J. Embryol. exp. Morph. 83, Supplement, 233-259 (1984)
Printed in Great Britain © The Company of Biologists Limited 1984
233
Computer modelling of cell division during
development using a topological approach
By ROBERT RANSOM AND RAYMOND J. MATELA
Department of Biology, The Open University, Walton Hall, Milton Keynes,
MK7 6AA, U.K.
TABLE OF CONTENTS
Summary
Introduction
Description of the framework
Simulating cell division with sorting
Boundary problems and three-dimensional representations
Graphs and mechanics
Conclusions
References
Appendix
SUMMARY
Development in multicellular animals consists of a constant progression of cell division,
differentiation and morphogenesis. Our understanding of the relationship between division
and the acquisition of shape and form is not well understood, and this paper describes a
computer representation of cell division processes with possible applications to the modelling
of developmental events. This representation is not itself a model in the true sense, but is a
scaffolding onto which a set of model assumptions and parameters can be built. We discuss one
such set of assumptions, used to model cell sorting, describe the extension of the framework
to represent sheets of cells in three dimensions, and make some observations on the incorporation of mechanical forces into the representation.
INTRODUCTION
We will begin by stating the obvious. Metazoan development involves a
progressive series of cell divisions from egg to adult. In some organisms there is
a relatively simple and well-studied cell lineage which usually gives the identical
multicullular end product from the egg. Development of this kind is seen in
nematodes, for example (Horvitz & Sulston, 1980). In other creatures, cell
divisions give rise not just to the creature itself, but also to the protective tissues
around the developing embryo. Also seen in more complex creatures are tissues
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in which cell division goes on after adulthood has been reached. Amongst these
are of course epithelia and sites of blood cell production.
Cell division is therefore a vital and all pervading phenomenon in the development and maintenance of multicellular life forms. However, studies of the role
of cell division in development have classically looked upon the phenomenon as
a passive process which is carried out in response to chemical prepatterning of
some kind. Thus regions of high cell division activity might be 'activated' by some
morphogen, relegating cell division as a consequence rather than a cause of
morphogenetic events.
Some studies have in contrast suggested that the mechanical activity of the cell
division process can itself direct changes in tissue shapes during morphogenesis,
without recourse to any underlying prepattern. A somewhat trivial but graphic
example of this was earlier given by Ransom (1975, 1977) who used a simple
computer tesselation model to show that the shapes of clones in Drosophila
imaginal structures marked by X-irradiation of larvae may be simulated by
simple rules of cell division coupled with mechanical constraints from surrounding tissues, so obviating the need to propose gradients of mitotic activity or other
phenomena.
Recent studies have also highlighted the importance of cell-internal mechanical forces in the generation of shapes and patterns during development. Odell,
Oster, Alberch & Burnside (1981) presented a mathematical model which suggested that mechanical forces may co-ordinate the shape changes of large cell
sheets without the intervention of long-range chemical signalling. This model
paid particular attention to the discrete nature of multicellular tissues: although
the equations for mechanical force were derived from classical laws of physics,
the propagation of force between cells is a function of the interactions between
neighbouring cells. Mittenthal & Mazo (1983) presented a model for eversion of
Drosophila imaginal disc segments which suggests that morphogenesis, in this
case an elongation process, results from differences in cell distributions in turn
brought about by changes in patterns of strain and adhesion.
Oster, Murray & Harris (1983) have formulated a model which attempts to
explain mesenchymal morphogenesis, particularly feather germ formation and
the condensation of skeletal rudiments, in terms of mechanical deformation. All
these studies, together with the papers by Oster and Odell in the present volume,
suggest that cell mechanics are important parameters in the control of morphogenetic processes.
Although these papers highlight the discrete cellular nature of multicellular
development, they ignore the process of cell division almost totally. This is no
criticism of the models themselves: imaginal disc eversion, for example, does not
involve a significant amount of cell division (Fristrom & Fristrom, 1975), and the
mathematics of mechanical forces in cells had to be worked out in the nondividing state first, for reasons of tractability.
But if cell division is an important driving force in the development of shapes
Computer modelling of cell division
235
and patterns, then some way must be found to integrate the mechanical
parameters derived by Oster, Odell and their co-workers into a model
framework which allows cell division to take place. In this paper we shall provide
the theory behind such a framework, and will then proceed to show how it can
be used to represent cell division in a two-dimensional cell sheet. We shall also
give a brief description of how the framework has been used to model cell sorting
in the maintenance of Drosophila compartment boundaries (Matela, Ransom &
Bowles, 1983), followed by a description of extension of the framework to
simulate growth and division of cell sheets in three-dimensional space. We shall
conclude by suggesting how mechanical forces may be integrated into the cell
division framework. The rules for the geometrical representation of cells used in
this study are given in the Appendix.
DESCRIPTION OF THE FRAMEWORK
Representations of cell division processes on the computer were first
developed by Eden (1960) who used a'simple tesselated space, a 'chessboard',
where cells are represented by spaces on the board, and division occurs by
placing a 'daughter cell' into an adjacent square orthogonal to the 'cell' which is
to divide (Fig. 1). This type of tesselation model has its origins in the automata
of vonNeuman (1953), which was followed by various other studies, notably Ede
& Law (1969) and Mitolo (1973) who used it to model elongation of the
vertebrate limb bud, Leith & Goel (1971) used it to simulate cell sorting, and
Ransom (1975, 1977) used a hexagonal tesselation to study clone shapes in the
Drosophila imaginal structures.
Models of this type are very limited, however. Cell movement can only occur
by movement from square to, square (or hexagon to hexagon) of the tesselated
space, and thus the 'shapes' of cells are confined to the shape of the individual
elements of the tesselation pattern. Cell deformations and changes in cell adjacency cannot be modelled adequately, and cells are all of the same size.
n
9
Fig. 1. Simulation of cell division on a square tesselated grid. The single cell (left)
'divides' by occupying a neighbouring square (centre). Successive divisions produce
a cluster of 'cells' (right).
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In order to overcome these shortcomings, Matela & Fletterick (1979, 1980)
proposed a novel approach to the representation of cells on the computer. This
approach involved the use of graph theory, a branch of mathematics concerned
with the topological relationship between elements (see for example Ore, 1963)
and should not be confused with the more usual 'graph' constructed geometrically. A graph is constructed from a set of points called vertices, and a set of line
segments called edges. The vertices represent the individual elements under
consideration, in our case cells, and the edges connecting pairs of vertices
represent some relationship between the elements, for example cell adjacency.
Fig. 2 shows an example of a small cluster of cells represented in this way. It is
important to realize that this representation gives information only about the
adjacencies between cells. Whilst it is clearly more accurate in this respect than
V1
Fig. 2. A triangulated graph may be used to model a cell system. Each vertex
(vl... vn) represents a cell, and each edge (el.. .em) represents an adjacency
between two cells. The mapping of the dual onto the original graph is also shown.
Each vertex on the dual represents a cell corner, so the shapes of the cells are
apparent.
Computer modelling of cell division
237
the rigid neighbour relationships of the regular tesselation models, no geometric
information is given by the graph per se: this is because the graph is a topological,
and not a geometric structure.
For any planar graph, however, a dual graph can be constructed. This dual is
formed by selecting a single point within each face of the graph, and then joining
those points which are in neighbouring faces (Fig. 2). In our case, because the
graph structure is triangulated, a face consists of the area bounded by three edges
which form a triangle. For example (el, e2, e3) define a face in the graph of Fig.
2. The points are therefore vertices in the dual graph, and the joining lines are
the edges on the dual graph. The dual graph representation of our cell system
connects not cells as such but cell corners, and so the dual does, in a sense, give
geometric information about the cell system.
Although Fig. 2 shows an arrangement of cells which looks 'realistic' in that
the cells are of different shapes and sizes, with different numbers of adjacent
neighbours, there is a further problem in using the graph approach as a
framework for cell modelling. This is the property of cell size. In order to build
a representation of size into the framework, the original graph and the dual have
to be mapped into a two- or three-dimensional space. The graph then becomes
not just an abstract entity giving information about cell adjacencies (Matela &
Fletterick, 1979,1980; Matela et al. 1983), but is also at the same time a kind of
cellular automaton (von Neuman, 1953), where at any time tl the position and
size of each cell is represented by the x, y (or x, y, z) co-ordinates of the corresponding graph vertex, together with the co-ordinates of the vertices to which
each cell is directly adjacent.
The geometric rules for constructing the dual graph are, at this time, completely dependent on the imposed geometrical nature of the graph. In our simulations,
each vertex on the dual is positioned at the centroid of each face on the original
graph. There is no hard and fast reason for the choice of the centroid, save that
of simplicity in calculation. Conversely, there is no biological evidence to suggest
the choice of an alternate pattern.
The graph representation is especially powerful for two reasons. Thefirstis that
single exchanges between the edges connecting neighbouring vertices represents
changes in cell adjacencies. Consider a group of four vertices and the corresponding cells, as shown in Fig. 3. If the edge between cells Bl and Wl is changed to
connect B2 and B3 then the net effect is to separate the two cells Bl and Wl and
to bring the cells B2 and B3 together. This simple exchange mechanism (Matela
& Fletterick, loc cit) provides the mechanism for cell sorting and motility in our
model framework. The second reason for the power of the graph representation
uses this exchange mechanism and is based on a theorem due to Lawson (1971).
Theorem: Let S be any finite set of points in the plane. Given any two
triangulations of S, say T and T', there exists a finite sequence of exchanges by which T can be transformed to T'.
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B3
B3
Fig. 3. The effect of the exchange rule in changing cell adjacencies. Cells marked by
vertices Wl and Bl are connected by an edge in the left diagram. Cells marked by
vertices B3 and B2 are not adjacent. If the edge across the quadrilateral of vertices
switches to connect B3 and B2, these cells become adjacent and Bl and Wl lose their
adjacency. This simple switch is used as the basis of the cell sorting driver used in the
graph framework (Matela & Fletterick, 1979).
This theorem provides us with a mechanism for achieving any desired pattern
from a given starting pattern by the repeated exchange of edges.
Having dealt with the conceptual aspects of the framework, we now justify our
choice of parameters within this topological framework. Firstly, we chose to use
a triangulated graph (see Fig. 2). This implies that our modelled cells always
meet at corners with two other cells. The justification for this is partly based on
the general assumption that hexagonal packing (or at least an average of about
six neighbours/cell) is most commonly seen in cell populations: hexagonal packing gives three cell corners, although individual instances of other corner
arrangements may be seen in living tissues.
This condition of triangularity also imposes restrictions on the way in which
cell divisions occur in our model framework. We chose to allow cells with from
four to nine immediate neighbours to divide, based on the assumption that the
thermodynamic instability of cells with more than nine neighbours would tend
to lower the bond number for such cells. Provision was made within the computer
program to flag for these cells and lower the bond number accordingly. A series
of 'division masks' were devised to simulate the division process on the graph
(Matela, Ransom & Bowles loc cit: see also Fig. 4) and the rules for division are
that
(1) division involves the formation of a new vertex
(2) the new vertex is positioned on the graph adjacent to the dividing vertex
(3) changes in the pattern of edges around these two vertices are made to reestablish the triangularity condition
(4) the number of neighbours around the new and dividing vertices are balanced as equally as possible.
Computer modelling of cell division
Fig. 4. Cell division rules used for cells with four to nine neighbours. In each case
the upper diagram represents adjacencies on graph and dual before division. The
lower diagram shows adjacencies after cell division.
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Implicit in our division rules is the assumption that cells divide into progeny
which are both of equal size, if we assume that equal-sized cells possess on
average the same number of neighbours. Further consideration of the relationship between cell size and cell adjacency patterns is given in the Appendix. Our
choice of division masks is based as follows, and depends on a purely empirical
line of reasoning. If a cell is, at the time of division, surrounded by n neighbours,
then as the volume of cytoplasm contained in the cell is not increased by the
division process, n should remain the same, at least until further growth takes
place, i.e. after the daughter cells have begun to increase in size. The n contacts
between parent cell and neighbours must therefore be redistributed to include
both daughter cells, and given the assumption of equal size of the daughters,
together with the triangulation condition, the number of neighbours nl and n2
around the daughters is arithmetically derivable (it is, for example possible to
arrive at division solutions other than those shown in Fig. 4, although there is in
fact a finite number of ways of partitioning n-gons). The partition must satisfy the
following equation
nl + n2 = 2n - (n-4)
suggesting a net decrease in adjacencies per cell after division. But as a result of
the division process, two of the immediate neighbours of the dividing cell become
adjacent to both of the daughter cells. This results in 'rescue' of the two edges
lost during the division process.
SIMULATING CELL DIVISION WITH SORTING
By implementing the rules outlined in the previous section into a set of
algorithms, it is possible to simulate cell division on the computer. In the first
tests of the framework (Matela & Ransom, 1982; Matela etal. 1983) geometrical
considerations were not included, and the only information obtained using the
exchange mechanism outlined above involved changes in cell adjacencies during
the course of the simulation. We showed that, given two initial 'cell types' A and
P, 'sorting out' could be modelled during cell multiplication by the use of a simple
exchange driver. The driver was that edges between A and P cells are exchanged
for either A - A or P-P edges, if possible, during the course of the simulation.
Boundaries between the two marked groups of cells with and without the sorting
drivers are shown in Fig. 5. It can be seen that sorting smooths the border to a
large extent, and we earlier suggested (Matela et al. 1983) that the maintenance
of compartment boundaries in Drosophila imaginal discs may involve a biological analogue of our sorting mechanism.
Fig. 5. An example of simulation of A-P compartment border maintenance in the
Drosophila wing disc between two groups of cells (P and A) grown from 10 initial
cells (5P+5A) to 410finalcells (205P+205A). The upper picture is the non-sorting
program version. P cells are shaded in each case.
Computer modelling of cell division
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BOUNDARY PROBLEMS AND THREE-DIMENSIONAL
REPRESENTATIONS
Consideration of planar groups of cells necessitates the presence of a boundary
around the cell mass. In the previously reported uses of the topological
framework we have represented boundary cells as having a lesser number of
contacts than internal cells, and have devised 'special case' rules for the division
of such cells (see Appendix Fig. 18). It is apparent, however, that most developing tissues do not consist of flat plates of cells, but are closed surfaces. The
advantage of a closed surface may lie in its mechanical properties: forces are
distributed over the whole surface, and there are no boundary effects to contend
with. Whatever the reason for sheet closure, it is only rarely that a flat, true
monolayer of cells is found in nature, the only example springing to mind being
that of the sea lettuce Ulva. Even insect imaginal discs, often modelled as 'two
dimensional', are really closed sheets with the epithelial layer continuous with
the overlying cellular peripodial membrane. Embryos which pass through a stage
of being a single layer of cells, for example Amphioxus (Heuttner, 1936), consist
of a hollow ball of cells. Consideration of closed sheet cell division systems
therefore presents a more sensible goal for the graph framework than the study
of open sheets. This approach entails consideration of cell sheets in threedimensional space. Although an additional level of complexity is introduced in
going from two to three dimensions, the way is opened for the modelling of
changes in the conformation of cell sheets, avoiding the limitations of the planar
framework, which is restricted to the study of clonal relationships.
We should emphasize here that the graph itself is able to represent structures
in a closed sheet more easily than in an open disc, because no boundary effects
have to be considered. The problem arises in the geometrical representation of
the dual. It is not difficult to construct any static dual from a graph in threedimensional space. But when the number of vertices in the graph is increased by
'cell division', a single planar cell is split into two cells, each of which may take
up a new orientation in space if the number of faces on the cell sheet is not to stay
the same throughout the simulation. The orientations taken up by the daughter
cells are a function of the mechanical forces acting in the sheet, and we will make
some observations on the way in which these forces could be incorporated into
our framework in the next section.
Changes in the conformation of the graph 'cell sheet' in three dimensions are
produced by three different activities (1) cell enlargement, due to growth or
relaxation of a contracted cell (2) cell contraction, or (3) cell division. It is
therefore necessary to decide on the rules which govern the effect of size or shape
changes in single cells on their neighbours. The conformational changes are
produced as a result of changes in shape of the whole cell and not the single face
that is displayed using the graph model. Some measure of the cell volume is
therefore necessary to simulate conformational changes.
Such a measure would ideally involve incorporation of mechanical forces into
Computer modelling of cell division
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p (x, y, z)
p (phi, theta, r)
Fig. 6. The spherical polar co-ordinate system. Any point p in three-dimensional
space may be defined by two angles, phi and theta and a radius r.
the framework, and this we have not yet carried out. It is, however, possible to
investigate the behaviour of the growing sheet of cells in three dimensions by the
use of an algorithm which simulates a rudimentary form of local growth. This is
accomplished by the use of a spherical polar coordinate system for keeping a
record of the positions of the graph vertices. In brief, spherical polar co-ordinates
define points in space by three parameters: distance from a central point, r,
together with angles phi and theta (see Fig. 6). The vertices of the graph may
therefore be represented by these parameters, and initially a constant value of
r is used: this puts all vertices onto the same spherical surface. In order to
increase the size of a cell, the vertex representing the cell simply has its r value
increased by a known amount. The increase in size achieved is similar to the
increase in size of a enclosed area drawn on the surface of a balloon after expansion. Although the angles of the edges of the area with respect to the centre point
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of the balloon are the same before and after expansion, the radius has increased.
In the cellular system, this increase does not alter the connections between the
grown cell and its nearest neighbours, which may have greater or lesser r values.
The repetitive application of this simple growth rule produces growth in a dividing sheet of cells.
Although all cells start out with constant r value, variation in r produces
Fig. 7. Simulation of the growth of a cell sheet in three dimensions: graph (top) and
cells (bottom), present in the starter pattern of 17 cells. 8 cells are marked by shading.
Computer modelling of cell division
245
deviation from the initial spherical surface. Phi and theta also vary as local cell
shape changes occur, and as the division process gives rise to new vertices. The
only invariant position throughout the simulation is the origin of the initial
sphere. This can be thought of as the position through which a 'pin' anchors the
growing cell sheet. The spherical sheet that we have chosen as the shape for our
starting pattern of cells is by no means the only pattern available. By choosing
appropriate r values for each vertex in the starting pattern, any pattern of cells
could be used.
Our first simulations using this system show that the rules outlined above can
produce a growing sheet of cells, if growth is isotropic in all cells. Figs 7 and 8
show the starting and finishing patterns when 17 initial cells (the same configuration was used in each case) were grown up to 300 cells for several different
random starters. The shape present after a number of divisions is highly dependent on the distribution of the cells in the starting pattern: as might be expected,
an elongated initial pattern produces an elongated cell mass after division.
The cell sheet in three-dimensional space has the same facilities for clone
marking, differential cell division and cell sorting as does the two-dimensional
simulation. As yet, we have only investigated differential growth and division,
using a starter pattern made up of two groups of clonally marked cells, one group
of which divides more slowly than the other group. If the difference in division
rate is fairly large (one type divides and grows at twice the rate of the other, for
instance), the convex nature of the growing cell sheet is perturbed (Fig. 8 bottom). We are at present investigating the behaviour of this three dimensional
growth system quantitatively.
GRAPHS AND MECHANICS
In this section we shall suggest how parameters of mechanical force may be
built into the graph framework. In doing this, we will assume that the mechanical
forces we wish to model are those produced by the cellular cytoskeleton (see for
example Clarke & Spudich, 1976). Odell etal. (1981) have already presented a
mathematical model of the effect of changes in shape of single cells over a
filament of cells. This model is based on an apical 'purse string' of contractile
elements within each cell which can actively deform the cell (Fig. 9). To simplify
the mathematics, each cell is studied only in cross section. A number of viscoelastic units are modelled within each cell as shown in Fig. 10, but only the apical
filament bundle is active, that is to say, only this bundle can initiate cell deformations. The model also assumes that all cells have constant volume. The most
important aspect of this work is the effect of perturbations in the apical contractile filaments. When stretched a small amount and then released, they behave
like ordinary springs. If the stretch is extended beyond a critical threshold, the
filaments contract to a new, shorter rest length, which is then returned to after
further perturbation. When the behaviour of the contractile filaments is
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Fig. 8
Computer modelling of cell division
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apical filament bundle
Fig. 9. Left: network of contractilefilamentsin the apical region of an epithelial cell.
Right: 'purse-string' contraction of the apical circumference by the apical bundle.
(After Odell et al. 1981.)
simulated over a line of cells, folding patterns such as those seen in gastrulation
and neurulation can be produced by these mechanical perturbations alone.
There is one immediate difference between the Odell model and our graph
theoretic framework. This is the plane in which the cells are represented. The
Odell model considers cells in cross section, whilst our representation deals with
a cell surface. The graph can simulate contractions and relaxations on this surface, but cannot account for changes in three-dimensional conformation between neighbouring cells, as such conformational changes are produced by
changes in the distribution of the cell contents throughout the depth of the cells.
First let us consider how the graph can model the behaviour of the apical 'purse
string' of contractile filaments. If we consider a single cell as a planar case (Fig.
11), we can apply forces along the edges between the apical surface of the cell
and its neighbours. These forces are envisaged to be viscoelastic in nature, and
may be isotropic or antisotropic. In each case both contraction and expansion
Fig. 8. Cells present after 300 divisions of the starter pattern in Fig. 7. Top and
centre: two different random starters for equal division rates in shaded and unshaded
cells. Bottom: shaded cells divided twice as fast as the unshaded cells.
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as
bs
Fig. 10. A cross section through an epithelial cell (dashed lines) showing the arrangement of viscoelastic units used by Odell et al. (1981) to model the cytoskeleton
(solid lines). The apical region (as) is the only unit to be mechanically excitable.
bs = basal surface.
forces are possible. Contraction of an apical surface in a cell is therefore achieved
by a reduction in the lengths of the edges between the vertex representing the cell
and its nearest neighbours.
In order to give depth to the cells in the graph framework, the following system
is proposed. Instead of a single triangulated graph, two graphs are used (Fig. 12).
The first graph represents the apical surfaces of the cells, and the second graph
represents the basal surfaces. Clearly, there is a one-to-one correspondence between the vertices in the apical and basal graphs. This correspondence is reflected
Fig. 11. Contraction on the 'apical graph'. Each edge between two vertices is made
up of two components, 11 and 12. These components refer to the parts of the edge
in the two cells. If the apex of the central cell contracts isotropically, all 12 components are proportionally shortened.
Fig. 12. Representation of the two graphs gl and g2 in three dimensions. Each
vertex on gl has a corresponding vertex in g2, and together they define the position
of a single cell. Edges on gl and g2 are here shown as solid lines, and edges between
vertices on gl and g2 for the same cell are shown as dashed lines.
Computer modelling of cell division
v2
Fig. 11
Fig. 12
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in that there is an edge joining every vertex in the apical graph to its counterpart
in the basal graph. The maintenance of these parallel surfaces is important in that
it eliminates from our framework the need to represent the diagonal and longitudinal 'struts' as used in the Odell model. These struts or their equivalent
representation are necessary to maintain the cuboidal nature of the cell sections
during the processes of contraction and relaxation. The maintenance of parallelism between apical and basal cell surfaces in the graph framework keeps the cells
rigid in the graph framework. Fig. 13 shows a cross-sectional comparison between the elements of the Odell model and our two-graph framework.
Growth is handled in the two-graph framework in the following manner. Each
cell grows from a volume Cv immediately after cell division to volume 2xCv
before the next cell division. The apical surface area when the cell is 'at rest' is
proportional to Cv, as is the basal surface area. Likewise, the lengths of the edges
on both the apical and basal graphs are proportional to each other. This proportionality in fact holds the key to the simplicity of this method of representation.
Fig. 14 (top) shows a cross section through three adjacent cuboidal cells. If one
cell contracts apically, then, given constant volume, it must expand basally. If the
two cells are 'free' in space, then the cells might be conformed as shown in the
centre part of the figure after the contraction. If, instead, the cell is held in a
sheet, then the 'anchoring' activity of the surrounding cells will result in a passive
change in the distribution of the contents of the nearest neighbours of the apically
as
bs
Fig. 13. A comparison of the representation of mechanical 'struts' in the Odell
model (left) and on the proposed two-graph framework (right). The behaviour of the
apical and basal surfaces (as and bs) on the graph framework are modelled by
applying force equations to the edges of graphs gl and g2. In both cases the outlines
of the cells are shown as dashed lines, whilst the lines representing mechanical 'struts'
are shown as solid lines. In the graph framework, these lines are the edges on the
graphs (each in turn made up two components from the two cells joined by the edge,
in this case el is composed of 11+12, e2 is composed of 13+14). The edge d, joining
vertices on gl and g2 for the same cell is also proposed as a mechanical 'strut'.
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251
1
1
t
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Fig. 14. Changes in conformation in a group of cells after apical contraction. The
dashed lines represent the approximate axes of the cells (i.e. the edge on the twograph model connecting vertices on gl and g2. The top diagram shows three cells
before contraction. The centre diagram shows the behaviour of the cells if they are
unanchored, and the bottom diagram shows the behaviour when the cells are
attached to neighbouring cells.
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contracted cell, as shown in the bottom part of Fig. 13. In the two-graph
framework, each edge between the apically contracting cell and its nearest neighbours is bisected at the point at which the cell boundary crosses each edge. On
contraction, the part of each edge (or a subset of the edges in the case of antisotropic contraction) 'belonging' to the contracting cell decreases. The overall
length of each edge stays the same. At the basal surface the opposite occurs. The
proportional decrease in length of the contracting cells' apical edge is mirrored
by a corresponding increase in length of the basal edge. The surrounding cells are
in contrast 'squeezed' basally.
The two-graph model can therefore account for transient local changes in cell
shape, but what if an active contraction takes place over a number of cells, a
process considered to fuel morphogenetic processes like gastrulation and neurulation by Odell, Oster and co-workers. In this case the local redistribution of
cell contents would affect the global conformation of the cell sheet, because the
buffering activity of the cells surrounding a single apically contracting cell would
be lost. In cases of this kind, geometric rules must be used to minimize the
disruption to the cell sheet at each single contraction.
The graphic representation of this kind of framework can be handled in several
different ways. Firstly, the duals of both apical and basal graphs may be reconstructed, and the 'depth' of the cell sheet may then be completely visualized.
Alternatively, a computationally simpler method of representation is to take the
midpoint of the constant distance between the apical and basal vertices for each
cell, and to reconstruct a single dual based on these 'average' vertex positions.
A picture like those shown in the previous section may then be obtained.
We should emphasize that at this time the two-graph methodology is only at
the planning stage. We include this rather full description of its features as a
pointer to the flexibility of the framework in modelling cell division and cell
mechanics, and we will report on the detailed analysis of the behaviour of the
framework when simulated in due course.
CONCLUSIONS
We have presented a framework for studying cell divisions and cell forces
during morphogenesis. The two-graph framework may be of especial use in
studies of cell conformations during particular developmental events. Three
possible applications of the system are (1) simulation of the evagination process
in insect imaginal discs during metamorphosis; (2) simulation of the shapes taken
up by cells along Drosophila compartment boundaries; and (3) invagination in
hollow blastulae. Evagination is a suitable process to model using the graph
framework because, although few cell divisions are involved, current models of
this process invoke adhesiveness gradients (see for example Mittenthal & Mazo,
1983). The ability of the graph framework to model variable cell shapes and
variable neighbour contacts between cells suggests that the system should be able
Computer modelling of cell division
253
to simulate evagination given suitable adhesiveness drivers (and the validity of
the adhesiveness model itself!).
Brower, Smith & Wilcox (1982) have observed an 'unusually shaped' (sic) line
of cells running across the presumptive site of the dorsoventral compartment
boundary in Drosophila imaginal discs. It would be of interest to see if these cell
shapes are also found in simulations of disc growth where the disc is conformed
in three-dimensional space. It is, of course, not possible to use the planar model
to observe cell shape changes in imaginal discs because of the folding seen in real
discs.
Study of invagination processes using the two-graph framework presents an
opportunity to examine the rules necessary for three-dimensional morphogenesis during early development. The system may be used to answer several
questions, namely, do the Odell rules work as well in three dimensions as in two,
and if not, what modifications are necessary to model three-dimensional
morphogenesis.
Although much work remains to be done, especially with regard to the incorporation of force equations into the framework, we believe that the graph
approach offers an elegant and tractable way of handling the simulation of sheets
in three dimensions.
We thank Mike Bowles for ably helping with the computer programming and the Nuffield
Foundation for partial financial assistance.
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Drosophila wing imaginal disc. /. Embryol. exp. Morph. 67, 137-151.
CLARKE, M. & SPUDICH, J. A. (1977). Nonmuscle contractile proteins: The role of actin and
myosin in cell mobility and shape determination. Ann. Rev. Biochem. 46, 797-822.
EDE, D. & LAW, J. T. (1969). Computer simulation of vertebrate limb morphogenesis. Nature
221, 244-248.
EDEN, M. (1960). A two dimensional growth process. In Proceedings of the 4th Berkeley
Symposium on Mathematical Statistics and Probability, pp. 223-228. University of California Press.
FRISTROM, D. & FRISTROM, J. W. (1975). The mechanism of evagination of imaginal discs of
Drosophila melanogaster. I. General considerations. DevlBiol. 43, 1.
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mutants of the nematode Caenorhabditis elegans. Genetics 96, 435-442.
HUETTNER, A. F. (1941). Fundamentals of Comparative Embryology of the Vertebrates.
Toronto: Macmillan.
LAWSON, C. L. (1971). Discrete Mathematics 3, 365.
LEITH, A. G. & GOEL, N. S. (1971). Simulation of movement of cells during self sorting.
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MATELA, R. J. & FLETTERICK, R. J. (1979). A topological exchange model for cell self sorting.
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MATELA, R. J. & FLETTERICK, R. J. (1980). Computer simulation of cellular self sorting: a
topological exchange model. J. theor. Biol. 84, 673-690.
MATELA, R. J. & RANSOM, R. (1982). Topological computer simulation of cell division. In
Proceedings International AMSE Symposium 'Modelling and Simulation'. Paris.
EMB 83S
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MATELA, R. J., RANSOM, R. & BOWLES, M. A. (1983). Computer simulation of compartment
maintenance in the Drosophila wing imaginal disc. J. theor. Biol. 103, 357-378.
MITOLO, V. (1973). A model approach to some problems of limb morphogenesis. Acta Embryologiae Experimentalis pp. 323-340.
MITTENTHAL, J. E. & MAZO, R. M. (1983). A model for shape generation by strain and cellcell adhesion in the epithelium of an arthropod leg segment. /. theor. Biol. 100, 443-483.
ODELL, G. M., OSTER, G. F., ALBERCH, P. & BURNSIDE, B. (1981). The mechanical basis of
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OSTER, G. F., MURRAY, J. D. & HARRIS, A. K. (1983). Mechanical aspects of mesenchymal
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RANSOM, R. (1975). Computer analysis of division patterns in the Drosophila head disc.
/. theor. Biol. 53, 445-462.
RANSOM, R. (1977). Computer analysis of cell division in Drosophila imaginal discs: model
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Urbana: University of Illinois Press.
APPENDIX
The geometry of the framework
We have developed the following rules for visualizing the growth of the planar
Fig. 15. Choice of the 'longest chord' (Ic) across the cell chosen for division.
tv = target vertex, te = target edge.
Computer modelling of cell division
255
graph. At present these rules are arbitrary, and make no detailed attempt to
model changes in cell size. We consider each vertex as bounded by an /z-gon
where n is the number of vertices connected to the given vertex. When a vertex
is picked for division, the longest chord (that is, the line between the most distant
neighbouring vertices) across the n-gon is first determined. As the division mask
procedure works by superimposing one of the edges of the division mask on an
edge between the chosen vertex and one of its neighbours (the 'target edge' - see
Fig. 15), this edge is chosen as one of the edges on the longest chord. Each vertex
is positioned as the centroid of the n-gon which surrounds it. After the division
mask has been applied, the two new vertices are positioned within the current
bounding rc-gon.
If the cell is 'concave', rather than 'convex' (see Fig. 16), it is usually possible
to produce two 'convex' cells by selection of an appropriate target edge. This
selection is necessary to regularize the shapes produced and to minimize
subsequent positive feedback of irregularities. The presence of a concavity is
detected simply by comparing the length of the edge from the centroid to a given
vertex with that from the centroid to the chord joining the vertices on either side
Fig. 16. Identification of 'concave' cells. If edge a is shorter than the constructed line
b, then the cell is assumed concave.
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R. RANSOM AND R. J. MATELA
sd
Fig. 17
new vertex
Fig. 18
Computer modelling of cell division
257
graph
boundary
Fig. 19. Closing up gaps on the boundary of the cell mass. If unconnected vertices
vl and v2 are too close together, as defined by the angle theta, a new edge (ne) is
inserted between them. This edge encloses vertex ev.
of the given vertex. If the second length is greater, a concavity is present.
The positions of the new centroids are selected in all cases so as to further
minimize the probability of concavity (Fig. 17). Two lines crossing therc-gonare
found: the first joining the two vertices shared by the daughters (sd) and the
second across the centre of the mask (/c). The point where these two lines
intersect is used with the surrounding vertex positions to calculate the centroids
of the new daughters, and hence decides where the daughter vertices are to be
placed.
Boundary cells are produced with two neighbouring cells (see Fig. 18). The
new vertex position is a point normal to the midpoint of the edge to which new
edges are to be added, at a distance equal to half the length of the edge (Fig. 18).
A check is made on the angle theta between the new edges and those of the edges
on the graph adjacent to them. If this angle is less than a given value, a further
new edge (ne) is added which closes off the two adjacent edges to form an
enclosed triangle. Overlap of added edges is thus prevented. When an edge is
Fig. 17. Positioning daughter vertices (dv) during 'cell division'. The lines between
the two vertices (sdl, sd2) which are adjacent to both daughters, and Ic (see Fig. 12)
are constructed. The daughter vertices are placed in the centres of the two polygons
either side of sd along the line Ic.
Fig. 18. Position of new vertices on the boundary, bv = existing vertices on boundary.
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R. RANSOM AND R. J. MATELA
added in this way, the enclosed vertex is recentred according to the mean of the
surrounding vertex coordinates. When a new edge is added to close off the
boundary in this way the new edge is also checked to see whether it causes
overlap with the next adjacent edge, and so on.
Two checks are also made on the graphics of edge rearrangements, for instance during sorting. If the new edge is more than one and a half times the old
one in length, the rearrangement is not allowed, as the connection represented
by the edge would produce elongated cells. When a rearrangement is allowed,
then the four vertices involved (those at either end of the old edge, and those at
either end of the new one) are recentred.
Every internal vertex in the graph is regularly recentred with relation to its
encircling neighbours after every ten divisions. In order to simulate growth, the
program increases the n-gon size before division, by extending the edges from
the centroid to the surrounding vertices by a given multiplier (Fig. 20). The
whole graph is then rearranged, by first searching for vertices at successively
l*m
Fig. 20. Cell 'growth' in the two-dimensional plane occurs by increasing the lengths
of all the edges for each growing cell. A simple multiplier is used, such that each
length increases from length / before growth to length l*m after growth.
Computer modelling of cell division
259
greater distances from the enlarged cell, and then recentring each vertex at a
given graph distance in a radial pattern.
The rules for three-dimensional representation are similar to those used in the
plane. Once a vertex has been selected for division, the nearest neighbours are
located, and the centroid of the n-gon defined by these neighbours on the 'best
fit' plane is determined. The daughter vertex positions are calculated on this
plane as in the two dimensional case. Having obtained the x,y,z Cartesian coordinates for the updated vertex positions, they are then converted to spherical
polar co-ordinates. The radii of the daughters are then set to the original radius
of the 'parent' vertex, and the storage arrays are then updated with these
positions.