Pattern Formation Models and Developmental Constraints GEORGE F

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 embry- how the chemical prepattern is laid down. This ologists with insight into how embryonic patterns can be modeled in one of two ways. 1) Simple may be laid down. In this chapter we shall illus- chemical gradients are established across tissues, trate the basic pattern-forming principles of these assuming that certain cells act as sources or sinks models using as an example the formation of for the chemicals, which diffuse from cell to cell bones in the vertebrate limb. The physics of the via junctions, or through the intercellular space. pattern forming process imposes constraints on 2) A chemical prepattern can arise by means of the possible cartilage patterns the limb may ex- “diffusion-driven instabilities,” a notion first pro- hibit. These constraints are reflected in the gen- posed by Turing (’52), and subsequently elabo- eral similarity one observes amongst all tretrapod rated and applied by numerous authors to a va- limbs. Indeed, one class of developmental con- riety of embryonic situations (e.g., Meinhardt, straints on limb evolution are a consequence of ’82; Murray, ’81). The mechanical form-shaping these developmental construction rules. We shall events that occur in embryogenesis are not taken illustrate how some of these construction rules into account by such chemical prepattern models. arise in the context of a particular model for limb Furthermore, the identity of the morphogens is morphogenesis. proving quite elusive. Two views of pattern formation have dominated The mechanochemical models take a quite the thinking of embryologists in the past few different approach. Pattern formation and years. The first might be called the chemical pre- morphogenesis are not regarded as separable pattern viewpoint, and the second could be called processes. Rather, chemical and mechanical the mechanochemical interaction viewpoint. They processes are presumed to interact continuously may be roughly characterized as follows. to produce, simultaneously, both the chemical Chemical prepattern models separate the pro- pattern and the form-shaping movements (e.g., cess of pattern formation and morphogenesis into Oster et al., ’83, ’85). Moreover, since these mod- several sequential steps. Embryonic patterns are els are framed in terms of measurable quantities first specified as distributions of chemical (“mor- such as forces and displacements, they focus at- phogens”) concentrations. Subsequently, these tention on the process of morphogenesis itself. chemical patterns are “read out” by the cells, and Despite their quite different assumptions about the appropriate changes in cell shape, differentia- the physical basis for embryonic architecture, the tions, and/or migrations are executed according to mathematical mechanisms that underlie the two the chemical blueprint. The notion of “positional types of models have similar characteristics. In information” (Wolpert, ’71) depends on such a our discussion here we shall focus on these sim- chemical prepattern. In this view, morphogenesis ilarities, rather than on the differences, for it 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 Received October 31, 1988; revision accepted March 3, 19S9. 0 1989 ALAN R. LISS, INC. MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 187 LIGCIT

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Fig. 1. The principle of local activation with lateral inhibition in a neural net. In (a), an 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 Fig. 2. Top: The Hermann grid produces the illusion of shown in (b). 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). turns out that there are common themes that im- Because its inhibitory surround is illuminated more, a cell pose constraints on the possible forms of embry- located at the intersection of two white strips experiences a onic structure. In this chapter we shall outline the greater inhibitory effect than one located in a single white main ideas that underlie models of pattern forma- strip between two black squares. Thus, a cell at the interesec- tion, using as an example a model for chon- tion fires more weakly than its neighbors, and that region appears darker than the surrounding white regions. drogenesis 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 be- the phenomena. cause of its proximity to unstimulated cells in the GENERAL PROPERTIES OF PATTERN dark region. Therefore, it will fire more strongly FORMATION MODELS than cells away from the boundary, and so con- tribute to the apparently bright Mach band. Simi- Pattern 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 Fig- known 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 illumina- sion 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. is, an excited neuron inhibits the firing of its This simple phenomenon of local excitation 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- 188 G.F. OSTER AND J.D. MURRAY 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 deforma- tions that obey Hooke’s law of elasticity. In order for equation (1) to generate spatial patterns, the spatial and temporal properties of the system z must conspire to 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 in- Fig. 3. An array of cells with lateral inhibitory fields will teraction term is simply Fickian diffusion. These produce bands of excitation. This pattern was produced by the models differ essentially only in the choice of neural shell model described in Appendix A.l. chemical kinetics between the reacting morphogens 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 to 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 reac- ing 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 re- inhibition 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 gener- from 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 fundamen- chemical 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 den- are 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 ac- cells. The “local dynamics” term accounts for the count for the mechanical forces between cells and Fig. 4. The left panel shows a divaricate line pattern gen- about the underlying mechanism, both models implement lo- erated by the neural secretion model of Ermentrout, et al. cal excitation coupled with lateral inhibition, and so they ('86) (see Appendix A.l). The middle pattern shows the same produce essentially the same patterns, The right panel shows pattern generated by a diffusion-reaction model (Meinhardt, a typical pattern of this sort on the mollusk Nerita turrita '84; Meinhardt and Klingler, '87) (see Appendix A.2). Al- (Ermentrout et al., '86). though they are based on dramatically different assumptions 190 G.F. OSTER AND J.D. MURRAY ACTlVATl ON INHI6I TION into a complex form due to bifurcations and epi- genetic feedback. We shall give an 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 to 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 FAST FAST DIFFUSION DIFFUSION region, and spreads laterally in a wave until the global pattern is established. As we shall discuss Fig. 5. A chemical reaction system can generate lateral below, most biological patterns originate locally, inhibition. An “activator” molecule [O],when it collides with in some “organizing tissue” and develop sequen- a substrate, releases several molecules of its own type (auto- catalysis), as well as inhibitory molecules [.]. The inhibitors tially thereafter. That is, wavelike pattern forma- can combine with the activators and prevent their subsequent tion is the rule in development. This might have reaction. If the inhibitor can diffuse more rapidly than the been anticipated from the study of pattern forma- activator, it will spread faster and can arrest the autocata- tion models for the following reasons. lytic reaction before it spreads too far. The equations for this scheme are given in Appendix A.2. Models in which patterns grow simultaneously over an entire field tend to 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 ul- the extracellular matrix material, as well as the timate steady-state patterns, although qualita- concentrations of regulatory chemical substances tively similar, are determined by the initial con- such as calcium (cf. Oster et al., ’83). ditions. In developing systems there are always Both of these types of model require some mech- inherent stochastic effects. This fact was ex- anism to prompt the cells to commence their mor- ploited by Murray (’79, ’81a,b) in a reaction- phogenetic activities. The chemical prepattern diffusion model for the patterns of animal coat models assume that the cells read the completed markings. Since the initial conditions for an indi- morphogen concentration profile, then execute vidual animal are unique, this implies that the their morphogenetic movements accordingly. ultimate coat pattern for each animal is unique. Thus a complex prepattern is required to generate Figure 6 shows some simulations of the reaction- a correspondingly complex form. The mechano- diffusion model for progressively larger domains. chemical models also require some prepattern, The distribution of light and dark patches de- but in contrast to the diffusion-reaction models it pends on the random starting conditions. While need only be a very simple one. A simple gradient the general scale and type of pattern varies with in cell type will do: for example, a cell lineage the starting conditions, the overall similarity of and/or cell aging mechanism could trigger a local the patterns is maintained. increase in cell traction, thus initiating the mor- This example illustrates one type of develop- phogenetic process. This process will then unfold mental constraint. The role of scale and geometry MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 191 a * b

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 progres- Fig. 7. (a)Spatial pattern generated by a reaction-diffu- sively larger. (b) represents the first bifurcation from a uni- sion mechanism for a tapering domain,t mimicking the inte- form color; successive panels show how the pattern becomes gument of the tail in a developing embryo (Murray, '81a,b). increasingly complex as the domain size increases. Note, how- Note how the spot-like pattern is forced into a stripe pattern ever, that the pattern effectively disappears for very large as the tail gets thinner. Also shown on the left (b)is the tail of domains. These results suggest that most small animals (a) a cheetah (Acinonyxjubatis), and on the right (c) the tail of a and most large animals (f) tend to be uniform in color-a jaguar (Panthem oncu),which illustrates the developmental common feature of mammalian coat patterns. constraint described in the text. are important parameters in determining the ul- However, the conclusions we shall draw about timate pattern in any lateral inhibition mecha- constraints on cell aggregation patterns are quite nism. In situations where the domain size varies, general, for it turns out that a large class of pat- such as the tail integument shown in Figure 7, tern formation models exhibit properties similar the thicker proximal part of the tail might be to this simple one. large enough to sustain a spot pattern, while the thinner distal part may only sustain stripes. Fig- Motile cells aggregate by directed migration ure 7a shows a numerical simulation of a reac- The beginning stages of organogenesis are fre- tion-diffusion mechanism that demonstrates this quently heralded by patterned aggregations of phenomenon. The model suggests a developmen- motile cells. These aggregations then differenti- tal constraint wherein a spotted animal can have ate, undergoing further morphogenetic transfor- a striped tail, but a striped animal cannot have a mations as the mature organ takes shape. How- spotted tail. Figure 7b,c shows that cheetahs and ever, the initial anlage determines the general jaguars do indeed conform to this constraint. geometry and size of the organ. There are several mechanisms that can produce such aggregations A MODEL FOR CHONDROGENIC of motile cells': CONDENSATIONS The general considerations discussed above are Chemotais: Cells may move toward the source easier to understand if we have a concrete exam- of a chemical attractant. For example, during the ple. In this section we outline a specific model that aggregation phase of the slime mold Dictyos- can generate the pattern of cartilage condensa- telium discoideum, motile cells move toward a tions in the vertebrate limb bud. We do not pre-

~ sume that this model is necessarily the correct 'Cell aggregations can also arise in a stationary cell population by one; our purpose is rather to illustrate the prin- localized differential cell division, However, it has been demonstrated that chondrogenic condensations do not involve extensive cell mitosis ciples that underlie pattern formation models. (Hinchliffe and Johnson, '80). 192 G.F. OSTER AND J.D. MURRAY specialized epidermal region called the “apical epidermal ridge.” Cells emerging from the progress zone are, for a time, not “competent” to aggregate. This could result from the action of an inhibitory chemical secreted by the apical epidermal ridge region, or it could be simply a conse- AER quence of cell maturation. At a certain distance 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 com- petence. 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, then aggregate into chondrogenic condensations (shown either chemotactic or haptotactic. Here we will cross-hatched). Cartilage condensations form by recruiting treat the case of chemotactic aggregation because cells from the surrounding tissue (shown dotted). it is simpler; however, the haptotactic mechanism 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 to 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 mi- toward 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 corre- are 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 of =-1+-1 itself being deformed by the mechanical action of (2) other cells, or by differences in osmotic swelling. ~~ cell densitv ~ I- 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. However, models for each mechanism have been (3) proposed, and, surprisingly, the patterns they predict are much the same. The reasons for this How the model generates spatial patterns will become clear later on; for now we shall pro- By 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 dis- assumption 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 den- cals that they secrete themselves. sity; then this region broadcasts more chemoattractant than neighboring regions, and so recruits A chemotactic model for 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 re- The 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 MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 193

Fig. 9. In the beginning stages of pattern formation, the tive growth rates increase in amplitude. In simple situations, cell density can be viewed as the summation of sine and the pattern with the maximum initial growth rate will even- cosine wave functions of different sizes (i.e., wavelengths). tually dominate. However, this is not always true: sometimes The dispersion relation gives the initial growth rate of the patterns that commence growing slowly eventually overtake different harmonics as a function of their size. Those harmon- and surpass patterns whose early growth rate was faster. ics with negative growth rates die out, while those with posi- larger. Counterbalancing the autocatalytic aggre- field becomes unstable and breaks up into a spa- gation is the effect of random cell motions (diffu- tial pattern, or an existing pattern becomes un- sion), which tends to smooth out inhomogeneities. stable and bifurcates into another spatial pattern. If the aggregation process predominates, cell re- Note that the term “bifurcation” has both a collo- cruitment depletes the immediate neighborhood quial meaning (to branch, or split) and a technical of cells. This creates an effective region of inhibi- meaning, which refers to the spatial instability of tion around each aggregation center, producing a a solution to the model equations (cf. Appendix characteristic spacing of the centers. B). Unfortunately, from such a verbal description it Determining the threshold values and the is not clear that the model will produce anything range of the model parameters tha lead to spatial more than random aggregations of cells (indeed, patterns can be a difficult task, which is usually Dictyostelium aggregations are not very regularly accomplished by a combination of mathematical spaced). To appreciate the fact that these pro- analysis and computer simulation (Murray, ’82). cesses can give rise to a regular spatial pattern However, a good predictor as to whether a model one must do a mathematical analysis; a mere ver- can generate spatial patterns is the “dispersion bal description of a process cannot delineate phe- relation” for the model. This is a plot of the nomena that depend on a quantitative balance of growth rates of different patterns as a function of competing effects. Appendix B presents this anal- the scale of the pattern, as shown in Figure 9. ysis in detail; here we shall only attempt to pro- Such plots are central to understanding how spa- vide an intuitive appreciation of the operation. tial patterns emerge in morphogenetic models, and Appendix B gives a quantitative example of MORPHOGENETIC PROPERTIES this important concept. OF THE MODEL In order to determine which parameter is cru- cial, it is frequently convenient to view a develop- Spatial patterns arise when a developmental mental process as occurring on two different time parameter exceeds a threshold value scales. Chemical prepattern models assume that Nonlinear models develop several different the cells remain essentially stationary while the kinds of spatial nonuniformities. This happens morphogen is secreted, for otherwise the convec- when one of the model parameters exceeds a tive effect of the cells’ motions would disrupt the threshold value, whereupon an initially uniform chemical pattern. Conversely, mechanochemical 194 G.F. OSTER AND J.D. MURRAY models assume that, while the cells are continu- largely migrates in a proximo-distal direction ously differentiating, on the time scale of mor- (Hinchliffe and Johnson, ’80). In vertebrates, phogenesis (i.e., cell movement), their state of dif- there is an apparent exception: the formation of ferentiation is almost constant. Thus, in each type the digital arch proceeds in an anterior-posterior of model some parameters are varying slowly in direction as well as proximo-distally (Shubin and comparison with the evolution of the spatial pat- Alberch, ’86). However, it is important to note terns. that the onset of differentiation of the digital arch In simplified models for a developing system is correlated with the sudden broadening and flat- one variable may be selected as the slowly vary- tening of the distal region of the limb bud into a ing developmental parameter (or “bifurcation pa- “paddle.” We shall see below that this change in rameter”): that which defines the property that geometry is the key to understanding this appar- slowly progresses until the uniform field becomes ent exception to the sequential development rule. unstable. This instability arises because, when the developmental parameter exceeds a threshold Cell recruitment zones define morphogenetic value, the system is no longer in mechanical or fields chemical equilibrium, and it evolves to a new- An important feature of models that explicitly spatially non-uniform-configuration. For ex- account for cell density is the creation of zones of ample, in the simple chemotaxis model sketched recruitment around the chondrogenic foci. That above it is clear that no pattern will arise if the is, an aggregation center autocatalytically en- cells’ chemotactic response is not strong enough to hances itself while depleting the surrounding tis- overcome the dispersive influence of their random sue of mesenchymal cells, thus setting up an motions. Only when the parameter stimulating effective lateral inhibition against further aggre- tactic motion is sufficiently large can a non- gation. Moreover, adjacent foci compete for cells, uniform distribution of cells come about. The pa- producing a nearly cell-free region between them. rameter space corresponding to the chemotactic Thus, a condensation establishes a “zone of influ- model is shown in Figure 11. This figure also ence,” which precludes formation of other foci; illustrates another general feature of morpho- this will play an important role in our subsequent genetic models: usually no single parameter de- discussion of branching. termines the bifurcation point. Rather, certain dimensionless groupings of parameters determine Tissue geometry is important in controlling the model’s behavior. We will return to this im- patterns portant point later. All models for chondrogenic condensations are strongly affected by the size and shape of the Embryonic patterns are generally laid down growing limb bud.2 Tissue geometry controls the sequentiallp expression of certain model parameters whose in- An important feature of morphogenetic pat- fluence on the growth of patterns is often decisive. terns is that they are usually laid down in a Bifurcations in a model are detected by investi- sequential fashion, rather than simultaneously gating when an integer number of sine or cosine over an entire tissue. Frequently, a pattern ap- waves can “fit”into a domain of a certain size (cf. pears to spread outward from an initiating region, Appendix B). The reason for this is that, at the or organizing center. A good example of this is the very onset of an instability, the solutions to the formation of feather germ in birds and scale ar- model equations look like sine or cosine waves. In rays in reptiles (Sengel, ’76; Davidson, 1983a,b; order to satisfy the conditions at the boundary of Oster et al., ’83). There the hexagonal patterns of the domain (e.g., there are fixed values at the primordia form first in a single row, then spread boundary, or it is impermeable to cells and chemi- laterally to form the final hexagonal array (Perel- cals), one must impose the condition that only an son et al., ’86). As we mentioned earlier, analyses integral number of wavelengths fit in the domain. of both diffusion-reaction and mechanochemical This determines the size of the patterne3 models for this process show that it is much easier to create reproducible and stable patterns if they ‘Indeed, Alberch and Gale (’83) have shown that the pattern of cartilage condensation in real limb buds is dramatically affected by are generated sequentially, rather than arising reductions in size. simultaneously over the entire field. 3Strictly speaking, the wavelength-fitting procedure reliably pre- This sequential aspect of development is clearly dicts the final pattern only in the one-dimensional case. In two and three dimensions, ambiguities creep in, and more sophisticated evident in the vertebrate limb, where the pattern methods, or computer simulation, must be employed. MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 195 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:

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 (B> bud. Thus, according to (equation 41, a decrease in the secretion rate, b, may be compensated, for ex- a, Fig. 10. The size of the domain must be large enough to ample, by an increased chemotactic sensitivity, “fit” an integral number of waves. In (A) the limb cross- by a decreased rate of degradation, F, or by a de- section is large enough to accommodate a single wavelength, crease in the domain size, L. These scaling ratios, corresponding to a single peak in cell density. If the domain which relate the system parameters and the do- size and/or shape changes sufficiently, so that two complete main size and shape, arise from the necessity of harmonics can fit into one dimension of the limb cross-section, rendering the model equations dimensionless, then the distribution in (A) will become unstable and evolve to the pattern shown in (B). 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 For example, suppose we have a situation such ratios from a given set of parameters. However, as in Figure 10A, which shows a limb cross- the bifurcation behavior that they control is an section in a proximal region. If the size of the intrinsic property of the underlying physics and domain were too small, a full sine wave could not chemistry of the system. “fit” into the domain, and so growth of any patterns would be suppressed. If, however, the do- There are mang routes through parameter main were to grow, when it reached a critical size, space that lead to a spatial pattern the uniform state would break up, and a single From the above discussion it is clear that vary- mode would commence to grow (i-e.,a single peak ing the domain size is but one way to trigger in cell density). In Figure 10A a spatial pattern bifurcations. Indeed, this lack of uniqueness of with a single peak-corresponding to sin(m/L)- the bifurcation parameter has important implica- has commenced to grow. The critical domain size tions for the interpretation of experiments, for it that causes the uniform state to become unstable means that there are many developmental paths is the “bifurcation” size, since at this value the that lead to the same bifurcation. For example, model equations switch from one solution (uni- suppose the parameter values are such that the form) to a new solution (one peak in cell density). system is at the point P, where no structures can If the domain geometry is changed so that two develop, as shown in Figure 11. if the cell diffu- complete sine waves can fit into the horizontal sional motility M decreases, the point in parame- dimension (cf. Fig. lOB), then the solution in ter space moves toward C. On crossing the bifur- 10(A) with a single peak becomes unstable, and cation curve the mechanism develops spatial the system evolves to that shown in 10(B) corre- structures. However, the system can arrive at the sponding to two peaks in cell density. same point C-and develop the same spatial structures-if the maximum rate of attractant se- Developmental parameters occur in cretion, b, increases appropriately or if the hap- dimensionless groupings totactic parameter, a, increases. In fact, the sys- In 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. 196 G.F. OSTER AND J.D. MURRAY

Bifurcation Curve t 1

Fig. 11. Bifurcation diagram for the chemotactic model, tial patterns can develop. However, if combinations of param- showing how different paths through developmental parame- eters change moving the system along path P + A or P - C, ter space can lead to spatial structures. The axes are dimen- then the uniform state becomes unstable, and a spatial pat- sionless ratios consisting of the following variables. M is the tern evolves. Interestingly, in this model, the spatial struc- random motility coefficient of the chondroblasts, n is the cell ture can disappear again if the parameter path follows P + A density, and h is a measure of the chemoattractant secretion + B. For example, a monotone decrease in cell density, n, first rate at low cell densities. p is the degradation rate of che- triggers pattern formation, but at very low densities the pat- moattractant, and b is the maximum secretion rate. When the tern disappears again. system is at point P, the uniform state is stable, and no spa-

The bifurcation space also highlights other as- rion for judging the validity of a model. For ex- pects of the modeling process. For example, sup- ample, Murray ('82 has shown that pattern for- pose the original cell density, no, decreases. The mation by some diffusion-reaction mechanisms path from P is toward A and patterns develop. If (e.g., the one employed in the shell model de- the cell density falls too low, the system moves scribed in Appendix A.2) is quite delicate: the along path A + B, whereupon the system bifur- range of parameters that correspond to spatial cates again, this time from a patterning state to patterns is very small. It is not likely that evolu- one in which no structure develops. Thus, a mono- tion would have stumbled onto such mechanisms, tonically changing cell parameter can lead the in comparison with others whose range of allow- system first through a pattern-forming episode, able parameters is much wider. followed by the disappearance of the pattern. Of particular importance is the fact that different models do make different predictions as to the Patterns do not distinguish between the types outcome of certain experimental interventions. of models These predictions are contained in the different Models that simulate different developmental dimensionless groupings of variables that govern mechanisms cannot be distinguished solely on the the operation of each type of model. For exam- basis of the predicted pattern. Both morphogen ple, interventions that affect cell tractions will and mechanochemical models predict similar disrupt chondrogenesis according to a mechano- types of bifurcations. Indeed, Figure 4 illustrates chemical model, but will not affect the patterns that both neural and diffusion-reaction models predicted by a morphogen model. Or, in the case can produce the same kinds of spatial patterns of the shell patterns, a neural model would sug- (since both are implementations of lateral inhibi- gest different experimental interventions than tion). Thus, it is generally not possible to deduce would the diffusion-reaction model. the mechanism underlying a spatial pattern from the pattern alone, since many mechanisms can CONSTRAINTS ON MORPHOGENETIC generate the same pattern. However, the sensitiv- PATTERNS ity of patterns to variations in the parameters is The physical and chemical processes underly- dependent on the model employed. Therefore, the ing morphogenetic models impose certain restric- robustness of the predicted patterns is one crite- tions on the geometry of the resulting pattern. MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 197

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

Fig. 13. An example of a branching diagram showing how That is, not everything is possible within the con- the limb of a salamander (Ambystoma mexicanurn) can be straints of mechanical and chemical interactions. built up from sequences of F, S, and B bifurcations. Note that this is an adult limb; frequently, the original condensation In this section we outline some of these restric- pattern may be obscured by subsequent growth and differ- tions on form (see also Alberch, this volume). entiation. There are three types of bifurcations We are now in a position to assert our cen- Segmental bifurcations can occur when the tral theoretical point: regardless of the underly- length of the domain exceeds a critical value. ing mechanism-chemical or mechanical-both That is, if a condensation field grows too long, its types predict that, in practice, the aggregation extremities may be able to establish independent patterns are limited to three types, illustrated in recruitment domains and divide the field into sub- Figure 12: domains. In terms of the scenario developed above 1. Focal condensations, which we denote by F, (Fig. lo), there is enough room to fit in another arise as isolated foci in a uniform field, provided wavelength. there is sufficient tissue volume and cell density. Branching bifurcations can occur when the do- 2. Branching bifurcations, denoted by B, main size broadens so that the existing condensa- wherein an existing condensation branches into a tion is too large, and its borders become unstable. Y-shaped configuration. Each region has the potential to set up its own 3. Segmental bifurcations, denoted by S,where- recruitment domain, and so the existing conden- in a condensation either buds off a posterior ele- sation branches in two. This is like segmental ment, or an existing element subdivides itself bifurcation, except that it takes place trans- longitudinally (i.e., proximo-distally) into two versely rather than longitudinally in the limb subsegments. bud domain. The explanation for this restricted list of pat- That these types of condensations are the most tern possibilities is given in Oster et al. (’83) (see likely patterns arises from the nature of the local also, Oster et al., ’87). Heuristically, we have seen activationilateral inhibition mechanism and the that in order for a pattern-uniform or nonuni- sequential nature of limb chondrogenesis. The form-to become unstable and give way to a new size dependence of the condensation domain (a pattern, one of the “slowly varying” model param- type F bifurcation) explains the phenomenon of eters must pass through a bifurcation threshold. digital arch formation in urodeles: an indepen- At that point, the balance between local activa- dent focal condensation can form providing the tion (Lea,the autocatalytic aggregation of chon- distal “paddle” is large enough so that two re- drocytes) and lateral inhibition (i.e., the depletion cruitment centers do not inhibit one another. of cells between aggregation centers) is upset. As a consequence of these restrictions on the Thus, focal condensations arise in situations possible kinds of cartilage condensation, we can when the balance between activation and lateral describe any vertebrate limb as a sequence of F, S, inhibition is such that the uniform field becomes or B bifurcations (cf. Fig. 13). Thus, Shubin and unstable. This requires, amongst other things, Alberch, (’86) are able to construct generalized that the condensation domain be sufficiently “branching diagrams” that describe all known large, so that the focus is effectively isolated from amphibian limb morphologies (see also Alberch, other competing foci. this volume; Oster et al., ’87). These restrictions 198 G.F. OSTER AND J.D. MURRAY on form constitute a “developmental constraint,” rameters,” which measure the aggregate effect of which can only be violated by gross alterations in competing influences. Therefore, it is usually not the geometry and/or cellular properties of the de- easy to isolate a single “cause” underlying a mor- veloping limb bud. phogenetic phenomenon: spatial patterns emerge as a consortium of a number of competing effects Certain patterns of chondrogenesis conspire to produce the phenomenon of local auto- are unlikelg catalysis and lateral inhibition. From the above discussion we can see that the- ACKNOWLEDGMENTS oretical models predict a restricted menu of possible chondrogenic patterns. For example, it is Most of the ideas contained in this paper were quite unlikely-although not impossible-to ob- formulated in collaboration with Pere Alberch. tain a “trifurcation,” i.e., a branching of one ele- Greg Kovacs did the simulations shown in Figure ment into three (or more) elements. Even though 3. G.F.O. was supported by NSF grant MCS subsequent growth may make it appear as if three 8110557. elements arose simultaneously at a common LITERATURE CITED branching point, the theory suggests that all Alberch, P. (1987) Orderly monsters: Evidence for internal branches are initially binary. Moreover, in the constraint in development and evolution. To appear. absence of any biasing factors, condensation pat- Alberch, P., and E. Gale (1983) Size dependency during the terns are symmetric and uniform: a condensation development of the amphibian foot. Colchicine induced digi- domain is broken into equally spaced subdomains tal loss and reduction. J. Embryol. Exp. Morphol., 76:177- whose number and spacing are determined by the 197. geometry of the region. Davidson, D. (1983a) The mechanism of feather pattern development in the chick I: The time determination of feather The presence of asymmetries in the limb bud, position. J Embrol. Exp. Morphol., 74:245-259. such as the anterior-posterior axis, reflects the Davidson, D. (1983b) The mechanism of feather pattern de- presence of asymmetrically situated influences velopment in the chick 11: Control of the sequence of pattern such as the “zone of polarizing activity.” This is a formation. J. Embryol. Exp. Morphol., 74:261-273. Ermentrout, B., J. Campbell, and G. Oster (1986) A model for differentiated region located near the distal poste- shell patterns based on neural activity. The Veliger, 28: rior margin of the limb bud, which appears to in- 369-388. duce an anterior-posterior gradient in the sizes of Hinchliffe, J., and D. Johnson (1980) The Development ofthe the cartilage condensations. The “airplane wing”- Vertebrate Limb. The Clarendon Press, Oxford. shaped cross-section of the growing limb bud en- Levine, M., and J. Shefner (1981) Fundamentals ofSensation sures that the central digits must be larger than and Perception. Addison-Wesley, Reading, MA. Mach, E. (1865) Uber die Wirkung der raumlichen Vert- the marginal ones, while the simple anterior- heilung del Lichtreizes auf die Netzhaut, I. Sitzungsbe- posterior gradient imposed by the zone of polariz- richte der mathematisch-naturwissenschaftlichen.Classe ing activity admits the possibility of an anterior- der kaiserlichen Akademie der Wissenschaften, 52303- posterior gradation in digit size. Indeed, only 332. rarely are limbs found with central digits smaller Meinhardt, H. (1982) Models of Biological Pattern Formation. Academic Press, New York. than the marginal ones. Meinhardt, H. (1984) Models for positional signalling, the threefold subdivision of segments and the pigmentation DISCUSSION pattern of mollusks. J. Embryol. Exp. Morphol. [Suppl.], Using a simple model for chondrogenesis we 83:289-311. have tried to show how spatial aggregation pat- Meinhardt, H., and H. Klingler (1987) A model for pattern formation on the shells of mollusks. J. Theor. Biol., 126:63- terns arise from a uniform population of mesen- 89. chymal cells. The common feature of most mor- Murray, J. (1979) A pattern formation mechanism and its phogenetic models is that they mimic the neural application to mammalian coat patterns. In: Vito Volterra phenomenon of local activation with lateral inhi- Symposium on Mathematical Models in Biology. Lecture bition. While this property confers the capacity to Notes in Biomathematics, Vol. 39. C. Barrigozzi, ed. Sprin- ger-Verlag, Heidelberg. generate spatial patterns, the variety of patterns Murray, J. (1981a) A pre-pattern formation mechanism for is constrained by the geometry of the domain and animal coat patterns. J. Theor. Biol. 88:161-99. the gradients in the model parameters. This is Murray, J. (1981b) On pattern formation mechanisms for because centers of aggregation must compete lepidopteran wing patterns and mammalian coat patterns. with adjacent centers to recruit cells from the sur- Philos. Trans. R. SOC.Lond. [Biol.], 295:473-496. Murray, J. (1982) Parameter space for Turing instabilities in rounding tissue. The cellular properties and the reaction-diffusion mechanisms: A comparison of models. J. tissue geometry combine in “dimensionless pa- Theor. Biol., 98;143-163. MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 199 Murray, J.,and G. Oster (1984a) Generation of biological pat- eled by the following equations: tern and form. IMA J. Math. Med. Biol., 1:1-25. Murray, J., and G. Oster (1984b) Cell traction models for Excitation: generating pattern and form in morphogenesis. J. Math. E,+,(x) = .frlW~(~’- x)P,(x‘)~x’ (3) Biol., 19:265-280. Oster, G., J. Murray, and A. Harris (1983) Mechanical as- Inhibition: pects of mesenchymal morphogenesis. J. Embryol. Exp. It+l(X) = J(LWI(X’ - x)P,(x’)dx’ (4) Morphol., 78:83-125. Oster, G., J. Murray, and P. Maini (1985) A model €or chon- Here the kernals WE(x’ - x) and W1(d - x) de- drogenic condensations in the developing limb: The role of fine the connectivity of the mantle neuron popu- extracellular matrix and cell tractions. J. Embryol. Exp. lation by weighting the effect of neural contacts Morphol., 89:93-112. between cells located at position x’ and a cell at x. Oster, G., P. Alberch, J. Murray, and N. Shubin (1987) Evolu- In general, the inhibitory kernal, W,(x‘ - XI is tion and morphogenetic rules. The shape of the vertebrate limb in ontogeny and phylogeny. Evolution (in press). broader than the excitatory kernal, WEfx’ - x); Perelson, J., A. Hyman, P. Maini, J. Murray, and G. Oster i.e., activation has a shorter range than inhibi- (1986) Nonlinear pattern selection in a mechanical model tion. 0 is the domain of the mantle; for most for morphogenesis. J. Math. Biol., 24:525-242. shells this is a finite interval, but it may be circu- Ratliff, F. (1972) Contour and contrast. Sci. Am., 226:90-101. lar in the case of mollusks such as limpets and Segel, L. (editor) (1980) Mathematical Models in Molecular and Cellular Biology. Cambridge University Press, Oxford. planar in cowries. Sengel, P. (1976) The Morphogenesis ofskin. Cambridge Uni- The equations that Meinhardt and Klingler versity Press, Oxford. (’87) employ to model the shell patterns assume Shubin, N., and P. Alberch (1986) A morphogenetic approach that within each cell an autocatalytically pro- to the origin and basic organization of‘the tetrapod limb. In: duced “activator” stimulates pigment secretion, Evolutionary Biology. M. Hecht, B. Wallace, and W. Steere, eds. Academic Press, New York, pp. 181-202. and that cells communicate with their adjacent Turing, A. (1952) The chemical basis of morphogenesis. neighbors by diffusion. The general form of the Philos. Trans. Roy. SOC.[Biol.] 237:37-72. equations are as follows: 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 forma- F(a, h), G(a, h) 2 0 (7) tion by lateral inhibition models. Here we give the model equations used to generate these pat- where the kinetic terms F(a, h) and G(a, h) vary terns. according to the model. The particular forms used Ermentrout et al. (’86)investigated a model for by Murray (’81a,b) and by Meinhardt and Kling- shell patterns based on the notion that the pig- ler (’87) can be found in the references cited. ment-secreting cells in the mantle were con- APPENDIX B: A TAXIS-BASED MODEL trolled by neural activity. They employed a neu- FOR CHONDROGENESIS ral net model of the following sort: The basic idea of the model is that cells aggre- P,,l(X) = S[P,(dl - Rt (1) gate into a prechondrogenic condensation under R,+l(X) = YP,(X) + 6R,(x) (2) the stimulus of a tactic factor. This can be either a chemoattractant secreted by each cell, or a hap- Here, P, is the amount of pigment secreted during totactic guidance cue created by cell tractions (cf. the time period t, and Rt is the amount of a refrac- Oster et al., ’83; Murray and Oster, ’84a,b). From tory substance produced during secretion. (Other the point of view of the patterns the model gener- forms of the model that incorporate pigment de- ates, chemotaxis is indistinguishable from hap- pletion and memory effects produce similar pat- totaxis. However, a chemotactic model is simpler terns.) The model parameters y and 6 control the to analyse because it involves one less equation production and metabolism of R. The connectivity than a haptotaxis model. Therefore, since both of the neural net was assumed to follow the same cases are covered by the analysis, we shall couch pattern as the retinal ganglion cells discussed in the model in terms of a chemotactic response, and “General Properties of Pattern Formation Mod- carry out the analysis in detail.4 els,’ above. That is, thev were connected so as to generate a lateral inhibitory field. This was mod- *This model is taken from Oster et al. (‘87) 200 G.F. OSTER AND J.D. MURRAY The model equations lated to the first order attractant degradation ki- Consider a population of cells migrating and se- netics, T = Up, then p* = 1. If we scale the creting a chemoattractant, c. Conservation equa- attractant concentration by C = bT = b/p, then tions for the cell density, n(x,t) (equation 8) and b* = 1. Finally, if we choose L to be the attractant concentration, c(x,t) (equation 9) can chemotaxis length scale, L = = G/p, be written as (cf. Segel, '80): then a* = 1.With these choices, equations 11 and 12 become __- M2n - aV.nVc at - - random motility chemotaxis d2C n --dc - C dC bn I)-+-- (14) - = DV2c - dt dXL n + 1 dt + n+h-E - - We have thus reduced the parameters from six in diffusion secretion decay equations 8 and 9 to two dimensionless parame- by cells ters or groupings, D and M: 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 pc is the degradation rate of c. For alge- vary reciprocally, the behavior of the system re- braic simplicity only, we shall only examine the mains unchanged. one dimensional form, so V = dldx. Making the equations nondimensional Linear btfurcation analgsis The first step is to render equations 8 and 9 Now we carry out the linear stability analysis dimensionless. This serves to reduce the parame- in detail, and determine the dispersion relation, ter count, to scale the equations for the subse- which gives an estimate of the size of the patterns quent analysis, and to remove the dependence on in terms of the parameters. any particular choice of units. The role of the di- First, we linearize the system about the homo- mensionless parameter groupings that arise in geneous non-zero steady state, which from equa- this procedure is discussed in "Morphogenetic tions 13 and 14 is Properties of the Model." To do this we introduce no, c, = -no a typical time scale, T, length scale, L, and con- no + 1 centration, C, all of which we shall choose later. where no is arbitrary, but c, is not. This intro- Then 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* = pT (10) a* = aTC/L2 D* = DTlL2 M" = MTIL2 the linear system corresponding to equations 13 and 14 becomes With these definitions, equations 8 and 9 become, on dropping the asterisks for algebraic conve- nience,

We now look for solutions in the form

At this stage we can reduce the number of dimensionless parameters by choosing the representa- where p is the dimensionless growth rate, and k is tive quantities T,L, and C appropriately to reflect the wave number, which measures the size of pat- the scales of biological relevance. For example, terns (i.e., k = ~TA,where X is the wavelength of from (equation 10) if we choose the time scale re- the pattern; thus k has dimensions lllength). If MORPHOGENETIC MODELS AND EMBRYONIC PATTERNS 201 we substitute equation 18 into equation 17 we get given by the following relation between p and k for equation 18 to be a solution:

F(p, k) = Q2 + p[1 + k2(M + D)] To be specific, suppose we take no as the slowly varying parameter. Then, as no increases 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 max- mines the growth rate p for each wave number k; imum growth rate is determined. From equation alternatively, with = 2rr/w, it gives the growth 23 it is k I 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 equa- should be positive for a band of wavelengths (cf., tions 24 and 10. The dimensional spacing wave- Fig. 9). This means that the larger of the two solu- length is tions of equation 19 must be such that p > 0 for I 1 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 Alternatively, we could have chosen either of the H(k2) = MDk4 + k2 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 in- In terms of the original dimensional variables, fluence of random cell motion (diffusion). The lat- this 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 of the parameters in (equation 25) there would be gives p vs. k2 or p vs. w2, as illustrated in Figure 9. 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 effect of a small cell motility (M) can also be The model can be elaborated to incorporate achieved by either a large chemoattractant effect, other variables. Here, we show how cell differentiation of mesenchymal cells into chondroblasts (a),or by an increased secretion rate, (b). There- fore, a system can be changed from a parameter can be included. We assume that, as aggregation domain where no pattern can form into a region proceeds, the cell density increases and mesen- in which spatial patterns evolve by varying one or chymal cells commence to differentiate into chon- more parameters independently. Thus by varying droblasts, capable of secreting cartilage. We can the number of cells we can arrange for p to be represent this by a density-dependent reaction that converts mesenchymal cells (denoted by n) positive for a range of k (and hence 0).From equation 20 the k2 which gives the minimum H deter- into chondrocytes (denoted by N): mines the k :with the maximum growth rate; it is neN (26) 202 G.F. OSTER AND J.D. MURRAY 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 not include this step in the model, for by then the * = DV2n - xV-(nVc) + KIN - &f(n) spatial pattern has been laid down. The immobil- at '---\---J - '71 u ity of the chondrocytes is taken to reflect their random ,&motaxis dedifferentiation differentiation motion mutual adhesion. We do not specifically model (28) what triggers the adhesion (presumably by cell Finally, the chemoattractant equation 9 must in- adhesion 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 com- dC - = D,V2c - pc Szn SIN mence to swell and assume a characteristic mor- + + dt - iy---, - - 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. The chondroblasts are immobile, and so their bal- (29) ance 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. looks like this: However, the patterns it generates are substan- tially the same as in the simpler model of equa- --aN - -KIN + Kzf(n)- yN(No - N) (27) tions 8 and 9 for parameters in the range, which at --- give a dispersion relation similar to that illus- dedifferentiation differentiation swelling trated in Figure 9.