Pattern formation models and developmental constraints

THE JOURNAL OF EXPERIMENTAL ZOOLOGY 251:186-202 (1989)

Pattern Formation Models and Developmental Constraints GEORGE F. OSTER AND JAMES D. MURRAY Departments of Biophysics, Entomology, and Zoology, University of California, Berkeley, California 94720 (G.F.0.); Centre for Mathematical Biology, Mathematical Institute, Oxford, OX1 3LB, England (J.D.M.) ABSTRACT Most schemes for embryonic pattern formation are built around the notion of lateral inhibition. Models of this type arise in many settings, and all share some common characteristics. In this paper we examine a number of pattern formation models and show how the phenomenon of lateral inhibition constrains the possible geometries that can arise.

Morphogenetic models have provided embryologists with insight into how embryonic patterns may be laid down. In this chapter we shall illustrate the basic pattern-forming principles of these models using as an example the formation of bones in the vertebrate limb. The physics of the pattern forming process imposes constraints on the possible cartilage patterns the limb may exhibit. These constraints are reflected in the general similarity one observes amongst all tretrapod limbs. Indeed, one class of developmental constraints on limb evolution are a consequence of these developmental construction rules. We shall illustrate how some of these construction rules arise in the context of a particular model for limb morphogenesis. Two views of pattern formation have dominated the thinking of embryologists in the past few years. The first might be called the chemical prepattern viewpoint, and the second could be called the mechanochemical interaction viewpoint. They may be roughly characterized as follows. Chemical prepattern models separate the process of pattern formation and morphogenesis into several sequential steps. Embryonic patterns are first specified as distributions of chemical (“morphogens”) concentrations. Subsequently, these chemical patterns are “read out” by the cells, and the appropriate changes in cell shape, differentiations, and/or migrations are executed according to the chemical blueprint. The notion of “positional information” (Wolpert, ’71) depends on such a chemical prepattern. In this view, morphogenesis is simply a slave process that is fully determined once the chemical pattern is established; therefore, models in this school focus on the problem of 0 1989 ALAN R. LISS, INC.

how the chemical prepattern is laid down. This can be modeled in one of two ways. 1) Simple chemical gradients are established across tissues, assuming that certain cells act as sources or sinks for the chemicals, which diffuse from cell to cell via junctions, or through the intercellular space. 2) A chemical prepattern can arise by means of “diffusion-driven instabilities,” a notion first proposed by Turing (’52), and subsequently elaborated and applied by numerous authors to a variety of embryonic situations (e.g., Meinhardt, ’82; Murray, ’81). The mechanical form-shaping events that occur in embryogenesis are not taken into account by such chemical prepattern models. Furthermore, the identity of the morphogens is proving quite elusive. The mechanochemical models take a quite different approach. Pattern formation and morphogenesis are not regarded as separable processes. Rather, chemical and mechanical processes are presumed to interact continuously t o produce, simultaneously, both the chemical pattern and the form-shaping movements (e.g., Oster et al., ’83, ’85).Moreover, since these models are framed in terms of measurable quantities such as forces and displacements, they focus attention on the process of morphogenesis itself. Despite their quite different assumptions about the physical basis for embryonic architecture, the mathematical mechanisms that underlie the two types of models have similar characteristics. In our discussion here we shall focus on these similarities, rather than on the differences, for it Received October 31, 1988; revision accepted March 3, 19S9.

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LIGCIT

ACT1VAT ION 1

INUlBlTlON

Fig. 1. The principle of local activation with lateral inhibition in a neural net. In (a), a n excited retinal ganglion cell inhibits the activity of its neighbors (the symbols k and -1 indicate inhibitory synapses). A large population of cells hooked up in this way produces the activation-inhibition field shown in (b).

Fig. 2. Top: The Hermann grid produces the illusion of dark spots at the intersections of the white bars. This is an example of pattern formation by lateral inhibition. Each excited retinal is surrounded by an inhibitory field (bottom). Because its inhibitory surround is illuminated more, a cell located at the intersection of two white strips experiences a greater inhibitory effect than one located in a single white strip between two black squares. Thus, a cell at the interesection fires more weakly than its neighbors, and that region appears darker than the surrounding white regions.

turns out that there are common themes that impose constraints on the possible forms of embryonic structure. In this chapter we shall outline the main ideas that underlie models of pattern formation, using as an example a model for chondrogenesis in the vertebrate limb. We shall keep our discussion qualitative; however, in Appendix tween light and dark regions. For example, a cell B we present one such model in mathematical de- just inside the light region is not inhibited as tail for those who wish a deeper understanding of strongly as cells further from the boundary bethe phenomena. cause of its proximity to unstimulated cells in the dark region. Therefore, it will fire more strongly GENERAL PROPERTIES OF PATTERN than cells away from the boundary, and so conFORMATION MODELS tribute t o the apparently bright Mach band. SimiPattern formation arises from local larly, cells in the dark region near the boundary activation with lateral inhibition are inhibited more strongly than cells deeper in In 1865, the Austrian physicist Ernst Mach pro- the dark region, so an apparently dark band is posed an explanation for the visual illusion now produced. In the Hermann illusion shown in Figknown as “Mach bands” (Mach, ’65; Levine and ure 2, cells at the intersections of the white strips Shefner, ’81; Ratliff, ’72). This well-known illu- between the black squares have more illuminasion is produced when a light and a dark field are tion in their inhibitory surrounding region than juxtaposed: near the boundary between the fields other cells in the white regions. Therefore, they there appear to be lighter and darker bands. are more strongly inhibited, and appear darker. Mach hypothesized that the neurons of the retina Thus lateral inhibition creates the illusion of a exhibit the phenomenon of Zateral inhibition; that spatial pattern. This simple phenomenon of local excitation is, an excited neuron inhibits the firing of its neighbors (Fig. 1).Lateral inhibition has the ef- with lateral inhibition characterizes the behavior fect of enhancing contrast at the boundary be- of many neural nets, and it can be used to gener-

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kinetics of the morphogens or the mechanical properties of cells. The spatial interaction term accounts for the ways in which neighboring cells or chemicals interact with one another in space; for instance, by diffusion according to Fick’s law, or by mechanical interactions, such as deformations that obey Hooke’s law of elasticity. In order for equation (1)to generate spatial patterns, the spatial and temporal properties of the system must conspire t o generate an analog of the neural phenomenon of local activation and lateral inhibition. How is this accomplished? First let us examine diffusion-reaction models. Chemical prepattern models built around diffusion-reaction instabilities have the general form given in equation 1.The morphogenetic variables are the morphogen concentrations, and the local dynamics are their reaction rates; the spatial interaction term is simply Fickian diffusion. These Fig. 3. An array of cells with lateral inhibitory fields will models differ essentially only in the choice of produce bands of excitation. This pattern was produced by the chemical kinetics between the reacting morphoneural shell model described in Appendix A.l. gens and the relative magnitudes of the diffusion coefficients, which are necessarily unequal. Local ate a large variety of other spatial patterns. For activation is achieved by making the kinetics example, Figure 3 shows how lateral inhibition autocatalytic (analogous t o excitation by light in can generate a series of parallel bars. Ermentrout the neural net). Lateral inhibition is produced by et al. (’86) studied the color patterns found on mol- introducing a chemical that inhibits activator lusk shells using a simple model whereby neural production and that can diffuse faster than the activity stimulated secretion of pigment. If the activator. Thus, the inhibiting morphogen can neural net controlling the secretion possessed the outrun the spread of the autocatalytic reaction property of lateral inhibition, the model was able and quench its spread, thus creating a zone of to reproduce many of the observed color patterns; lateral inhibition around the excited zone (cf. Fig. Figure 4 shows one example. Appendix A gives 5). Alternatively, one can introduce a substrate the equations used to generate these patterns. We that is consumed by the production of activator, so shall use this model below as the basis for discuss- that its depletion quenches the autocatalytic reacing other pattern formation models. tion. Examples of such diffusion-reaction equations are given in Appendix A.l. Other kinetic Morphogenetic models depend upon lateral schemes that produce spatial patterns are reinhibition viewed by Murray (’82), who shows that they are The phenomenon of local activation and lateral all mathematically equivalent. Figure 4 shows inhibition underlies most models for morpho- that the same patterns produced by a neurally genetic pattern formation. This can be understood implemented lateral inhibition can also be generfrom an examination of the equations that govern ated by a diffusion-reaction mechanism. the development of spatial patterns in both the As we mentioned above, mechanochemical chemical prepattern and many of the mechano- models simulate morphogenesis in a fundamenchemical models. They have the general form: tally different way from the morphogen models. For example, rather than modeling how a presumed chemical prepattern fully specifies the form of the cartilage anlagen, they model the (1) morphogenetic movements themselves. In these The morphogenetic variables in this equation models, the morphogenetic variables are cell denare such things as chemical concentrations, rates sities and geometric displacements of the cells of cell division, or mechanical displacements of from their initial positions. The equations accells. The “local dynamics” term accounts for the count for the mechanical forces between cells and

z

Fig. 4. The left panel shows a divaricate line pattern generated by the neural secretion model of Ermentrout, et al. ('86) (see Appendix A.l). The middle pattern shows the same pattern generated by a diffusion-reaction model (Meinhardt, '84; Meinhardt and Klingler, '87) (see Appendix A.2). Although they are based on dramatically different assumptions

about the underlying mechanism, both models implement local excitation coupled with lateral inhibition, and so they produce essentially the same patterns, The right panel shows a typical pattern of this sort on the mollusk Nerita turrita (Ermentrout et al., '86).

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FAST DIFFUSION

INHI6ITION

FAST DIFFUSION

Fig. 5. A chemical reaction system can generate lateral inhibition. An “activator” molecule [O],when it collides with a substrate, releases several molecules of its own type (autoThe inhibitors catalysis), as well as inhibitory molecules .].[ can combine with the activators and prevent their subsequent reaction. If the inhibitor can diffuse more rapidly than the activator, it will spread faster and can arrest the autocatalytic reaction before it spreads too far. The equations for this scheme are given in Appendix A.2.

the extracellular matrix material, as well as the concentrations of regulatory chemical substances such as calcium (cf. Oster et al., ’83). Both of these types of model require some mechanism to prompt the cells t o commence their morphogenetic activities. The chemical prepattern models assume that the cells read the completed morphogen concentration profile, then execute their morphogenetic movements accordingly. Thus a complex prepattern is required to generate a correspondingly complex form. The mechanochemical models also require some prepattern, but in contrast t o the diffusion-reaction models it need only be a very simple one. A simple gradient in cell type will do: for example, a cell lineage and/or cell aging mechanism could trigger a local increase in cell traction, thus initiating the morphogenetic process. This process will then unfold

into a complex form due to bifurcations and epigenetic feedback. We shall give a n example of this in Section 3. The mechanochemical and diffusion-reaction models achieve local activation and lateral inhibition in different ways. However, in both cases, the emergence of spatial patterns unfolds when the spatially uniform state becomes unstable t o spatial perturbations of a given size, causing it to break up into spatial patterns corresponding to that size. Patterns can form simultaneouslg or sequentiallg There is one more general property of pattern formation models which we should mention before proceeding to examine a models of limb formation. Spatial patterns may develop from a uniform state in one of two ways. First, the pattern may grow more or less simultaneously over an entire field. Second, the pattern may appear sequentially; that is, it commences in a particular region, and spreads laterally in a wave until the global pattern is established. As we shall discuss below, most biological patterns originate locally, in some “organizing tissue” and develop sequentially thereafter. That is, wavelike pattern formation is the rule in development. This might have been anticipated from the study of pattern formation models for the following reasons. Models in which patterns grow simultaneously over an entire field tend t o be less reproducible than those that form sequentially (cf. Murray, ’81a,b; Meinhardt, ’82; Perelson et al., ’86).When complex patterns form over a large field, the ultimate steady-state patterns, although qualitatively similar, are determined by the initial conditions. In developing systems there are always inherent stochastic effects. This fact was exploited by Murray (’79, ’81a,b) in a reactiondiffusion model for the patterns of animal coat markings. Since the initial conditions for an individual animal are unique, this implies that the ultimate coat pattern for each animal is unique. Figure 6 shows some simulations of the reactiondiffusion model for progressively larger domains. The distribution of light and dark patches depends on the random starting conditions. While the general scale and type of pattern varies with the starting conditions, the overall similarity of the patterns is maintained. This example illustrates one type of developmental constraint. The role of scale and geometry

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Fig. 6. Patterns generated by a reaction-diffusion mechanism showing the effect of size on the patterns (Murray, '81a,b). For simplicity, the domain illustrated shown the same size, but from (a)to ( f ) the actual domain is progressively larger. (b) represents the first bifurcation from a uniform color; successive panels show how the pattern becomes increasingly complex as the domain size increases. Note, however, that the pattern effectively disappears for very large domains. These results suggest that most small animals (a) and most large animals (f) tend to be uniform in color-a common feature of mammalian coat patterns.

are important parameters in determining the ultimate pattern in any lateral inhibition mechanism. In situations where the domain size varies, such as the tail integument shown in Figure 7, the thicker proximal part of the tail might be large enough to sustain a spot pattern, while the thinner distal part may only sustain stripes. Figure 7a shows a numerical simulation of a reaction-diffusion mechanism that demonstrates this phenomenon. The model suggests a developmental constraint wherein a spotted animal can have a striped tail, but a striped animal cannot have a spotted tail. Figure 7b,c shows that cheetahs and jaguars do indeed conform to this constraint.

a

t

b

Fig. 7. (a)Spatial pattern generated by a reaction-diffusion mechanism for a tapering domain, mimicking the integument of the tail in a developing embryo (Murray, '81a,b). Note how the spot-like pattern is forced into a stripe pattern as the tail gets thinner. Also shown on the left (b)is the tail of a cheetah (Acinonyxjubatis), and on the right ( c ) the tail of a jaguar (Panthem oncu),which illustrates the developmental constraint described in the text.

However, the conclusions we shall draw about constraints on cell aggregation patterns are quite general, for it turns out that a large class of pattern formation models exhibit properties similar to this simple one. Motile cells aggregate by directed migration The beginning stages of organogenesis are frequently heralded by patterned aggregations of motile cells. These aggregations then differentiate, undergoing further morphogenetic transformations as the mature organ takes shape. However, the initial anlage determines the general geometry and size of the organ. There are several mechanisms that can produce such aggregations of motile cells':

A MODEL FOR CHONDROGENIC CONDENSATIONS Chemotais: Cells may move toward the source The general considerations discussed above are of a chemical attractant. For example, during the easier to understand if we have a concrete examaggregation phase of the slime mold Dictyosple. In this section we outline a specific model that telium discoideum, motile cells move toward a can generate the pattern of cartilage condensations in the vertebrate limb bud. We do not presume that this model is necessarily the correct 'Cell aggregations can also arise in a stationary cell population by localized differential cell division, However, it has been demonstrated one; our purpose is rather to illustrate the prin- that chondrogenic condensations do not involve extensive cell mitosis ciples that underlie pattern formation models. (Hinchliffe and Johnson, '80). ~

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specialized epidermal region called the “apical epidermal ridge.” Cells emerging from the progress zone are, for a time, not “competent” t o aggregate. This could result from the action of an inhibitory chemical secreted by the apical epidermal ridge region, or it could be simply a conseof cell maturation. At a certain distance quence AER from the progress zone the cells become competent to aggregate. Since we are concerned only with the generation of the spatial pattern itself, the model need not address what initiates competence. When cells become competent to aggregate and start to move, we assume that their aggregation Fig. 8. A model limb bud. Cells under the influence of the is guided by some kind of taxis. The signal may be apical epidermal ridge (AER) proliferate in the progress zone, either chemotactic or haptotactic. Here we will then aggregate into chondrogenic condensations (shown treat the case of chemotactic aggregation because cross-hatched). Cartilage condensations form by recruiting it is simpler; however, the haptotactic mechanism cells from the surrounding tissue (shown dotted). will produce the same patterns (Oster et al., ’83). Therefore, we shall assume that at a certain point chemoattractant that is thought to be secreted by after leaving the progress zone cells commence t o a small group of “pacemaker” cells. Likewise, ver- secrete a chemoattractant. This will cause them tebrate lymphocytes will move chemotactically to begin to aggregate as each cell attempts to mitoward the site of an inflammation. grate toward nearby concentrations of attractant. Haptotuxis: Motile cells exert strong tractions The components of the model can be understood on the extracellular matrix through which they from the following “word equations”; the correare crawling. These tractions compress the ma- sponding mathematical expressions are given in trix, creating a gradient in adhesive sites that Appendix B. bases the cells’ motions so that they move up the We must write two conservation equations for adhesive gradient. the concentrations of mesenchymal cells, n, and Convection: Cells may be moved passively, rid- the chemoattractant, c. The equation for the ing on other moving cells, or they may be pushed motile mesenchymal cells has the form: or pulled by the extracellular matrix, which is rate of change 1of - +1 - = itself being deformed by the mechanical action of (2) cell densitv other cells, or by differences in osmotic swelling. Although the dominant effect may be different in different situations, in most cases, several of The equation governing the chemotactic chemical is given by these mechanisms operate at the same time. It is not known which, if any, of these mechanisms generates the condensations of chondroblasts that form embryonic bone prepatterns. (3) However, models for each mechanism have been proposed, and, surprisingly, the patterns they How the model generates spatial patterns predict are much the same. The reasons for this will become clear later on; for now we shall proBy the following reasoning, it is clear that cells ceed to formulate a simple model based on the moving according to the above rules may not disassumption that, like Dictyosteliurn amoebae, the tribute themselves uniformly in space. Suppose a mesenchymal cells respond to chemotatic chemi- small fluctuation produces a local rise in cell dencals that they secrete themselves. sity; then this region broadcasts more chemoattractant than neighboring regions, and so recruits A chemotactic model f o r chondrogenesis more cells. This process is autocatalytic, for the Consider the model limb bud shown in Figure 8. more cells that aggregate the stronger the reThe limb grows by adding cells to the “progress cruiting signal becomes, as larger aggregates of zone” at its distal end, which is capped by the a cells emit more attractant, and so grow even ~~

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Fig. 9. In the beginning stages of pattern formation, the cell density can be viewed as the summation of sine and cosine wave functions of different sizes (i.e., wavelengths). The dispersion relation gives the initial growth rate of the different harmonics as a function of their size. Those harmonics with negative growth rates die out, while those with posi-

tive growth rates increase in amplitude. In simple situations, the pattern with the maximum initial growth rate will eventually dominate. However, this is not always true: sometimes patterns that commence growing slowly eventually overtake and surpass patterns whose early growth rate was faster.

larger. Counterbalancing the autocatalytic aggregation is the effect of random cell motions (diffusion), which tends to smooth out inhomogeneities. If the aggregation process predominates, cell recruitment depletes the immediate neighborhood of cells. This creates an effective region of inhibition around each aggregation center, producing a characteristic spacing of the centers. Unfortunately, from such a verbal description it is not clear that the model will produce anything more than random aggregations of cells (indeed, Dictyostelium aggregations are not very regularly spaced). To appreciate the fact that these processes can give rise to a regular spatial pattern one must do a mathematical analysis; a mere verbal description of a process cannot delineate phenomena that depend on a quantitative balance of competing effects. Appendix B presents this analysis in detail; here we shall only attempt to provide an intuitive appreciation of the operation.

field becomes unstable and breaks up into a spatial pattern, or an existing pattern becomes unstable and bifurcates into another spatial pattern. Note that the term “bifurcation” has both a colloquial meaning (to branch, or split) and a technical meaning, which refers to the spatial instability of a solution to the model equations (cf. Appendix

MORPHOGENETIC PROPERTIES OF THE MODEL Spatial patterns arise when a developmental parameter exceeds a threshold value Nonlinear models develop several different kinds of spatial nonuniformities. This happens when one of the model parameters exceeds a threshold value, whereupon an initially uniform

B). Determining the threshold values and the range of the model parameters tha lead to spatial patterns can be a difficult task, which is usually accomplished by a combination of mathematical analysis and computer simulation (Murray, ’82). However, a good predictor as to whether a model can generate spatial patterns is the “dispersion relation” for the model. This is a plot of the growth rates of different patterns as a function of the scale of the pattern, as shown in Figure 9. Such plots are central to understanding how spatial patterns emerge in morphogenetic models, and Appendix B gives a quantitative example of this important concept. In order t o determine which parameter is crucial, it is frequently convenient t o view a developmental process as occurring on two different time scales. Chemical prepattern models assume that the cells remain essentially stationary while the morphogen is secreted, for otherwise the convective effect of the cells’ motions would disrupt the chemical pattern. Conversely, mechanochemical

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models assume that, while the cells are continuously differentiating, on the time scale of morphogenesis (i.e., cell movement),their state of differentiation is almost constant. Thus, in each type of model some parameters are varying slowly in comparison with the evolution of the spatial patterns. In simplified models for a developing system one variable may be selected as the slowly varying developmental parameter (or “bifurcation parameter”): that which defines the property that slowly progresses until the uniform field becomes unstable. This instability arises because, when the developmental parameter exceeds a threshold value, the system is no longer in mechanical or chemical equilibrium, and it evolves to a newspatially non-uniform-configuration. For example, in the simple chemotaxis model sketched above it is clear that no pattern will arise if the cells’ chemotactic response is not strong enough to overcome the dispersive influence of their random motions. Only when the parameter stimulating tactic motion is sufficiently large can a nonuniform distribution of cells come about. The parameter space corresponding to the chemotactic model is shown in Figure 11. This figure also illustrates another general feature of morphogenetic models: usually no single parameter determines the bifurcation point. Rather, certain dimensionless groupings of parameters determine the model’s behavior. We will return t o this important point later.

Embryonic patterns are generally laid down sequentiallp An important feature of morphogenetic patterns is that they are usually laid down in a sequential fashion, rather than simultaneously over an entire tissue. Frequently, a pattern appears t o spread outward from an initiating region, or organizing center. A good example of this is the formation of feather germ in birds and scale arrays in reptiles (Sengel, ’76; Davidson, 1983a,b; Oster et al., ’83). There the hexagonal patterns of primordia form first in a single row, then spread laterally to form the final hexagonal array (Perelson et al., ’86). As we mentioned earlier, analyses of both diffusion-reaction and mechanochemical models for this process show that it is much easier t o create reproducible and stable patterns if they are generated sequentially, rather than arising simultaneously over the entire field. This sequential aspect of development is clearly evident in the vertebrate limb, where the pattern

largely migrates in a proximo-distal direction (Hinchliffe and Johnson, ’80). In vertebrates, there is an apparent exception: the formation of the digital arch proceeds in an anterior-posterior direction as well as proximo-distally (Shubin and Alberch, ’86). However, it is important to note that the onset of differentiation of the digital arch is correlated with the sudden broadening and flattening of the distal region of the limb bud into a “paddle.” We shall see below that this change in geometry is the key to understanding this apparent exception to the sequential development rule.

Cell recruitment zones define morphogenetic fields An important feature of models that explicitly account for cell density is the creation of zones of recruitment around the chondrogenic foci. That is, an aggregation center autocatalytically enhances itself while depleting the surrounding tissue of mesenchymal cells, thus setting up an effective lateral inhibition against further aggregation. Moreover, adjacent foci compete for cells, producing a nearly cell-free region between them. Thus, a condensation establishes a “zone of influence,” which precludes formation of other foci; this will play an important role in our subsequent discussion of branching. Tissue geometry is important in controlling patterns All models for chondrogenic condensations are strongly affected by the size and shape of the growing limb bud.2 Tissue geometry controls the expression of certain model parameters whose influence on the growth of patterns is often decisive. Bifurcations in a model are detected by investigating when an integer number of sine or cosine waves can “fit”into a domain of a certain size (cf. Appendix B). The reason for this is that, at the very onset of an instability, the solutions to the model equations look like sine or cosine waves. In order to satisfy the conditions at the boundary of the domain (e.g., there are fixed values at the boundary, or it is impermeable t o cells and chemicals), one must impose the condition that only an integral number of wavelengths fit in the domain. This determines the size of the patterne3 ‘Indeed, Alberch and Gale (’83) have shown that the pattern of cartilage condensation in real limb buds is dramatically affected by reductions in size. 3Strictly speaking, the wavelength-fitting procedure reliably predicts the final pattern only in the one-dimensional case. In two and three dimensions, ambiguities creep in, and more sophisticated methods, or computer simulation, must be employed.

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by tissue geometry alone, but by dimensionless ratios of the various parameters. For example, a dimensionless ratio appearing in the chemotaxis model is (cf. Appendix B) as follows:

(B> Fig. 10. The size of the domain must be large enough to “fit” an integral number of waves. In (A) the limb crosssection is large enough to accommodate a single wavelength, corresponding to a single peak in cell density. If the domain size and/or shape changes sufficiently, so that two complete harmonics can fit into one dimension of the limb cross-section, then the distribution in (A) will become unstable and evolve to the pattern shown in (B).

For example, suppose we have a situation such as in Figure 10A, which shows a limb crosssection in a proximal region. If the size of the domain were too small, a full sine wave could not “fit”into the domain, and so growth of any patterns would be suppressed. If, however, the domain were to grow, when it reached a critical size, the uniform state would break up, and a single mode would commence to grow (i-e.,a single peak in cell density). In Figure 10A a spatial pattern with a single peak-corresponding to sin(m/L)has commenced t o grow. The critical domain size that causes the uniform state to become unstable is the “bifurcation” size, since at this value the model equations switch from one solution (uniform) to a new solution (one peak in cell density). If the domain geometry is changed so that two complete sine waves can fit into the horizontal dimension (cf. Fig. lOB), then the solution in 10(A) with a single peak becomes unstable, and the system evolves to that shown in 10(B) corresponding to two peaks in cell density.

where OL is the haptotactic motility, b is the maximum secretion rate of chemoattractant, 1.1.is the decay rate of the chemoattractant, and L is a characteristic dimension of the system, e.g., the “diameter” of the proximal section of the limb bud. Thus, according to (equation 41, a decrease in the secretion rate, b, may be compensated, for example, by an increased chemotactic sensitivity, a, by a decreased rate of degradation, F, or by a decrease in the domain size, L. These scaling ratios, which relate the system parameters and the domain size and shape, arise from the necessity of rendering the model equations dimensionless, that is, independent of a particular choice of units (e.g., centimeters or inches). They are not unique, for there are many ways to create dimensionless ratios from a given set of parameters. However, the bifurcation behavior that they control is an intrinsic property of the underlying physics and chemistry of the system.

There are mang routes through parameter space that lead to a spatial pattern From the above discussion it is clear that varying the domain size is but one way to trigger bifurcations. Indeed, this lack of uniqueness of the bifurcation parameter has important implications for the interpretation of experiments, for it means that there are many developmental paths that lead to the same bifurcation. For example, suppose the parameter values are such that the system is at the point P, where no structures can develop, as shown in Figure 11. if the cell diffusional motility M decreases, the point in parameter space moves toward C. On crossing the bifurcation curve the mechanism develops spatial structures. However, the system can arrive at the same point C-and develop the same spatial structures-if the maximum rate of attractant seDevelopmental parameters occur in cretion, b, increases appropriately or if the hapdimensionless groupings totactic parameter, a, increases. In fact, the sysIn our discussion of branching patterns so far tem can move into the spatial structure region by we have focused our attention on the role of limb a variety of paths (e.g., from P to A). The central geometry (i.e., cross-sectional shape). However, point is that, although there are certain features analyses of the various models reveals that the whose presence is essential for pattern formation, system parameters arise in natural groupings. there is no unique property that is responsible That is, the bifurcation behavior is controlled not for it.

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t

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Fig. 11. Bifurcation diagram for the chemotactic model, showing how different paths through developmental parameter space can lead to spatial structures. The axes are dimensionless ratios consisting of the following variables. M is the random motility coefficient of the chondroblasts, n is the cell density, and h is a measure of the chemoattractant secretion rate at low cell densities. p is the degradation rate of chemoattractant, and b is the maximum secretion rate. When the system is at point P, the uniform state is stable, and no spa-

tial patterns can develop. However, if combinations of parameters change moving the system along path P + A or P C, then the uniform state becomes unstable, and a spatial pattern evolves. Interestingly, in this model, the spatial structure can disappear again if the parameter path follows P + A + B. For example, a monotone decrease in cell density, n, first triggers pattern formation, but at very low densities the pattern disappears again.

The bifurcation space also highlights other aspects of the modeling process. For example, suppose the original cell density, no,decreases. The path from P is toward A and patterns develop. If the cell density falls too low, the system moves along path A + B , whereupon the system bifurcates again, this time from a patterning state t o one in which no structure develops. Thus, a monotonically changing cell parameter can lead the system first through a pattern-forming episode, followed by the disappearance of the pattern.

rion for judging the validity of a model. For example, Murray ('82 has shown that pattern formation by some diffusion-reaction mechanisms (e.g., the one employed in the shell model described in Appendix A.2) is quite delicate: the range of parameters that correspond to spatial patterns is very small. It is not likely that evolution would have stumbled onto such mechanisms, in comparison with others whose range of allowable parameters is much wider. Of particular importance is the fact that different models do make different predictions as to the outcome of certain experimental interventions. These predictions are contained in the different dimensionless groupings of variables that govern the operation of each type of model. For example, interventions that affect cell tractions will disrupt chondrogenesis according to a mechanochemical model, but will not affect the patterns predicted by a morphogen model. Or, in the case of the shell patterns, a neural model would suggest different experimental interventions than would the diffusion-reaction model.

Patterns d o not distinguish between the types of models Models that simulate different developmental mechanisms cannot be distinguished solely on the basis of the predicted pattern. Both morphogen and mechanochemical models predict similar types of bifurcations. Indeed, Figure 4 illustrates that both neural and diffusion-reaction models can produce the same kinds of spatial patterns (since both are implementations of lateral inhibition). Thus, it is generally not possible to deduce the mechanism underlying a spatial pattern from CONSTRAINTS ON MORPHOGENETIC the pattern alone, since many mechanisms can PATTERNS generate the same pattern. However, the sensitivity of patterns to variations in the parameters is The physical and chemical processes underlydependent on the model employed. Therefore, the ing morphogenetic models impose certain restricrobustness of the predicted patterns is one crite- tions on the geometry of the resulting pattern.

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Fig. 12. The three possible types of chondrogenic condensations: (a)focal condensation, F; (b) branching bifurcation, B; (c) segmental bifurcation, S.

That is, not everything is possible within the constraints of mechanical and chemical interactions. In this section we outline some of these restrictions on form (see also Alberch, this volume).

There are three types of bifurcations We are now in a position t o assert our central theoretical point: regardless of the underlying mechanism-chemical or mechanical-both types predict that, in practice, the aggregation patterns are limited to three types, illustrated in Figure 12: 1. Focal condensations, which we denote by F, arise as isolated foci in a uniform field, provided there is sufficient tissue volume and cell density. 2. Branching bifurcations, denoted by B, wherein an existing condensation branches into a Y-shaped configuration. 3. Segmental bifurcations, denoted by S , wherein a condensation either buds off a posterior element, or an existing element subdivides itself longitudinally (i.e., proximo-distally) into two subsegments. The explanation for this restricted list of pattern possibilities is given in Oster et al. (’83) (see also, Oster et al., ’87). Heuristically, we have seen that in order for a pattern-uniform or nonuniform-to become unstable and give way to a new pattern, one of the “slowly varying” model parameters must pass through a bifurcation threshold. At that point, the balance between local activation (Lea,the autocatalytic aggregation of chondrocytes) and lateral inhibition (i.e., the depletion of cells between aggregation centers) is upset. Thus, focal condensations arise in situations when the balance between activation and lateral inhibition is such that the uniform field becomes unstable. This requires, amongst other things, that the condensation domain be sufficiently large, so that the focus is effectively isolated from other competing foci.

Fig. 13. An example of a branching diagram showing how the limb of a salamander (Ambystoma mexicanurn) can be built up from sequences of F, S, and B bifurcations. Note that this is an adult limb; frequently, the original condensation pattern may be obscured by subsequent growth and differentiation.

Segmental bifurcations can occur when the length of the domain exceeds a critical value. That is, if a condensation field grows too long, its extremities may be able to establish independent recruitment domains and divide the field into subdomains. In terms of the scenario developed above (Fig. lo), there is enough room to fit in another wavelength. Branching bifurcations can occur when the domain size broadens so that the existing condensation is too large, and its borders become unstable. Each region has the potential to set up its own recruitment domain, and so the existing condensation branches in two. This is like segmental bifurcation, except that it takes place transversely rather than longitudinally in the limb bud domain. That these types of condensations are the most likely patterns arises from the nature of the local activationilateral inhibition mechanism and the sequential nature of limb chondrogenesis. The size dependence of the condensation domain (a type F bifurcation) explains the phenomenon of digital arch formation in urodeles: an independent focal condensation can form providing the distal “paddle” is large enough so that two recruitment centers do not inhibit one another. As a consequence of these restrictions on the possible kinds of cartilage condensation, we can describe any vertebrate limb as a sequence of F, S, or B bifurcations (cf. Fig. 13). Thus, Shubin and Alberch, (’86) are able to construct generalized “branching diagrams” that describe all known amphibian limb morphologies (see also Alberch, this volume; Oster et al., ’87). These restrictions

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on form constitute a “developmental constraint,” which can only be violated by gross alterations in the geometry and/or cellular properties of the developing limb bud.

Certain patterns of chondrogenesis are unlikelg From the above discussion we can see that theoretical models predict a restricted menu of possible chondrogenic patterns. For example, it is quite unlikely-although not impossible-to obtain a “trifurcation,” i.e., a branching of one element into three (or more) elements. Even though subsequent growth may make it appear as if three elements arose simultaneously at a common branching point, the theory suggests that all branches are initially binary. Moreover, in the absence of any biasing factors, condensation patterns are symmetric and uniform: a condensation domain is broken into equally spaced subdomains whose number and spacing are determined by the geometry of the region. The presence of asymmetries in the limb bud, such as the anterior-posterior axis, reflects the presence of asymmetrically situated influences such as the “zone of polarizing activity.” This is a differentiated region located near the distal posterior margin of the limb bud, which appears to induce an anterior-posterior gradient in the sizes of the cartilage condensations.The “airplane wing”shaped cross-section of the growing limb bud ensures that the central digits must be larger than the marginal ones, while the simple anteriorposterior gradient imposed by the zone of polarizing activity admits the possibility of an anteriorposterior gradation in digit size. Indeed, only rarely are limbs found with central digits smaller than the marginal ones.

DISCUSSION Using a simple model for chondrogenesis we have tried to show how spatial aggregation patterns arise from a uniform population of mesenchymal cells. The common feature of most morphogenetic models is that they mimic the neural phenomenon of local activation with lateral inhibition. While this property confers the capacity to generate spatial patterns, the variety of patterns is constrained by the geometry of the domain and the gradients in the model parameters. This is because centers of aggregation must compete with adjacent centers to recruit cells from the surrounding tissue. The cellular properties and the tissue geometry combine in “dimensionless pa-

rameters,” which measure the aggregate effect of competing influences. Therefore, it is usually not easy t o isolate a single “cause” underlying a morphogenetic phenomenon: spatial patterns emerge as a consortium of a number of competing effects conspire to produce the phenomenon of local autocatalysis and lateral inhibition.

ACKNOWLEDGMENTS Most of the ideas contained in this paper were formulated in collaboration with Pere Alberch. Greg Kovacs did the simulations shown in Figure 3. G.F.O. was supported by NSF grant MCS 8110557. LITERATURE CITED Alberch, P. (1987) Orderly monsters: Evidence for internal constraint in development and evolution. To appear. Alberch, P., and E. Gale (1983) Size dependency during the development of the amphibian foot. Colchicine induced digital loss and reduction. J. Embryol. Exp. Morphol., 76:177197. Davidson, D. (1983a) The mechanism of feather pattern development in the chick I: The time determination of feather position. J Embrol. Exp. Morphol., 74:245-259. Davidson, D. (1983b) The mechanism of feather pattern development in the chick 11: Control of the sequence of pattern formation. J. Embryol. Exp. Morphol., 74:261-273. Ermentrout, B., J. Campbell, and G. Oster (1986) A model for shell patterns based on neural activity. The Veliger, 28: 369-388. Hinchliffe, J., and D. Johnson (1980) The Development o f t h e Vertebrate Limb. The Clarendon Press, Oxford. Levine, M., and J. Shefner (1981) Fundamentals ofSensation and Perception. Addison-Wesley, Reading, MA. Mach, E. (1865) Uber die Wirkung der raumlichen Vertheilung del Lichtreizes auf die Netzhaut, I. Sitzungsberichte der mathematisch-naturwissenschaftlichen.Classe der kaiserlichen Akademie der Wissenschaften, 52303332. Meinhardt, H. (1982) Models of Biological Pattern Formation. Academic Press, New York. Meinhardt, H. (1984) Models for positional signalling, the threefold subdivision of segments and the pigmentation pattern of mollusks. J. Embryol. Exp. Morphol. [Suppl.], 83:289-311. Meinhardt, H., and H. Klingler (1987) A model for pattern formation on the shells of mollusks. J. Theor. Biol., 126:6389. Murray, J. (1979) A pattern formation mechanism and its application t o mammalian coat patterns. In: Vito Volterra Symposium on Mathematical Models in Biology. Lecture Notes in Biomathematics, Vol. 39. C. Barrigozzi, ed. Springer-Verlag, Heidelberg. Murray, J. (1981a) A pre-pattern formation mechanism for animal coat patterns. J. Theor. Biol. 88:161-99. Murray, J. (1981b) On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat patterns. Philos. Trans. R. SOC. Lond. [Biol.], 295:473-496. Murray, J. (1982) Parameter space for Turing instabilities in reaction-diffusion mechanisms: A comparison of models. J. Theor. Biol., 98;143-163.

MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS Murray, J.,and G. Oster (1984a) Generation of biological pattern and form. IMA J. Math. Med. Biol., 1:1-25. Murray, J., and G. Oster (1984b) Cell traction models for generating pattern and form in morphogenesis. J. Math. Biol., 19:265-280. Oster, G., J. Murray, and A. Harris (1983) Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morphol., 78:83-125. Oster, G., J. Murray, and P. Maini (1985) A model €or chondrogenic condensations in the developing limb: The role of extracellular matrix and cell tractions. J. Embryol. Exp. Morphol., 89:93-112. Oster, G., P. Alberch, J. Murray, and N. Shubin (1987) Evolution and morphogenetic rules. The shape of the vertebrate limb in ontogeny and phylogeny. Evolution (in press). Perelson, J., A. Hyman, P. Maini, J. Murray, and G. Oster (1986) Nonlinear pattern selection in a mechanical model for morphogenesis. J. Math. Biol., 24:525-242. Ratliff, F. (1972) Contour and contrast. Sci. Am., 226:90-101. Segel, L. (editor) (1980) Mathematical Models in Molecular and Cellular Biology. Cambridge University Press, Oxford. Sengel, P. (1976) The Morphogenesis ofskin. Cambridge University Press, Oxford. Shubin, N., and P. Alberch (1986) A morphogenetic approach t o the origin and basic organization of‘the tetrapod limb. In: Evolutionary Biology. M. Hecht, B. Wallace, and W. Steere, eds. Academic Press, New York, pp. 181-202. Turing, A. (1952) The chemical basis of morphogenesis. Philos. Trans. Roy. SOC.[Biol.] 237:37-72. Wolpert, L. (1971) Positional information and pattern formation. Curr. Top. Dev. Biol., 6:183-224.

APPENDIX A: COLOR PATTERNS ON MOLLUSK SHELLS: NEURAL AND CHEMICAL MODELS Figure 4 shows two examples of pattern formation by lateral inhibition models. Here we give the model equations used to generate these patterns. Ermentrout et al. (’86)investigated a model for shell patterns based on the notion that the pigment-secreting cells in the mantle were controlled by neural activity. They employed a neural net model of the following sort: (1) P,,l(X) = S[P,(dl - Rt (2) R,+l(X) = YP,(X) + 6R,(x) Here, P , is the amount of pigment secreted during the time period t, and Rt is the amount of a refractory substance produced during secretion. (Other forms of the model that incorporate pigment depletion and memory effects produce similar patterns.) The model parameters y and 6 control the production and metabolism of R . The connectivity of the neural net was assumed t o follow the same pattern as the retinal ganglion cells discussed in “General Properties of Pattern Formation Models,’ above. That is, thev were connected so as to generate a lateral inhibitory field. This was mod-

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eled by the following equations:

Excitation: E,+,(x) = . f r l W ~ (-~ x)P,(x‘)~x’ ’ (3) Inhibition: I t + l ( X ) = J(LWI(X’ - x)P,(x’)dx’ (4) Here the kernals WE(x’ - x) and W1(d - x ) define the connectivity of the mantle neuron population by weighting the effect of neural contacts between cells located at position x’ and a cell at x. In general, the inhibitory kernal, W,(x‘ - XI is broader than the excitatory kernal, WEfx’ - x); i.e., activation has a shorter range than inhibition. 0 is the domain of the mantle; for most shells this is a finite interval, but it may be circular in the case of mollusks such as limpets and planar in cowries. The equations that Meinhardt and Klingler (’87) employ to model the shell patterns assume that within each cell an autocatalytically produced “activator” stimulates pigment secretion, and that cells communicate with their adjacent neighbors by diffusion. The general form of the equations are as follows:

F(a, h), G(a, h) 2 0

(7)

where the kinetic terms F(a, h) and G(a, h) vary according to the model. The particular forms used by Murray (’81a,b)and by Meinhardt and Klingler (’87) can be found in the references cited.

APPENDIX B: A TAXIS-BASED MODEL FOR CHONDROGENESIS The basic idea of the model is that cells aggregate into a prechondrogenic condensation under the stimulus of a tactic factor. This can be either a chemoattractant secreted by each cell, or a haptotactic guidance cue created by cell tractions (cf. Oster et al., ’83;Murray and Oster, ’84a,b). From the point of view of the patterns the model generates, chemotaxis is indistinguishable from haptotaxis. However, a chemotactic model is simpler to analyse because it involves one less equation than a haptotaxis model. Therefore, since both cases are covered by the analysis, we shall couch the model in terms of a chemotactic response, and carry out the analysis in detail.4 *This model is taken from Oster et al. (‘87)

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The model equations Consider a population of cells migrating and secreting a chemoattractant, c. Conservation equations for the cell density, n(x,t) (equation 8) and attractant concentration, c(x,t) (equation 9) can be written as (cf. Segel, '80):

--+ -

_ _- M 2 n at

-

aV.nVc

random motility chemotaxis

= DV2c dt

bn n+h-E

diffusion

secretion decay by cells

dC

lated to the first order attractant degradation kinetics, T = U p , then p* = 1. If we scale the attractant concentration by C = bT = b / p , then b* = 1. Finally, if we choose L to be the = G / p , chemotaxis length scale, L = then a* = 1.With these choices, equations 11and 12 become

n C (14) n + 1 We have thus reduced the parameters from six in equations 8 and 9 to two dimensionless parameters or groupings, D and M: d2C -dc- - I)-+-dt dXL

where M > 0 and a > 0 are the cell motility and chemotactic parameters respectively, and D is the diffusion coefficient of c. The secretion rate is a saturating function, bnl(n + h), with b and h posi- Equation 15 shows that if the quantities a and b tive, and p c is the degradation rate of c . For alge- vary reciprocally, the behavior of the system rebraic simplicity only, we shall only examine the mains unchanged. one dimensional form, so V = dldx. Linear btfurcation analgsis Making the equations nondimensional Now we carry out the linear stability analysis The first step is to render equations 8 and 9 in detail, and determine the dispersion relation, dimensionless. This serves to reduce the paramewhich gives an estimate of the size of the patterns ter count, to scale the equations for the subsein terms of the parameters. quent analysis, and to remove the dependence on First, we linearize the system about the homoany particular choice of units. The role of the digeneous non-zero steady state, which from equamensionless parameter groupings that arise in 13 and 14 is tions this procedure is discussed in "Morphogenetic no Properties of the Model." To do this we introduce no, c, = no +1 a typical time scale, T, length scale, L, and concentration, C, all of which we shall choose later. where no is arbitrary, but c, is not. This introThen we define the following dimensionless quan- duces one more parameter into the model, the tities background cell population no. By writing n* = nlh c* = c1C x* = x1L u=n-n0, v=c-c, t* = tT b* = bT1C p* = p T (10) D* = DTlL2 M" = MTIL2 the linear system corresponding to equations 13 a* = aTC/L2 and 14 becomes With these definitions, equations 8 and 9 become, on dropping the asterisks for algebraic convenience,

We now look for solutions in the form At this stage we can reduce the number of dimensionless parameters by choosing the representative quantities T ,L, and C appropriately to reflect the scales of biological relevance. For example, from (equation 10) if we choose the time scale re-

where p is the dimensionless growth rate, and k is the wave number, which measures the size of patterns (i.e., k = ~ T Awhere , X is the wavelength of the pattern; thus k has dimensions lllength). If

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201

we substitute equation 18 into equation 17 we get given by the following relation between p and k for equation 18 t o be a solution: F(p, k) =

+ p[1 + k2(M + D)]

To be specific, suppose we take no as the slowly varying parameter. Then, as noincreases there is a critical value n, when Mmin = 0. At this value the dispersion curve just touches the p = 0 axis. This is a curve in the (p, k ) plane which deter- For larger values of no wavelength w with maxmines the growth rate p for each wave number k; imum growth rate is determined. From equation alternatively, with k = 2rr/w, it gives the growth 23 it is rate in terms of the wavelength w.The quadratic equation, F(p, k) is called the dispersion relation, so named because of its origins in the physics of wave propagation. The basic requirements for a mechanism to gen- The spacing pattern picked out by this model in erate spatial patterns is that the growth rate terms of the original parameters is given by equashould be positive for a band of wavelengths (cf., tions 24 and 10. The dimensional spacing waveFig. 9). This means that the larger of the two solu- length is I 1 tions of equation 19 must be such that p > 0 for some finite range of k > 0. Since equation 16 is simply a quadratic equation in p, the only way p can be positive is if Q2

I

+ k2

Alternatively, we could have chosen either of the dimensionless motility parameters (equation 15) as the bifurcation parameter. for some k2. By inspection this can clearly always In this model the chemical pattern is laid down be achieved if simuZtuneousZy with the cell aggregations. The autocatalytic step is provided by the increased chemoattractant concentration that accompanies cell aggregation, which opposes the stabilizing inIn terms of the original dimensional variables, fluence of random cell motion (diffusion). The latthis condition is (cf. Fig. 11) eral inhibition is provided by the depletion of cells in the vicinity of an aggregation focus. As mentioned at the beginning of this section, haptotaxis If this holds, the dispersion relation equation 19 would produce comparable patterns, but in place gives p vs. k2 or p vs. w2, as illustrated in Figure 9. of the parameters in (equation 25) there would be Clearly, if the cells' motility or dispersal is mechanical and chemical parameters involving small enough and/or the cell density, no,is large cells and matrix properties. enough, but not too large, the cells can form Cell dwerentiation can be included aggregations. From equation 22 we see the use of in the model these dimensional groupings. For example, the The model can be elaborated to incorporate effect of a small cell motility (M) can also be other variables. Here, we show how cell differachieved by either a large chemoattractant effect, entiation of mesenchymal cells into chondroblasts (a), or by an increased secretion rate, (b). Therecan be included. We assume that, as aggregation fore, a system can be changed from a parameter proceeds, the cell density increases and mesendomain where no pattern can form into a region chymal cells commence to differentiate into chonin which spatial patterns evolve by varying one or droblasts, capable of secreting cartilage. We can more parameters independently. Thus by varying represent this by a density-dependent reaction the number of cells we can arrange for p to be that converts mesenchymal cells (denoted by n) positive for a range of k (and hence 0).From equaN): into chondrocytes (denoted by tion 20 the k 2 which gives the minimum H determines the k with the maximum growth rate; it is n e N (26)

H(k2) = M D k 4

:

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The chondrocytes are not mobile, but continue to where f(n>is a monotonic, saturating function of secrete chemoattractant, although at a lower rate n, with f ( n ) = 0. than the chondroblasts. A subsequent differentia- Equation (8)for n must be modified to include the tion step commits the chondroblasts irreversibly gain and loss of cells by differentiation: to chondrocytes (i.e., cartilage cells), but we need = DV2n - xV-(nVc) + KIN - &f(n) not include this step in the model, for by then the ' 7 1 u at spatial pattern has been laid down. The immobilrandom ,&motaxis dedifferentiation differentiation ity of the chondrocytes is taken t o reflect their motion (28) mutual adhesion. We do not specifically model what triggers the adhesion (presumably by cell Finally, the chemoattractant equation 9 must inadhesion molecules) but simply specify it as a clude secretion by the chondrocytes, and possibly characteristic accompanying increased cell den- by the chondroblasts: sity. Following differentiation, chondrocytes comdC = D,V2c - pc + Szn + SIN mence t o swell and assume a characteristic mordt phology. This swelling is probably osmotic, but for diffusion degradation secretion by secretion by the purposes of this model it simply appears as a chondroblasts chondrocytes decrease in cell density following differentiation. (29) The chondroblasts are immobile, and so their balance equation need not contain any cell motion This model consists of three equations rather terms. Therefore, the conservation equation for N than two, and so its analysis is more difficult. However, the patterns it generates are substanlooks like this: tially the same as in the simpler model of equations 8 and 9 for parameters in the range, which -aN - - -KIN + K z f ( n )- yN(No - N ) (27) at give a dispersion relation similar to that illustrated in Figure 9. dedifferentiation differentiation swelling

*

'---\---J

-

---

-

iy---,

-

-