A computer graphic simulation of squamous epithelium

J. theor. Biol. (1995) 175, 283-293

A Computer Graphic Simulation of Squamous Epithelium D. SmEKEL,J. RASHBASSt AND E. D. WILLIAMS

Department of Histopathology, University of Cambridge, Addenbrooke's Hospital, Cambridge CB2 2QQ, U.K. (RecehJed on 16 March 1994, Accepted & revisedform on 9 May 1995)

An epithelium maintains its integrity through the organized growth and orderly differentiation of a transient cell population derived from stem cells. This organization is dependent upon both physical mechanisms such as cell adhesion and attraction and the relationship between differentiation and cell division. The interactions between these processes are complex and difficult to conceptualize from a purely mathematical approach. We have therefore set out to develop a graphic model of an epithelium controlled by rules that can be modified. We have chosen to model epidermis, the most superficial part of skin, with cells differentiating from a stem cell population and being lost from the surface of the model. The model is novel not only in the rules that govern cell behaviour, but also because it does not require a predefined lattice to assign the position of cells. Each cell assumes a position depending upon the balance of adhesive and repulsive forces that it experiences. Chemical factors which affect the differentiation of individual cell types are assumed to be produced both by cells within the model and externally from the underlying connective tissue. These "chemical factors" diffuse through the model with a concentration that declines as an inverse square with distance from the source. The rules allow the model to grow from a single stem cell to reach a steady state. At steady state the pattern and clonal structure is strikingly similar to that seen in a range of normal epithelia. Furthermore, if part of the model is removed it is capable of regenerating itself without additional rules. The model allows the visualization of the effects of introducing new rules and modifying the interaction between chosen rules. This study demonstrates that a set of simple rules can be used to make a dynamic flexible model resembling skin. © 1995 Academic Press Limited

differentiation and a consistent movement o f cells through the layers o f the epidermis (Wright & Alison, 1984). In addition, the structure must be capable of regeneration if the organization is partly destroyed. Some o f the individual control mechanisms involved are known, while others can be inferred or surmised. The number involved is such and the potential interactions so numerous that it can be difficult to predict the contribution o f any one process to the overall structure. In an attempt to understand better the effects o f the interactions o f the mechanisms controlling a stem cell tissue in the maintenance o f the morphological structure o f skin, we have set out to model the processes involved. The squamous epithelium o f the skin is a multilayered sheet o f cells. The uppermost layer is in contact with the exterior, the undulating, deepest layer is separated from the underlying connective tissue

Introduction The generation o f a multi-million-cell organism from a single cell involves the generation o f a multiplicity of cell types. Fundamental to this process are the complex mechanisms that control cell division, cell differentiation and organisation (Edlemann, 1988). In the mature organism the same processes o f cell division, differentiation and the maintenance o f organization take place in a variety o f stem cell tissues: for example the s q u a m o u s epithelium o f the skin (Wright & Alison, 1984; Hall & Watt, 1989). Here these processes under normal conditions maintain a steady state with a permanent ordered structure. The maintenance o f this steady state requires that cell loss balances cell production, and that there is an ordered pattern o f

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by a limiting membrane, the basement membrane. Experimental evidence suggests that the epithelium is maintained by stem cells which lie at the deepest part of the epidermis (Withers, 1967; Potten, 1974)immediately above the basement membrane. The stem cells generate directly or indirectly all the other cells in the skin (Potten & Morris, 1988). The basal cells which lie along the basement membrane are also capable of cell division and the successive layers of cells that they generate, differentiate, keratinize, die and are ultimately shed. The ordered progression of cellular development in the epidermis lends itself to computer simulation and many models of epithelial patterning have been produced (Cowan & Morris, 1988; Cruywagen et al., 1992; Morris, 1992; Pilz et al., 1993). Most of these place the cells within a restraining lattice that may be square, triangular or hexagonal (Meakin, 1986; Smolle et al., 1992a, b). This, however, has the limitation that cells are relatively constrained during the simulation in a way that does not apply biologically and produces a pattern that is inevitably limited by the lattice. In an attempt to simulate skin growth and differentiation without a preconceived lattice array, we have modelled several biological parameters that determine how cells interact. Most of the parameters have been chosen as they have been found to be important in empirical studies of epithelial differentiation. Normal skin cells express a variety of cell surface adhesion molecules that function both to bind cells together and to tie cells to the underlying connective tissue (Adams & Watt 1991; Hertle et al., 1991; Jones & Watt, 1993). As the cells mature and differentiate so these adhesive forces change (Jones & Watt, 1993). In addition, cells change their ability to divide and differentiate depending on where they are within the epithelium and their age (Wright & Alison, 1984). In this paper we have taken the novel approach of combining both physical and biological interactions to produce a graphic simulation of skin growth and development. We show that a model with an architecture simulating normal epidermis can be produced using a few simple rules and the parameters can be chosen in such a way as to generate a structure closely similar to normal skin from a single progenitor stem cell. The model will reach and maintain a steady state and shows the clonal structure found in experimental analyses of epithelia (Seddon et al., 1992). Furthermore, the same rules used to grow normal skin will allow the model to heal itself if damaged. The model can be used to generate experimentally testable hypotheses of the interactions between different controlling mechanisms governing cell interactions in normal and neoplastic epithelia.

Construction of the Model

A representation of a histological section through the stratified squamous epithelium of the skin has been produced using a program written in C on a Sun SPARCclassic computer. The X11 libraries have been used to produce a graphical output. CELL TYPES

To take account of the different histological layers observed in normal skin we have defined five different categories of cell in the model; stem, basal, intermediate, mature and dead cells. Each cell is regarded as spherical of unit radius, but to suit the characteristics of the display window may be represented as an ellipse. All cell types are assumed to have the same physical properties and to occupy a position within a predefined two dimensional box. A cell will move during the simulation as a result of interactions with other cells in the model. The properties of each cell are outlined below. The basement membrane is defined by the program as the interface between the row of cells on the bottom of the model and the underlying space representing the connective tissue of the dermis (see below). Stem cells lie on the basement membrane. They do not lie next to other stem cells, they divide to give rise to one stem cell and one basal cell. There is a presumed interaction between stem cells, basal cells, and the underlying connective tissue which leads to undulations in the basement membrane. This is represented in the model by a downwards attractive force exerted on the stem cells from the underlying connective tissue. This connective tissue attractive force (CTAF) can be varied, but during any one simulation is assumed to be uniform throughout the model. Similarly, the basal cells respond to a connective tissue repulsive force (CTRF) by moving away from the base of the model. This can also be varied. Basal cells line the basement membrane on either side of stem cells. They divide to give basal cells but when a basal cell loses contact with the basement membrane it will differentiate into an intermediate cell. Basal cells may also differentiate to take on the property of a stem cell if the level of a stem cell factor falls below a specified threshold (see below). Intermediate cells form several layers of cells above basal cells; they are not able to divide, they arise from basal cells and give rise to mature cells. If, unusually, an intermediate cell comes into contact with the basement membrane it will once again become a basal cell. Mature cells form one or more layers of cells above intermediate cells. They are derived from intermediate cells; they are not able to divide and they give rise to

A S I M U L A T I O N OF E P I T H E L I A L G R O W T H

dead cells. They are terminally differentiated but differ from dead cells (see below) in that they cannot be lost from the model. Dead cells form the uppermost cell layers of the epidermis. They are derived from mature cells and eventually desquamate and are lost from the model. As dead cells they are not capable of giving rise to any other cells under any circumstance. The possible transitions between the different cell types are illustrated in Fig. 1.

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respond to it, either as a threshold or on a linear basis. The underlying connective tissue also produces chemical factors but the response to this is all or none and it is assumed that the level in the normal model is always above the threshold.

Connective tissue attractive force (CTAF ) The underlying connective tissue secretes a chemical to which stem cells respond by moving downward towards the bottom of the limiting box. The value is uniform throughout the model.

CELL ACTIVITIESAND CONTROL

Connective tissue repulsive force (CTRF )

There are six different categories of cell activity: division, movement, differentiation, ageing, death and desquamation. The activities are controlled by chemical factors, physical factors and the lapse of time. Each factor is separately variable.

The underlying connective tissue secretes a chemical to which the basal cells respond by moving upwards. The value of this force is uniform throughout the model.

Chemical factors

Stem cell factor

Cells are assumed to secrete chemical factors which are distributed throughout the model with the concentration of the factor secreted by one cell being distributed according to the inverse square law. The diffusion in the model takes place at a much faster rate than the cell motion, so that the factor is assumed to be in a steady state. Cells have the ability to recognize the concentration of the appropriate factor and to

Stem cells in the model secrete a chemical to which other stem cells are sensitive. The cells react to this factor by movement in the direction down the concentration gradient. This factor also effects basal cells. If the concentration of this factor falls below a threshold level the basal cell will differentiate into a stem cell.

Stem and basal cell factors A factor produced from both the stem and basal cells acts on the intermediate cells and prevents their differentiation to mature cells. The factor decays with distance which is calculated by summing 1/R 2 contributions from all the stem cells, and 1/10R 2 contributions from all the basal cells.

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Stem Cell FIG. 1. A schematic diagram showing the direction of cell differentiation. Arrows vertically upwards represent differentiation while those downward are dedifferentiation. Dashed arrows refer to infrequent events.

The physical forces result from the pressure arising from cell contact and from cell separation. It is assumed that adjacent cells are connected by fine processes which exert an adhesive force when stretched up to a defined limit of distance at which point they rupture. The range over which the force acts depends on the type of cell (see Appendix, Table A1) the forces being proportional to the separation of the cells (as in H o o k ' s law) so that there is no adhesive force when cells are in contact. The sum of all the forces acting on a cell including those the cell exerts as a result of its response to chemical factors determines its movement. The connective tissue also interacts physically with the basal cells. If a basal cell experiences a downwards force, then the basal cell will exert this force on the underlying connective tissue. The connective tissue will then exert an upwards reaction on the basal cell, which is a fixed proportion of the downwards force.

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D. STEKEL E T AL. identical in position to the parent, the other is placed with its centre on the edge of the parent cell. Both are of unit size and will thus push each other apart. The cell divides upwards at an angle between 0 and 90 degrees from the horizontal. The angle is chosen at random, the direction, left or right is also randomly chosen but the probability is biased so that cells tend to divide away from the nearest stem cell. This probability is calculated according to the ratio of the stem cell factor that the cell receives from the left and right. Cell division is followed by an absolute refractory period consisting of a set number of iterations during which newly formed cells cannot divide. Cell movement

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An additional physical force is the squash force. This describes the force exerted by two overlapping cells on each other. If the distance between the centres of any two neighbouring cells falls below the sum of the two cell radii, each cell will experience an outwards force as shown in Fig. 2. TIME

The passage of time in the model is calculated by counting the number of biological iterations. CELLACTIVITIES Cell division

In healthy skin only stem cells and basal cells are capable of cell division. The timing of cell division in any one stem or basal cell is determined by a set probability. The process of cell division involves the formation of two daughter cells, in practice one is

The motion of each cell in the model is calculated by treating the cell as a physical object, essentially a fluid-filled balloon. The cells exert forces on one another either physically for example by squashing against one another (the squash force) or chemically (by secreting and reacting to chemicals that influence the direction of movement). A cell will move according to the sum of the vectors of the forces that it experiences, but as it lacks inertia it will immediately be brought to a standstill by friction and adhesive forces from its neighbours. DIFFERENTIATION

Differentiation of individual cells is determined by the age, position and neighbours of that cell. Stem cells it is assumed require information from the underlying connective tissue and providing they remain in contact with the basement membrane do not differentiate. However, stem cells are also sensitive to neighbouring intermediate cells such that if a newly formed stem cell finds itself totally surrounded by intermediate cells the level of an 'intermediate cell substance' exceeds a threshold and the stem cell differentiates into an intermediate cell. Stem cells normally divide to produce a basal cell and a stem cell. The stem cell is assumed to produce a factor that acts on basal cells and prevents their differentiation to another stem cell. When the sum of the concentration of this inhibitor from the stem cells to the right or left falls below a threshold value the basal cell will differentiate to a stem cell. Basal cells, like the stem cells, also respond to signals from the underlying connective tissue. Providing they remain in contact with the basement membrane they do not differentiate, once they leave the basement membrane they can differentiate to become intermediate cells. Intermediate cells will differentiate to form either basal cells or mature cells. If unusually, an inter-

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A S I M U L A T I O N OF E P I T H E L I A L G R O W T H

mediate cell moves to lie on the basement membrane and is therefore not entirely surrounded by other cells, it will differentiate to become a basal cell once again. Mature cells are formed by differentiation of intermediate cells. Once an intermediate cell reaches a given age, and providing the level of the intermediate cell maturation factor is less than a threshold value it will become a mature cell. AGEING

Stem cells and basal cells do not age. Once a cell leaves the basement membrane and becomes an intermediate cell, it will start to age. Age is calculated by the number of biological iterations since leaving the basement membrane; age is inherited by daughter cells and plays a role in determining cell maturation and death. The only exception is when an intermediate cell differentiates back into a basal cell, at which time the internal clock is reset. The cell or its progeny will start to senesce again only when they leave the basement membrane.

used are either one stem cell, or a number of evenly spaced stem cells. ALGORITHM

The model has two types of iteration: physical iterations and biological iterations. Biological iteration The model first checks which cells are on the basement membrane. It then checks all the cells for differentiation, death and desquamation. Finally, the cells are checked for any mitoses. Physical iteration The forces between all the cells are calculated (simultaneously), and the cells are then moved (simultaneously). The model will repeat physical iterations until the distance between any two cells in the model is less than a threshold value and slightly less than the radius of a cell. At this point, the model is assumed to be in a physical steady state, and a biological iteration is performed.

DEATH

A dead cell for the purposes of the model is one capable of desquamation and from the model. The death of a mature cell is dependent upon its age and position. The threshold of various factors that the cell perceives are set so that it will not die unless there are three or more neighbouring mature or dead cells.

DISPLAY

The program produces a graphical display of the skin, at each biological iteration, immediately before the mitoses. The display shows the following information: Cell type

DESQUAMATION

Dead cells will desquamate if the level of a signal they receive from the intermediate cells and mature cells falls below a threshold value. Desquamated cells are lost from the model. This signal is calculated on the basis of the inverse square law.

The ceils are drawn to fit the size and shape of the window in which they are being displayed. Thus even though the model treats the cells as circular, they may be drawn as ellipses. The colour or shading of the ellipse gives the category of the cell. Clonal unit

BASEMENTMEMBRANE The underlying connective tissue and basement membrane, while they are thought of as interacting with the cells of the model, are not separately modelled. Since the basement membrane is not separately defined its position depends upon the rules defining the position of the stem and basal cells. The algorithm used to define the basement membrane is given in the Appendix. INITIALCONFIGURATION One of the intended features of the model is that it should be able to generate a stable representation of skin from a single stem cell. The starting conditions

In some instances it is of value to be able to define the progeny derived from a single stem cell (the clonal unit). A nucleus is drawn inside the ellipse and the colour of the nucleus indicates the clonal unit of the cell. Recent mitoses Both daughter cells of the mitoses in the previous biological iteration are drawn as filled rather than hollow ellipses. The drawing routines usually used by the program have not been used to produce the figures for this paper, but have been replaced by grey-scale drawing routines so as to obtain black and white output.

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D. STEKEL E T A L .

Results SKIN GROWTH FROM A SINGLE STEM CELL

When started from a single stem cell, placed halfway along the bottom of the screen, the model grows to reach an apparent steady state. At first, the stem cell divides laterally to form basal cells. These basal cells start to move upwards, while remaining adherent to the stem cell which themselves are located at the bottom of the model. The basal cells divide so that the skin consists of a stem cell, with a layer of basal cells on each side, rising upwards, and intermediate cells sitting on top of the basal layer. As the intermediate layer thickens and ages, the intermediate cells start to mature [Fig. 3(a)]. Meanwhile, the basal layer at the bottom extends sideways until the most lateral basal cell passes the point at which the level of stem cell inhibitory factor prevents differentiation to another stem cell. This new stem cell is subject to the connective tissue attractive forces and move downwards. The shapes of the rete ridges begin to appear [Fig. 3(b)].

Mature cells in the centre of the model start to die and the layered structure of the epithelium can be seen in the older portions of the skin. Growth continues, with the basement layer extending laterally, the rete ridges forming, and the layered structure of the skin forming above the ridges. The dead cells at the top of the model start to desquamate and are lost from the model, leaving the fully developed section of the epithelium of approximately constant thickness [Fig. 4(a)]. There is a striking similarity to normal skin [Figure 4(b)]. THE MODEL REACHES A "STEADY STATE"

The steady state can be defined as the point at which the numbers of the different types of cells remain essentially constant (Fig. 5) and the skin qualitatively maintains the same structure at each iteration. At this point the skin consists of a basal layer, with stem cells at the bottom of each rete peg with intermediate cells, followed by a thin layer--often one cell t h i c k - - o f mature cells, and above them a thicker layer of dead cells. The model is dynamic and stem cells and basal cells continue to divide, sometimes to form new basal cells, sometimes to form new intermediate cells. The number of stem cells in the model remains fixed. The number of basal cells show little fluctuation. The rate of desquamation on average matches the rate of formation of new cells and the parameters that determine this balance are considered in more detail below. No attempts have been made to prove any formal mathematical results about convergence. CLONAL UNITS

FIG. 3. (a) The early stage of the model showing a single stem cell (filled black) and the basal cell progeny derived from it (black nucleus). The stem cell remains at the base of the model due to the attraction of the underlying connective tissue while the basal cells with a lesser degree of attraction move upwards. Some basal cells have differentiated to form intermediate cells (grey stippling) and a single mature cell (empty black circle) has formed at the top of the growing cells. (b) When basal cells become sufficiently distant from their parent stem cell one will de-differentiate to become a stem cell (filled black). This is subject to the increased attraction of the underlying connective tissue and moves downwards, forming a rete ridge.

Experimental studies in animals suggest that stratified epithelia consist of well-defined non-overlapping clonal units of cells each derived from a single progenitor stem cell (Seddon et al., 1992). When the model is grown from a single stem cell, progeny from neighbouring older stems can be seen intermingled with the newer clones. When the model first reaches the steady state, there are progeny from neighbouring stem cells in the younger clonal units [Fig. 6(a)]. However, as the model continues to grow, new cells from the newly formed stem cell and the new basal cells that they produce gradually displace the cells derived from the older stem cells upwards, these age, die and desquamate. The model sorts into distinct clonal columns above each stem cell [see Fig. 6(b)]. It is interesting to note that the boundaries of the clonal units show some mixing of cells; a feature that is seen in the biological analysis of clonality [Fig. 6(c)]. HEXAGONALPACKING It was found that the cells essentially pack themselves into a hexagonal lattice. This result was largely

J. theor. Biol.

i V FIG. 4. (a) Fully formed epithelium showing the layered structure with stem cells (filled black) at the base of each rete ridge. The basal cells (black nucleus) line the lower margin of the epithelium with a layer of intermediate cells (grey stippling) and then mature cells (empty black circle) above them. The mature cells die to produce the most superficial layer of dead cells (empty stippled circle). The processes in the model take place in a box which is 100 cell radii wide, and 60 cell radii high. The different boundaries of the box are handled very differently and the rules defining these are given in the Appendix.

FIG. 4. (b) A light micrograph of a piece of human skin showing the epidermis and two complete rete ridge. The layers of differentiated epidermal cells which correspond to the cell types assigned in the model can be distinguished.

D . STEKEL ET AL.

( f a c i n g p. 288)

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FIG. 6. (a) A picture of the model showing a single clonal unit of cells at the point when steady state is first achieved. There is considerable intermingling of cells from the marked clone with those derived from the neighbouring younger clone shown to the right. The stem cells have not been marked, but are at the base of each of the five rete ridges. (b) With further biological iterations the clonal units become well demarcated from each other and form into vertical columns of cells derived from each stem cell. There is very little intermingling of cells in neighbouring clones.

FIG. 6. (c) A photomicrograph of the ventral surface of the tongue in which a histochemical technique for the enzyme glucose-6-phosphate dehydrogenase has been use to identify a clonal unit within the epithelium (Seddon et al., 1992). One clone in which a stem cell and all its progeny lack the enzyme is shown clearly. The irregular border of the unstained region reflects the overlap with cells from neighbouring clones. The clonal structure and borders are strikingly similar to that shown in Figs 6(a) and (b).

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A SIMULATION OF E P I T H E L I A L G R O W T H

independent of the relative strengths of the forces in the model. The hexagonal packing is not particularly surprising, as the cells in the model are circular. What was interesting is that the packing was not perfect, and that often there were faults or gaps in the structure. While this observation might not seem "biological", it could be that fluid nature of cells in reality allows them to stretch to fill these gaps (Thompson, 1942). EFFECTS OF VARYING THE CONDITIONS OF THE MODEL

Wound regeneration

One test of the model is to remove part and observe the ability of the remainder to regenerate the whole. The left half, the right half, or the middle third of the steady state epithelium was removed in different experiments. After the section of skin was removed, new stem cells were formed from basal cells at the edge of the wound. These start dividing asymmetrically to produce new basal cells, which begin to fill up the gap. Eventually the gap in the model fills itself, and the model returns to the steady state. Parameter sensitivity

There are a number of parameters in the program (see Appendix) which may potentially affect the structure of the model. All these parameters have been adjusted to test the effects they have on the overall appearance of the skin at steady state. The results are summarized in Table A2 in the Appendix. Mitosis rates. There are two mitosis rates in the model. The mitosis rate for stem cells, and the mitosis rate for basal cells. It was found that changing the stem cell mitosis rate had little effect on the steady state of the model. Changing the mitosis rate of basal cells had

a marked effect on the total number of cells in the model. However, the shape and structure of the skin at the steady state was essentially the same, even though there may have been fewer or more cells (Fig. 7). Time to maturation and death. The length of time that a cell remains as a mature cell was found to be essential to the maintenance of the mature layer below the layer of dead cells. If the time was too short, then in many places along the skin, the dead layer of cells would form immediately above the intermediate, not the terminally differentiated, mature layer. Increasing this time period simply thickened both the mature and dead layers of the skin. Simply increasing or decreasing the lifetime of the cells (as intermediate cells) merely increased the number of cells in the model, but had little effect on the overall structure of the skin.

Discussion

Many groups have used computer models of cell patterning as a method of investigating how biological parameters may combine to produce a recognized morphology. The method is a powerful one, but is limited both by a lack of knowledge of the biological processes in any system and by the subjective involvement of the operator in determining the rules to be incorporated in the model. In most previous systems the cells are allowed to distribute themselves in a predefined lattice. This has been used to look at how tumour cells might grow and spread in an epithelium (Williams & Bjerknes, 1972). In this system random cell division was combined with competition between benign and malignant cells, the latter were assumed

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D. STEKEL E T .4L. similar to the arrangement of cells seen in human skin. This is an observation similar to that first made and explained by D'Arcy Thompson in 1917 (Thompson, 1942). Second the balance of forces attracting stem cells toward the underlying connective tissue is greater than that acting on the basal cells. This produces the undulations seen in the basal layer. It is unlikely that this is the sole biological mechanism responsible for the wavy interface seen between the dermis and epidermis in some parts of normal skin. However, the experimental evidence suggests that a population of cells with the replicative characteristics of stem cells express molecules on their surface that allows rapid adhesion to the basement membrane. As their proliferative capacity is lost so the affinity they have for the basement membrane declines (Jones & Watt, 1993). We have not taken account of the changes in cell size that occur as cells move up the epidermis. This is clearly an important process, but varies extensively in different regions of the epidermis and in different epithelia. One reason for this change in shape may well be the lateral physical forces in the epithelium. We consider that these forces are likely to vary extensively over different regions of the skin and be dependent upon the movement and folding that the epidermis experiences locally in conjunction with genetic patterning influences. In addition changes in cell shape, similar to that seen in the stones of an arch, may also play a role

to have acquired a "carcinogenic advantage". The patterns produced in this situation were compared to the distribution of tumour as it spreads in an epithelium. Subsequent developments based upon this type of model have incorporated the action of either intrinsic or extrinsic chemical factors that alter the behaviour of individual cells (Meakin, 1986; Smolle et al., 1992a, b. In these models, however, the position of the cells is constrained by the lattice while in the actual tissue there is a flexible organisation. In an attempt to overcome the need for a predefined lattice and to incorporate the biological mechanisms of cell adhesion, we have used an approach in which cells distribute themselves on the basis of forces between individual cells and between cells and the base of the model. A similar technique has been used by Bodenstein (1986) who developed a program to investigate cell mixing in experimental chimaeras. Here, cells were allowed to repel each other if they overlapped, but the adhesive properties between all cells were the same. In our model the range over which the adhesive force acts varies depending upon the cell type while the forces generated by the underlying connective tissue act differently on different types of cell. The effect of these forces in the model is two-fold. First, this produces an arrangement of cells that pack together in a "best-fit" that approximates to hexagonal as the ceils are circular, and shows fault lines very

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FIG.7. A graph comparing the observedeffecton the total cell number of cells in the model at steady state of changing either the probability that a stem cell or a basal cell will divide in any biological iteration. In one case the probability that a stem cell will divide is fixed at 0.5 while the probability of basal cell division is varied. In the other case the probability that a basal cell will divide is fixed at 0.25 while the probability of a stem cell dividing is varied. When the total cell number exceeds 1200 there is no longer space in the window and cells are lost from the model.

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A SIMULATION OF EPITHELIAL GROWTH

in generating the curvature of the basal layer of cells. However, even in the absence of such features, our model suggests that interactions between individual cells and the underlying connective tissue can generate a two dimensional simulacrum of epidermis. Intracellular adhesion and attachment to the basement membrane is known to be perturbed in a range of blistering skin disorders (Lever & SchaumburgLever, 1990). In subsequent experiments we will alter the forces that act between individual cells to investigate whether we can mimic the morphological changes seen in these disorders. One of the fundamental processes of malignant change is the acquisition of a mutation in the D N A of a stem cell that affects all subsequent progeny. In order that we might use our model to study the effects of such a mutation we have identified the clones of cells that are derived from each stem cell. The results that we obtain show that in the steady state the cells derived from one stem cell form themselves into columns above the stem and its daughter basal cells. A little intermingling is seen at the lateral margins of each clone. This columnar arrangement of a clone is very similar to that seen in other epithelia and is little affected when the stem cells in the model are allowed to move laterally rather than remain fixed. Hypotheses of stem cell systems are numerous (for reviews, see Wright & Alison, 1984; Hall & Watt, 1989) and the present model has assumed only a single stem cell in each clone and arbitrarily spaced them equally. Another possibility that has been suggested is a stem cell niche (Potten et al., 1979). This hypothesis was based initially on observations in the haemopoeitic system but has subsequently been applied to other systems including epithelia. In the niche model there are several stem cells in a localized microenvironment. Division of the cells is followed by random or partly random loss of one daughter cell from the niche, with loss of its stem-cell properties. A further modification of our model would alter the rules that determine a stem cell to incorporate the concepts of a localized environment that defines the niche. While the introduction of a stem cell niche might not be expected to alter the overall structure of the epithelium, it does have important implications on the rate at which a somatic mutation affecting one stem cell might affect the model. The model has been stylized in several ways and only takes account of the cellular interactions that occur in the epidermis; no dynamic interaction has been produced that involves the underlying connective tissue of the dermis. This is clearly a limitation as the role of the dermis in determining epidermal patterning has been shown both during development and in the steady state (Martin, 1990; Sengel, 1990). Similarly it

may be that the distance from dermal structures such as blood vessels is important in producing the overlying epidermal morphology. In addition, the images that we have produced are based upon rules that are applied in two dimensions and the model needs to be developed to incorporate three-dimensional interactions. One of the most interesting uses of the model is in the generation of neoplasia, although of course it is also of considerable value as a teaching aid. Neoplasia in the skin takes a variety of forms from the flat plaques of Bowen's disease to the hyperplastic papillae of solar keratoses. Having created a set of rules in a model which by their interaction maintain a normal skin-like structure we hope to introduce into one stem cell modifications of individual rules, either singly or in combination, and compare the form of the abnormal structures produced with those of human epidermal neoplasia. Through these studies we will generate hypotheses of the types of mechanisms that are necessary to maintain a steady state of an epithelium and the alterations in these mechanisms or their interactions that take place in the various forms of epithelial neoplasia. REFERENCES ADAMS,J. C. & WATT,F. M. ( 1991 ). Expression of beta l, beta 3, beta 4, and beta 5 integrins by human epidermal keratinocytes and non-differentiating keratinocytes. J. Cell Biol. 115, 829-841. BODENSTEIN,L. (1986). A dynamic simulation model of tissue growth and cell patterning. Cell Diff. 19, 19-33. COWAN, R. & MORRIS, V. B. (1988). Division rules for polygonal cells. J. theor. Biol. 131, 33--42. CRUYWAGEN,G. C., MAINI,P. K. & MURRAV,J. D. (1992). Sequential pattern formation in a model for skin morphogenesis. IMA J. math. appl. Med. Biol. 9, 227-248. EDLEMANN,G. M. (1988). In: Topobiology. New York: BasicBooks.

HALL,P. A. & WAVY,F. M. (1989). Stem cells: the generation and maintenance of cellular diversity. Development 106, 619-633. HERTLE, M. D., ADAMS, J. C. & WATT, F. M. (1991). lntegrin expression during human epidermal development in vivo and in vitro. Development 112, 193-206. JONES, P. H. & WATT, F. M. (1993). Separation of human epidermal stem cells from transit amplifying cells on the basis of differences in integrin function and expression. Cell 73, 713-724. LEVER, W. F. & SCHAUMBURG-LEVER,G. (1990). In: Histopathology of the Skin. MARTIN, P. (1990). Tissue patterning in the developing mouse limb. Int. J. Devl Biol. 34, 323-326. MEAKIN, P. (1986). A new model of biological pattern formation. J. theor. Biol. 118, 101-113. MORRIS, V. B. (1992). Spatial patterns of cells in dividing epithelia. J. theor. Biol. 159, 343-360. PILZ, L., RITTGEN, W. & TAUTU, P. (1993). Modelling cellular morpho- and carcinogenisis. In: Stochastic Spatial Processes. Berlin: Springer-Verlag. POVYEN,C. S. (1974). The epidermal proliferative unit: the possible role for the central basal cell. Cell Tissue Kinet. 7, 77-86. POTTEN, C. S. & MORRIS, R. J. (1988). Epithelial stem cells in vivo. J. Cell Sci. (Suppl.). 10, 45~52. POTTEN, C. S., SCHOFIELD, R. & LAJTHA,L. G. (1979). A comparison of cell replacement in bone marrow, and three regions of surface epithelium. Biochim. biophys. Aeta. 560, 281-299.

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SEDDON,S. V., WALKER,D. M., WILLIAMS,G. T. & WILLIAMS,E. D.

(I 992). The clonal organisation of the squamous epithelium of the tongue. Cell Prolif. 25, 115-124. SENGEL,P. (1990). Pattern formation in skin development. Int. J. Devl Biol. 34, 33-50. SMOLLE; J., SMOLLE, J. F. STETTNER, H. & KERL, H. (1992a). Relationship of tumor cell motility and morphologic patterns. Part I. Melanocytic skin tumors. Am. J. Dermatopathol. 14, 231-237. SMOLLE,J., TANIGUCHI,S. & KERL, H. (1992b). Relationship of tumor cell motility and morphologic patterns. Part 2. Analysis of tumor cell sublines with different motility in vitro. Am. J. Dermatopathol. 14, 315-318. THOMPSON,D'A. W. (1942) In: On Growth and Form, Vol. II, 2nd edn. Cambridge: Cambridge University Press. WILLIAMS,T. RrBJERKNES,R. (1972). Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 19-21. WZTHERS,H. R. (1967). Recovery and repopulation in vivo by mouse skin epithelial cells during fractionated irradiation. Radiat. Res. 32, 227-2239. WRIGHT,N. & ALISON, M. (1984). In: The Biology of Epithelial Cell Populations. Oxford: Clarendon Press.

APPENDIX A1. The following a l g o r i t h m is used to define the basement m e m b r a n e : A I . I . The b a s e m e n t m e m b r a n e can never be m o r e than 20 cell radii a b o v e the b o t t o m o f the screen. A1.2. I f a cell detects a n o t h e r cell below it, a n d within 3 cell radii from it, then it a d d s 2 to a p a r t i c u l a r flag (basal_flag). A1.3. I f a cell detects a n o t h e r cell that is below it, between 3 and 5 cell radii from it, a n d at an angle less than n/4 from the vertical, then it a d d s 1 to basal_flag. If basal_flag is f o u n d to be greater than o r equal to 6, then the cell is assumed to be not on the basement. Otherwise, it is assumed to be on the basement. A2. The b o u n d a r i e s o f the m o d e l are defined as follows: A2. I. T h e b o t t o m : only stem cells and the occasional basal cell are expected ever to be in c o n t a c t with

TABLE A1

The adhesive f o r c e s between cells act at different ranges. The table shows the distance that the f o r c e acts as a proportion o f each cell's radius Type of cell Stem cell Basal cell Intermediate cell Mature cell Dead cell

Adhesion range 0.4 0.2 0.0 0.0 0.0

the b o t t o m o f the box. I f a cell is p u s h e d below the b o t t o m o f the box, it is m o v e d up so that it sits inside the box. A2.2. The top: cell that m o v e a b o v e the t o p o f the box are lost to the model. In healthy skin, the m o d e l is set up so that this does n o t occur. A2.3. T h e edges: the a s s u m p t i o n is m a d e that the skin continues outside the confines o f the box. This is i m p o r t a n t in calculating effects due to chemical secretions, otherwise an i m b a l a n c e will occur at the edges o f the model. T h u s the following rules apply: A2.3.1. A cell that w o u l d move off the edge o f the box is forced to r e m a i n inside the box. This m a k e s the a s s u m p t i o n that for every cell m i g r a t i n g in one direction across any vertical line, there will be a n o t h e r cell m i g r a t i n g the o t h e r way. A2.3.2. The m o d e l assumes the existence o f a stem cell outside each edge o f the box, in a m i r r o r image p o s i t i o n to the nearest stem cells to the edge. A2.3.3. In calculating the effect o f i n t e r m e d i a t e a n d m a t u r e cells on d e a d cells for the d e s q u a m a t i o n process, the m o d e l assumes the existence o f intermediate a n d m a t u r e cells in m i r r o r image p o s i t i o n s to those cells within a distance o f 20 cells o f each edge.

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TABLEA2 A list of the variables defined in the model. The feature that each parameter defines in the model is given and the qualitative effects on the overall structure of increasing or decreasing each variable is described Variable Connective tissue force (CTAF) Connective tissue force (CTRF) CTRF range

Description

Effect of Reduction

Effect of Increase

attractive

Downwards force on stem cells

repulsive

Upwards force on basal cells

Skin floats up to top of screen and rete ridges shorten No rete ridges

Stem cells become disconnection from rest of skin Stem cells become disconnected from rest of skin Basal cells form too high up in wound regeneration No rete ridges

basaldown_mult

Mitosis probability Basal probability Stem threshold Maturation threshold

Desquamation threshold

Upwards range for which CTRF is felt Proportion of downwards force that a basal cell actually feels

Probability of stem cell mitosing in any biological iteration Probability of basal cell dividing in any biological iteration Signal threshold for a basal cell to become a stem cell Signal threshold for an intermediate cell to become a mature cell Signal level for a dead cell to desquamate

Constant in squash force Constant in adhesion forces

Glue range

Range over which adhesion forces act

Shorter rete ridges Longer rete ridges. Stem cells lose contact with neighbouring basal cells during generation of the model See Figure 4

See Figure 4

See Figure 4

See Figure 4

Fewer stem cells on screen. Lower density of cells Mature layer in contact with basement membrane

More stem cells on screen. Higher density of cells Thicker intermediate layer

Thinner dead cell layer

Thicker dead cell layer

Equilibrium reached in which cells are overlapping: program crashes Neighbouring ceils become disconnected under influence of CTAF or CTRF

Cells no longer pack together and become disconnected

Neighbouring cells become disconnected under influence of CTAF or CTRF

Skin floats up to top of screen. Equilibrium reached in which cells are overlapping: program crashes Equilibrium reached in which cells are overlapping: program crashes