The mechanical basi osf cell rearrangement

Development 109, 373-386 (1990)Printed in Great Britain © The Company of Biologists Limited 1990

373

The mechanical basis of cell rearrangement

I. Epithelial morphogenesis during Fundulus epiboly

MICHAEL WELIKY1 and GEORGE OSTER2*

'Croup in Neurobiology and 2Departments of Molecular and Cell Biology, and Entomology, University of California, Berkeley, CA 94720,USA

•Author for correspondence at Department of Entomology, University of California, 201 Wellman Hall, Berkeley, CA 94720, USA.

Summary

Many morphogenetic processes are accomplished bycoordinated cell rearrangements. These rearrangementsare accompanied by substantial shifts in the neighborrelationships between cells. Here we propose a model forstudying morphogenesis in epithelial sheets by directedcell neighbor change. Our model describes cell re-arrangements by accounting for the balance of forcesbetween neighboring cells within an epithelium. Cellrearrangement and cell shape changes occur when theseforces are not in mechanical equilibrium. We will showthat cell rearrangement within the epidermal envelopinglayer (EVL) of the teleost fish Fundulus during epibolycan be explained solely in terms of the balance of forcesgenerated among constituent epithelial cells.

Within a cell, we account for circumferential elasticforces and the force generated by hydrostatic andosmotic pressure. The model treats epithelial cells astwo-dimensional polygons where the mechanical forcesare applied to the polygonal nodes. A cell node protrudesor contracts when the nodal forces are not in mechanicalequilibrium. In an epithelial sheet, adjacent cells sharecommon boundary nodes; in this way, mechanical forceis transmitted from cell to cell, mimicking junctionalcoupling. These junctional nodes can slide, and nodes

may appear or disappear, so that the number of polyg-onal sides is variable.

Computer graphics allows us to compare numericalsimulations of the model with time-lapse cinemicroscopyof cell rearrangements in the living embryo, and dataobtained from fixed and silver stained embryos. Bymanipulating the mechanical properties of the modelcells we can study the conditions necessary to reproducenormal cell behavior during Fundulus epiboly. We findthat simple stress relaxation is sufficient to account forcell rearrangements among interior cells of the EVLwhen they are isotropically contractile. Experimentalobservations show that the number of EVL marginalcells continuously decreases throughout epiboly. Inorder for the simulation to reproduce this behavior, cellsat the EVL boundary must generate protrusive forcesrather than contractile tension forces. Therefore, thesimulation results suggest that the mechanical propertiesof EVL marginal cells at their leading edge must be quitedifferent from EVL interior cells.

Key words: cell rearrangement, morphogenesis, modeling,mechanics.

Introduction

Cell rearrangement is an important process drivingepithelial morphogenesis in many developmental set-tings, including imaginal disc evagination in Drosophlla(Fristrom, 1976), epiboly in Fundulus (Keller andTrinkaus, 1987), and the almost universal vertebratephenomenon of 'convergent extension', documented inXenopus by Keller (1978, 1984). For a description ofother examples, see Fristrom (1988). In this paper, weshow how a mechanical model can provide insight intothe dynamics of cell rearrangement within an epithelialsheet. We apply the model to the specific case of epibolyin the teleost fish Fundulus heteroditus and show thatnumerical simulations can reproduce the essential fea-

tures of cell shape change and cell rearrangementobserved during normal development.

The model we propose is based upon evidence thatcells are capable of exerting substantial forces upontheir surroundings. For example, moving fibroblastscan exert enough tension to wrinkle a sheet-rubbersubstratum (Harris etal. 1980). Wounds made at andbehind the marginal attachment site of the epithelialenveloping layer in Fundulus cause the epithelium toretract immediately showing that the cell sheet is underconsiderable tension (Trinkaus, 1951). We will showthat cell rearrangement during Fundulus epiboly can beunderstood in terms of forces arising from two sources:(1) mechanical stress resulting from stretching or defor-mation of the cell sheet by external forces and (2)

374 M. Weliky and G. Oster

internally generated protrusive forces generated by aspecifically activated cell subpopulation.

Each cell in an epithelial sheet generates protrusiveand contractile forces and, following the laws of mech-anics, exerts equal and opposite forces on neighboringcells. By coupling a large number of cells together, wecan simulate cell rearrangement in a two-dimensionalepithelial sheet. Once the mechanical force balanceequations are solved, computer graphics allows us tocompare directly the results of simulations with photo-micrographs and time-lapse films of cell shape changesand rearrangements. Most importantly, we can storeand keep track of all forces generated by each simulatedcell throughout all phases of the rearrangement pro-cess. This gives us the opportunity to relate directly theobserved phenomenon of cell rearrangement to theunderlying forces generated by each cell. We thenattempt to reproduce the experimental observations bymodifying the model cell properties. This allows us tohypothesize various scenarios for how different cellsmay generate different forces dependent upon theirpositions within the cell sheet. In so doing, we not onlyvalidate the model but gain insight into the pattern ofcellular force generation that drives cell rearrangementand cell shape change within an epithelial sheet under-going morphogenesis.

The first section discusses those features of epithelialcells that contribute to the generation of forces. Next,these features are incorporated into a mechanicalmodel of an epithelial cell which is capable of generat-ing contractile and protrusive forces. Finally, the modelis applied to the particular problem of epiboly inFundulus; we discuss the mechanical conditions necess-ary to reproduce the normal pattern of cell rearrange-ments and shape changes that are experimentally ob-served.

A mechanical model for cell rearrangement

Sources of mechanical force within an epithelial cellCell motion in vitro is driven by protrusive and contrac-tile forces generated within a cell and exerted upon asuitably adherent substratum (Oster, 1984, 1988; Osterand Perelson, 1988). Cell movement and subsequentcell rearrangement within an epithelial sheet dependupon the ability of a cell to adhere to and exert forcesupon neighboring cells. The cellular cytoskeleton isresponsible for providing the mechanical stability of cellshape and for generating the forces that drive cellprotrusion and contraction. The cytoskeleton consistsof actin filaments, microtubules, intermediate filamentsand a host of associated molecular components. Thesize, distribution and organization of these componentsare under dynamic cellular control. With regard to cellmotility, the most important cytoskeletal componentsare the actomyosin filamentous structures. Indeed,cellular regulation of the cortical actomyosin cytogelaccounts, at least in part, for control of membranedeformability, cell rigidity and cytoplasmic viscosity,and consequently is a major factor in controlling cell

shape changes. For the purposes of this model, we willconsider the cortical actomyosin cytogel as the pivotalcomponent controlling mechanical force generationleading to cellular contraction and protrusion.

ContractionCell contraction arises from the same sliding filamentmechanism responsible for contraction in muscle fibers.Many types of epithelial cells contain circumferentialactin microfilament bundles associated with junctionsthat attach each cell in an epithelium to its neighbors.Isolated circumferential microfilament bundles fromretinal pigmented epithelium form polygonal arraysattached to zonula adherens junctional sites. Uponexposure to ATP, these isolated polygonal networkscontract (Owaribe and Masuda, 1982; Owaribe et al.1981). Therefore, circumferential actin microfilamentbundles provide a means for cell contraction, and theirattachment to zonula adherens junctions allows thisforce to be transmitted to adjacent cells.

ProtrusionProtrusive activity arises within the cell cortex in theform of cellular projections containing actin fibers (i.e.lamellipodia, filopodia). The cortical gel is composed oflong polymer chains of actin which are crosslinked intoa three-dimensional network. The forces generated byan actin gel are the hydrostatic and osmotic pressureswithin the gel, and elastic forces due to the actinpolymer fibers (Oster, 1984). Force generation is de-pendent upon ionic control of the gel's elastic andosmotic properties. The swelling pressure of the gel isthe difference between the hydrostatic/osmotic press-ure, Posm tending to expand the gel and the restrainingelastic pressure, Pctas generated by the gel fibers(Fig. 1A). The gel is in mechanical equilibrium whenthe hydrostatic/osmotic pressure is in balance with theelastic pressure:

P _ p nswell ' osm 'das (1)

By altering the balance between Posm and Pd^ thecortical cytogel can expand and contract. This is ac-complished by ionically regulating solation and gelationfactors as well as actomyosin fiber contractions. (Korn,1982; Weeds, 1982). Solating factors affect the gel in anumber of ways including severing network fibers andbreaking crosslinks between fibers as well as depoly-merizing fibers. Solation has the general effect ofbreaking up the network, which reduces the elasticmodulus of the gel (Fig. IB). Suppose the gel is inmechanical equilibrium, then according to equation (1),if Peias is reduced the result will be a net positiveswelling pressure and the gel will expand. Thus,hydrostatic/osmotic pressure within actin cytogel cantheoretically provide the necessary mechanical force toaccount for local cell protrusion: when solation factorslocally weaken the gel, hydrostatic/osmotic forcescause the cytogel to locally swell. We emphasize thatthe swelling of the cortical gel is a local phenomenon.Thus gel osmotic pressure is different from membraneosmotic pressure: puncturing a membrane osmometer

Epithelial morphogenesis during Fundulus epiboly 375

posm

VOLUME

VOLUME

Gel swells to newequilibrium

Solatlon reduceselastic modulus

Fig. 1. (A) The swelling pressure, Pswen, is the differencebetween the hydrostatic/osmotic pressure, Pojm tending toexpand the gel and the restraining elastic pressure, Petas,produced by the gel fibers. The gel is in mechanicalequilibrium at a volume Vo when the hydrostatic/osmoticand elastic pressures just balance. (B) Solation of the gel bybreaking crosslinks and/or severing gel fibers reduces thegel elastic modulus so that the hydrostatic/osmotic pressurecan swell the gel to a new equilibrium volume Vt.

will deflate the osmotic pressure, but puncturing a gelhas no global effect. This activity may be triggered, forexample, by an ionic leak through the membrane. Inturn, this may be caused by a variety of stimuli such asmechanical stress at adhesion sites, binding of ligands tomembrane receptors, or insertion of new membrane(Oster, 1984, 1988; Oster and Perelson, 1988).

Modeling an epthelial sheetIn most cases, an epithelium is composed of a singlelayer of cells. Each cell is tightly coupled to its neigh-bors by a ring of intercellular junctions which, inaddition to maintaining the cohesion between cells, alsoforms a tight permeability barrier between cells (Fris-trom, 1988). In addition, the membrane boundaries ofindividual cells within an epithelium are, to a goodapproximation, polygonal in shape (Honda, 1978;Keller and Trinkaus, 1987). These features will allow usto model an epithelium as a two-dimensional sheet ofcontiguous, coupled polygonal cells. We begin bydescribing a mechanical model for a single cell. Tomodel an epithelium, it will then only be necessary tomechanically couple a suitable number of cells into acontinuous two-dimensional sheet.

Our mechanical model for a cell must be capable ofboth contractile and protrusive behavior consistent withthe cell mechanisms described in the previous section.This is accomplished by mechanically describing the

corticalactln cytogel

circumferentialmlcrofl lamentbundle

nod*3

nodei nod*4

Fig. 2. (A) Polygonal model of a single epithelial cell. Theforces within a cell are modeled by describing the balanceof nodal pressure and tension. Here, edges are labeled bygiving the nodes they connect as subscripts; e.g. edge 23connects node 2 to node 3. (B) The elastic tension forcesacting on a node are due to the circumferential apicalmicrofllament bundles and the cortical actomyosin gel. Thetangential elastic tension forces around the cell perimeterare resolved into nodal forces as shown; the net tensileforce T2 on node 2 is the vector sum of the edge tensionsT21+T23. (C) Pressure acts outward and normal to thepolygonal cell surface at each node. The nodal pressure, Pi,is the polygonal approximation to the average pressurewithin a local region of the cell cortex. The average corticalpressure acts in a direction normal to the local cell surfacecurvature. Similarly, Pi acts normal to the polygonalsurface curvature at a node, in a direction that bisects thenodal angle.

local balance of elastic forces and pressure withindifferent regions of a cell. Our model is a simplificationof the actual cellular machinery responsible for thegeneration of these forces, but we believe it capturesthe essential mechanical features.

The central features of the model are illustrated inFig. 2. In order to simulate the model on a computer,these features must be translated into a mathematicalform, and Appendix A gives a comprehensive descrip-tion of the model. We begin by representing eachepithelial cell by a two-dimensional polygon (i.e. as acollection of nodes connected by line segments) with a


variable number of sides. At each node we calculate thebalance of mechanical forces.

Elastic tension forcesBy making each polygon side an elastic element,circumferential tension forces encircle the cell per-imeter. The tension can be varied independently indifferent regions of the cell. This represents both thecontractile machinery of circumferential actomyosinmicrofilament bundles and the elastic forces of thecortical cytogel. For instance, we can represent circum-ferential microfilament bundle contraction by increas-ing the elastic tension in some or all sides of the cellularpolygon. Reducing the elastic tension forces modelssolation of the cortical actin cytogel without ac-companying microfilament contraction. We shall rep-resent these elastic forces by a pair of vectors, orientedalong the cell perimeter, applied to each node(Fig. 2A,B). The magnitude of these vectors representsthe local elastic tension at each cell node. The tension ateach node can be independently altered by reducing orincreasing the magnitude of the two elastic vectors.

The net elastic force at each node is the vector sum ofthe two elastic tension vectors at that node (Fig. 2B).Note that the net elastic force vector varies with theangle subtended by the edge tensions. This anglerepresents the local polygonal approximation to the cellsurface curvature. The more acute the angle, the higherthe surface curvature; as the angle approaches 180degrees, the surface curvature approaches zero. Forconvex curvature, the vector sum is directed inward andopposite to the pressure vector force, while for concavecurvature, the net elastic force points outward. There-fore, the elastic forces at a node where the surfacecurvature is concave will tend to push the cell surfaceoutwards and straighten while convex regions willcontract and be pulled inwards. In practice, very fewconcave cell shapes are observed in epithelia.

PressureThe contributions of both cell hydrostatic and osmoticpressure are combined into a single pressure vector, P,which is applied outward and normal to the polygonalcell surface at each node (Fig. 2C). The magnitude ofthe pressure can be varied independently at differentnodes. In this way, the model allows for the localmodulation of osmotic and hydrostatic pressure withindifferent regions of a cell. However, for the simulationsdiscussed in this paper, we have assumed that thepressure is the same at each node, and varies inverselywith the cell area. Thus if the cell area increases, thepressure will fall, which will tend to restore the cell backto its original surface area. In this way, the cell area willtend to remain almost constant.

Cell shape depends on the balance of contractiveand expansive forces

The net force at each cell node is the sum of the nodaltensions and pressure. Thus the shape of the cellboundary reflects the relative balance of the net elasticforce and pressure at each node. If the elastic forces at

O solation expansion © gelation

\protrusionnode

Fig. 3. Cell protrusion is a local phenomenon. Panel Ashows a cellular protrusion, and Panel B shows thepolygonal approximation used in the model. (1) Protrusioncommences when a local site of cortical solation decreasesthe actin gel elasticity, so that the local cortical pressure isgreater than the elastic pressure. (2) The local corticalpressure exceeds the elastic pressure and the cell protrudes.(3) The cortical actin network then regels, and the cyclerepeats (Oster and Perelson, 1988). Over a number ofcycles, the net displacement of the protruding cell surfacewill be in the direction of the local average pressure, whichacts normal to the local cell surface. Therefore, aprotruding cortical region tends to move in a directionnormal to the local surface curvature. In the polygonalmodel, we mimic this cycle by introducing two newprotrusion nodes very close to the protrusion apex andreducing the elastic tensions at the protrusion apex. Thisallows a localized region of the poygonal cell surface toexpand until the local nodal pressure is counterbalanced bythe nodal tensions. The nodal pressure, which is equivalentto the local average pressure in Panel A, acts in a directionthat bisects the nodal angle; that is, the expansion is alwaysnormal to the polygonal surface curvature.

all nodes are equal (i.e. the cell is contracting equally allaround its periphery), the cell will relax to a regularpolyhedral form. For a polygonal cell with a largenumber of faces, the cell will approach a circular shape.The cell shape will remain stable so long as all nodaltensions and pressures are in mechanical equilibrium.The nodal pressure drives cell protrusion, while thenodal tensions drive contraction (Fig. 2). As long as themechanical forces are balanced at a cell node, that nodewill not move. Upsetting the balance of forces at a nodecan lead either to local cell protrusion or contraction.For example, reducing the elastic tension forces at aparticular cell node (e.g. by solation) will cause thatnode to protrude due to the now dominant pressureforces (Fig. 3). Increasing the elastic tension forces at a


node (e.g. by contraction of the actomyosin microfila-ment bundles) will cause that node to retract due to thedominant circumferential elastic forces.

The model epitheliumWe model an epithelial sheet by allowing adjacent cellsto share common boundary nodes (Fig. 4). In this case,a single node may experience forces generated by morethan one cell. In addition to forces generated by cells,external vector forces can be applied to particularcommon boundary nodes if necessary. This allows themodel to simulate stretching and deformation of the cellsheet by forces extrinsic to the epithelium. This will beimportant when we discuss the application of the model

Force balanceat a common vertex

B

Fig. 4. (A) Adjacent cells share a common junctionalvertex node. The node is in mechanical equilibrium whenthe pressure and elastic forces from all cells are balanced.(B) Cell 1 protrudes when the elastic modulus of its cortexdecreases, resulting in an imbalance of forces at thecommon vertex node. Since protrusion is a localphenomenon, we insert two new nodes close to, and oneither side of, the protrusion node; the node then slides inthe direction of the net force imbalance, which in this caseis in the direction of the common boundary of cells 2 and 3(see Fig. 3).

to Fundulus epiboly. The total vector force acting atsuch a node is simply the sum of all forces from the cellssharing this node plus the external forces.

Fig. 4A shows a typical stable configuration of threecells sharing a central common junctional node. Whenthe nodal tensions and pressures are in mechanicalequilibrium, the node position will not change. Fig. 4Bshows that unbalanced forces can result in cell pro-trusion and subsequent cell rearrangement. All nodalpressures and tensions are initially in mechanical equi-librium. Reducing the nodal tension for cell 1 at thecentral shared node results in internally unbalancedforces. For cell 1, nodal pressure now dominates andconsequently the forces between cell 1 and the othertwo cells are not in mechanical equilibrium. There nowexists a net force in the direction of the outwardpressure vector of cell 1. Therefore, cell 1 protrudesforward and the shared junctional node moves in thedirection of the net force. If cell 1 continues to protrudeforward, it will eventually completely separate theother two cells. Note that at protrusion sites weintroduce a pair of new nodes so as to model the localnature of cell protrusion.

Modeling cell rearrangementIn our model, nodes represent the junctional contactpoints between three cells, the nodes move according tothe balance of cellular forces applied to them. Cellrearrangement occurs when two nodes meet (Fig. 5).At this time, one pair of cells previously in contactseparate while the second pair of cells, which werepreviously separate, establish contact, (c.f. AppendixC).

Simulating the force balance equationsIn order to mechanically simulate the model epithelialsheet, we use an iterative technique called the finitedifference method that solves the force balance at everynode during successive time steps. Each time steprepresents a snapshot of all current cell positions and allvector forces (Gould and Tobochnik, 1988). The finitedifference method determines the successive position(xn+l,yn+l) of a node given the position (xnyn) at theprevious iteration and the total vector force (F(x),F(y))applied to that node during the current time step:

xn+I =xn + F (x )—A*

yn+i = yn + F(y)—

(2)

(3)

where At is the time step interval between successiveiterations, tn+i=t,,+At, and JX is the viscous drag force.

In other words, during each iteration a node moves adistance proportional to the total force applied to thatnode. The positions of all nodes are simultaneouslyupdated each iteration. The time scale factor At must bevery small in order to generate a stable solution,otherwise, oscillations and instabilities may develop;Appendix D gives a more comprehensive discussion ofthe numerical simulation of the force balanceequations.


Fig. 5. All cell rearrangements can be constructed from thecanonical situation illustrated above. The pair of cells (cells2 and 4), which are separated in the initial configuration(A), establish contact with one another. The remaining pair(cells 1 and 3), which are initially in contact with oneanother, separate in (B). Cell rearrangement occurs whentwo nodes meet, and the junctions 'change allegiance'.Here, nodes A and B move toward one another in (A).During junctional rearrangement, new nodes C and D arecreated which subsequently separate (C). In the processedge AB has shortened and vanished, to be replaced byedge CD. The insert shows how the net force imbalance atnode B causes the node to slide to the left (c.f. Section:Interior EVL cell rearrangement).

Application of the model to eplboly in Fundulus

The pattern of cell rearrangements and cell shapechanges that occur during epiboly has been well docu-mented by Trinkaus and coworkers for the teleost fishFundulus (Keller and Trinkaus, 1987; Betchaku andTrinkaus, 1978) and by Kageyama for Oryzias latipes(Kageyama, 1982). In Fundulus, the enveloping layer(EVL) is an epithelium that begins as a monolayeredcell sheet covering the upper portion of the animalhemisphere (Fig. 7A). It is attached at its margin to theyolk syncytial layer (YSL) which lies underneath.During epiboly, the YSL and EVL together expand andspread over the surface of the egg. Much of theexperimental evidence indicates that all cells of theEVL, including marginal cells, are under tension. Forinstance, a slit made through the EVL gapes andwidens. Moreover, no evidence for protrusive marginalcell activity could be found during normal epiboly. Itappears that the EVL is passively pulled down over thespherical egg by an active contraction wave in the YSLwhich spreads slightly ahead of the EVL. Spreadingceases with the closure of the yolk plug at the vegetalpole when the EVL completely covers the egg surface.

As the EVL spreads vegetally, it conforms to thespherical geometry of the egg. This means that, in theanimal hemisphere, the EVL margin must lengthen toaccommodate the increasing circumference as it spreadstoward the equator. Once past the equator, the marginnow must decrease as it spreads toward the vegetal pole(Fig. 7C). Surprisingly, the experimental observationsshow that the number of marginal cells continuallydecreases throughout all stages of epiboly (Fig. 6). It ispuzzling that the number of marginal cells decreaseseven in the animal hemisphere where the EVL marginis increasing in circumference. Our simulations havebeen able to reproduce this behavior only when certainmechanical conditions are satisfied. This has led us topropose a mechanical explanation for this phenom-enon.

Initial conditions for the simulationWe have begun our simulations of EVL expansion at

5 0 -

1 \\

0 90 180

Stage of epiboly In polar angle (degrees)

Normalized data fromKeller & Trinkaus (1987)

Marginal nod* forces nInterior node forces.

Marginal node forces =18 % Interior node forces.

Marginal nod* forces =12 % Interior node forces.

Fig. 6. Changing the circumferential elasticforce in EVL marginal cells affects the rate atwhich cells leave the EVL/YSL boundary.When the initial circumferential force is equalto that of interior cells, marginal cells arealways in tension around their entireperimeter, including their leading edges, andthe number of marginal cells does not changeuntil near the very end of epiboly. As theelastic force is reduced at nodes along theEVL/YSL boundary, the marginal cellsgenerate a net "protrusive' mechanical force.This condition of reduced cortical tension isnecessary to cause marginal cell retraction

away from the EVL/YSL boundary. When the circumferential force is quite low, the simulation results match theexperimentally observed data. However, when the circumferential force is too low, the marginal cell protrusive force is sostrong that the rate of marginal cell reduction is much faster than is experimentally observed.

Fig. 7. The model reproduces the essential features of Fundulus epiboly. Color represents the balance of pressure andcircumferential elastic tension forces within each cell. The cell force is calculated by summing the pressure and elastic forces at eachcell node. When elastic forces dominate at a node, the cell is 'contractile' in that region. When pressure dominates at a node, thecell is generating 'protrusive' force there. Increasing outward 'protrusive' force (+) is represented by dark blue to purple. Increasinginward 'contractile' force is represented by light blue to red. (A) Early stage of epiboly. (B) Middle stage of epiboly. (C) Late stageof epiboly. (D) End of epiboly.

Fig. 8. The pattern of marginal cell rearrangements is shown during the middle stage of epiboly. Cells a and b leave the EVLmargin by shortening their marginal boundary while adjacent cells establish contact. Cell a retracts from the margin in (B) while cellb pulls away in (C) The portions of marginal cells along the EVL marginal boundary are purple showing that they are 'protrusive'.The interior portions of these cells as well as the other fully interior cells are in various states of tension shown by the range ofgreen to red shades of color. The upper and lower tapered ends of elongated cell c become progressively red indicating increasingcontractile force.Fig. 9. The pattern of marginal and interior cell rearrangements during the later stages of epiboly. Cells a and b narrow theirmarginal boundaries before retracting away from the EVL margin. Cell a leaves the margin in (B) while cell b pulls away in (C).Interior cell rearrangement is biased such that cell pairs along the animal-vegetal axis (c and d, e and f) separate allowing medio-lateral cells to establish contact. Cells c and d separate in (B) while cells e and f separate in (C).


the equivalent of stage 14 of normal embryonic devel-opment. We chose this as our starting point since celldivision ceases at this stage (Betchaku and Trinkaus,1978) (cell division is easy to incorporate into themodel, and will be reported on in a subsequent publi-cation). The EVL is a circular cap composed of about3000 cells covering the animal hemisphere of the egg,with its margins at about a 45 degree polar angle to theanimal pole. In order to conserve computational time,we model the EVL using only 300 cells, since simu-lations with larger numbers of cells show that this doesnot introduce any artifacts into the results. Inspectionof stage 14 embryos reveals that EVL cells appear to bearranged in a roughly regular polyhedral packed array.Voronoi polygons have been shown to closely approxi-mate the regular polyhedral packing of cells within anepithelium (Honda, 1978); thus we established theinitial distribution of cell shapes from a random distri-bution of 300 Voronoi polygons of nearly equal area.Appendix B describes how Voronoi polygons are con-structed.

In the model, all nodes of non-marginal cells areassigned equal initial values for their circumferentialtension forces. The nodes of marginal cells that do notlie on the EVL boundary are assigned the same initialcircumferential tension force as non-marginal cells.Nodes of marginal cells that do lie on the EVLboundary are assigned equal, but reduced, circumfer-ential tension forces; that is, they have a net positive gelswelling pressure that produces an outwardly directedprotrusive force. The spreading of the EVL is driven bya contraction wave within the YSL that propagatesvegetally. The YSL mechanically pulls on the marginsof EVL boundary cells where they are junctionallyattached to the YSL. Inspection of embryos reveals thatthe EVL marginal cell ring remains roughly a planecircle (Keller and Trinkaus, 1987). During each ite-ration of the simulation, all cell nodes that lie along theEVL boundary move a specified amount toward thevegetal pole. The velocity at which the margin advancesis not constant throughout all stages, but constantlyaccelerates (Betchaku and Trinkaus, 1978). There is a2.5 times increase in the marginal velocity during thelater stages of epiboly than at earlier stages. We haveincluded this factor into the model simulations.

Cells are constrained throughout the simulation tomove upon the surface of a three-dimensional sphererepresenting the shape of the egg upon which the EVLspreads. Cells are still treated as two-dimensionalpolygons, but they generate three-dimensional forcevectors as they move upon the curved surface of thesphere. In reality, the egg is not rigid, but deformssomewhat as epiboly proceeds; however, we haveneglected that effect in our simulations and taken thesphere as rigid.

The model reproduces the essential features ofFundulus epibolyThe model simulations are able to reproduce theessential features of epiboly including cell shapechanges, reduction in the number of marginal cells, and

the dynamic pattern of cell rearrangements (c.f. Kellerand Trinkaus, 1987, and Betchaku and Trinkaus 1978for the original experimental work).

Early stage of epibolyCells are arranged in a roughly regular polyhedralpacked array (Fig. 7A). The blue color of most cellsshows that they are under mild tension (Fig. 7 describesthe stress color coding system we have used). Note thatthe portions of marginal cells that lie along the EVLboundary are colored purple which indicates that theyare in a protrusive state.

Middle stage of epibolyThe color coding in Fig. 7B shows that there is a stressgradient from the EVL boundary to the animal pole.Cells closer to the EVL boundary experience a muchlarger mechanical stress (indicated by red) than cellscloser to the animal pole (indicated by yellow/green).This is due to the rate at which mechanical stress can bedispersed through the sheet relative to the strength ofthe external force being applied to the EVL boundary.If the rate at which cells are able to relax stresses byrearrangement and shape change is slow relative to thespeed at which the sheet is deformed by the contractionwave, then mechanical stress will build up in cells closerto the site of the external force faster than it can bedispersed throughout the sheet. It will take cells furtheraway from the boundary longer to experience themechanical stress as it spreads through the sheet fromthe site of its application at the EVL boundary. Theslower the sheet is deformed, the more evenly will stressbe distributed throughout the sheet, since the rate ofstress dispersion will be relatively faster than the rate ofstress buildup near the site of the applied externalmechanical force at the sheet boundary (one can thinkof stress as a conserved quantity that diffuses from thesource at the EVL boundary down the stress gradienttowards the animal pole). Fig. 8 shows marginal cellrearrangement during this stage of epiboly.

Late stage of epibolyInterior EVL cells in the lower vegetal hemisphereelongate perpendicular to the EVL boundary whilecells in the upper animal hemisphere remain in a morecompact regular hexagonal packing (Fig. 7C). EVLmarginal cells also begin to change shape: at first theyare elongated parallel to the EVL boundary, buteventually they become elongated perpendicular to theboundary. These shape changes are primarily due to thespherical geometry, since cells in different regions ofthe EVL sheet experience different proportions ofazimuthal and circumferential stresses. Cells in both theupper animal hemisphere and lower vegetal hemisphereare being stretched along the animal-vegetal axis andare moving downwards toward the vegetal pole. How-ever, cells in the upper animal hemisphere move towardthe equator and therefore are moving from a region ofsmaller to larger circumference. Therefore they experi-ence progressively larger 'hoop stresses' in the circum-ferential direction that tend to balance the azimuthal


Fig. 10. Line tracings of cell boundaries located in the EVLmarginal region taken from a sequence of 35 mmmicrographs of a living Fundulus embryo. The lower end ofcell b lies along the EVL margin. Cell b narrows itssubmarginal boundary with cell f (A-C) until they separatein (D). Cell b narrows its marginal boundary as it retractsaway from the EVL boundary (A-D). (From Keller andTrinkaus, 1987.)

stress imposed by the contraction wave. Thus they willbe stretched in two orthogonal directions and will tendto increase in surface area in a fairly isotropic manner.In contrast, cells in the lower vegetal hemisphere will bemoving from regions of larger to smaller circumference.Therefore, their medio-lateral stresses are beingrelaxed, while the azimuthal stresses remain strong.This will tend to elongate these cells perpendicular tothe EVL boundary. Fig. 9 shows marginal and interiorEVL cell rearrangement during a late stage of epiboly.Compare this sequence of frames to that of Fig. 10which shows cell rearrangement at an equivalent stagein the living embryo (Keller and Trinkaus, 1987). Notethat in both the simulated and actual sequences, inter-ior EVL rearrangement occurs in a manner that leads tothe separation of pairs of cells that lie along theanimal-vegetal axis (cells C and D, and E and F inFig. 9) and subsequent establishment of contact be-tween pairs of cells along the medio-lateral axis.

End of epibolyYolk plug closure occurs at the vegetal pole as the EVLcompletely covers the egg. In Fig. 7D note the redstreaks that are parallel to the animal-vegetal axis insome of the more elongated cells in the vegetal hemi-sphere. These are due to the large contractile forcesassociated with the vertical orientation of these cells. InFig. 9, cells C, E and F relax their circumferentialtension after they separate from neighboring cells along

the animal-vegetal axis. This is indicated by a change intheir overall color: note that these cells change fromred/yellow, indicating high tension, to yellow/green,indicating reduced tension.

The mechanical conditions necessary toreproduce cell rearrangement and cell shapechange

Interior EVL cell rearrangementAll interior sub-marginal EVL cells have strong circum-ferential contractile tension forces. As the EVL ispulled down over the egg, the cells tend to elongateperpendicular to the EVL/YSL boundary in the lowervegetal hemisphere. The circumferential forces try torestore each cell shape to a more isodiametric circularform.

Fig. 5 shows a typical group of interior EVL cellsundergoing cell rearrangement. Elongation of cells 1and 3 decreases the angle between circumferentialtension vectors at node B, since this node is located atthe tapered ends of these cells. For cell 2, the angleincreases as the cell elongates, since this node lieswithin the flatter middle region of the elongating cell.Smaller angles between edges (i.e. high surface curva-ture), generates a larger inward net elastic force vector,which is the sum of the two circumferential elasticvectors. Therefore cells 1 and 3 will generate a largernet elastic force at node B than cell 2. Adding togetherthe vector forces from all cells produces a net vectorresultant to the left; consequently node B will move tothe left in the direction of the net force imbalance. Thesame effect causes node A to move to the right. As aresult, the upper and lower ends of the elongated cellsin the middle will contract pulling the two medio-lateralcells on either side towards one another. Eventually,the two medio-lateral cells, 2 and 4, establish contactwith one another while the central elongated cells, 1and 3, that were originally in contact separate.

Cell rearrangement at the EVL/YSL boundaryIn order for the simulation to reproduce the observedreduction in marginal cell number, the properties ofEVL marginal cells must be different from those of theinterior EVL cells. As Fig. 6 shows, it is necessary toreduce the circumferential elastic tension force at thenodes of cells that lie on the EVL boundary. Thebalance between the pressure and circumferential elas-tic tension forces at these nodes must be such that thereexists a net outward directed protrusive force.

Marginal EVL cell attachment to the YSL preventsthese cells from retracting except by rearrangement.Cell nodes along the EVL margin can only move medio-laterally (or horizontally in Fig. 11 and Fig. 13). This isdue to the strong junctional adhesions that keep mar-ginal cells firmly attached to the YSL, and preventsthese cells from moving in a direction along the animal-vegetal axis away from their point of attachment to theYSL. Therefore, only the horizontal component ofvector forces generated by marginal cells need be


YOLK SYNCYTIAL LAYER

YOLK SYNCYTIAL LAYER

Fig. 11. Reduced circumferential elastic forces in boundarycells cause cells to intercalate away from the boundarythroughout epiboly. Cell protrusion along the EVL marginis driven by the dominant internal pressure forces at eachnode along the margin. The horizontal component of thisvector force is larger in a cell that makes an acute anglewith the EVL margin. Cell nodes move in the direction ofthe resultant force imbalance.

considered. Looking at Fig. 11, note that cell 3 makesan acute angle with the EVL boundary while cell 2makes a larger obtuse angle with the boundary. Fromthe figure, it can be seen that the outward directedpressure vector of cell 3 points in a more horizontaldirection than that of cell 2. Therefore, the vector forceof cell 3 has a larger horizontal component than that ofcell 2. If the two vectors are added, the sum will be a netvector resultant directed towards the left, in the direc-tion of the pressure vector of cell 3 with the dominanthorizontal component. Consequently, the junctionalnode shared by cell 3 and 2 will move to the left. Fromthis analysis, it appears that cells making an acute anglewith the EVL/YSL boundary tend to encroach uponother cells making a more obtuse angle. Analysis oftime lapse photography shows this also to be the case inthe living embryo; a cell pulls away from the boundary

Fig. 12. A comparison of EVL marginal cell boundaryshapes during the middle stage of epiboly. Marginal cellsare darkly shaded, interior cells are more lightly shaded.(A) Tracing of marginal cell boundaries from a fixed andsilver-stained Fundulus embryo (Keller and Trinkaus, 1987).Marginal cells make random angles with the EVLboundary. Some marginal cells elongate parallel to theboundary. (B) Result from model simulation. When'protrusive' forces exist at cell nodes along EVL margin,cells make random angles with the EVL margin. This issimilar to the actual distribution of cell boundary shapes inthe real embryo. Some marginal cells elongate parallel tothe boundary as seen in the real embryo. (C) Result frommodel simulation. When marginal cells are exclusively intension, including nodes along the EVL margin, cells onlymake 90 degree angles with the EVL margin. This does notmatch the random angles seen in the real embryo. Inaddition, none of the marginal cells elongate parallel to theboundary.

when the two neighboring cells extend toward oneanother and meet.

The shapes of marginal cells do not appear normalwhen net circumferential tension forces exist at EVLboundary nodes. Under these conditions, notice thatcell boundaries of marginal cells adjacent to the EVLboundary are all aligned perpendicular to the EVLboundary (Fig. 12C). The normal condition shows amixture of randomly oriented cell boundaries(Fig. 12A,B). This can be understood by analyzing themechanical forces generated by these exclusively con-tractile marginal cells. In Fig. 13, cell 3 makes a smallerangle with the EVL boundary than cell 2. Thus, the netelastic force generated by cell 3 at the circled node islarger than that of cell 2. Since marginal cell nodes canonly slide medio-laterally along the EVL/YSL junc-tional attachment zone, the force resultant is a vectortowards the right in the direction of the larger net elasticforce generated by cell 3. When both cell 2 and cell 3make a 90 degree angle with the EVL boundary, the netelastic force generated by each cell is equal. The node isnow in mechanical equilibrium and stable.


RESULTANTFORCE RESULTANT > 0.

B

Fig. 13. Increasing the circumferential elastic forces in EVLboundary cell nodes results in marginal cells that make 90degree angles with the EVL/YSL boundary.(A) Unbalanced forces exist at a node when cells subtendunequal angles with the EVL boundary. The circled nodemoves in the direction of the resultant vector force. (B) Thecircled node is in mechanical equilibrium and stationarywhen both cells subtend equal angles with the EVLboundary.

Discussion

Models have been frequently used in conjunction withexperimental observations to provide insight into theunderlying factors that may drive cell rearrangements.They are especially helpful to corroborate theories in aquantitative manner or when these factors are notamenable to current experimental techniques. Forexample, Mittenthal and Mazo (1983) investigateddifferential cell adhesion and bending stress as factorsin imaginal disc evagination. Honda etal. (1982) studiedcell rearrangement during corneal epithelium woundhealing, based upon the principle of boundary shorten-ing, and Sulsky (1982) modeled cell reaggregationarising from differences in surface tension between celltypes.

When attempting to formulate mathematical modelsof biological phenomenon, one must be aware that themodel will be a simplification, or abstraction, of the realsystem. Here, one such simplification involves themodeling of cells as two-dimensional polygons ratherthan as three-dimensional volumes. Experimental ob-servation in Fundulus shows that at the beginning ofstage 14, cells are quite thick. As the surface area ofeach cell increases during the expansion and spreadingof the EVL, the cells' thickness reduces dramatically(Betchaku and Trinkaus, 1978), and by the end ofepiboly, each cell is extremely thin. Since cell volume isapproximately conserved, we have allowed the cellsurface area to increase when the cell is under tensioncorresponding to the assumption that increasing area isachieved at the expense of cell thickness.

Our model also does not explicitly address the issueof how intercellular junctions are modulated during cellrearrangement. We have simply assumed that junc-

tional rearrangements accommodate the balance offorces at cell contact sites. Though active junctionalremodeling may play an additional role during cellintercalation and rearrangement, we did not find itnecessary to include this in the model in order toreproduce the essential features of Fundulus epiboly.

Other animal species undergo a developmental pro-cess similar to epiboly in Fundulus. For example, cellsat the leading marginal edge of the chick blastoderm(equivalent to the Fundulus enveloping layer), activelycrawl over the embryonic surface (New, 1959; Trinkaus,1984b). These cells provide the driving force thatstretches and deforms the epithelial sheet. No directevidence exists for active marginal cell crawling duringFundulus epiboly; for instance, no lamellipodial projec-tions or other such cell organelles associated with activecell motility have been observed in marginal cells(Betchaku and Trinkaus, 1978). On the other hand, ifthe EVL marginal cell attachments with the YSL aresevered, the EVL first retracts due to mechanicaltension within the cell sheet (Trinkaus, 1951). Sub-sequently, the marginal cells will begin to actively crawltowards that site of previous attachment with the YSLand will cease this behavior when they reach the site.This was believed to be evidence for active cell motilitydriving epiboly. However, it is characteristic of epi-thelial cell sheets that cells at an exposed wound marginexhibit active cell motility even though this may not bepart of their normal behavioral repertoire in the undis-turbed epithelium (Honda et al. 1982). Thus the pres-ence of protrusive activity in severed EVL cells could bean artifact of wound healing behavior and not anintegral mechanism necessary for normal epiboly. Wehave shown that in order to reproduce normal cellshape changes and cell rearrangements at the EVLboundary, the properties of marginal cells in oursimulation must be such that they generate net protru-sive forces at nodes along the boundary. That is, theircircumferential tensions at the leading margin must besmaller than at interior boundaries. This appears tosupport the hypothesis that marginal cells do generateactive cell tractions that drive marginal cell rearrange-ment during normal Fundulus epiboly. Alternatively,unlike interior cells, EVL marginal cells are attached tothe YSL all along their leading edge. These attachmentstransmit the stress generated by the contraction wave inthe YSL. This arrangement may supply an externalprotrusive force component to the leading edges of themarginal cells that is distinct from internally generatedprotrusive forces. Since our model applies forces only atnodes, and not along edges, we cannot distinguishbetween these two possibilities.

Our simulation study shows simple stress relaxation issufficient to account for cell rearrangement amonginterior EVL cells. Cells simply rearrange so as toregain their original isodiametric form. While we havemodeled this as a passive process, we cannot excludethe possibility that cells may also actively move in such away as to redistribute internal stress. Indeed, Betchakuand Trinkaus (1978) observed long cellular projectionsamong interior EVL cells extending under adjacent


cells in the direction of the EVL boundary. While theremay be active cell interdigitation, this is not necessaryto explain the observations.

The importance of mechanical stress in guiding orinfluencing cell movement during embryonic develop-ment is still unclear. In vitro experiments show arelationship between cell motility and mechanical ten-sion, such as tension-induced inhibition of protrusiveactivity (Kolega, 1986), tractional structuring of col-lagen fibrils by fibroblasts (Stopak and Harris, 1982),and tension-induced cell spreading within an epithelialsheet (Takeuchi, 1979, 1983). Beloussov et al. (1975)studied the relationship between mechanical stress andmorphology in developing Rana temporaria embryos bydissecting embryos and observing the shape alterationsof the various parts. By analyzing these deformations,he constructed maps of mechanical stress, and showedthat, during several periods, redistribution of the stresspatterns occurs.

During Fundulus epiboly, submarginal cells near theEVL boundary have been observed to elongate parallelto the margin along with the marginal cells (see Fig. 6,Keller and Trinkaus, 1987). If these cells were beingpassively drawn out, they would elongate perpendicularto the margin, or as seen in our simulation, at leastremain somewhat isodiametric in shape. Similar cellshape changes have been observed in spreading cul-tured cornea epithelia. Cells located a few cell diam-eters away from the cornea epithelial margin elongateparallel to the margin, but do so only when marginalcells can adhere to the substratum and spread, expand-ing and stretching the epithelium (c.f. Fig. 7, Takeuchi,1983). Submarginal cell elongation is not observedwhen marginal cells cannot adhere to and spread uponthe substratum. Under this condition, the marginal cellsnow elongate parallel to the epithelial boundary, appar-ently spreading laterally upon their neighbors (c.f.Fig. 5, op. cit.). Cells deeper within the epitheliumretain a more typical regular polyhedral shape in allcases. Takeuchi hypothesizes that submarginal cellsnear the corneal epithelial border become elongatedbecause they are stimulated to change their shapeactively. Driven by marginal cell spreading, whichexpands the epithelial sheet, submarginal cells arestretched and the resulting increased mechanical ten-sion forces trigger the submarginal cells to change froma stationary to an active motile state.

Similarly, as the EVL is pulled down over theembryonic yolk sphere by YSL contractions duringepiboly, cell stretching and increased mechanical ten-sion forces could, as Takeuchi hypothesizes, stimulateprotrusive activity of both EVL marginal and submar-ginal cells, causing them to change their shape andelongate. The color coding in Fig. 7B shows that thereis a stress gradient from the EVL boundary to theanimal pole. Cells near and at the EVL boundary aresubjected to higher tension forces (indicated by red)than more interior cells within the epithelium (indicatedby yellow/green). This could confirm Takeuchi's hy-pothesis that tensile stress can induce active cell shapechange and motility. Since our simulation shows that

the highest stress is experienced by cells at and near theEVL boundary, these cells would be expected to beactivated first and would show behavior that deviatesfrom cell shape changes due to passive stretching of thesheet.

Though in our simulation, only EVL marginal cellsare actively generating protrusive mechanical forces,we could next try to make motility dependent upon thedegree of tensile stress experienced by each cell. In thiscase, marginal cells would have identical initial mechan-ical properties as interior cells of the EVL sheet. Thehigh mechanical stress experienced by cells at and nearthe EVL margin would stimulate them to change to amotile state. We would see if this is sufficient toreproduce submarginal cell elongation parallel to theEVL, as well as the marginal cell behavior discussed inthis paper. This suggests an interesting experimentaltest: inhibiting the YSL contraction wave (e.g. withcytochalasin) should prevent the spreading of the EVL.This would have the result of relieving marginal cells,and cells near the margin, from the high mechanicalstress normally experienced during epiboly as the EVLis stretched and deformed. Under these conditions, willmarginal cells still rearrange and will submarginal andmarginal cell elongation still be observed?

A second experiment could test the model's predic-tion of a gradient of cell shape from the animal captoward the vegetal pole (Fig. 7C,D). Cells in the animalhemisphere are isodiametric in shape while cells in thevegetal pole are narrow and elongated. This phenom-enon results from two mechanisms: (1) cells in the lowervegetal hemisphere are moving from regions of largerto smaller circumference which tends to elongate andnarrow their shape along the animal-vegetal axis; (2)the EVL margin is drawn at a rate faster than cells can'flow' to equilibrate stress due to viscous resistance.Inhibiting YSL contraction, and subsequent EVL mar-ginal advancement, during the later stages of epibolyshould allow stress to equilibrate within the tissue andlead to the gradual disappearance of the gradient of cellshapes.

Finally, the epithelial model presented here permitsus to investigate cell rearrangements in a variety ofsettings. In a future publication we will report onstudies of epithelia whose morphogenesis is driven byactive and directed cell rearrangements, rather thanpassive stress relaxation of externally applied forces.

Appendices

(A) Modeling the forces in an epithelial sheetOur model epithelium consists of a sheet of2-dimensional polygons whose sides are shared byneighboring cells. The polygonal nodes are placed ateach apex, which forms the junction between threecells. Each polygon has a variable number of sides andnodes can slide so that the length of a side is alsovariable. To turn these finite elements into a mechanicalbody, we assign elastic tensions to each side, and aninternal pressure to each polygonal area. In this Appen-


dix, we give the details of how these forces arecomputed. A complete program listing is availableupon request.

Elastic tension forcesThe elastic tension forces at cell node, n, are rep-resented by a pair of two-dimensional vectors, T(,,,„_/)and T(n n + / ) , applied to the node. These vectors aredirected along the line segments connecting this node toeach of the two adjacent nodes, where T(n,,_7) denotesthe vector directed towards the node in the counter-clockwise direction and T(,,,,+7) towards the clockwisenode (Fie. 2B). We will use the two-dimensional unit

tnode (Fie. 2B). We will use the two-dimensional unitvectors TV,,,,,_7) and t(n,n+7) to describe the tensionvectors' direction. We can describe the elastic vectorforces applied to each cell node as follows

T(rt,n-/) = T(,,,,,_/) X kcias X P (1)

T(n.n-/) = T(n „_/) X ke/as X P (2)

Here P is the cell perimeter, and fce/as is the elasticmodulus of the cell cortex (i.e. the circumferentialmicrofilament bundle and the cortical actin gel matrix).Larger values of kc/as reflect increasing contractionand/or density of microfilament bundles, while lowervalues reflect solation, which reduces the elastic modu-lus of the cortical actin gel matrix. Increasing the cellperimeter, P, stretches cortical fibers thus increasingtheir elastic restoring forces.

PressureThe pressure at each node is represented by the two-dimensional vector P directed outward from the cellboundary and bisecting the angle between the two linesegments joined at a node (Fig. 2C); that is, thepressure acts normal to the polygonal surface at eachnode. The two-dimensional unit surface normal vectorP describes the direction of the nodal pressure, and itsmagnitude varies inversely with the area of the cell:

PP = constant x (3)

cell area

The net force at a nodeThe net force acting at a cell node, n, is the vector sumof the nodal tensions and pressure:

(4)— P + T („,„_/) + •1 Jl+J) •

The total force acting at an epithelial node is the sumof all net forces from all cells snaring this node plus theexternal forces:

N

F = (5)

where N is the number of cells sharing the commonjunctional node.

(B) Tessellation of a plane into Voronoi polygonsThe starting configuration of cell shapes and sizes isdetermined by constructing Dirichlet, or Voronoi, poly-

Voronot polygon

Perpendicular bisector

Fig. 14. Construction of Voronoi polygons. First a set ofpoints is laid down on the plane (these could correspondroughly to the location of cell nuclei). From each point acircular wave propagates out with the same constant speed.Where the wavefronts collide a chord is drawn between thetwo intersection points; this is the perpendicular bisector ofthe centerlines. The bisector lines are extended to form apolygon about each point. The set of polygons so generatedconstitute a Voronoi covering of the plane.

gons for a set of points distributed in a plane. Thistechnique has been used by several workers to modelepithelial cell packing, and appears to be quite realisticwith respect to the overall geometry (Honda, 1983).The idea behind this construction is shown in Fig. 14.There are several methods of calculating the polygons(e.g. Lord and Wilson, 1984; Honda, 1978, 1983); wehave employed the following algorithm to set up theinitial distribution of epithelial cells. In fact, the initialcellular tessellation is not very important in determiningthe final tissue shape.

We begin by setting down a number of points equal tothe number of desired cells; these can be thought of asthe location of cell nuclei. The points are distributed insuch a way that each point is roughly equidistant fromall nearest neighbor points. A Voronoi region for apoint P is a polygon such that every point in its interioris closer to P than to any other point in the set (Lord andWilson, 1984). This tessellates the plane into convexpolygons where each polygon represents a cell. Acomputer program generates the initial cell configur-ation by tessellating a plane composed of a grid of 512by 512 elements into regions. First, points are distrib-uted within the plane according to the criteria above.Next, the distance from each grid element to each pointis calculated; the search is confined to a circular areawith a radius about twice the average nearest point-to-point distance, centered upon the current grid element.The current grid element is tagged with the identifyingnumber of the closest point. In this way, polygonalregions are constructed around each point: a Voronoiregion is composed of all grid elements that are closer topoint P than to any other point in the plane.


B

Fig. 15. (A) When nodes a and b move closer than athreshold distance to each other node exchangecommences. (B) New nodes, c and d, are created on theperpendicular bisector of line ab. (C) Cell 1 now takes nodec as its apex, and cell 2 takes node d as its apex. Thus cells3 and 4 now share edge cd.

(C) Algorithm for cell junction rearrangementCell rearrangement occurs when two nodes are lessthan a minimum threshold distance apart (the thresholddistance is made very small relative to the size of thecells). When this situation occurs, the following algor-ithm is applied:

1. First determine the pair of cells that are common toboth nodes A and B and the remaining two cells that areindividually junctioned at one or the other node(Fig. 15A). In this case, cells 1 and 2 have nodes A andB in common, while cell 3 is junctioned only to node Aand 4 to node B.

2. Calculate new nodes C and D (Fig. 15B). Placenode C and D on either side of edge AB such that theline segment connecting nodes C and D is the perpen-dicular bisector of edge AB. The distance betweennodes C and D is made slightly larger than the minimumthreshold distance for determining if cell rearrangementshould occur.

3. Remove nodes A and B.4. Establish junctional connectivity at new nodes C

and D (Fig. 15C). The two cells that were previouslycommon to nodes A and B now are individuallyjunctioned to the appropriate node, either C or D. Inthis case, cell 1 is junctioned at node C (since node Clies in the direction of cell 1 relative to edge AB inFig. 15B) and cell 3 at node D. Cells that wereindividually junctioned to either nodes A or B are nowcommonly junctioned to both new nodes C and D.

Note that this algorithm begs the question of thedetails of how junctions 'change allegiance', a processwhich remains unclear at the molecular level (Fristrom,1982). When the mechanics of juctional redistributionare elucidated it will be possible to incorporate this intothe neighbor change model.

(D) Numerical solution of the equations of motionThe configuration of the model epithelium is specifiedby giving the location of each node. Therefore, theequations of motion describing the changing configur-ation have the form

(6)

where (i is a viscous coefficient-,-x is the position vectorof the nodes, and F(x) is the vector of forces acting onthe nodes. In writing this, we have assumed that inertialforces are negligible (Odell et al. 1981). This is a set ofcoupled differential equations where the number ofequations is equal to the number of nodes in oursimulation, which can be many hundreds. Since theequations are nonlinear, an analytical solution to thissystem is not possible, so we must apply numericalmethods.

The standard technique for the numerical solution ofdifferential equations involves the conversion of thedifferential equation to a finite difference equation. Weused the Euler-Cromer algorithm or 'last point' ap-proximation for solving the equations of motion (Gouldand Tobochnik, 1988). This is an iterative procedurewhich determines the successive position \(,+/) of anode at time t+1 given the position x, at the previousiteration and the calculated change in position Ax,:

x,+/ = x, + Ax,

where Ax, is calculated from eqn.(6) as:

Ax = F(x)—

(7)

(8)

and At is the time step between successive iterations.Since we are dealing with a sheet of cells which can

deform in three- dimensions, we must express theposition of each node in terms'of a three-dimensionalCartesian coordinate system .fixed at the center of theembryonic yolk sphere. The equations of motion thentake the form:

x,+/ = x, + F(x)—

y,+7 = y, + F(y)—

z,+ , = z, + F(z)—

(9)

(10)

(11)

These equations relate the successive positions of anode to the total force applied to that node at each timestep. Thus at every time step, the new position of eachnode is determined by first calculating the total vectorforce applied to that node. This vector is then multi-plied by the time scale, At divided by the viscosity, /z.The result is the displacement of the node during thecurrent iteration. The new position is then calculated byadding the newly determined displacements to theprevious position. The positions of all nodes are simul-taneously updated each iteration. The time scale Atmust be chosen to be very small in order to generate astable solution, otherwise, oscillations and instabilitiesmay develop. More sophisticated numerical integrationmethods exist. Though more complicated to implementthan the technique described in this paper, they exhibitgreater stability and efficiency. See Gould and Toboch-nik (1988) for an overview and discussion of numericalintegration techniques.

This research was supported by NSF Grant No. MCS-


8110557 to GFO and NIH Training Grant No. GMO7048 toMW. The authors would like to thank Diane Fristrom, RayKeller, Gary Odell, John Trinkaus, Norman Weliky, and PaulWilson for a critical reading of the manuscript. Thanks also toBrian Barsky and the Berkeley Computer Graphics Labora-tory.

References

BELOUSSOV, L. V. , DORFMAN, J. G. AND CHERDANTZEV, V. G .

(1975). Mechanical stresses and morphological patterns inampibian embryos. J. Embryol. exp. Morph. 34, 559-574.

BETCHAKU, T. AND TRINKAUS, J. P. (1978). Contact relations,surface activity, and cortical microfilaments of marginal cells ofthe enveloping layer and of the yolk syncytial and yolkcytoplasmic layers of Fundulus before and during epiboly. J. exp.Zool. 206, 381-426.

FRISTROM, D. K. (1976). The Mechanism of evagination of imaginaldiscs of Drosophila melanogaster, III. Evidence for cellrearrangement. Devi Biol. 54, 163-171.

FRJSTROM, D. K. (1982). Septate junctions in imaginal disks ofDrosophila: a model for the redistribution of septa during cellrearrangement. J. Cell Biol. 94, 77-87.

FRISTROM, D. K. (1988). The cellular basis of epithelialmorphogenesis. A review. Tissue and Cell 20, 645-690.

GOULD, H. AND TOBOCHNIK, J. (1988). An Introduction toComputer Simulation Methods. Part 1. Addison-Wesley,Reading, Mass.

HARRIS, A. K., WILD, P. AND STOPAK, D. (1980). Silicone rubbersubstrata: a new wrinkle in the study of cell locomation. Science208, 177-79.

HONDA, H. (1978). Description of cellular patterns by Dirichletdomains: the two-dimensional case. J. theor. Biol. 72, 523-543.

HONDA, H. (1983). Geometrical models for cells in tissue. Int. Rev.Cytol. 81, 191-248.

HONDA, H., OGITA, Y., HIGUCHI, S. AND KANI, K. (1982). Cell

movements in a living mammalian tissue: long term observationof individual cells in wounded corneal endothelia of cats. /.Morph. 174, 25-39.

JACOBSON, A., OSTER, G. F., ODELL, G. AND CHENG, L. (1986).Neurulation and the cortical tractor model for epithelial folding.J. Embryol. exp. Morph. 96, 19-49.

KAGEYAMA, T. (1982). Cellular basis of epiboly of the envelopinglayer in the embryo of the Medaja, Oryzias latipes. II. Evidencefor cell rearrangement. J. exp. Zool. 214, 257-266.

KELLER, R. E. (1978). Time-lapse cinemicrographic analysis ofsuperficial cell behavior during and prior to gastrulation inXenopus laevis. J. Morph. 157, 223-248.

KELLER, R. E. (1984). The cellular basis of gastrulation in Xenopuslaevis: active, postinvolution convergence and extension bymediolateral interdigitation. Am. Zool. 24, 589-603.

KELLER, R. E. AND TRINKAUS, J. P. (1987). Rearrangement ofenveloping layer cells without disruption of the epithelialpermeability barrier as a factor in Fundulus Epiboly. Devi Biol.120, 12-24.

KOLEGA, J. (1986). Effects of mechanical tension on protrusiveactivity and microfilament and intermediate filament organizationin an epidermal epithelium moving in culture. J. Cell Biol. 102,1400-1411.

KORN, E. (1982). Actin polymerization and its regulation byproteins from nonmuscle cells. Physiol. Rev. 62, 672-737.

LORD, E. A. AND WILSON, C. B. (1984). The MathematicalDescription of Shape and Form, John Wiley and Sons, NewYork, NY.

MITTENTHAL, J. E. AND MAZO, R. M. (1983). A model for shapegeneration by strain and cell-cell adhesion in the epithelium of anarthropod leg segment. J. theor. Biol. 100, 443-483.

NEW, D. A. T. (1959). Adhesive properties and expansion of thechick blastoderm. J. Embryol. exp. Morph. 7, 146-164.

ODELL, G. M., OSTER, G., ALBERCH, P. AND BURNSIDE, B. (1981).

The mechanical basis of morphogenesis. I. Epithelial folding andinvagination. Devi Biol. 85, 446-462.

OSTER, G. (1984). On the crawling of cells. J. Embryol. exp.Morph. 83, 327-364.

OSTER, G. (1988). Biophysics of the leading lamella. Cell Motil.Cytoskel. 10, 164-171.

OSTER, G. AND PERELSON, A. (1988). The physics of cell motihty.In /. Cell Sci. Suppl.: Cell Behavior: Shape, Adhesion andMotility. C. M. J. Heaysman and F. Watt (ed), pp. 35-54.

OWARIBE, K., KODOMA, R. AND EGUCHI, G. (1981). Demonstration

of contractility of circumferential actin bundles and itsmorphogenetic significance in pigmented epithelium in vitro andin vivo. J. Cell. Biol. 90, 507-514.

OWARIBE, K. AND MASUDA, H. (1982). Isolation andcharacterization of circumferential microfilament bundles fromretinal pigmented epithelial cells. J. Cell Biol. 95, 310-315.

STOPAK, D. AND HARRIS, A. K. (1982). Connective tissuemorphogenesis by fibroblast traction. I. Tissue cultureobservations. Devi Biol. 90, 383-398.

SULSKY, D. (1982). Models of Cell and Tissue Movement. Ph. D.dissertation, Dept. Math., New York Univ.

TAKEUCHI, S. (1979). Wound healing in the cornea of the chickembryo. IV. Promotion of the migratory activity of isolatedcorneal epithelium in culture by the application of tension. DeviBiol. 70, 232-240. •

TAKEUCHI, S. (1983). Wound healing in the cornea of the chickembryo. V. An observation and quantitative assessment of thecell shapes in the isolated corneal epithelium during spreading invitro. Cell Tissue Res 229, 109-127.

TRINKAUS, J. P. (1951). A study of the mechanism of epiboly in theegg of Fundulus heteroclitus. J. exp. Zool. 118, 269-320.

TRINKAUS, J. P. (1984a). Mechanism of Fundulus epiboly - acurrent view. Am. Zool. 24, 673-688.

TRINKAUS, J. P. (1984b). Cells Into Organs. The Forces That Shapethe Embryo. 2nd ed. Prentice-Hall, Englewood Cliffs, NJ.

WEEDS, A. (1982). Actin-binding proteins - regulators of cellarchitecture and motility. Nature 296, 811-816.

(Accepted 13 March 1990)

The mechanical basi osf cell rearrangement

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