la-

ser ablation, we have been able to estimate the relativemagnitude of the coefficients describing cell elasticity,contractility, and line tension in the wing disc. Thesedata show that line tension L on individual cell bound-aries is positive (i.e., contractility predominates over ad-hesive cell-cell interactions at cell boundaries). Further-more, we find that a general contractility described by

the perimeter elasticity G is also required to accountfor our experimental observations. A perfect hexagonallattice represents a global minimum of the energy func-tion for epithelia with these physical properties; thedisordered packing geometry induced by proliferationreflects a local minimum with a higher value of theenergy function.

Discussion

Many genes that play a role in epithelial junction remod-eling have been identified [3, 9, 20]. Their gene productsinfluence physical cellular properties; this in turn triggerscellular rearrangements. However, the physical mecha-nisms involved in rearrangement of epithelial cell pack-ing are not understood. Here, we describe and quantifyfor the first time the force balances in the apical junc-tional network of an epithelium: the proliferating wingdisc epithelium of Drosophila. We combine experimentand theory to understand how the physical propertiesof these cells influence cell packing geometry andmorphology.To investigate the contributions of cell mechanics,

adhesion, and cortical contractility to the developmentof specific packing geometries, we have developeda vertex model for the junctional network. Configura-tions of the junctional network observed in tissuescorrespond to network configurations for which forcesare balanced. Two different types of relaxed networkconfigurations, corresponding to global minima of anenergy function, occur: a solid-like hexagonal networkand a liquid-like soft network, which is irregular (Fig-ure 1). Because of its liquid nature, this state of thenetwork can be easily remodeled. This suggests thatincreased adhesion (leading to negative boundary ten-sion) might in some instances actually facilitate rear-rangement—it will be interesting to investigate whethersuch a situation occurs in vivo, for example during con-vergent extension.Packing geometries observed in vivo correspond in

our vertex model to local minima of an energy function.We have shown that irregular tissue morphologies inthe wing disc can be accounted for by changes intro-duced by proliferation. We have simulated network

Figure 3. Parameter Regions MatchingObserved Tissue Properties

Properties of cell packing for different valuesof the parameters L and G line tension andcontractility; K is the area elastic coefficientand A(0) is the preferred area of cells. Simula-tions of tissue growth were performed fordifferent parameter values (black dots). Inthe red outlined region, the distribution Pn

of n sided cells in simulations is similar tothe one observed in the wing disk of Dro-sophila (DP < 0.004, see Supplemental Exper-imental Procedures). In the blue outlinedregion, the relative areas of n sided cellsmatch those observed in experiments(DA < 0.07). Comparing simulations of laserablation with experiments, we find quantita-tive agreement in the green outlined region

(DL < 0.07). Yellow dots indicate cases I–V. The straight solid line indicates the transition between the soft lattice and the hexagonal groundstate. It is not possible to simulate proliferation to the left of the dashed red line with our current proliferation algorithm. The hatched regioncorresponds to that shown in Figure 1.

Cell Division and Epithelial Cell Packing Geometry2101

Only  case  I  agrees    with  experiment  

Fahradifar  et  al,  Curr.  Biol  (2007)  

Tissue  growth  in  the  vertex  model  :  relaxa)on  a]er  laser  abla)on  

Fahradifar  et  al,  Curr.  Biol  (2007)  

Tissue  growth  in  the  vertex  model  :  relaxa)on  a]er  laser  abla)on  

Simula)ons  :  Lij,  Ga=Gb=0  

-­‐Exponen)al  relaxa)on  

-­‐Myosin  par)cipates  in  Tension.  

Fahradifar  et  al,  Curr.  Biol  (2007)  

Tissue  growth  in  the  vertex  model  :  relaxa)on  a]er  laser  abla)on  

to the cut boundary can be compared to those obtainedin simulations for estimating parameter values of thevertex model. In the vertex model, we can study the dis-placements that follow the removal of individual bondsin a stable and stationary network configuration. Wecan thus mimic bond-cutting experiments startingfrom a stable network morphology generated by thegrowth algorithm described in the last section. To simu-late laser ablation, we randomly select bonds for re-moval. We set to zero the normalized line tension Lij ofthe cut bond as well as the normalized contractility Ga

of the cells sharing the cut bond. Removal of both theseterms is necessary for achieving quantitative agreementbetween simulations and experiments (see Supplemen-tal Experimental Procedures). We then numerically de-termine the new relaxed network configuration afterthese local parameter changes by using the conjugategradient method described previously. For a given pairof parameters G and L for all other cells, we determinethe resulting distributions of area and perimeterchanges of the pair of cells sharing the removed bonds.The gray points in Figures 5C and 5D show normalizedarea and perimeter changes, with respect to the normal-ized change in bond length, for many different removedbonds in the case that L = 0.12 and G = 0.04 for cellssurrounding the cut cells (case I). Here, experiment andtheory match quantitatively. As we move away from theparameter values of case I, the calculations and experi-ment begin to differ (for example Figures 5E and 5F).From these calculations, we conclude that there isa small parameter region near case I (outlined in greenin Figure 3) where theorymatches the experiments. Sim-ulations with these parameters also accurately describethe displacement of vertices in the field of cells sur-rounding the cut bond (Supplemental Data and FiguresS6 and S7).By performing simulations of both proliferation and la-

ser ablation, we have been able to estimate the relativemagnitude of the coefficients describing cell elasticity,contractility, and line tension in the wing disc. Thesedata show that line tension L on individual cell bound-aries is positive (i.e., contractility predominates over ad-hesive cell-cell interactions at cell boundaries). Further-more, we find that a general contractility described by

the perimeter elasticity G is also required to accountfor our experimental observations. A perfect hexagonallattice represents a global minimum of the energy func-tion for epithelia with these physical properties; thedisordered packing geometry induced by proliferationreflects a local minimum with a higher value of theenergy function.

Discussion

Many genes that play a role in epithelial junction remod-eling have been identified [3, 9, 20]. Their gene productsinfluence physical cellular properties; this in turn triggerscellular rearrangements. However, the physical mecha-nisms involved in rearrangement of epithelial cell pack-ing are not understood. Here, we describe and quantifyfor the first time the force balances in the apical junc-tional network of an epithelium: the proliferating wingdisc epithelium of Drosophila. We combine experimentand theory to understand how the physical propertiesof these cells influence cell packing geometry andmorphology.To investigate the contributions of cell mechanics,

adhesion, and cortical contractility to the developmentof specific packing geometries, we have developeda vertex model for the junctional network. Configura-tions of the junctional network observed in tissuescorrespond to network configurations for which forcesare balanced. Two different types of relaxed networkconfigurations, corresponding to global minima of anenergy function, occur: a solid-like hexagonal networkand a liquid-like soft network, which is irregular (Fig-ure 1). Because of its liquid nature, this state of thenetwork can be easily remodeled. This suggests thatincreased adhesion (leading to negative boundary ten-sion) might in some instances actually facilitate rear-rangement—it will be interesting to investigate whethersuch a situation occurs in vivo, for example during con-vergent extension.Packing geometries observed in vivo correspond in

our vertex model to local minima of an energy function.We have shown that irregular tissue morphologies inthe wing disc can be accounted for by changes intro-duced by proliferation. We have simulated network

Figure 3. Parameter Regions MatchingObserved Tissue Properties

Properties of cell packing for different valuesof the parameters L and G line tension andcontractility; K is the area elastic coefficientand A(0) is the preferred area of cells. Simula-tions of tissue growth were performed fordifferent parameter values (black dots). Inthe red outlined region, the distribution Pn

of n sided cells in simulations is similar tothe one observed in the wing disk of Dro-sophila (DP < 0.004, see Supplemental Exper-imental Procedures). In the blue outlinedregion, the relative areas of n sided cellsmatch those observed in experiments(DA < 0.07). Comparing simulations of laserablation with experiments, we find quantita-tive agreement in the green outlined region

(DL < 0.07). Yellow dots indicate cases I–V. The straight solid line indicates the transition between the soft lattice and the hexagonal groundstate. It is not possible to simulate proliferation to the left of the dashed red line with our current proliferation algorithm. The hatched regioncorresponds to that shown in Figure 1.

Cell Division and Epithelial Cell Packing Geometry2101

Simula)ons  :  Lij,  Ga=Gb=0  

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Figure 1 - Mechanics of local cell rearrangements during tissue elongation in

Drosophila embryos.From the following article

Nature and anisotropy of cortical forces orienting Drosophila tissue morphogenesisMatteo Rauzi, Pascale Verant, Thomas Lecuit & Pierre-François LenneNature Cell Biology 10, 1401 - 1410 (2008) Published online: 2 November 2008

doi:10.1038/ncb1798

(a) Germband at the onset and after about 40 min of elongation. Cell junctions are marked by an E-cadherin–GFPfusion protein and cells outlined in orange were tracked during intercalation. Scale bar, 20 m (b) Duringintercalation, cells lose contacts with anterior (A) and posterior (P) neighbours and gain new contacts along theperpendicular axis. Myosin II is enriched at v-junctions. Intercalation brings two 3-way vertices into contact (bluearrowheads) and produces a 4-way vertex (pink). Expansion of a new junction results in a topological change calledT1 transition. (c) Myosin II is specifically enriched at shrinking v-junctions. Scale bar, 20 m. (c') Relative myosin IIconcentration as a function of junction orientation (mean s.e.m., n = 434 junctions in one embryo) and werenormalized to 1 for 'horizontal' junctions.

© 2012 Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.

partner of AGORA, HINARI, OARE, INASP, ORCID, CrossRef and COUNTER

Figure 1 : Nature and anisotropy of cortical forces orienting :... http://www.nature.com.gate1.inist.fr/ncb/journal/v10/n12/fig_...

1 sur 1 17/05/12 23:14

M  Rauzi,  …,  PF  Lenne,  Nat    Cel  Biol  (2008)  

Tissue  elonga)on  from  oriented  T1  

Tissue  elonga)on  from  oriented  T1  

M  Rauzi,  …,  PF  Lenne,  Nat    Cel  Biol  (2008)  

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M. Osterfield,…, S Shvartsman, Dev Cell (2013)

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The end!