Notochord morphogenesis in Xenopus laevis: simulation of...

Development 113, 1231-1244 (1991)Printed in Great Britain © The Company of Biologists Limited 1991

1231

Notochord morphogenesis in Xenopus laevis: simulation of cell behavior

underlying tissue convergence and extension

MICHAEL WELIKY1'* STEVE MINSUK2, RAY KELLER2 and GEORGE OSTER2'3

1 Group in Neurobiology, 2 Department of Molecular and Cell Biology, and 3 Department of Entomology, University of California, Berkeley,CA 94720, USA

•Address correspondence to: M. Weliky, c/o G. Oster, 201 Wellman Hall, University of California, Berkeley, CA 94720, USA

Summary

Cell intercalation and cell shape changes drive noto-chord morphogenesis in the African frog, Xenopuslaevis. Experimental observations show that cellselongate mediolaterally and intercalate between oneanother, causing the notochord to lengthen and narrow.Descriptive observations provide few clues as to themechanisms that coordinate and drive these cellmovements. It is possible that a few rules governing cellbehavior could orchestrate the shaping of the entiretissue. We test this hypothesis by constructing acomputer model of the tissue to investigate how rulesgoverning cell motility and cell-cell interactions canaccount for the major features of notochord morphogen-esis. These rules are drawn from the literature on in vitrocell studies and experimental observations of notochordcell behavior. The following types of motility rules areinvestigated: (1) refractory tissue boundaries that inhibitcell motility, (2) statistical persistence of motion, (3)contact inhibition of protrusion between cells, and (4)

polarized and nonpolarized protrusive activity. We showthat only the combination of refractory boundaries,contact inhibition and polarized protrusive activityreproduces normal notochord development. Guided bythese rules, cells spontaneously align into a parallelarray of elongating cells. Self alignment optimizes thegeometric conditions for polarized protrusive activity byprogressively minimizing contact inhibition betweencells. Cell polarization, initiated at refractory tissueboundaries, spreads along successive cell rows into thetissue interior as cells restrict and constrain theirneighbors' directional bias. The model demonstratesthat several experimentally observed intrinsic cellbehaviors, operating simultaneously, may underlie thegeneration of coordinated cell movements within thedeveloping notochord.

Key words: computer simulation, mathematical model,morphogenesis, cell motility, notochord.

Introduction

Morphogenesis of embryonic tissues is driven by avariety of mechanisms including cell rearrangement andshape change. In many instances, directed cell move-ments cause tissues to extend and narrow, so called'convergence and extension'. Examples include gastru-lation in Xenopus laevis (Keller, 1978, 1984; Wilson etal. 1989) and the zebra fish (Warga and Kimmel, 1990),notochord development in Xenopus laevis (Keller et al.1989), archenteron elongation during sea urchin gastru-lation (Hardin, 1989), and imaginal disk evagination inDrosophila (Fristrom, 1976). As a result of active cellintercalation, the number of cells along one axis of thetissue decreases while the number of cells along anorthogonal axis increases. During this process, it seemscertain that cell motility must be somehow coordinated,biased or constrained so that it leads to the requiredchanges in tissue shape. Advances in video microscopynow allow detailed observations of these cell move-

ments (Keller et al. 1989; Hardin, 1989). However, it isdifficult to infer from observations of cell motions theforces that drive these motions, or the principles thatcoordinate and guide the cells. In this respect, modelsare helpful in deducing the conditions under which avariety of factors could drive morphogenesis. Examplesinclude the mechanical forces driving cell chondrogen-esis (Oster et al. 1985), neural plate folding (Odell etal.1981; Jacobson et al. 1986), and epithelial cell rearrange-ment (Honda et al. 1982; Weliky and Oster, 1990); cellsignalling during Dictyostelium aggregation (MacKay,1978); and differential cell adhesion driving imaginaldisk evagination (Mittenthal and Mazo, 1983), and celland tissue sorting (Sulsky et al. 1984).

In this paper, computer simulation is used toinvestigate the specific contribution of different cellbehaviors to notochord morphogenesis. We propose totest a set of rules governing the behavior of motile cells,drawn from the literature on in vitro cell motilitystudies. These rules include: (1) persistence of direc-

1232 M. Weliky and others

tional movement, (2) polarization due to tensioiinduced inhibition of protrusion (Kolega, 1986), and (3contact inhibition (Erickson, 1978a,b). We shall alscinclude the experimentally observed phenomenon thacell protrusion is inhibited at the notochord-somiteboundary (Keller et al. 1989). These behavioral rule;are incorporated into a simulation model, described irWeliky and Oster (1990), that accounts for the balanaof mechanical forces within cells, and between medianically coupled cells within a tissue. The rules of eelbehavior control the mechanics of individual cells, scthat when a large number of cells are coupled to oneanother, we can compare the development of thesimulated tissue to morphometric data from scanningelectron micrographs and time-lapse recordings of eelbehavior in cultured explants.

Two features distinguish this work from previousmodeling efforts. First, we explicitly describe a tissue asa collection of discrete, junctionally coupled cells,rather than treating the tissue as a continuous material(Mittenthal and Mazo, 1983; Hardin and Cheng, 1986).This enables us to account directly for the cell shapechanges, cell movements and rearrangements thatunderlie tissue morphogenesis. Second, we explicitlymodel the mechanical forces that cells generate andexert on one another. Thus, not only is cell behaviorsimulated in a mechanically rigorous manner, but wecan also analyze the contribution of mechanics to cellmotility and behavior within the tissue.

First, we review some of the important features ofnotochord development and describe how cells behaveduring the late gastrula and early neurula stages. Next,we describe the mechanical cell model and the set ofmotility rules that we shall test. Finally, the simulationsdemonstrate how these motility rules can direct cellbehavior and tissue shape change during notochorddevelopment.

Cell behavior during notochord development

During the late gastrula stage, the cells of the notochordbecome distinct from those of the surrounding somiticmesoderm tissue, and the boundary between the twotissues begins to straighten and align (Keller et al. 1989).As the boundary forms, the cells cease their protrusiveactivity at the lateral surfaces of the notochord.Subsequently, when motile notochord cells intercalateinto the boundary, their protrusive activity also ceasesalong the lateral notochord surface.

From the late gastrula stage to early neurula stage,the notochord narrows in width and extends in length.During this time, the number of cells spanning thewidth of the notochord is reduced from about five or sixcells to one. Time-lapse video microscopy reveals thatactive cell intercalation is primarily responsible forthese changes (Keller et al. 1989). Fig. 1 shows a patchof fluorescein-labeled cells, grafted into the pre-notochord region of a stage 10+ embryo. The patchbreaks up into many fragments due to widespread cellmovements that mix the labeled and nonlabeled cells

Fig. 1. (A) A fluorescein-labeled patch of cells is graftedinto the pre-notochord region of a stage 10+ embryo.(B) The developed notochord shows breakup of the labeledpatch caused by cell intercalation within the tissue. Thenotochord (labeled n) extends vertically along theanterior-posterior axis of the embryo and is flanked bysomitic mesoderm (labeled s) (from Keller and Tibbetts,1989, with permission).

(Keller and Tibbetts, 1989). Fig. 1 also shows that cellselongate mediolaterally across the width of the noto-chord as they rearrange. Protrusive activity becomesincreasingly polarized, so that opposite ends of theelongated cells bear large lamelliform and filiformprotrusions, while the flattened cell sides bear shortstubby filiform protrusions. Recent work has demon-strated that elongation begins with cells at thenotochord-somite boundaries, then sweeps toward thetissue interior along successive cell rows (Shih andKeller, 1992). In this way, cells in the middle of thenotochord are the last to polarize and elongate.

The mechanical model

Here we briefly describe the important features of thecomputer simulation model. For a more completedescription, see Weliky and Oster (1990). Cells arerepresented by two-dimensional polygons, and each cellis capable of generating contractile and protrusivemechanical forces. To model a tissue, a large number ofpolygonal cells are mechanically coupled into a continu-ous two-dimensional sheet. Adjacent cells share com-mon boundaries and vertices; this allows mechanicalforces to be transferred from one cell to another,mimicking coupling by junctional adhesions (Fig. 2). Inaddition to vertices at all three-cell junctions, verticesare also introduced along the common boundariesbetween cells to allow for curved surfaces. Alljunctional nodes can slide, and nodes may disappear or

B

ARaaultantT fores

'A2

Fig. 2. The notochord tissue is represented as an array ofpolygonal cells in which adjacent cells share commonvertex nodes. The upper inset shows that vertices areintroduced along the common boundaries between cells toallow for curved surfaces. The mechanical force balancebetween cells is computed at each node, shown in the lefthalf of the lower inset. Elastic tension forces, T, actcircumferentially around the cell perimeter while pressureforces, P, act outward and normal to the cell surface.When the mechanical forces at a node are unbalanced, thenode slides in the direction of the net vector force shownin the right half of the insert. By reducing the corticaltension forces in cell 3, the swelling pressure nowdominates the cortical tension forces and the node slidesupward in the direction of the net unbalanced force.(A) Under the influence of the protrusive forces, nodes Aand B slide between cells 2 and 4. (B) When the two nodesmeet, cell rearrangement occurs. At this time, new nodesC and D are created such that cells 1 and 3 now contacteach other while cells 2 and 4 separate. Cells 1 and 3redirect their protrusive activity to separate nodes C andD. (C) Cells 1 and 3 continue to protrude in oppositedirections and move past one another.

appear, so that the number of polygonal sides defining acell is variable. Thus the polygonal model can mimic theshape of virtually any cell quite closely. The modelaccounts for the balance of mechanical forces withinindividual cells, and between mechanically coupledcells in the tissue. Cell rearrangement and cell shapechanges take place when the forces between neighbor-ing cells deviate from mechanical equilibrium. Thus,morphogenetic cell movements reflect the presence ofunbalanced mechanical forces within a tissue; move-

Notochord morphogenesis In Xenopus laevis 1233

ment will continue until mechanical equilibrium isrestored.

The forces generated by all cells at a commonjunctional node are vectorially summed to generate anet nodal force (Fig. 2A). A node is displaced by anamount proportional to the net nodal force (i.e., as ifthe node were subjected to linear frictional drag). Thetwo intracellular forces acting at a node are theosmotic/hydrostatic pressure, and the elastic tension inthe cortical actin gel (Oster, 1988). Two mechanisms foractive cell intercalation are modeled. The first allowscells to 'push' their way in between neighboring cells,and is used when directional persistence is incorporatedinto the simulations. When a cell is activated to move,its cortical tension drops at the node where protrusionwill take place, reflecting solation of the cortical actingel. Cortical pressure then drives the node in thedirection of the net force imbalance and the cellprotrudes forward (Fig. 2B). The second mechanismallows for cell protrusion into interstitial spaces createdby adjacent cell retraction, and is used when contactinhibition of protrusion is incorporated into thesimulations. When activated, a cortical protrusionextends outward to fill the available interstitial spacebetween cells. Upon contacting an adjacent cell, theprotrusion is inhibited. Since cells share commonboundaries, we cannot explicitly model interstitialspaces; however, we can simulate this protrusivebehavior by realizing that in order to create aninterstitial space into which a cell can crawl, adjacentcells had to contract their boundaries. Therefore, wecan model a cell moving into a space by having a nodemove only when it experiences forces from neighboringcontractile cells. For example, at a node where one cellis protrusively active while the remaining two cells arecontractile, the node will be pulled in the direction ofthe net contractile forces produced by the twononprotruding cells. In this way, the protruding cellsurface expands outward, simulating its movement intothe space vacated by its neighbors.

We use an iterative finite difference method to solvethe force balance at every node during successive timesteps. Each time step represents a snapshot of allcurrent mechanical forces and cell geometries. Asnodes move according to the applied balance of forces,cell rearrangement occurs when two nodes meet(Fig. 2B). At this time, exchange of cell neighborsoccurs: the pair of cells initially separated from oneanother establish contact while the pair which wereinitially in contact separate. If two of the rearrangingcells are protruding, their activity is redirected such thatafter rearrangement, they continue to protrude inopposite directions (Fig. 2C). Protruding nodes areselected randomly. This reproduces the experimentallyobserved behavior of rearranging notochord cells.

Rules governing motile cell behavior

In this section we describe how the cell motility rulesare modeled and explain their experimental justifi-


cation. Three classes of rules have been investigated:(1) how cells behave at the tissue boundaries; (2) howcells polarize and (3) how cells change their direction ofmovement.

Cell behavior at the tissue boundariesProtrusions are absent in notochord cells where theycontact adjacent somitic mesoderm (Keller et al. 1989).In order to model this behavior, we simply inhibitprotrusive activity at nodes along the boundaries of thecell sheet representing the border between notochordand somitic tissue (we will use the term, refractoryboundary, to refer to a tissue boundary at whichprotrusive activity is inhibited). Therefore, marginalcells at these boundaries protrude only at interiornodes; when an intercalating cell reaches this boundary,protrusive activity ceases at nodes which touch theboundary. When the simulations incorporate bothpolarized protrusive activity and refractory boundaries,marginal cell protrusion is restricted to the interiornode most directly pointing away from the boundarytoward the tissue interior.

Cell polarizationMotile cells are modeled either as unidirectionally orbidirectionally protrusive. Unidirectionally protrusivecells are modeled by randomly selecting a single cell

node to protrude; these cells generally tend to remainisodiametric. Bidirectionally protruding cells aremodeled by restricting protrusive activity to two nodeslocated at opposite ends of the cell. Bidirectionallyprotrusive cells will intrinsically elongate into a spindleshape.

In addition to notochord cells, many other types ofmotile cells in vitro and in vivo exhibit bipolar shapes.On a fibronectin-coated substratum, Xenopus prospec-tive head mesoderm cells elongate into a spindle-shaped form and extend two or more lamelliformprotrusions (Winklbauer, 1990). Fundulus deep cells ofa midgastrula stage show bipolarity with a leadinglamella and a trailing edge that is firmly attached to thesubstratum (Trinkaus, 1984). In vitro crawling fibro-blasts also tend to become progressively elongated andbipolar, forming dominant adhesive contacts to thesubstratum at opposite ends of the cell (Fig. 3A). Thecell exerts the tractional forces on the substratum atthese contact points.

Mechanical tension increases in the lateral cellsurfaces of an elongating cell and this tension appears toinhibit protrusive activity in those regions (Kolega,1986). We use this observation to model the develop-ment of polarized protrusive activity in a cell byimposing a dependence between protrusive activity andthe degree of cell elongation (Fig. 3B). To do this we

TENSION

0) (2) (3) (4)

INHIBITED NODES

(1)(2) (3) (4)

BFig. 3. Comparison of a polarizing cell to the model approximation. [A] Motile cell elongation. (1) An initial nonpolarizedcell. (2) An extending protrusion generates mechanical tension that inhibits protrusive activity in adjacent cell surfaces.(3) Continued extension leads to progressive lengthening of the lateral surfaces under tension. (4) Final polarized cell withlarge lamellipodial regions at opposite ends of the cell. [B] Model. (1) An intially nonpolarized cell has equal probabilitiesof protruding at all nodes. (2) The probability of protruding at a node increases as the node successfully extends (shown bylarge arrow). (3) When the cell reaches a minimum threshold of elongation, protrusive nodes that lie along the lengtheninglateral cell surfaces are progressively inhibited. The node that is most opposite to the maximally protruding node is notinhibited. (5) Final polarized cell with two opposing protruding nodes.

Notochord morphogenesis in Xenopus laevis 1235

Fig. 4. The results of asimulation incorporatingrefractory tissue boundaries alongthe entire tissue perimeter, 99 %directional persistence, and cellpolarization. When cellprotrusion is inhibited along theentire tissue perimeter - incontrast to only at the right andleft boundaries as in subsequentsimulations - the tissue contractsinto a circular shape. Almost allmarginal cells have elongatedtowards the tissue interior by theend of the simulation.

define a dimensionless 'cell shape index' as the ratio ofthe square of the cell perimeter to its area. The shapeindex is smallest for an isodiametric cell, and increasesthe more elongated a cell becomes. Therefore, theshape index can be used to measure the extent to whichprotrusive activity will be inhibited on lateral cellsurfaces (Appendix A describes the algorithm indetail). A polarized cell will intrinsically elongate anddevelop protrusions at its opposite ends. In somesimulations (Figs 4 and 6), a simpler model for cellpolarization has been used.. In these cases, twoopposing .nodes are simply chosen to be protrusivewhile all other nodes are inhibited.

i

How cells change their direction of motionWe model two mechanisms by which cells change theirdirection of movement, (a) In the absence of cell-cellinteractions, cells exhibit directional persistence suchthat they tend to maintain their direction of motion,albeit with some degree of randomness, (b) Cellsinteract by contact inhibition of protrusive activity.

Directional persistence is observed in motile cells invitro, as well as in vivo. We model persistence of motionby assigning probabilities to each cell node, whichallows any cell to change direction on any iteration. Adirectional persistence of 100 % means that a cell willnever change direction; a cell with 0 % persistence willchange direction at each iteration. Intermediate valuesmodel varying degrees of directional persistence. Forexample, a cell with 95% persistence will changedirection, on average, once every twenty iterations. It isnot obvious what numerical level of persistence bestreflects the behavior of living cells, nor is it obviouswhat are the implications of various levels of persist-ence for the behavior of the whole tissue. Therefore,each of the simulations was run using several differentlevels of persistence varying from 0 % to 100 %.

Contact inhibition of protrusive activity is commonlyobserved during in vitro studies of cell motility(Abercrombie and Ambrose, 1958; Erickson, 1978a,b).Inhibition of localized protrusive activity occurs whenthe lamella of one cell contacts the surface of a secondcell, whereupon the leading lamella ceases its forwardmotion, and often redirects its movement along thelateral sides of the protrusion (Erickson, 1978a).

Contacting cells can protrude at their free margins,implying that contact inhibition is a local phenomenonwhich does not necessarily paralyze protrusive activitythroughout the entire cell.

Unlike epithelial cells, cells of the notochord do notappear to be continuously joined to one another aroundtheir entire surfaces. SEM reveals interstitial spacesbetween cells (Keller et al. 1989). Therefore, it ispossible that, as cells contract and pull away from theirneighbors, spaces open up into which an active cell cancrawl. Our hypothesis is that protruding cells do not'push' their way between adjacent cells but rather crawlinto vacated spaces created by withdrawal of adjacentcells. When a protrusion mechanically contacts astationary or advancing neighboring cell, that pro-trusion is inhibited. This behavior is analogous to cellsprotruding in the direction of least resistance. In ourmodel for contact inhibition, nodes are assigned aprobability of protruding in the range of zero to one.The nodal probability increases by a constant amount,Pacuvc, during each iteration that the node is activated toprotrude (Appendix A). When contact inhibitionoccurs between two cells, the probability of protrudingat both nodes is reduced by a constant amount, Pinhib,during each iteration that the contact is sustained(Appendix B). In order to determine when contactinhibition has occurred, the direction of mechanicalforces within cells, or their protrusive activity, indicateswhether they are stationary, withdrawing or advancingtowards their neighbors (Appendix D).

An alternative mechanism would allow protrudingcells to mechanically 'push' their way between neigh-bors without being contact inhibited, as long as adjacentcells 'yield'. If adjacent cells mechanically deform orshift their position, then protrusion continues. Other-wise, if the mechanical resistance of adjacent cells isabove a prescribed amount, then the cell ceases toprotrude in that particular direction and is 'blocked'.Like classical contact inhibition, this behavior wouldlead to cell protrusion in the direction of leastmechanical resistance. Though this alternative modelcould have been implemented, we wanted to restrictourselves to investigating the morphogenetic conse-quences of previously described in vitro contactinhibition behavior.


The contribution of specific cell behaviors tonotochord development

In this section we illustrate the effect of differentmotility rules on tissue development by presenting aseries of computer simulations. In order to keep thecomputation manageable, we use a two-dimensionalcell sheet to model a small, representative region ofnotochord tissue (Figs 4-7). Our two-dimensionalmodel is equivalent to a tissue explant, removed fromthe embryo, which has been reduced in depth to a singlecell layer lying parallel to the epithelium. Boundaryconditions are applied to the cell sheet which simulate anumber of those found in the intact embryo. Constantdrag forces are applied to cells on the upper and lowerboundaries of the cell sheet, mimicking the mechanicalproperties of tissue lying above and below the modeledregion. One drag component is applied to nodalmovement along the vertical tissue axis, slowing therate of tissue elongation. The second component acts asviscous drag on horizontal nodal movement along theboundaries, and is chosen so that cell rearrangementoccurs at the same rate along the boundary and withinthe tissue interior. Under these conditions, the modeltissue is not subjected to explicit external compressiveforces. The left and right boundaries of the cell sheetrepresent the mediolateral notochord-somite tissueboundaries, at which cell protrusive activity is inhibited.In all simulations, except Fig. 4, unrestricted protrusiveactivity is allowed at cell nodes that lie on the upper andlower boundaries of the cell sheet where additionalnotochord tissue would extend along the anterior-posterior axis of the embryo. Appendix E describesconditions that maintain the rectangular tissue shape.

In each simulation we incorporate three motilityrules, one selected from each of the three classesdescribed above. All simulations include inhibition ofcell protrusion along the right and left tissue bound-aries. In addition, we select a mechanism by which cellschange their direction of motion and whether they areunidirectionally or bidirectionally protrusive. We willfocus our attention on the contributions of thesebehavioral rules to five important features of notochorddevelopment: (1) tissue extension and narrowing, (2)cell rearrangement, (3) incorporation of cells into thenotochord-somite tissue boundaries, (3) cellelongation, and (4) parallel cell alignment.

All simulations begin with the same cell configur-ation. The cells are initially isodiametric, correspondingto the unelongated, nonpolarized, early notochord cellsobserved during the late midgastrula stage. At the startof each simulation, one or more nodes in each cell arerandomly chosen for protrusive activity, subject to theconstraints of the particular behavioral rules beingtested.

Tissue extension and narrowingThe most important factor responsible for tissueextension and narrowing is the inhibition of cellprotrusive activity along the left and right tissueboundaries while protrusive activity is permitted along

the top and bottom boundaries. The simulations showthat such biasing of cell protrusive activity alwaysproduces tissue extension and narrowing regardless ofthe motility behavior within the tissue interior(Figs 5-7). Cell protrusive activity at the top andbottom tissue boundaries is directed away from thetissue interior and contributes to vertical tissue exten-sion. Simultaneously, refractory boundaries constrainthe motility of marginal cells along the right and leftmargins of the tissue, biasing their protrusive activityand movement towards the interior of the tissue.Therefore, marginal cells converge towards oneanother, pulling the opposite sides of the tissue closertogether and compressing interior cells. This compres-sion is relieved by outward expansion of marginal cellsat the top and bottom tissue boundaries, which furthercontributes to vertical tissue extension.

The generation of compressive forces is easy tounderstand when a cell 'pushes' in between itsneighbors and squeezes them. However, when cells areonly able to crawl into the open spaces created byadjacent cell withdrawal, compressive forces that drivetissue extension can arise from differences in surfacecurvature along the cell surface. Consider the marginalcells at the top and bottom tissue boundaries. Theyhave large flat regions along the tissue boundary andwith individual adjacent cells. In addition, they makesharp acute angles where three cells meet. Because thenet nodal tension varies with surface curvature,stronger contractile forces are generated at three-celljunctional nodes than at nodes along the flat surfaces.Cells will have a tendency to expand outward alongtheir flattened surfaces which offer the least resistanceto internal pressure. For marginal cells, expansionoccurs primarily along their flattened surfaces whichform the tissue boundaries. The tissue verticallyextends because external forces along the top andbottom boundaries are smaller than the forces resistingcompression within the tissue.

If cell protrusive activity is inhibited along the entiretissue perimeter, rather than just along the right and leftboundaries, and the boundary conditions are modifiedso that the rectangular tissue shape is not rigidlyenforced, the tissue contracts into a circular shape withelongated cells encircling the perimeter (Fig. 4, and seeAppendix E).

Cell rearrangementIn the absence of contact inhibition, the frequency withwhich cells change their direction of movement isdetermined by their strength of directional persistence.When directional persistence is strong enough, exten-sive cell movements and cell rearrangements occurthroughout the tissue (Figs 5B and 6). This is revealedby the breakup of the patch of darkened cells. Comparethe similarity of the breakup of the simulated cell patchin Figs 5B and 6 with the breakup of the fluorescein-labeled cell patch in the actual notochord (Fig. 1). Inboth cases, labeled and unlabeled cells mix togetherand most of the labeled cells are eventually incorpor-ated into the right and left tissue boundaries.


Fig. 5. The results ofsimulations incorporating rightand left refractory tissueboundaries, varying degrees ofdirectional cell persistence, andunidirectional cell protrusion.In both A and B, tissueelongation and narrowingoccur but with varying degreesand patterns of cell movementand rearrangement. The shapeof interior cells remainprimarily isodiametricthroughout both simulations.(A) 0 % directional persistence(cells change direction everyiteration). Similar results areobserved for persistence valuesup to about 99 %. Tissueelongation drives 'passive' cellrearrangements along thevertical tissue axis. This leadsto an increase in the numberof interior cells along thetissue length (from an averageof 13 to 17) while reducing thenumber of cells spanning thetissue width (from an averageof 8 to 6). This process isreflected in the darkened cellpatch which changes indimension from 4 by 3 cells to2 by 5 cells. The patch doesnot break up and none of thelabeled cells intercalate intothe tissue boundaries. Thenumber of marginal cellsincreases only slightly from 14to 15 on the right tissueboundary and from 16 to 18 onthe left. (B)99.8% directionalpersistence (cells changedirection on average onceevery 500 iterations). Increaseddirectional persistence leads towidespread cell rearrangements as revealed by the breakup of the darkened cell patch. Most of the labeled cells migrateinto the right and left tissue boundaries. Cell intercalation is responsible for the dramatic increase in the number ofmarginal cells along the right and left tissue boundaries. The number of cells on the right boundary increases from 13 to38, while on the left from 16 to 43. Since the number of marginal cells increases faster than the rate of tissue elongation,they elongate horizontally due to simple mechanical compaction. Note that interior cells remain primarily isodiametric. Thenumber of interior cells spanning the width of the tissue is reduced from 8 to 2 cells as a result of the cell migration intothe tissue boundaries.

Directional persistence must be above 99% (e.g.,cells change direction less than once every hundrediterations) in order to produce cell rearrangement. Cellprotrusion must persist at a three-cell junctional nodelong enough for a cell to traverse the distance to thenext adjacent three-cell node. When two nodes meet,cell neighbor change occurs (Fig. 2). Under thiscondition, active cell protrusion will drive cell re-arrangement regardless of the ability of the tissue toextend. Figs 6A and B show that the degree of cellrearrangement in a fully or weakly extending tissue isroughly the same.

In contrast, if directional persistence falls below 99 %(i.e. cells change direction on average more frequentlythan once every hundred iterations), motile cells will beunable to reach the next three-cell nodal junctionbefore changing their direction. In this case, cellrearrangement becomes strongly dependent upon thedegree of tissue elongation (Fig. 5A). As the cell sheetvertically lengthens, cells are stretched and elongate inthe direction of tissue extension. Cell rearrangementoccurs as elongating cells attempt to return to theirinitial isodiametric shape. Cells aligned along thevertical axis of extension separate, allowing adjacent


cells to establish contact. These cell rearrangementscause the darkened cell patch to elongate and narrow.The patch does not break up, so that labeled andunlabeled cells do not mix. If the tissue is mechanicallyprevented from lengthening, no cell rearrangementswill occur, (see Oster and Weliky, 1990; Weliky andOster, 1990 for other examples and a complete

Fig. 6. The results ofsimulations incorporating rightand left refractory tissueboundaries, high directional cellpersistence, and cellpolarization. These simulationsuse the simplified cellpolarization model in whichinterior cells always protrude attwo opposing nodes. Marginalcells protrude at the node mostdirectly pointing toward thetissue interior. In all casesextensive cell intercalation leadsto the breakup of the labeledcell patch as labeled andunlabeled cells randomly mixtogether. In A and B, extensivecell elongation occurs in manydifferent directions. (A) 100 %directional persistence (cellsnever change direction). Asmall drag force, slightlyresisting tissue extension, isapplied to cell nodes at theupper and lower tissueboundaries. In this case thetissue lengthens by a factor ofabout 1.6. (B) 100% directionalpersistence. A stronger dragforce is applied to cell nodes atthe upper and lower tissueboundaries, which results inreduced ability of thenotochord to extend. The tissueis therefore shorter and fatterthan in A (both simulations arerun for an equal number ofiterations). (C)99% directionalpersistence (cells changedirection on average once every100 iterations). Interior cellelongation is not as extensive asin B since cells occasionallychange their protrusiondirection. Note that mostinterior cells tend to remainisodiametric; however, interiorcells that abut marginal cellstend to be horizontallyelongated as seen in the lastthree panels. The same dragforce is used as in B. Forpersistence values less than99% (not shown), resultssimilar to Fig. 5A are observed.

discussion of the mechanical forces underlying 'passivecell rearrangements).

Fig. 7 shows that when contact inhibition is incorpor-ated into the simulation, net cell movement is initiallyblocked within the interior of the tissue. This is due tothe random, uncoordinated protrusive activity ofneighboring cells which mutually inhibit each other. As

Fig. 7. The results of a simulation incorporating right and left refractory tissue boundaries, contact inhibition of protrusionand cell polarization. Color represents the probability that a cell node will protrude during each iteration. Blue representslow probability while red represents maximum probability. Below each frame the iteration number is shown. At iteration 0,all nodes of interior cells have an initial probability of 0.8. Throughout the simulation, marginal cell protrusion is inhibitedat all nodes except for the interior-most node whose probability for protrusion is 1.0. Beginning at the right and left tissueboundaries, polarization and elongation spread along successive cell rows into the tissue interior. As cells elongate, nodeslocated at the extending regions of the cell turn red indicating that these nodes are becoming strongly protrusive, whilenodes lying along the lengthening cell sides turn blue indicating that they are becoming inhibited. This can be appreciatedby following the behavior of the numbered cells. In iteration 0, all interior cells are isodiametric and nonpolar. By iteration80, the first row of interior cells start to polarize and elongate by protruding between the initially polarized marginal cells(note the color change of cell 5). By iteration 140, the second row of interior cells are polarizing by intercalating betweenthe already polarized first row cells (note cell 7). By iteration 180, the third cell row begins to polarize, which is seen bythe color change and elongation of cell 6. Note that the centrally located cells 1 and 2 remain nonpolarized andisodiametric while their neighbors to the right and left have already polarized and started elongating. By iteration 700, allcells have horizontally polarized and elongated with strong, bipolar protrusions at their extending regions. Betweeniteration 700 and 28000, intercalation decreases the number of cells spanning the tissue width from an average of 6 to 2cells, and increases the average number of cells along the right or left tissue boundary from 21 to 44 cells. Note that byiteration 28000, cell intercalation has caused the previously adjacent cells 3 and 4 to migrate to opposite tissue boundaries,and cells 5 and 7, which were neighbors in iteration 700, to separate.


BFig. 8. Comparison of the repetitive extension and retraction cycles of a Xenopus deep cell and a modeled cell. The solidline indicates the cell shape in the previous frame of the sequence. (A) The changing shape of a single deep cell is seen intracings from time-lapse recordings of a cultured Xenopus laevis explant of the involuting marginal zone of a midgastnila(from Keller and Hardin, 1987, with permission). (B) Modeled cell behavior is shown during a simulation incorporatingintrinsic cell polarization and contact inhibition. The protrusive cell nodes at each frame are marked with short arrows. Thecell changes its protrusion direction many times until it eventually becomes aligned horizontally, with protrusive activity ateach end of the cell. The horizontal cell orientation corresponds to mediolateral elongation within the notochord tissue.

a result, cells remain nonpolarized and continue toprotrude and contract in many directions without anynet cell movement (Fig. 8B); this behavior reflects thecells' inability to establish a stable and persistentprotrusion direction. Fig. 8A shows that Xenopus deepcells exhibit a similar 'kneading' or jostling behaviorconsisting of repeated cycles of protrusion and retrac-tion, also without any net cell movement (Keller andHardin, 1987). This cell activity is also experimentallyobserved during notochord development (Keller et al.1989). As cells in Fig. 7 polarize horizontally, stable andpersistent bipolar protrusions develop at theirelongating tips which drive cell movement and re-arrangement within the model tissue.

Incorporation of cells into the tissue boundariesWhether cell motility within the tissue is random(Figs 5B and 6) or directed horizontally (Fig. 7), cellmovement eventually brings interior cells in contactwith the left or right tissue boundaries. Once in contactwith these boundaries, cells rarely pull away. This leadsto a progressive increase in the number of cells alongthe right and left boundaries while the number of cellswithin the tissue interior consequently decreases.Eventually, almost all interior cells become incorpor-ated into the tissue boundaries.

Once cells reach the boundary, not only do they'stick' and rarely pull away, but they expand theirregion of contact with the boundary. Without invokingany special adhesive boundary conditions, there is asimple mechanical explanation for this phenomenon.The inhibition of protrusive activity along the right andleft tissue boundaries implies that only contractilemechanical forces exist at cell nodes lying along theseboundaries. Therefore, this behavior can be understoodby analyzing the balance of mechanical forces at theseexclusively contractile marginal cell nodes. A cell that

has newly intercalated into the tissue boundary initiallyforms a tapered point with the boundary. Nodal forces"cause the tapered point to expand to a blunt edge untilthis cell, and its two marginal cell neighbors, make rightangles with the boundary. When adjacent marginal cellsat a common junctional node make 90 degree angleswith the boundary, both cells generate net elastic nodalforces of equal magnitude (the elastic nodal force iscurvature dependent; therefore when cells have equalcurvature at a node, or equivalently, make equal anglesat a node, the magnitudes of their elastic forces areequal). The node is now in mechanical equilibrium andstable. See Weliky and Oster (1990) for a more detailedmechanical description of this behavior. Of course,specifically adhesive boundaries would have producedthe same cellular configurations. However, it turns outthat the mechanical balance of forces at an inhibitoryboundary produces the same effect as an adhesiveboundary.

Active cell elongationWhen cells protrude unidirectionally with strongdirectional persistence, interior cells do not elongatebut remain mostly isodiametric throughout the simu-lation (Fig. 5B). This is because nonprotrusive cellregions retract as the protruding node extends forward,thus maintaining the cell's isodiametric shape. Cellrearrangement plays a crucial role in this process byallowing cells to accommodate their neighbors byadjusting their shapes. Fig. 5B also shows that, unlikeinterior cells, marginal cells elongate. This occurs whenthe entire tissue does not lengthen fast enough toaccommodate the increase in the number of marginalcells. As interior cells migrate and crowd into thelimited tissue boundaries, they will elongate perpen-dicularly to the boundaries as a result of simplemechanical compaction. When unidirectional pro-


trusion is replaced by bidirectional protrusion, theresulting intrinsic cell polarity produces elongatedinterior cells (Figs 6 and 7). Thus the simulationssuggest that elongation of cells within the interior of thenotochord tissue must be actively generated by the cellsthemselves, and does not result from passive mechan-ical compaction, as is true for marginal cells along thetissue boundaries.

Fig. 6B shows that when cells protrude bidirec-tionally with strong directional persistence, not all cellselongate to the same extent. One sees a wide range ofelongations, from near isodiametric cells to longspindle-shaped cells. This is because at each node, one,two or three cells can be protruding simultaneously.When only one cell protrudes at a node, this cell willelongate much more than when multiple cells protrudeat that node. At nodes where all three cells areprotruding, the protrusive forces can cancel.

Parallel cell alignmentFig. 6B reveals that when interior cells polarize withstrong directional persistence, and with no cell-cellcontact inhibition, cell elongation is extensive, butrandomly oriented. Close inspection shows that cellalignment occurs in local domains or groups of cells, butit does not extend globally. As the simulation pro-gresses, additional neighboring cells elongate and alignwithin these domains, and finally, as obliquely orientedcells intercalate between already elongated marginalcells, they rotate and asssume the horizontal orientationimposed by their marginal cell neighbors. At the end ofthe simulation, most cells have elongated and arehorizontally aligned. However, even during later stagesof the simulation a number of vertically orientedinterior cells still remain (fourth panel in Fig. 6B).

When interior cells polarize with only slightlyreduced directional persistence, a different pattern ofcell elongation results (Fig. 6C). In this case mostinterior cells remain roughly isodiametric. However,horizontal elongation is seen among many of theinterior cells that abut marginal cells. This is seen mostclearly during the middle to later stages of thesimulation (last three panels of Fig. 6C). The reason forthis behavior is that occasional direction changes allowinterior cells to sample the surrounding mechanicalconditions imposed by adjacent cells. As described inthe previous section, cell elongation occurs morerapidly when only one cell protrudes at a junctionalnode. Since polarized marginal cells always protrude atthe node which is furthest from the tissue boundary,abutting interior cells can rapidly elongate horizontallyby protruding between the contractile or nonprotrusiveregions of two neighboring marginal cells. In contrast,cells located deeper within the tissue are surrounded byshifting mechanical conditions caused by the randomdirectional changes of their neighbors. Thus theycannot elongate as well and so remain roughlyisodiametric.

Not all interior cells abutting marginal cells elongatehorizontally; some remain more isodiametric (see lasttwo frames of Fig. 6C). Most of these cells are

protruding along the vertical tissue axis, and areprevented from elongating by mechanical restrictionsimposed by the already horizontally polarized marginalcells. A more efficient strategy would be to inhibit cellprotrusion in a direction which is mechanicallyobstructed by adjacent cells, while allowing continuedprotrusion in an unrestricted direction. This form ofcell-cell interaction can be incorporated into thesimulation in the form of contact inhibition ofprotrusive activity. The results are shown in Fig. 7.When contact inhibition of protrusive activity iscombined with cell polarization, all interior cellselongate exclusively in a horizontal direction. Cellpolarization and elongation occurs in a wave of activitythat sweeps from the right and left tissue boundariesinto the tissue interior. Starting with biased protrusiveactivity of marginal cells, successive cell rows polarizeby a process of restricting and constraining theirdirectional bias (Fig. 9). Initial marginal cell protrusiveactivity must be biased such that protrusion occurs onlyat the interior most node; when marginal cells areallowed to protrude randomly at any interior node,interior cells do not elongate exclusively in a horizontaldirection. All cells in Fig. 7 have horizontally polarizedwithin the first three hundred iterations. During thistime the number of instances of contact inhibitionbetween cells decreases to near zero (Fig. 10). Duringthe remainder of the simulation, the cells maintain thisaligned configuration and continue to intercalatebetween one another until most cells have beenincorporated into the right and left tissue boundaries.In order that interior cells do not prematurely polarizein random directions during the early stages of thesimulation, the magnitude of contact inhibition must beseven times larger than the rate at which the nodalprotrusion strength increases due to protrusive activity.This ensures that random variations of protrusionstrength will be dampened by contact inhibition.

Discussion

Multiple cell behaviors drive morphogenesisWithin the embryo, there are many different cellbehaviors operating simultaneously, so it is difficult todetermine how each contributes to the overall morpho-genetic movements. Computer simulation allows us totest the specific role that individual cell behaviors, orcombinations of cell behaviors, play during tissuemorphogenesis. The mechanical cell model described inWeliky and Oster (1990) is designed to simulate theforces generated by cells within a tissue. In this paper,we have proposed a set of cell motility rules whichregulate the cell mechanics. Using the model, we caninvestigate the morphogenetic consequences of theserules in a mechanically correct manner.

Our simulations show that random, nonpolarized cellmotility, combined with refractory tissue boundaries,produces elongation of the tissue and of marginal cells,but leaves interior cells isodiametric (Fig. 5B). Whenintrinsic cell polarity is combined with high directional


oGOQDC

ooo

Fig. 9. Polarized cellprotrusive activity spreadsprogressively towards thetissue interior. (A) At thesimulation start, polarizedmarginal cells [white] lyingalong the notochord boundaryare constrained to protrudeonly at their interior-mostnode (protrusion probability=1.0). Protrusion is inhibited atall other marginal cell nodesby assigning to them aprotrusion probability of 0.0.In contrast, interior cells[shaded] have an initialuniform protrusion probability of 0.8 at all nodes (a nonzero nodal probability of protrusion is indicated by an arrow).Initially, cell movement is blocked within the tissue interior by contact inhibition As each marginal cell extends towardsthe tissue interior at its protrusively active node, nonprotruding regions retract and pull away from adjacent submarginalcells at the circled nodes. This allows the first row of submarginal cells [light shading] to extend towards the notochordboundary, by protruding at the circled nodes in between the retracting regions of marginal cells. (B) As these interior cellselongate, protrusive activity in nodes along their lateral surfaces is inhibited. This in turn allows the second row of interiorcells [dark shading] to elongate by protruding at the circled nodes. (C) Row by row, polarization spreads towards theinterior of the tissue.

B

total call protrusions

— total Instance* of contactInhibition

60 120 180 240 300 360 360

Iteration Number

14,000 28,000

Fig. 10. As cells polarize across the width of thenotochord, the total number of inhibitory contacts betweenprotruding cell nodes and adjacent cells rapidly decreasesto near zero within the first 300 iterations. The reduction inthe number of total inhibitory contacts parallels theemergence of a parallel aligned cell array. In addition, thetotal number of cell protrusions slowly decreasesthroughout the simulation due to lateral inhibition ofprotrusive activity and cell migration into the notochordboundaries.

persistence, cells initially elongate in random orien-tations throughout the entire tissue interior (Fig. 6Aand B); reducing directional persistence causes onlythose interior cells that abut marginal cells to horizon-tally elongate (Fig. 6C). Finally, when contact inhi-bition of protrusive activity is combined with intrinsiccell polarity and refractory tissue boundaries, thesimulation reproduces all of the major features ofnotochord development: cell rearrangement extendsand narrows the tissue, while transverse cell elongationbegins at the refractory notochord-somite boundariesand spreads along successive cell rows into the tissue

interior (Fig. 7). These results support the notion thatseveral rules for cell motility and cell-cell interaction,operating simultaneously, may drive notochord mor-phogenesis.

Coordinated cell behaviors arise from local cellinteractionsMany problems in biology can be formulated asoptimization or constraint satisfaction problems.Examples include the organization of social insectcolonies (Oster and Wilson, 1978), population geneticsand neural network dynamics (Bounds, 1987). One wayto solve large optimization problems is to implementthe constraints as excitatory or inhibitory interactionsamong a system of interacting cells. The system isallowed to iteratively converge, from an intitial randomconfiguration, to a stable solution that satisfies thelargest set of compatible constraints (Hopfield andTank, 1986).

Similarly, the emergence of organized cell motionwithin developing tissues can be treated as an optimiza-tion or constraint satisfaction problem. In the modelnotochord shown in Fig. 7, constraints upon cell motionare implemented through local inhibition of cellprotrusion between contacting cells. Initially, randomprotrusive activity allows each cell to explore its localenvironment in an attempt to find an admissibledirection to move. But because this activity is tem-porally and spatially uncoordinated among neighboringcells, contact inhibition forces cells to constantly changetheir protrusion direction. This results in the jostling or'kneading' behavior of cells without any net cellmovement (Fig. 8B). As cell protrusive activity be-comes increasingly biased in a horizontal direction, thetotal number of contact inhibitions between neighbor-ing cells is minimized (Fig. 10). This allows cells toelongate into a parallel array by intercalating between


one another without further changing their protrusivedirection. An array of parallel elongating cells providesthe most geometrically favorable conditions for effec-tive protrusion: wherever three cells meet only one cellis protrusively active while the two remaining cells arecontractile. Thus, the simulations reveal that localconstraints upon cell motility can force cells within atissue to organize into a specific geometrical configur-ation.

Two-dimensional simulation of a three-dimensionaltissueThe in vivo notochord is a three-dimensional structurein which constituent cells undergo three-dimensionalmovements and shape changes. Alhough the notochordtissue and cells change shape in three dimensions -especially in the later stages of development - we havefocused only upon those features that lend themselvesto a two-dimensional analysis; i.e., that have asignificant planar component, parallel to the epi-thelium. These features include mediolateral cellelongation and tissue convergence and extension, whichhave already started during the earlier stages ofnotochord development (Keller et al. 1989). Our modeltissue should not be literally interpreted as a crosssection through the three-dimensional notochord, for inthis case, cell areas would not remain constant as cellsmove in and out of the modeled plane. Rather themodel should be viewed as a 'reduced' notochord inwhich the tissue depth has been reduced to one cell.Recent work with shaved notochord explants in whichthe depth of the tissue is diminished by removing anumber of cell layers, more closely corresponds to ourtwo-dimensional model (Shih and Keller, 1992). Inthese explants, the cells and tissue are restricted toprimarily planar movements but still retain many of theessential features of notochord development as de-scribed in previous studies and reproduced in this paper(Keller et al 1989).

Mechanisms for biasing cell motilityThe simulations demonstrate the importance of cellbehavior at a tissue boundary in constraining the tissueto change shape in specific ways. Though protrusiveactivity is biased only in cells that lie along the right andleft tissue boundaries, the effect this has upon globaltissue morphogenesis is very strong. For instance,refractory boundaries always produce tissue elongationand narrowing regardless of the cell behavior within thetissue. In addition, we have shown that specific cellbehaviors, such as cell polarization, can arise alongdeveloping tissue boundaries and subsequently spreadthroughout the tissue. However, this raises the questionof how cell motility is biased in tissues where no obviousboundaries exist, such as archenteron elongation in thesea urchin (Hardin, 1989).

Parallel arrays of cultured fibroblasts form by aprocess of self alignment, mediated by cell-cellinteractions in the form of contact inhibtion ofprotrusion (Elsdale and Wasoff, 1976; Erickson, 1978a,1978/J). Our simulations show that contact inhibition of

protrusive activity is a sufficient constraint on cellmotion to force successive cell rows to polarizetransversely across the width of the notochord. Analternative hypothesis is that cells transmit an inductivesignal, initiated by the marginal notochord cells. Nochemical signal has yet been found; however, the resultsof this paper cannot be used to prove or disprove theexact nature of the chemical or mechanical eventsunderlying notochord morphogenesis. Rather, theseresults can suggest potential mechanisms that mayunderlie the observed cell behaviors and suggest newexperiments. In the absence of any direct evidence for achemically transmitted polarization signal, our resultssuggest that a few simple, well characterized in vitro cellmotility rules can account for the experimentallyobserved mediolateral cell polarization within thenotochord and the elongation of the tissue.

Appendices

[A] Modeling cell polarizationThe following describes the cell model for generatingpolarized protrusive activity (see Fig. 3B):

1. Initially, protrusion will occur at each of the cellnodes with the same initial probability PnOde- We use 0.8as the initial value, i.e. there is an 80% chance that anode will protrude during each iteration. With thisinitial condition, the direction of cell protrusion is notbiased.

2. During each iteration, the nodal probability Pnodeis used to determine whether or not a node will beactivated to protude. The computer generates a randomnumber in the range of 0 to 1 for each cell node. If thisnumber is smaller than the nodal probability, then thenode will be activated during that iteration. Eachiteration that a node is activated to protrude, Pnode isincreased by a constant amount APac/Il,e=0.01. Nodeswith high protrusion probabilities will be more likely tobe protrusively active during subsequent iterations thannodes with lower values. Therefore, the larger Pnode is,the faster its rate of increase.

3. We use the shape parameter index, INDEX=PERIMETER2/AREA, to calculate the degree of cellelongation. The higher the index, the more elongated isthe cell's shape. We normalize the value of INDEX bythe initial cell index value calculated at the start of thesimulation. In this way, each cell initially has an indexvalue of 1.

4. When the shape index increases above a minimumthreshold value=1.05, the protrusion probability at allnodes, except the maximally protruding node and itsmost opposing node, is reduced as a function ofincreasing elongation (i.e. this would indicate that thecell has elongated to a point where sufficient mechan-ical tension is being generated by an advancinglamellipodium). For an inhibited node, the protrusionprobability becomes

Pnode = O.8~(C* INDEX) (1)

where 0.8 is the intitial spontaneous protrusion pro-


bability and c=70.0 is a constant. By differentiaing eq. 1we can calculate the amount of change in PnOde duringeach iteration as a function of the change in INDEX asfollows

AP,NDEX =-(c* AINDEX) (2)

where APINDEX is the change in the nodal probabilitywhich results from a change in the cell shape parameterindex, AINDEX, during each iteration.

[B] Reducing the nodal protrusion probability due tocontact inhibitionWhen a protruding cell node detects opposing forces inadjacent cells, this reflects mechanical contact betweenthe two cells as described in appendix D. Under thiscondition, the nodal probability is reduced by aconstant amount APinhib=0.075, which represents thereduced capacity of the node to protrude as a result ofcontact inhibition. The value of APinhib is chosen to besubstantially larger than the value of APactive, such thatinhibition strongly reduces the protrusive capacity of anode at a rate faster than the nodal protrusionprobabality can increase due to protrusive activity.

[C] Calculating the total change in nodal protrusionprobabilityFor each iteration, the total change in the nodalprotrusion probability AP^de is simply the sum of allcontributing probability changes as calculated in appen-dices A and B:

ctive + INDEX (3)

With the intitial protrusion probability of each cellnode set at 0.8,

Pnode = 0.8 (4)

the subsequent value of Pnode at time step, t+1, iscalculated by simply adding the current value of APnode

to the value of Pnode at the current time step, t:

inodcf,+i) "nodelt) ' ^^node(l) \p)

If Pnode is below 0.8 at a node which is free fromcontact inhibition with adjacent cells, Pnode returns toeither 0.8 or an appropriate value as determined by thecell shape index. If the cell has not yet polarized, thenall nodes will return to a nodal probability of 0.8. If theshape index is above the minimum threshold forinhibition, then the probability of protrusion at nodesthat are internally inhibited by cell elongation is

Pnode = 0.8- (C* INDEX) (6)

In this way, the spontaneous level of protrusion forinternally inhibited nodes will be reduced from thenormal value of 0.8.

[D] Modeling contact inhibitionContact inhibition occurs when neighboring cellsattempt to protrude or advance towards one another, orstationary cells block the advance of an adjacent cell.The direction of motion of a cell node is determined bythe direction of the net nodal vector forces. If the nodalvector points outward from the cell surface, then the

node is protruding or advancing. If the nodal vectorpoints inward, then the node is contracting or with-drawing. Therefore, for each cell at a junctional node,the program checks the direction of the nodal vectors inneighboring cells. If any of these vectors are zero, orpointing outward, then Pnode is reduced by an amountAPinhib as described in appendix B for other protruding,advancing or stationary adjacent cells.

In order to implement this algorithm, the pressure/area relationship must be modified to allow for'negative' cell pressure as follows:

Pressure = c * (area original — areajiew) (7)

where c is a scaling constant, area_original is the cellarea at the start of the simulation and areajiew is thecell area during each successive iteration. If areajiewincreases above areajyriginal, the internal pressure fallsand becomes negative. This mimics the behavior of anelastic cell which conserves volume by generatingrestoring tension forces when it is stretched. If oneregion of the cell protrudes and causes the cell area toexpand, other nonprotruding nodes will retract until thecell area returns to its original size. Conversely, whenthe cell is compressed, internal pressure increases untilthe cell area returns to its original value. We have madethis modification for the following reason. When thecell pressure is only allowed to be positive, the nodalforces are dependent upon two factors: (1) thecurvature of the local cell surface, which determines themagnitude of the net contractile elastic nodal forces,and (2) the magnitude of the outward cell pressurewhich varies inversely with the cell area. Differentcombinations of local curvature and cell pressure canresult in the same net mechanical forces at a node. Forthis reason, a drop in cell pressure may or may notalways lead to subsequent nodal retraction at anonprotrusive node, depending upon the local cellsurface curvature. When cell pressure is allowed tobecome negative, nonprotruding nodes will alwaysretract when the cell area expands in other locations. Inthis way, the original cell area is more accuratelymaintained by the retraction of nonprotruding nodes.

At the start of the simulation, areajoriginal is slightlylarger than the equilibrium cell area (the equilibriumcell area is reached when the sum of all forces at eachnode equals zero). The additional area accounts for theextracellular spaces between cells.

[E] Conditions that maintain the rectangularnotochord tissue shapeIn order to enforce the rectangular shape of thenotochord tissue, additional bounding boxes are placedalong the right and left sides of the tissue, of equalvertical height. This represents the attached somiticmesoderm. The important features of these boxes aretheir upper and lower horizontal surfaces which liecontiguous with those of the notochord tissue. Theelastic forces at cell nodes that lie along the notochordboundaries are increased above that used for cell nodeswithin the tissue interior. This tends to straighten thenotochord boundaries. The magnitude of the elastic


forces along both the horizontal notochord andbounding boxes are equal and larger than the elasticforces along the vertical notochord boundaries. Thisensures that the horizontal boundaries remain straightand do not bend where the notochord tissue andbounding boxes are attached. There are a number ofalternative sets of condtitions that would also tend toenforce the rectangular notochord shape. One exampleis to increase the elastic tension at cell nodes along thevertical notochord boundaries over that of the horizon-tal boundaries, and dispense with the bounding boxes.This will have the tendency of maintaining straightvertical tissue boundaries. Therefore, the rectangulartissue shape can be maintained by asymmetric mechan-ical forces acting along its horizontal and vertical axes.When this asymmetry is lost, for instance by removingthe bounding boxes and having elastic forces of equalmagnitude encircle the entire notochord perimiter, thetissue rounds up into a circle (Fig. 4).

This research was supported by NSF Grant No. MCS-8110557 to GO, NIH Training Grant No. GMO7048 to MW,NIH Grant No. HD25594 and NSF Grant No. DCB89052 toRK, and NIH Training Grant No. HD7375 to SM. Theauthors would like to thank Dianne Fristrom, LanceDavidson, Nicolas Cordova, and John Shih for a criticalreading of the manuscript, and Paul Tibbetts for photographicwork.

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(Accepted 5 September 1991)

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