Download Notochord morphogenesis in Xenopus laevis

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 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.
Introduction
ments (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-
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 move-
Key words: computer simulation, mathematical model,
morphogenesis, cell motility, notochord.
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
Notochord morphogenesis In Xenopus laevis
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-
1233
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-
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)
(4)
(3)
(2)
INHIBITED NODES
(1)
(3)
(2)
(4)
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.
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.
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.
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
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
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.
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-
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
Notochord morphogenesis in Xenopus laevis 1241
Fig. 9. Polarized cell
protrusive activity spreads
progressively towards the
tissue interior. (A) At the
simulation start, polarized
o
marginal cells [white] lying
GO
Q
along the notochord boundary
DC
are constrained to protrude
o
only at their interior-most
o
node (protrusion probability=
o
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
B
[shaded] have an initial
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.
total call protrusions
—
60
120
180
240
interior (Fig. 7). These results support the notion that
several rules for cell motility and cell-cell interaction,
operating simultaneously, may drive notochord morphogenesis.
total Instance* of contact
Inhibition
300
360
360
14,000
28,000
Iteration Number
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
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 PnOde- 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, Pnode is
increased by a constant amount APac/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 Pnode is,
the faster its rate of increase.
3. We use the shape parameter index, INDEX=
PERIMETER2/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 1.
4. 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-
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 PnOde during
each iteration as a function of the change in INDEX as
follows
AP,NDEX =-(c* AINDEX)
(2)
where APINDEX 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 APinhib=0.075, which represents the
reduced capacity of the node to protrude as a result of
contact inhibition. The value of APinhib is chosen to be
substantially larger than the value of APactive, 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 Pnode at time step, t+1, is
calculated 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 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 Pnode 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
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. MCS8110557 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.
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(Accepted 5 September 1991)