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 character- istics. In this paper we examine a number of pattern formation models and show how the phenome- non 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; there- fore, 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 ACT1VAT ION 1 INUlBlTlON Fig. 1. The principle of local activation with lateral inhi- bition 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 ex- cited 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 inhibi- tion. How is this accomplished? First let us exam- ine diffusion-reaction models. Chemical prepattern models built around diffu- sion-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 morpho- 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 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 equa- tions 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.