Notochord morphogenesis in Xenopus laevis: simulation of cell behavior underlying tissue convergence and extension

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1 Development 113, (1991) Printed in Great Britain The Company of Biologists Limited Notochord morphogenesis in Xenopus laevis: simulation of cell behavior underlying tissue convergence and extension MICHAEL WELIKY 1 '* STEVE MINSUK 2, RAY KELLER 2 and GEORGE OSTER 2 ' 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 notochord morphogenesis in the African frog, Xenopus laevis. Experimental observations show that cells elongate mediolaterally and intercalate between one another, causing the notochord to lengthen and narrow. Descriptive observations provide few clues as to the mechanisms that coordinate and drive these cell movements. It is possible that a few rules governing cell behavior could orchestrate the shaping of the entire tissue. We test this hypothesis by constructing a computer model of the tissue to investigate how rules governing cell motility and cell-cell interactions can account for the major features of notochord morphogenesis. These rules are drawn from the literature on in vitro cell studies and experimental observations of notochord cell behavior. The following types of motility rules are investigated: (1) refractory tissue boundaries that inhibit cell motility, (2) statistical persistence of motion, (3) contact inhibition of protrusion between cells, and (4) polarized and nonpolarized protrusive activity. We show that only the combination of refractory boundaries, contact inhibition and polarized protrusive activity reproduces normal notochord development. Guided by these rules, cells spontaneously align into a parallel array of elongating cells. Self alignment optimizes the geometric conditions for polarized protrusive activity by progressively minimizing contact inhibition between cells. Cell polarization, initiated at refractory tissue boundaries, spreads along successive cell rows into the tissue interior as cells restrict and constrain their neighbors' directional bias. The model demonstrates that several experimentally observed intrinsic cell behaviors, operating simultaneously, may underlie the generation of coordinated cell movements within the developing notochord. Key words: computer simulation, mathematical model, morphogenesis, cell motility, notochord. Introduction Morphogenesis of embryonic tissues is driven by a variety of mechanisms including cell rearrangement and shape change. In many instances, directed cell movements cause tissues to extend and narrow, so called 'convergence and extension'. Examples include gastrulation in Xenopus laevis (Keller, 1978, 1984; Wilson et al. 1989) and the zebra fish (Warga and Kimmel, 1990), notochord development in Xenopus laevis (Keller et al. 1989), archenteron elongation during sea urchin gastrulation (Hardin, 1989), and imaginal disk evagination in Drosophila (Fristrom, 1976). As a result of active cell intercalation, the number of cells along one axis of the tissue decreases while the number of cells along an orthogonal axis increases. During this process, it seems certain that cell motility must be somehow coordinated, biased or constrained so that it leads to the required changes in tissue shape. Advances in video microscopy now allow detailed observations of these cell movements (Keller et al. 1989; Hardin, 1989). However, it is difficult to infer from observations of cell motions the forces that drive these motions, or the principles that coordinate and guide the cells. In this respect, models are helpful in deducing the conditions under which a variety of factors could drive morphogenesis. Examples include the mechanical forces driving cell chondrogenesis (Oster et al. 1985), neural plate folding (Odell etal. 1981; Jacobson et al. 1986), and epithelial cell rearrangement (Honda et al. 1982; Weliky and Oster, 1990); cell signalling during Dictyostelium aggregation (MacKay, 1978); and differential cell adhesion driving imaginal disk evagination (Mittenthal and Mazo, 1983), and cell and tissue sorting (Sulsky et al. 1984). In this paper, computer simulation is used to investigate the specific contribution of different cell behaviors to notochord morphogenesis. We propose to test a set of rules governing the behavior of motile cells, drawn from the literature on in vitro cell motility studies. These rules include: (1) persistence of direc-

2 1232 M. Weliky and others tional movement, (2) polarization due to tensioi induced inhibition of protrusion (Kolega, 1986), and (3 contact inhibition (Erickson, 1978a,b). We shall alsc include the experimentally observed phenomenon tha cell protrusion is inhibited at the notochord-somite boundary (Keller et al. 1989). These behavioral rule; are incorporated into a simulation model, described ir Weliky and Oster (1990), that accounts for the balana of mechanical forces within cells, and between median ically coupled cells within a tissue. The rules of eel behavior control the mechanics of individual cells, sc that when a large number of cells are coupled to one another, we can compare the development of the simulated tissue to morphometric data from scanning electron micrographs and time-lapse recordings of eel behavior in cultured explants. Two features distinguish this work from previous modeling efforts. First, we explicitly describe a tissue as a 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 shape changes, cell movements and rearrangements that underlie tissue morphogenesis. Second, we explicitly model the mechanical forces that cells generate and exert on one another. Thus, not only is cell behavior simulated in a mechanically rigorous manner, but we can also analyze the contribution of mechanics to cell motility and behavior within the tissue. First, we review some of the important features of notochord development and describe how cells behave during the late gastrula and early neurula stages. Next, we describe the mechanical cell model and the set of motility rules that we shall test. Finally, the simulations demonstrate how these motility rules can direct cell behavior and tissue shape change during notochord development. Cell behavior during notochord development During the late gastrula stage, the cells of the notochord become distinct from those of the surrounding somitic mesoderm tissue, and the boundary between the two tissues begins to straighten and align (Keller et al. 1989). As the boundary forms, the cells cease their protrusive activity at the lateral surfaces of the notochord. Subsequently, when motile notochord cells intercalate into the boundary, their protrusive activity also ceases along 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 the width of the notochord is reduced from about five or six cells to one. Time-lapse video microscopy reveals that active cell intercalation is primarily responsible for these changes (Keller et al. 1989). Fig. 1 shows a patch of fluorescein-labeled cells, grafted into the prenotochord region of a stage 10+ embryo. The patch breaks up into many fragments due to widespread cell movements that mix the labeled and nonlabeled cells Fig. 1. (A) Afluorescein-labeledpatch of cells is grafted into the pre-notochord region of a stage 10+ embryo. (B) The developed notochord shows breakup of the labeled patch caused by cell intercalation within the tissue. The notochord (labeled n) extends vertically along the anterior-posterior axis of the embryo and isflankedby somitic mesoderm (labeled s) (from Keller and Tibbetts, 1989, with permission). (Keller and Tibbetts, 1989). Fig. 1 also shows that cells elongate mediolaterally across the width of the notochord as they rearrange. Protrusive activity becomes increasingly polarized, so that opposite ends of the elongated cells bear large lamelliform and filiform protrusions, while the flattened cell sides bear short stubby filiform protrusions. Recent work has demonstrated that elongation begins with cells at the notochord-somite boundaries, then sweeps toward the tissue interior along successive cell rows (Shih and Keller, 1992). In this way, cells in the middle of the notochord are the last to polarize and elongate. The mechanical model Here we briefly describe the important features of the computer simulation model. For a more complete description, see Weliky and Oster (1990). Cells are represented by two-dimensional polygons, and each cell is capable of generating contractile and protrusive mechanical forces. To model a tissue, a large number of polygonal cells are mechanically coupled into a continuous two-dimensional sheet. Adjacent cells share common boundaries and vertices; this allows mechanical forces to be transferred from one cell to another, mimicking coupling by junctional adhesions (Fig. 2). In addition to vertices at all three-cell junctions, vertices are also introduced along the common boundaries between cells to allow for curved surfaces. All junctional nodes can slide, and nodes may disappear or

3 Notochord morphogenesis In Xenopus laevis 1233 B ARaaultant T fores 'A 2 Fig. 2. The notochord tissue is represented as an array of polygonal cells in which adjacent cells share common vertex nodes. The upper inset shows that vertices are introduced along the common boundaries between cells to allow for curved surfaces. The mechanical force balance between cells is computed at each node, shown in the left half of the lower inset. Elastic tension forces, T, act circumferentially around the cell perimeter while pressure forces, P, act outward and normal to the cell surface. When the mechanical forces at a node are unbalanced, the node slides in the direction of the net vector force shown in the right half of the insert. By reducing the cortical tension forces in cell 3, the swelling pressure now dominates the cortical tension forces and the node slides upward in the direction of the net unbalanced force. (A) Under the influence of the protrusive forces, nodes A and B slide between cells 2 and 4. (B) When the two nodes meet, cell rearrangement occurs. At this time, new nodes C and D are created such that cells 1 and 3 now contact each other while cells 2 and 4 separate. Cells 1 and 3 redirect their protrusive activity to separate nodes C and D. (C) Cells 1 and 3 continue to protrude in opposite directions and move past one another. appear, so that the number of polygonal sides defining a cell is variable. Thus the polygonal model can mimic the shape of virtually any cell quite closely. The model accounts for the balance of mechanical forces within individual cells, and between mechanically coupled cells in the tissue. Cell rearrangement and cell shape changes take place when the forces between neighboring cells deviate from mechanical equilibrium. Thus, morphogenetic cell movements reflect the presence of unbalanced mechanical forces within a tissue; move- ment will continue until mechanical equilibrium is restored. The forces generated by all cells at a common junctional node are vectorially summed to generate a net nodal force (Fig. 2A). A node is displaced by an amount proportional to the net nodal force (i.e., as if the node were subjected to linear frictional drag). The two intracellular forces acting at a node are the osmotic/hydrostatic pressure, and the elastic tension in the cortical actin gel (Oster, 1988). Two mechanisms for active cell intercalation are modeled. The first allows cells to 'push' their way in between neighboring cells, and is used when directional persistence is incorporated into the simulations. When a cell is activated to move, its cortical tension drops at the node where protrusion will take place, reflecting solation of the cortical actin gel. Cortical pressure then drives the node in the direction of the net force imbalance and the cell protrudes forward (Fig. 2B). The second mechanism allows for cell protrusion into interstitial spaces created by adjacent cell retraction, and is used when contact inhibition of protrusion is incorporated into the simulations. When activated, a cortical protrusion extends outward to fill the available interstitial space between cells. Upon contacting an adjacent cell, the protrusion is inhibited. Since cells share common boundaries, we cannot explicitly model interstitial spaces; however, we can simulate this protrusive behavior by realizing that in order to create an interstitial space into which a cell can crawl, adjacent cells had to contract their boundaries. Therefore, we can model a cell moving into a space by having a node move only when it experiences forces from neighboring contractile cells. For example, at a node where one cell is protrusively active while the remaining two cells are contractile, the node will be pulled in the direction of the net contractile forces produced by the two nonprotruding cells. In this way, the protruding cell surface expands outward, simulating its movement into the space vacated by its neighbors. We use an iterative finite difference method to solve the force balance at every node during successive time steps. Each time step represents a snapshot of all current mechanical forces and cell geometries. As nodes move according to the applied balance of forces, cell rearrangement occurs when two nodes meet (Fig. 2B). At this time, exchange of cell neighbors occurs: the pair of cells initially separated from one another establish contact while the pair which were initially in contact separate. If two of the rearranging cells are protruding, their activity is redirected such that after rearrangement, they continue to protrude in opposite directions (Fig. 2C). Protruding nodes are selected randomly. This reproduces the experimentally observed behavior of rearranging notochord cells. Rules governing motile cell behavior In this section we describe how the cell motility rules are modeled and explain their experimental justifi-

4 1234 M. Weliky and others cation. Three classes of rules have been investigated: (1) how cells behave at the tissue boundaries; (2) how cells polarize and (3) how cells change their direction of movement. Cell behavior at the tissue boundaries Protrusions are absent in notochord cells where they contact adjacent somitic mesoderm (Keller et al. 1989). In order to model this behavior, we simply inhibit protrusive activity at nodes along the boundaries of the cell sheet representing the border between notochord and somitic tissue (we will use the term, refractory boundary, to refer to a tissue boundary at which protrusive activity is inhibited). Therefore, marginal cells at these boundaries protrude only at interior nodes; when an intercalating cell reaches this boundary, protrusive activity ceases at nodes which touch the boundary. When the simulations incorporate both polarized protrusive activity and refractory boundaries, marginal cell protrusion is restricted to the interior node most directly pointing away from the boundary toward the tissue interior. Cell polarization Motile cells are modeled either as unidirectionally or bidirectionally protrusive. Unidirectionally protrusive cells are modeled by randomly selecting a single cell node to protrude; these cells generally tend to remain isodiametric. Bidirectionally protruding cells are modeled by restricting protrusive activity to two nodes located at opposite ends of the cell. Bidirectionally protrusive cells will intrinsically elongate into a spindle shape. In addition to notochord cells, many other types of motile cells in vitro and in vivo exhibit bipolar shapes. On a fibronectin-coated substratum, Xenopus prospective head mesoderm cells elongate into a spindleshaped form and extend two or more lamelliform protrusions (Winklbauer, 1990). Fundulus deep cells of a midgastrula stage show bipolarity with a leading lamella and a trailing edge that isfirmlyattached to the substratum (Trinkaus, 1984). In vitro crawling fibroblasts also tend to become progressively elongated and bipolar, forming dominant adhesive contacts to the substratum at opposite ends of the cell (Fig. 3A). The cell exerts the tractional forces on the substratum at these contact points. Mechanical tension increases in the lateral cell surfaces of an elongating cell and this tension appears to inhibit protrusive activity in those regions (Kolega, 1986). We use this observation to model the development of polarized protrusive activity in a cell by imposing a dependence between protrusive activity and the degree of cell elongation (Fig. 3B). To do this we TENSION 0) (2) (3) (4) INHIBITED NODES (2) (3) (4) (1) B Fig. 3. Comparison of a polarizing cell to the model approximation. [A] Motile cell elongation. (1) An initial nonpolarized cell. (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 with large lamellipodial regions at opposite ends of the cell. [B] Model. (1) An intially nonpolarized cell has equal probabilities of protruding at all nodes. (2) The probability of protruding at a node increases as the node successfully extends (shown by large arrow). (3) When the cell reaches a minimum threshold of elongation, protrusive nodes that lie along the lengthening lateral cell surfaces are progressively inhibited. The node that is most opposite to the maximally protruding node is not inhibited. (5) Final polarized cell with two opposing protruding nodes.

5 Notochord morphogenesis in Xenopus laevis 1235 Fig. 4. The results of a simulation incorporating refractory tissue boundaries along the entire tissue perimeter, 99 % directional persistence, and cell polarization. When cell protrusion is inhibited along the entire tissue perimeter - in contrast to only at the right and left boundaries as in subsequent simulations - the tissue contracts into a circular shape. Almost all marginal cells have elongated towards the tissue interior by the end of the simulation. define a dimensionless 'cell shape index' as the ratio of the square of the cell perimeter to its area. The shape index is smallest for an isodiametric cell, and increases the more elongated a cell becomes. Therefore, the shape index can be used to measure the extent to which protrusive activity will be inhibited on lateral cell surfaces (Appendix A describes the algorithm in detail). A polarized cell will intrinsically elongate and develop protrusions at its opposite ends. In some simulations (Figs 4 and 6), a simpler model for cell polarization has been used.. In these cases, two opposing.nodes are simply chosen to be protrusive while all other nodes are inhibited. i How cells change their direction of motion We model two mechanisms by which cells change their direction of movement, (a) In the absence of cell-cell interactions, cells exhibit directional persistence such that they tend to maintain their direction of motion, albeit with some degree of randomness, (b) Cells interact by contact inhibition of protrusive activity. Directional persistence is observed in motile cells in vitro, as well as in vivo. We model persistence of motion by assigning probabilities to each cell node, which allows any cell to change direction on any iteration. A directional persistence of 100 % means that a cell will never change direction; a cell with 0 % persistence will change direction at each iteration. Intermediate values model varying degrees of directional persistence. For example, a cell with 95% persistence will change direction, on average, once every twenty iterations. It is not obvious what numerical level of persistence best reflects the behavior of living cells, nor is it obvious what are the implications of various levels of persistence for the behavior of the whole tissue. Therefore, each of the simulations was run using several different levels of persistence varying from 0 % to 100 %. Contact inhibition of protrusive activity is commonly observed during in vitro studies of cell motility (Abercrombie and Ambrose, 1958; Erickson, 1978a,b). Inhibition of localized protrusive activity occurs when the lamella of one cell contacts the surface of a second cell, whereupon the leading lamella ceases its forward motion, and often redirects its movement along the lateral sides of the protrusion (Erickson, 1978a). Contacting cells can protrude at their free margins, implying that contact inhibition is a local phenomenon which does not necessarily paralyze protrusive activity throughout the entire cell. Unlike epithelial cells, cells of the notochord do not appear to be continuously joined to one another around their entire surfaces. SEM reveals interstitial spaces between cells (Keller et al. 1989). Therefore, it is possible that, as cells contract and pull away from their neighbors, spaces open up into which an active cell can crawl. Our hypothesis is that protruding cells do not 'push' their way between adjacent cells but rather crawl into vacated spaces created by withdrawal of adjacent cells. When a protrusion mechanically contacts a stationary or advancing neighboring cell, that protrusion is inhibited. This behavior is analogous to cells protruding in the direction of least resistance. In our model for contact inhibition, nodes are assigned a probability 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 to protrude (Appendix A). When contact inhibition occurs between two cells, the probability of protruding at both nodes is reduced by a constant amount, Pinhib, during each iteration that the contact is sustained (Appendix B). In order to determine when contact inhibition has occurred, the direction of mechanical forces within cells, or their protrusive activity, indicates whether they are stationary, withdrawing or advancing towards their neighbors (Appendix D). An alternative mechanism would allow protruding cells to mechanically 'push' their way between neighbors without being contact inhibited, as long as adjacent cells 'yield'. If adjacent cells mechanically deform or shift their position, then protrusion continues. Otherwise, if the mechanical resistance of adjacent cells is above a prescribed amount, then the cell ceases to protrude in that particular direction and is 'blocked'. Like classical contact inhibition, this behavior would lead to cell protrusion in the direction of least mechanical resistance. Though this alternative model could have been implemented, we wanted to restrict ourselves to investigating the morphogenetic consequences of previously described in vitro contact inhibition behavior.

6 1236 M. Weliky and others The contribution of specific cell behaviors to notochord development In this section we illustrate the effect of different motility rules on tissue development by presenting a series of computer simulations. In order to keep the computation manageable, we use a two-dimensional cell sheet to model a small, representative region of notochord tissue (Figs 4-7). Our two-dimensional model is equivalent to a tissue explant, removed from the embryo, which has been reduced in depth to a single cell layer lying parallel to the epithelium. Boundary conditions are applied to the cell sheet which simulate a number of those found in the intact embryo. Constant drag forces are applied to cells on the upper and lower boundaries of the cell sheet, mimicking the mechanical properties of tissue lying above and below the modeled region. One drag component is applied to nodal movement along the vertical tissue axis, slowing the rate of tissue elongation. The second component acts as viscous drag on horizontal nodal movement along the boundaries, and is chosen so that cell rearrangement occurs at the same rate along the boundary and within the tissue interior. Under these conditions, the model tissue is not subjected to explicit external compressive forces. The left and right boundaries of the cell sheet represent the mediolateral notochord-somite tissue boundaries, at which cell protrusive activity is inhibited. In all simulations, except Fig. 4, unrestricted protrusive activity is allowed at cell nodes that lie on the upper and lower boundaries of the cell sheet where additional notochord tissue would extend along the anteriorposterior axis of the embryo. Appendix E describes conditions that maintain the rectangular tissue shape. In each simulation we incorporate three motility rules, one selected from each of the three classes described above. All simulations include inhibition of cell protrusion along the right and left tissue boundaries. In addition, we select a mechanism by which cells change their direction of motion and whether they are unidirectionally or bidirectionally protrusive. We will focus our attention on the contributions of these behavioral rules to five important features of notochord development: (1) tissue extension and narrowing, (2) cell rearrangement, (3) incorporation of cells into the notochord-somite tissue boundaries, (3) cell elongation, and (4) parallel cell alignment. All simulations begin with the same cell configuration. The cells are initially isodiametric, corresponding to the unelongated, nonpolarized, early notochord cells observed during the late midgastrula stage. At the start of each simulation, one or more nodes in each cell are randomly chosen for protrusive activity, subject to the constraints of the particular behavioral rules being tested. Tissue extension and narrowing The most important factor responsible for tissue extension and narrowing is the inhibition of cell protrusive activity along the left and right tissue boundaries while protrusive activity is permitted along the top and bottom boundaries. The simulations show that such biasing of cell protrusive activity always produces tissue extension and narrowing regardless of the motility behavior within the tissue interior (Figs 5-7). Cell protrusive activity at the top and bottom tissue boundaries is directed away from the tissue interior and contributes to vertical tissue extension. Simultaneously, refractory boundaries constrain the motility of marginal cells along the right and left margins of the tissue, biasing their protrusive activity and movement towards the interior of the tissue. Therefore, marginal cells converge towards one another, pulling the opposite sides of the tissue closer together and compressing interior cells. This compression is relieved by outward expansion of marginal cells at the top and bottom tissue boundaries, which further contributes to vertical tissue extension. The generation of compressive forces is easy to understand when a cell 'pushes' in between its neighbors and squeezes them. However, when cells are only able to crawl into the open spaces created by adjacent cell withdrawal, compressive forces that drive tissue extension can arise from differences in surface curvature along the cell surface. Consider the marginal cells at the top and bottom tissue boundaries. They have large flat regions along the tissue boundary and with individual adjacent cells. In addition, they make sharp acute angles where three cells meet. Because the net nodal tension varies with surface curvature, stronger contractile forces are generated at three-cell junctional nodes than at nodes along the flat surfaces. Cells will have a tendency to expand outward along their flattened surfaces which offer the least resistance to internal pressure. For marginal cells, expansion occurs primarily along their flattened surfaces which form the tissue boundaries. The tissue vertically extends because external forces along the top and bottom boundaries are smaller than the forces resisting compression within the tissue. If cell protrusive activity is inhibited along the entire tissue perimeter, rather than just along the right and left boundaries, and the boundary conditions are modified so that the rectangular tissue shape is not rigidly enforced, the tissue contracts into a circular shape with elongated cells encircling the perimeter (Fig. 4, and see Appendix E). Cell rearrangement In the absence of contact inhibition, the frequency with which cells change their direction of movement is determined by their strength of directional persistence. When directional persistence is strong enough, extensive cell movements and cell rearrangements occur throughout the tissue (Figs 5B and 6). This is revealed by the breakup of the patch of darkened cells. Compare the similarity of the breakup of the simulated cell patch in Figs 5B and 6 with the breakup of the fluoresceinlabeled cell patch in the actual notochord (Fig. 1). In both cases, labeled and unlabeled cells mix together and most of the labeled cells are eventually incorporated into the right and left tissue boundaries.

7 Notochord morphogenesis in Xenopus laevis 1237 Fig. 5. The results of simulations incorporating right and left refractory tissue boundaries, varying degrees of directional cell persistence, and unidirectional cell protrusion. In both A and B, tissue elongation and narrowing occur but with varying degrees and patterns of cell movement and rearrangement. The shape of interior cells remain primarily isodiametric throughout both simulations. (A) 0 % directional persistence (cells change direction every iteration). Similar results are observed for persistence values up to about 99 %. Tissue elongation drives 'passive' cell rearrangements along the vertical tissue axis. This leads to an increase in the number of interior cells along the tissue length (from an average of 13 to 17) while reducing the number of cells spanning the tissue width (from an average of 8 to 6). This process is reflected in the darkened cell patch which changes in dimension from 4 by 3 cells to 2 by 5 cells. The patch does not break up and none of the labeled cells intercalate into the tissue boundaries. The number of marginal cells increases only slightly from 14 to 15 on the right tissue boundary and from 16 to 18 on the left. (B)99.8% directional persistence (cells change direction on average once every 500 iterations). Increased directional persistence leads to widespread cell rearrangements as revealed by the breakup of the darkened cell patch. Most of the labeled cells migrate into the right and left tissue boundaries. Cell intercalation is responsible for the dramatic increase in the number of marginal cells along the right and left tissue boundaries. The number of cells on the right boundary increases from 13 to 38, 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. The number of interior cells spanning the width of the tissue is reduced from 8 to 2 cells as a result of the cell migration into the tissue boundaries. Directional persistence must be above 99% (e.g., cells change direction less than once every hundred iterations) in order to produce cell rearrangement. Cell protrusion must persist at a three-cell junctional node long enough for a cell to traverse the distance to the next adjacent three-cell node. When two nodes meet, cell neighbor change occurs (Fig. 2). Under this condition, active cell protrusion will drive cell rearrangement regardless of the ability of the tissue to extend. Figs 6A and B show that the degree of cell rearrangement in a fully or weakly extending tissue is roughly the same. In contrast, if directional persistence falls below 99 % (i.e. cells change direction on average more frequently than once every hundred iterations), motile cells will be unable to reach the next three-cell nodal junction before changing their direction. In this case, cell rearrangement becomes strongly dependent upon the degree of tissue elongation (Fig. 5A). As the cell sheet vertically lengthens, cells are stretched and elongate in the direction of tissue extension. Cell rearrangement occurs as elongating cells attempt to return to their initial isodiametric shape. Cells aligned along the vertical axis of extension separate, allowing adjacent

8 1238 M. Weliky and others Fig. 6. The results of simulations incorporating right and left refractory tissue boundaries, high directional cell persistence, and cell polarization. These simulations use the simplified cell polarization model in which interior cells always protrude at two opposing nodes. Marginal cells protrude at the node most directly pointing toward the tissue interior. In all cases extensive cell intercalation leads to the breakup of the labeled cell patch as labeled and unlabeled cells randomly mix together. In A and B, extensive cell elongation occurs in many different directions. (A) 100 % directional persistence (cells never change direction). A small drag force, slightly resisting tissue extension, is applied to cell nodes at the upper and lower tissue boundaries. In this case the tissue lengthens by a factor of about 1.6. (B) 100% directional persistence. A stronger drag force is applied to cell nodes at the upper and lower tissue boundaries, which results in reduced ability of the notochord to extend. The tissue is therefore shorter and fatter than in A (both simulations are run for an equal number of iterations). (C)99% directional persistence (cells change direction on average once every 100 iterations). Interior cell elongation is not as extensive as in B since cells occasionally change their protrusion direction. Note that most interior cells tend to remain isodiametric; however, interior cells that abut marginal cells tend to be horizontally elongated as seen in the last three panels. The same drag force is used as in B. For persistence values less than 99% (not shown), results similar to Fig. 5A are observed. cells to establish contact. These cell rearrangements cause the darkened cell patch to elongate and narrow. The patch does not break up, so that labeled and unlabeled cells do not mix. If the tissue is mechanically prevented from lengthening, no cell rearrangements will occur, (see Oster and Weliky, 1990; Weliky and Oster, 1990 for other examples and a complete discussion of the mechanical forces underlying 'passive cell rearrangements). Fig. 7 shows that when contact inhibition is incorporated into the simulation, net cell movement is initially blocked within the interior of the tissue. This is due to the random, uncoordinated protrusive activity of neighboring cells which mutually inhibit each other. As

9

10 Fig. 7. The results of a simulation incorporating right and left refractory tissue boundaries, contact inhibition of protrusion and cell polarization. Color represents the probability that a cell node will protrude during each iteration. Blue represents low 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 inhibited at all nodes except for the interior-most node whose probability for protrusion is 1.0. Beginning at the right and left tissue boundaries, polarization and elongation spread along successive cell rows into the tissue interior. As cells elongate, nodes located at the extending regions of the cell turn red indicating that these nodes are becoming strongly protrusive, while nodes lying along the lengthening cell sides turn blue indicating that they are becoming inhibited. This can be appreciated by following the behavior of the numbered cells. In iteration 0, all interior cells are isodiametric and nonpolar. By iteration 80, 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 between the already polarized first row cells (note cell 7). By iteration 180, the third cell row begins to polarize, which is seen by the color change and elongation of cell 6. Note that the centrally located cells 1 and 2 remain nonpolarized and isodiametric while their neighbors to the right and left have already polarized and started elongating. By iteration 700, all cells have horizontally polarized and elongated with strong, bipolar protrusions at their extending regions. Between iteration 700 and 28000, intercalation decreases the number of cells spanning the tissue width from an average of 6 to 2 cells, and increases the average number of cells along the right or left tissue boundary from 21 to 44 cells. Note that by iteration 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.

11 Notochord morphogenesis in Xenopus laevis 1239 B Fig. 8. Comparison of the repetitive extension and retraction cycles of a Xenopus deep cell and a modeled cell. The solid line indicates the cell shape in the previous frame of the sequence. (A) The changing shape of a single deep cell is seen in tracings 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 incorporating intrinsic cell polarization and contact inhibition. The protrusive cell nodes at each frame are marked with short arrows. The cell changes its protrusion direction many times until it eventually becomes aligned horizontally, with protrusive activity at each end of the cell. The horizontal cell orientation corresponds to mediolateral elongation within the notochord tissue. a result, cells remain nonpolarized and continue to protrude and contract in many directions without any net cell movement (Fig. 8B); this behavior reflects the cells' inability to establish a stable and persistent protrusion direction. Fig. 8A shows that Xenopus deep cells exhibit a similar 'kneading' or jostling behavior consisting of repeated cycles of protrusion and retraction, also without any net cell movement (Keller and Hardin, 1987). This cell activity is also experimentally observed during notochord development (Keller et al. 1989). As cells in Fig. 7 polarize horizontally, stable and persistent bipolar protrusions develop at their elongating tips which drive cell movement and rearrangement within the model tissue. Incorporation of cells into the tissue boundaries Whether cell motility within the tissue is random (Figs 5B and 6) or directed horizontally (Fig. 7), cell movement eventually brings interior cells in contact with the left or right tissue boundaries. Once in contact with these boundaries, cells rarely pull away. This leads to a progressive increase in the number of cells along the right and left boundaries while the number of cells within the tissue interior consequently decreases. Eventually, almost all interior cells become incorporated into the tissue boundaries. Once cells reach the boundary, not only do they 'stick' and rarely pull away, but they expand their region of contact with the boundary. Without invoking any special adhesive boundary conditions, there is a simple mechanical explanation for this phenomenon. The inhibition of protrusive activity along the right and left tissue boundaries implies that only contractile mechanical forces exist at cell nodes lying along these boundaries. Therefore, this behavior can be understood by analyzing the balance of mechanical forces at these exclusively contractile marginal cell nodes. A cell that has newly intercalated into the tissue boundary initially forms a tapered point with the boundary. Nodal forces" cause the tapered point to expand to a blunt edge until this cell, and its two marginal cell neighbors, make right angles with the boundary. When adjacent marginal cells at a common junctional node make 90 degree angles with the boundary, both cells generate net elastic nodal forces of equal magnitude (the elastic nodal force is curvature dependent; therefore when cells have equal curvature at a node, or equivalently, make equal angles at a node, the magnitudes of their elastic forces are equal). The node is now in mechanical equilibrium and stable. See Weliky and Oster (1990) for a more detailed mechanical description of this behavior. Of course, specifically adhesive boundaries would have produced the same cellular configurations. However, it turns out that the mechanical balance of forces at an inhibitory boundary produces the same effect as an adhesive boundary. Active cell elongation When cells protrude unidirectionally with strong directional persistence, interior cells do not elongate but remain mostly isodiametric throughout the simulation (Fig. 5B). This is because nonprotrusive cell regions retract as the protruding node extends forward, thus maintaining the cell's isodiametric shape. Cell rearrangement plays a crucial role in this process by allowing cells to accommodate their neighbors by adjusting their shapes. Fig. 5B also shows that, unlike interior cells, marginal cells elongate. This occurs when the entire tissue does not lengthen fast enough to accommodate the increase in the number of marginal cells. As interior cells migrate and crowd into the limited tissue boundaries, they will elongate perpendicularly to the boundaries as a result of simple mechanical compaction. When unidirectional pro-

12 1240 M. Weliky and others trusion is replaced by bidirectional protrusion, the resulting intrinsic cell polarity produces elongated interior cells (Figs 6 and 7). Thus the simulations suggest that elongation of cells within the interior of the notochord tissue must be actively generated by the cells themselves, and does not result from passive mechanical compaction, as is true for marginal cells along the tissue boundaries. Fig. 6B shows that when cells protrude bidirectionally with strong directional persistence, not all cells elongate to the same extent. One sees a wide range of elongations, from near isodiametric cells to long spindle-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 will elongate much more than when multiple cells protrude at that node. At nodes where all three cells are protruding, the protrusive forces can cancel. Parallel cell alignment Fig. 6B reveals that when interior cells polarize with strong directional persistence, and with no cell-cell contact inhibition, cell elongation is extensive, but randomly oriented. Close inspection shows that cell alignment occurs in local domains or groups of cells, but it does not extend globally. As the simulation progresses, additional neighboring cells elongate and align within these domains, and finally, as obliquely oriented cells intercalate between already elongated marginal cells, they rotate and asssume the horizontal orientation imposed by their marginal cell neighbors. At the end of the simulation, most cells have elongated and are horizontally aligned. However, even during later stages of the simulation a number of vertically oriented interior cells still remain (fourth panel in Fig. 6B). When interior cells polarize with only slightly reduced directional persistence, a different pattern of cell elongation results (Fig. 6C). In this case most interior cells remain roughly isodiametric. However, horizontal elongation is seen among many of the interior cells that abut marginal cells. This is seen most clearly during the middle to later stages of the simulation (last three panels of Fig. 6C). The reason for this behavior is that occasional direction changes allow interior cells to sample the surrounding mechanical conditions imposed by adjacent cells. As described in the previous section, cell elongation occurs more rapidly when only one cell protrudes at a junctional node. Since polarized marginal cells always protrude at the node which is furthest from the tissue boundary, abutting interior cells can rapidly elongate horizontally by protruding between the contractile or nonprotrusive regions of two neighboring marginal cells. In contrast, cells located deeper within the tissue are surrounded by shifting mechanical conditions caused by the random directional changes of their neighbors. Thus they cannot elongate as well and so remain roughly isodiametric. Not all interior cells abutting marginal cells elongate horizontally; some remain more isodiametric (see last two frames of Fig. 6C). Most of these cells are protruding along the vertical tissue axis, and are prevented from elongating by mechanical restrictions imposed by the already horizontally polarized marginal cells. A more efficient strategy would be to inhibit cell protrusion in a direction which is mechanically obstructed by adjacent cells, while allowing continued protrusion in an unrestricted direction. This form of cell-cell interaction can be incorporated into the simulation in the form of contact inhibition of protrusive activity. The results are shown in Fig. 7. When contact inhibition of protrusive activity is combined with cell polarization, all interior cells elongate exclusively in a horizontal direction. Cell polarization and elongation occurs in a wave of activity that sweeps from the right and left tissue boundaries into the tissue interior. Starting with biased protrusive activity of marginal cells, successive cell rows polarize by a process of restricting and constraining their directional bias (Fig. 9). Initial marginal cell protrusive activity must be biased such that protrusion occurs only at the interior most node; when marginal cells are allowed to protrude randomly at any interior node, interior cells do not elongate exclusively in a horizontal direction. All cells in Fig. 7 have horizontally polarized within the first three hundred iterations. During this time the number of instances of contact inhibition between cells decreases to near zero (Fig. 10). During the remainder of the simulation, the cells maintain this aligned configuration and continue to intercalate between one another until most cells have been incorporated into the right and left tissue boundaries. In order that interior cells do not prematurely polarize in random directions during the early stages of the simulation, the magnitude of contact inhibition must be seven times larger than the rate at which the nodal protrusion strength increases due to protrusive activity. This ensures that random variations of protrusion strength will be dampened by contact inhibition. Discussion Multiple cell behaviors drive morphogenesis Within the embryo, there are many different cell behaviors operating simultaneously, so it is difficult to determine how each contributes to the overall morphogenetic movements. Computer simulation allows us to test the specific role that individual cell behaviors, or combinations of cell behaviors, play during tissue morphogenesis. The mechanical cell model described in Weliky and Oster (1990) is designed to simulate the forces generated by cells within a tissue. In this paper, we have proposed a set of cell motility rules which regulate the cell mechanics. Using the model, we can investigate the morphogenetic consequences of these rules in a mechanically correct manner. Our simulations show that random, nonpolarized cell motility, combined with refractory tissue boundaries, produces elongation of the tissue and of marginal cells, but leaves interior cells isodiametric (Fig. 5B). When intrinsic cell polarity is combined with high directional

13 Fig. 9. Polarized cell protrusive activity spreads progressively towards the tissue interior. (A) At the simulation start, polarized marginal cells [white] lying along the notochord boundary are constrained to protrude only at their interior-most node (protrusion probability= 1.0). Protrusion is inhibited at all other marginal cell nodes by assigning to them a protrusion probability of 0.0. In contrast, interior cells [shaded] have an initial o GO Q DC o Notochord morphogenesis in Xenopus laevis 1241 uniform 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 towards the tissue interior at its protrusively active node, nonprotruding regions retract and pull away from adjacent submarginal cells at the circled nodes. This allows the first row of submarginal cells [light shading] to extend towards the notochord boundary, by protruding at the circled nodes in between the retracting regions of marginal cells. (B) As these interior cells elongate, protrusive activity in nodes along their lateral surfaces is inhibited. This in turn allows the second row of interior cells [dark shading] to elongate by protruding at the circled nodes. (C) Row by row, polarization spreads towards the interior of the tissue. B total call protrusions total Instance* of contact Inhibition interior (Fig. 7). These results support the notion that several rules for cell motility and cell-cell interaction, operating simultaneously, may drive notochord morphogenesis Iteration Number 14,000 28,000 Fig. 10. As cells polarize across the width of the notochord, the total number of inhibitory contacts between protruding cell nodes and adjacent cells rapidly decreases to near zero within the first 300 iterations. The reduction in the number of total inhibitory contacts parallels the emergence of a parallel aligned cell array. In addition, the total number of cell protrusions slowly decreases throughout the simulation due to lateral inhibition of protrusive activity and cell migration into the notochord boundaries. persistence, cells initially elongate in random orientations throughout the entire tissue interior (Fig. 6A and B); reducing directional persistence causes only those interior cells that abut marginal cells to horizontally elongate (Fig. 6C). Finally, when contact inhibition of protrusive activity is combined with intrinsic cell polarity and refractory tissue boundaries, the simulation reproduces all of the major features of notochord development: cell rearrangement extends and narrows the tissue, while transverse cell elongation begins at the refractory notochord-somite boundaries and spreads along successive cell rows into the tissue Coordinated cell behaviors arise from local cell interactions Many problems in biology can be formulated as optimization or constraint satisfaction problems. Examples include the organization of social insect colonies (Oster and Wilson, 1978), population genetics and neural network dynamics (Bounds, 1987). One way to solve large optimization problems is to implement the constraints as excitatory or inhibitory interactions among a system of interacting cells. The system is allowed to iteratively converge, from an intitial random configuration, to a stable solution that satisfies the largest set of compatible constraints (Hopfield and Tank, 1986). Similarly, the emergence of organized cell motion within developing tissues can be treated as an optimization or constraint satisfaction problem. In the model notochord shown in Fig. 7, constraints upon cell motion are implemented through local inhibition of cell protrusion between contacting cells. Initially, random protrusive activity allows each cell to explore its local environment in an attempt to find an admissible direction to move. But because this activity is temporally and spatially uncoordinated among neighboring cells, contact inhibition forces cells to constantly change their protrusion direction. This results in the jostling or 'kneading' behavior of cells without any net cell movement (Fig. 8B). As cell protrusive activity becomes increasingly biased in a horizontal direction, the total number of contact inhibitions between neighboring cells is minimized (Fig. 10). This allows cells to elongate into a parallel array by intercalating between

14 1242 M. Weliky and others one another without further changing their protrusive direction. An array of parallel elongating cells provides the most geometrically favorable conditions for effective protrusion: wherever three cells meet only one cell is protrusively active while the two remaining cells are contractile. Thus, the simulations reveal that local constraints upon cell motility can force cells within a tissue to organize into a specific geometrical configuration. Two-dimensional simulation of a three-dimensional tissue The in vivo notochord is a three-dimensional structure in which constituent cells undergo three-dimensional movements and shape changes. Alhough the notochord tissue and cells change shape in three dimensions - especially in the later stages of development - we have focused only upon those features that lend themselves to a two-dimensional analysis; i.e., that have a significant planar component, parallel to the epithelium. These features include mediolateral cell elongation and tissue convergence and extension, which have already started during the earlier stages of notochord development (Keller et al. 1989). Our model tissue should not be literally interpreted as a cross section through the three-dimensional notochord, for in this case, cell areas would not remain constant as cells move in and out of the modeled plane. Rather the model should be viewed as a 'reduced' notochord in which the tissue depth has been reduced to one cell. Recent work with shaved notochord explants in which the depth of the tissue is diminished by removing a number of cell layers, more closely corresponds to our two-dimensional model (Shih and Keller, 1992). In these explants, the cells and tissue are restricted to primarily planar movements but still retain many of the essential features of notochord development as described in previous studies and reproduced in this paper (Keller et al 1989). Mechanisms for biasing cell motility The simulations demonstrate the importance of cell behavior at a tissue boundary in constraining the tissue to change shape in specific ways. Though protrusive activity is biased only in cells that lie along the right and left tissue boundaries, the effect this has upon global tissue morphogenesis is very strong. For instance, refractory boundaries always produce tissue elongation and narrowing regardless of the cell behavior within the tissue. In addition, we have shown that specific cell behaviors, such as cell polarization, can arise along developing tissue boundaries and subsequently spread throughout the tissue. However, this raises the question of how cell motility is biased in tissues where no obvious boundaries exist, such as archenteron elongation in the sea urchin (Hardin, 1989). Parallel arrays of cultured fibroblasts form by a process of self alignment, mediated by cell-cell interactions in the form of contact inhibtion of protrusion (Elsdale and Wasoff, 1976; Erickson, 1978a, 1978/J). Our simulations show that contact inhibition of protrusive activity is a sufficient constraint on cell motion to force successive cell rows to polarize transversely across the width of the notochord. An alternative hypothesis is that cells transmit an inductive signal, initiated by the marginal notochord cells. No chemical signal has yet been found; however, the results of this paper cannot be used to prove or disprove the exact nature of the chemical or mechanical events underlying notochord morphogenesis. Rather, these results can suggest potential mechanisms that may underlie the observed cell behaviors and suggest new experiments. In the absence of any direct evidence for a chemically transmitted polarization signal, our results suggest that a few simple, well characterized in vitro cell motility rules can account for the experimentally observed mediolateral cell polarization within the notochord and the elongation of the tissue. Appendices [A] Modeling cell polarization The following describes the cell model for generating polarized protrusive activity (see Fig. 3B): 1. Initially, protrusion will occur at each of the cell nodes with the same initial probability P no de- We use 0.8 as the initial value, i.e. there is an 80% chance that a node will protrude during each iteration. With this initial condition, the direction of cell protrusion is not biased. 2. During each iteration, the nodal probability Pnode is used to determine whether or not a node will be activated to protude. The computer generates a random number in the range of 0 to 1 for each cell node. If this number is smaller than the nodal probability, then the node will be activated during that iteration. Each iteration that a node is activated to protrude, P no de is increased by a constant amount AP ac/il, e =0.01. Nodes with high protrusion probabilities will be more likely to be protrusively active during subsequent iterations than nodes with lower values. Therefore, the larger P no de is, the faster its rate of increase. 3. We use the shape parameter index, INDEX= PERIMETER 2 /AREA, to calculate the degree of cell elongation. The higher the index, the more elongated is the cell's shape. We normalize the value of INDEX by the initial cell index value calculated at the start of the simulation. In this way, each cell initially has an index value of When the shape index increases above a minimum threshold value=1.05, the protrusion probability at all nodes, except the maximally protruding node and its most opposing node, is reduced as a function of increasing elongation (i.e. this would indicate that the cell has elongated to a point where sufficient mechanical tension is being generated by an advancing lamellipodium). For an inhibited node, the protrusion probability becomes Pnode = O.8~(C* INDEX) (1) where 0.8 is the intitial spontaneous protrusion pro-

15 Notochord morphogenesis in Xenopus laevis 1243 bability and c=70.0 is a constant. By differentiaing eq. 1 we can calculate the amount of change in P no d e during each iteration as a function of the change in INDEX as follows AP,NDEX =-(c* AINDEX) (2) where AP INDEX is the change in the nodal probability which results from a change in the cell shape parameter index, AINDEX, during each iteration. [B] Reducing the nodal protrusion probability due to contact inhibition When a protruding cell node detects opposing forces in adjacent cells, this reflects mechanical contact between the two cells as described in appendix D. Under this condition, the nodal probability is reduced by a constant amount AP in hib = 0.075, which represents the reduced capacity of the node to protrude as a result of contact inhibition. The value of AP inhib is chosen to be substantially larger than the value of AP active, such that inhibition strongly reduces the protrusive capacity of a node at a rate faster than the nodal protrusion probabality can increase due to protrusive activity. [C] Calculating the total change in nodal protrusion probability For each iteration, the total change in the nodal protrusion probability AP^de is simply the sum of all contributing probability changes as calculated in appendices A and B: ctive + INDEX (3) With the intitial protrusion probability of each cell node set at 0.8, Pnode = 0.8 (4) the subsequent value of P node at time step, t+1, is calculated by simply adding the current value of AP node to the value of P no de at the current time step, t: inodc f, + i) "node lt) ' ^^node (l) \p) If Pnode is below 0.8 at a node which is free from contact inhibition with adjacent cells, Pnode returns to either 0.8 or an appropriate value as determined by the cell shape index. If the cell has not yet polarized, then all nodes will return to a nodal probability of 0.8. If the shape index is above the minimum threshold for inhibition, then the probability of protrusion at nodes that are internally inhibited by cell elongation is Pnode = 0.8- (C* INDEX) (6) In this way, the spontaneous level of protrusion for internally inhibited nodes will be reduced from the normal value of 0.8. [D] Modeling contact inhibition Contact inhibition occurs when neighboring cells attempt to protrude or advance towards one another, or stationary cells block the advance of an adjacent cell. The direction of motion of a cell node is determined by the direction of the net nodal vector forces. If the nodal vector points outward from the cell surface, then the node is protruding or advancing. If the nodal vector points inward, then the node is contracting or withdrawing. Therefore, for each cell at a junctional node, the program checks the direction of the nodal vectors in neighboring cells. If any of these vectors are zero, or pointing outward, then P no d e is reduced by an amount APinhib 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 cell area at the start of the simulation and areajiew is the cell area during each successive iteration. If areajiew increases above areajyriginal, the internal pressure falls and becomes negative. This mimics the behavior of an elastic cell which conserves volume by generating restoring tension forces when it is stretched. If one region of the cell protrudes and causes the cell area to expand, other nonprotruding nodes will retract until the cell area returns to its original size. Conversely, when the cell is compressed, internal pressure increases until the cell area returns to its original value. We have made this modification for the following reason. When the cell pressure is only allowed to be positive, the nodal forces are dependent upon two factors: (1) the curvature of the local cell surface, which determines the magnitude of the net contractile elastic nodal forces, and (2) the magnitude of the outward cell pressure which varies inversely with the cell area. Different combinations of local curvature and cell pressure can result in the same net mechanical forces at a node. For this reason, a drop in cell pressure may or may not always lead to subsequent nodal retraction at a nonprotrusive node, depending upon the local cell surface curvature. When cell pressure is allowed to become negative, nonprotruding nodes will always retract when the cell area expands in other locations. In this way, the original cell area is more accurately maintained by the retraction of nonprotruding nodes. At the start of the simulation, areajoriginal is slightly larger than the equilibrium cell area (the equilibrium cell area is reached when the sum of all forces at each node equals zero). The additional area accounts for the extracellular spaces between cells. [E] Conditions that maintain the rectangular notochord tissue shape In order to enforce the rectangular shape of the notochord tissue, additional bounding boxes are placed along the right and left sides of the tissue, of equal vertical height. This represents the attached somitic mesoderm. The important features of these boxes are their upper and lower horizontal surfaces which lie contiguous with those of the notochord tissue. The elastic forces at cell nodes that lie along the notochord boundaries are increased above that used for cell nodes within the tissue interior. This tends to straighten the notochord boundaries. The magnitude of the elastic

16 1244 M. Weliky and others forces along both the horizontal notochord and bounding boxes are equal and larger than the elastic forces along the vertical notochord boundaries. This ensures that the horizontal boundaries remain straight and do not bend where the notochord tissue and bounding boxes are attached. There are a number of alternative sets of condtitions that would also tend to enforce the rectangular notochord shape. One example is to increase the elastic tension at cell nodes along the vertical notochord boundaries over that of the horizontal boundaries, and dispense with the bounding boxes. This will have the tendency of maintaining straight vertical tissue boundaries. Therefore, the rectangular tissue shape can be maintained by asymmetric mechanical forces acting along its horizontal and vertical axes. When this asymmetry is lost, for instance by removing the bounding boxes and having elastic forces of equal magnitude encircle the entire notochord perimiter, the tissue rounds up into a circle (Fig. 4). This research was supported by NSF Grant No. MCS to GO, NIH Training Grant No. GMO7048 to MW, NIH Grant No. HD25594 and NSF Grant No. DCB89052 to RK, and NIH Training Grant No. HD7375 to SM. The authors would like to thank Dianne Fristrom, Lance Davidson, Nicolas Cordova, and John Shih for a critical reading of the manuscript, and Paul Tibbetts for photographic work. References ABERCROMBIE, M. AND AMBROSE, E. J. (1958). Interference microscopic studies of cell contacts in tissue culture. Expl Cell Res. 15, BOUNDS, D. G. (1987). New optimization methods from physics and biology. Nature 329, ELSDALE, T. AND WASOFF, F. (1976). Fibroblast cultures and dermatoglyphs: The topology of two planar patterns. Roux Arch. Devi Biol. 180, ERJCKSON, C. A. (1978a). Analysis of the formation of parallel arrays of BHK cells in vitro. Expl Cell Res. 115, ERICKSON, C. A. (1978ft). Contact behavior and pattern formation of BHK and polyoma virus-transformed BHK fibroblasts in culture. /. Cell Sci. 33, FRISTROM, D. K. (1976). The mechanism of evagination of imaginal discs of Drosophila melanogaster, in. Evidence for cell rearrangement. Devi Biol. 54, HARDIN, J. (1989). Local shifts in position and polarized motility drive cell rearrangement during sea urchin gastrulation. Devi Biol. 136, HARBIN, J. AND CHENG, L. (1986). The mechanisms and mechanics of archenteron elongation during sea urchin gastrulation. Devi Biol. 115, HARRIS, A. K., WILD, P. AND STOPAK, D. (1980). Silicone rubber substrata: a new wrinkle in the study of cell locomotion. Science 208, HONDA, H., OGITA, Y., HIGUCHT, S. AND KANI, K. (1982). Cell movements in a living mammalian tissue: long term observation of individual cells in wounded corneal endothelia of cats. /. Morph. 174, HOPFIELD, J. J. AND TANK, D. W. (1986). Computing with neural circuits: a model. Science 233, JACOBSON, A., OSTEJI, G. F., ODELL, G. AND CHENG, L. (1986). Neurulation and the cortical tractor model for epithelial folding. J. Embryol. exp. Morph. 96, KELLER, R. E. (1978). Time-lapse cinemicrographic analysis of superficial cell behavior during and prior to gastrulation in Xenopus laevis. J. Morph. 157, KELLER, R. E. (1984). The cellular basis of gastrulation in Xenopus laevis: active, postinvolution convergence and extension by mediolateral interdigitation. Am. Zool. 24, KELLER, R. E., COOPER, M. S., DANILCHEK, M., TIBBETTS, P. AND WILSON, P. A. (1989). Cell intercalation during notochord development in Xenopus laevis. J. exp. Zool. 251, KELLER, R. E. AND HARDIN, J. (1987). Cell behavior during active cell rearrangement: evidence and speculation. J. cell Sci. Suppl. 8, KELLER, R. E. AND TmBETTS, P. (1989). Mediolateral cell intercalation in the dorsal, axial mesoderm of Xenopus laevis. Devi Biol. 131, KOLEGA, J. (1986). Effects of mechanical tension on protrusive activity and micronlament and intermediate filament organization in an epidermal epithelium moving in culture. J. Cell Biol. 102, MACKAY, S. A. (1978). Computer simulation of aggregation in Dictyosielium discoideum. J. Cell Sci. 33, MTTTENTHAL, J. E. AND MAZO, R. M. (1983). A model for shape generation by strain and cell cell adhesion in the epithelium of an arthropod leg segment. /. theor. Biol. 100, ODELL, G. M., OSTER, G., ALBERCH, P. AND BURNSIDE, B. (1981). The mechanical basis of morphogenesis. I. Epithelial folding and invagination. Devi Biol. 85, OSTER, G. F. (1988). Biophysics of the leading lamella. Cell Motil. Cytoskel. 10, OSTER, G. F., MURRAY, J. D. AND MAINI, P. K. (1985). A model for chrondrogenic condensations in the developing limb: the role of extracellular matrix and cell tractions. /. Embryol. exp. Morph. 89, OSTER, G. F. AND WELIKY, M. (1990). Morphogenesis by cell rearrangement: a computer simulation approach. Seminars in Devi. Biol. 1, OSTER, G. F. AND WILSON, E. O. (1978). Caste and Ecology in the Social Insects. Princeton Univ. Press, Princeton, NJ. SHIH, J. AND KELLER, R. E. (1992). Cell behavioral patterns in the organizer region of Xenopus laevis. in preparation. SULSKY, D., CHILDRESS, S. AND PERCUS, J. (1984). A model of cell sorting. J. theor. Biol. 106, TRINKAUS, J. P. (1984). Cells Into Organs. The Forces That Shape the Embryo 2nd ed. Prentice-Hall, Englewood Cliffs, NJ. WARGA, R. M. AND KIMMEL, C. B. (1990). Cell movement during epiboly and gastrulation in zebrafish. Development 108(4), WEUKY, M. AND OSTER, G. F. (1990). The mechanical basis of cell rearrangement. I. Epithelial morphogenesis during Fundulus epiboly. Development 109, WILSON, P. A., OSTER, G. F. AND KELLER, R. E. (1989). Cell rearrangement and segmentation in Xenopus: direct observation of cultured explants. Development 105, WINKLBAUER, R. (1990). Mesodermal cell migration during Xenopus gastrulation. Devi Biol. 142, (Accepted 5 September 1991)