Modeling and experiment in developmental biology Edward C. Cox P r i n c e t o n University, Princeton, N e w Jersey, USA Models in developmental biology continue to yield valuable insights, yet do not play a strong role as guides to experiment. This may be because of the largely unexplored complexity of most developing organisms, and the fact that the most powerful models work at a very abstract level. In Dictyostelium discoideum, however, a fully formed model incorporating detailed experimental results is now available. Current Opinion in Genetics and Development 1992, 2:647-650 /
Introduction "I hope 1 shall not shock ... too much if I add that it is also a good rule not to put overmuch cordidence in the observational results that are put forward until they have been confirmed by theory" [1].
Models in general The goal of mathematical modeling in the natural sciences is twofold; to summarize observations, and to make practical experimental predictions. Generally, the less obvious the predictions, the more successful the theory. Familiar examples of successful models in this sense are those used to explain allosteric enzyme behavior [2,3], and the Hodgkin-Huxley model for nerve transmission [4]. Both rationalize a great deal of puzzling data and are formally rigorous and detailed, making nontrivial experimental predictions that can and have been tested. Models with similar power for developmental biologists are only now beginning to emerge. Qualitative models permeate the developmental biolog,y literature. Ideas about gradients [5], positional information [6], and lateral inllibition [7] are now commonplace. Qualitative models often summarize experimental data, their purpose being not so much to explore what is feasible as to create a useful representation of the data. Despite their undoubted importance, both as thumbnail sketches and for heuristic purposes, they tend to fail when called upon to represent anything but the simplest phenomena. This is because realistic representations of most biological data inevitably lead to models that contain strongly non-linear terms, even for relatively simple cases, such as the behavior of allosteric enzymes [2,3]. Thus, in the long run quantitative models cannot be successful, as animals and plants are not the simple sum of their parts.
Quantitative models, on the other hand, attempt to take into account the full richness of behavior expected of multi-component interacting systems. They explore in a quantitative manner what is physically possible and realistic. Most are non-linear, and set constraints on the general properties of the behavior of the entire system, be they interactions between different cell types through receptors and ligands, or the mechanical forces that arise and are modulated during gastrulation. One consequence of their generality is that they cannot yet act as guides for the experimentalist who is interested in biochemical mechanisms. They can, however, distinguish between what is possible and what is not, a distinction that is notoriously difficult to make in complex systems (for a good example see [8.]). Quantitative models are now sufficiently advanced for us to confidently state, for example, that repeating spatial patterns are ultimately the products of short-range activating and long-range inhibiting forces, although we cannot confidently predict what those forces might be, or their biochemical basis. Just why this is so is explored in this review.
Lateral-inhibition models as examples Lateral-inhibition models establish spatial patterns by inhibiting the growth of nearby patterning elements. There is a large literature dealing with them, beginning with Turing's classic 1952 reaction-diffusion paper [9]. Turing realized that periodic patterns could emerge from a uniform distribution of two (or more) morphogens if they interacted in certain ways, usually one of them an activator and the other an inhibitor, and if their diffusion coefficients were appropriately chosen. One of his essential ideas was that the reacting and diffusing morphogens were everywhere, their synthesis and degradation coupled both kinetically and by diffusion. To illustrate his
Abbreviation cAMP--cyclic-AMP. (~) Current Biology Ltd ISSN 0959-437X
647
648
Pattern formation and developmental mechanisms ideas, he modeled the emergence of tentacles around the Hydra mouth, an exanlple he chose from Child's 1941 book [5]. Since Turing's paper, reaction-diffusion models have been used to explain a wide variety of spatial and temporal patterns, and the properties of Tufing-like models have been explored in great detail [10,11,12"]. These studies provide us with five quite general insights into the properties of patterning systems that are somewhat akin to conservation laws in physics. First, two or more morphogens (or their equivalent) are necessary for the establishment of stable patterns. Second, linear systems lead to unbounded growth of the pattern. Third, geometry and size are crucial deten'ninants of the final pattern. Fourth, the same basic fomlalism can be used to generate traveling waves of morphogens as well as stable and chaotic structures. Finally, self-regulating gradients and prepattems can be built into the pattern-generating mechanism. Models basecl on reacting and diffusing morphogens are not the only way to achieve lateral inl'tibition. For example, buckling forces generated by cell adhesion and movement can be used tO explain epithelial folding [13-15], and neural models have been created to explain patterns on sea shells [16] and optical dominance columns in the visual cortex [17]. It is not widely appreciated that these examples, and others like them, are mathematically similar. Thus, as Fig. 1 illustrates, models based on such different principles as diffusing morphogens, chemotaxing cells, haptotaxing fibroblasts, and neural activity are formally equivalent.
Reaction-Diffusion Hydra; Animal coat patterns; Shell patterns; Segmentation
Mechanical Tissue invagination
Neural Activity
-
-
Mthough this lesson might appear discouraging for the experimentalist - - and often provokes a response similar to Eigen's "With a fifth order polynomial you can fit an elephant" - - the detmled formal analysis of these models has provided important general insights that should not be ignored. First and foremost, the fact that lateralinhibition models can be realized in many different ways is a central result. It is not obvious how it could have been brought home without modeling. Second, mechanical models may work in the absence of time-dependent diffusing chemicals, yet also yield periodic spatial patterns. Unlike reaction-diffusion mechanisms, where a prepattern can precede morphological change, form emerges continuously; in other words, intermediate patterns, not just the final patterns, are visible [15,18]. Mechanical models also suggest that shape change can precede and perhaps even induce the regulation of new genes, and this is a provocative idea - - we usually think of things happening the other w W around. Third, most lateral-inhibition models are self-regulating [19]. Mutual interaction between activating and inhibiting elements ensures this feature, providing stability and robustness to a developing system [20]. Non-regulating mechanisms - - for exan~ple, the inteq)retation of positional values on linear gradients that are themselves unregulated - - invite error [21-].
Specific examples
Short-range activation and Lateral inhibition
Shell patterns; Cortical stripes
The equivalence follows directly from the mathematics used to model the separate systems, and can be reduced to the statement that short-range activation and long-range inl~ibition are necessary and sufficient conditions for the emergence of stable periodic structures in space, regardless of the physical mechanisms underlying the model. In fact, the statement is stronger than this: if one chooses biologically reasonable mechanisms receptors and ligands, neural circuits, and mechanical forces - - we are restricted to models that exhibit local activation and lateral inhibition.
Two ex,'unples, chosen from the slime-mold literature, illustrate the fruitful combination of theory and experiment in a relatively simple morphogenetic context.
Chemotaxis Slime molds
Fig. 1. Four different classes of models whose equilibrium behavior is governed by solutions to the same general eigenvalue problem. Throughout the text they are referred to as lateral-inhibition models, although in many contexts, particularly in the original Turing sense, this nomenclature is somewhat misleading. In all cases the models contain two major terms, one governing spatial information, the other kinetic or dynamic information, governing, for example, the synthesis and decay of morphogens. A detailed comparative discussion of these models may be found in [10,12.*1. Specific references for reaction-diffusion, mechanical, neural and chemotactic models are [11,15,17,22], respectively.
Dictyostelium discoideum When slime-mold anloebae starve they signal to each other, forming an aggregation territory. Each territory will become a coordinated and mobile mass of cells (the slug) with tile abili W to crawl vast distances (relative to their length). Both the signaling process [22] and slug motion [23] have been modeled in detail. Signaling is easier to model because the biochemistry of the cyclicAMP (cAMP) system is fairly well lmown, and an attractive model for cell-cell communication has been available for several years [24]. One of the observational results is that aggregating amoebae form concentric circles of signaling and responding cells [25,26]. This is what one would
Modeling and experimentin developmentalbiology Cox naively expect when founding cells send out a concentric pulse of cAMP. However, the same population of cells also contains spirals of signaling and responding cells. Both the circles and spirals collide with and extinguish each other. What mechanism might give rise to these behaviors, especiaUy the establishment of spirals? One might argue qualitatively that a spiral is simply a broken circle, and that once broken 'non-finear effects take over'; indeed, this is a good starting point. However, understanding why both circles and spirals arise naturally from a single biochemical mechanism that incorporates the biochemical details of the cAMP response, as well as diffusion of cAMP, requires taking into account both the experimental literature and the mathematical properties of a particular non-linear system (for a review see [22]). These and other theoretical insights make signaling in D. discoideum arguably the most thoroughly modeled and best understood system in the developmental literature. This 'standard model', by extension, might also be used to explain the apparent propagation of scroll waves from the anterior to the posterior of migrating slugs [27]. The unforced accommodation of experimental observations is the hallmark of good modeling. At the moment, the model is complex in detail. One challenge for the future is to reduce the complexity by identifying and emphasising the details that capture the essential features. For a related, yet still complex inorganic system, the Belousov-Zhabotinsky reaction, great progress along these lines continues to be made [28], and it is reasonable to assume that this will also happen in DicO,ostelium.
genes acting in cells arrayed on different surfaces yield different patterns. Lateral-inhibition models explain this in an unforced manner, and it is perhaps in this sense that Eddington was led to his dictum, as quoted at the head of this review.
Prospects Why, then, haven't quantitative models had a greater impact on experiment? Because, with few exceptions [8.], experimentalists do not interact closely with modelers, and ideas about lateral inhibition and the properties expected of models embodying them are often rediscovered empirically. Part of the explanation for this phenomenon must surely lie in the enormous power of modern experimental tools. Progress continues at an explosive rate and in such an environment there is no perceived need for theory. At the same time, most quantitative mathematical models are currendy too general and, in a sense, too powerful, although the wedding of experiment and theory is now highly advanced in Dictyostelium signaling. Realistic models, again of the kind devised to describe D. discoideum signaling, also require well defined biochemical data, and this is only beginning to become available in metazoans. In the long run, however, the growing complexity of the experimental data and the multidimensional and non-linear interactions between the various parts of the system, will require quantitative models, both to serve as useful representations of the experimental facts, and to discover unexpected behavior of the system.
Polysphondylium pallidum A second example of how morphogenetic models have been used in both the explanatory and predictive sense involves the work of McNally and Cox [29] on patterning in P. pallidum. Normally, the P. pallidum fruiting body appears similar to a small delicate pine tree, with whorls of spore-bearing arms arranged regularly about a central trunk. The radial arrangement of arms within whorls is symmetrical, arising from a radial prepattern on the surface of a spherical cell mass [30]. Both the qualitative and the quantitative spacing between adjacent arms, as well as the emergence of the prepattern on the mass, can be modeled by reaction-diff-usion theory [31]. Monster whorls, which form rarely from large eUipsoidal cell masses rather than spheres, are often preceded by striped, as opposed to spotted, prepatterns. This is at first sight surprising. However, striped patterns are also outcomes of reaction-diffusion models, and here theory is an invaluable guide, not only in explaining the range of patterns observed in this system, but also the kinetics of their appearance. The realization that reaction-diffusion models tend to yield unstable stripes has further stimulated the modeling of segment formation during Drosophila embryogenesis [32,33].
Acknowledgments
In both of the above examples, geometry plays a major role. Of surprise for the experimentalist is that the same
I thank members of my laboratory and Lionel Harrison for many useful comments on earlier versions of this essay.
References and recommended reading Papers of particularinterest, publishedwithin the annual period review, have been highlightedas: • of special interest °° of outstanding interest 1. EDDINGTON A: New Pathways in Science, 1st edn. New York: MacMillan; 1935. 2.
KOSHLANDDE, NEMETHY G, FILMERD: Comparison of Experimental Binding Data and Theoretical Models in Proteins Containing Subunits. Biochemistry 1966, 5:365-385.
3.
MONODJ, WYMANJ, CHANGEUXJ-P: On the Nature of AIIosteric Transitions: A Plausible Model. J Mol Biol 1965, 12:88--118.
4.
HODGKINAL, HUXLEYAF: A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. J Physiol 1952, 117:500-544.
5.
CHILD CM: Patterns and Problems in DevelopmenL Chicago: University of Chicago Press; 1941.
649
650
Pattern formation and developmental mechanisms 6.
WOLPERTL: Positional Information and the Spatial Pattern
of Cellular Differentiation. J Tbeor Biol 1969, 25:1--47. 7.
WIGGLESWORTHVB: Local and General Factors in the Develo p m e n t of 'Pattern' in Rhodnius prolixus (Hemiptera). J E.xp Biol 1940, 17:180-200.
8. •
EDGARBA, ODELLGM, SCHUBIGERG: A Genetic Switch, Based on Negative Regulation, Sharpens Stripes in Drosophila Embryos. Dev Genet 1989, 10:124-142. This paper illustrates how the coupling of several non-linear interacUons leads to complex spatial patterns in Drosophila. Moreover, modeling goes hand in hand with experiment. 9.
TURINGAM: The Chemical Basis of Morphogenesis. Philc~ Trans R Soc Lond [Biol] 1952, B237:37-32.
10.
HARRISONLG: Kinetic 77JeoO, of Living Pattertl. Cambridge: Cambridge University Press; 1992, in press.
11.
MEINHARDTH: Models of Biological Pattert, Formation. New York: Academic Press; 1982.
12. MURRAYJD: Mathennatical Biolog}~, Biomathematics Text& 1st •• edn. Edited by Levin SA. New York: Springer-Verlag; 1989, 19. An excellent survey of modeling strategy. ParUcularly strong on qualitative and quantitative reasoning, and unusual in the number of well described biological examples.
A good rex~ew of the arguments for regulated gradients. 22.
TYSONJJ, MURRAYJD: Cyclic AMP Waves During Aggregation of Dictyostelium Amoebae. Development 1989, 106:421-426.
23.
ODEU. G, BONNER J: HOW the Dicgyostelium discoideum grex crawls. Phil~ Trans R Soc Lond [Biol] 1986, B312:487.
24.
MARTIEI.J, GOLDBETERA: A Model Based on Receptor Desensitization for Cyclic AMP Signaling in Dictyostelium Cells. Biopbys J 1987, 52:807.
25.
AI.C,V,ar,~O,AF, MONK M: Signal Propagation During Aggregation in the Slime Mould Dictyostelium discoideum. J Gen Microbiol 1974, 85:321-334.
26.
DEVREOTESP, POTEI. M, MACKA','S: Quantitative Analysis of Cyclic AMP Waves Mediating Aggregation in DicO~ostelium discoideum. Dev Biol 1983, 96:405.
27.
SIEGERTF, WEIJERCJ: Analysis of Optical Density Wave Propagation and Cell Movement in the CeUular Slime Mold Dictyostelium discoideum. PIJFsica D 1991, 49:224-232.
28.
G','ORGY1L, FIELD RJ: A Three-variable Model of Deterministic Chaos in the Belousov-Zhabotinslo, Reaction. Nature 1992, 355:806--810.
29.
McNAI.I.',' JG, COX EC: Spots and Stripes: the Patterning Spectrum in the Cellular Slime Mold Polysphondylium pallidum. Development 1989, 105:323-333.
30.
BYRNEG, COX EC: Spatial Patterning in Polysphondylium: Monoclonal Antibodies Specific for Whorl Prepatterns. Det, Biol 1986, 117:442-455.
13.
GMITROJI, SCRIVEN LE: A Physicochemical Basis for Pattern and Rhythm. In lntracelhdar TralLg~ort. Edited by Warren KB. Academic Press: New York; 1966:221-255.
14.
GIERERA: Physical A s p e c t s of Tissue Evagination and Biological Form. Q Rev Bioplo,s 1977, 10:529--593.
15.
OSTER GF, MtlRRA'¢ JD, HARRIS AK: Mechanical aspects of mesenchymal morphogenesis. J Embool Ex~ Morphol 1983, 78:83-125.
31.
BYRNEG, Cox E: Genesis of a Spatial Pattern in the Cellular Slime Mold Polysphondylium pallidum. Proc Natl Acad Sci U S A 1987, 84:4140-4144.
16.
ERMENTROUTB, CAMPBELLJ, OSTER G: A Model for Shell Patterns Based on Neural Activity. The Veliger 1986, 28:369-388.
32.
17.
SWINDALENV: A Model for the Formation of Occular Dom. inance Stripes. Proc R Soc Lond [Biol] 1980, B208:243-264.
LACALLITC, WILKINSONDA, HARRISON LG: Theoretical Aspects of Stripe Formation in Relation to Drosophila Segmentation. Det,elopment 1988, 104:105-113.
33.
18.
ODEU.GM, OSTER G, ALBERCHP, BURNSIDEB: The Mechanical Basis of Morphogenesis I. Epithelial Folding and lnvaginadon. Dev Biol 1981, 85:446-462.
LYONSMJ, HARRISON LG: A Class of Reaction-diffusion Mechanisms w h i c h Preferentially Select Striped Patterns. Cbem PI~, Lett 1991, 183:158-160.
19.
GIERERA, MEINHARDTH: A Theory of Biological Pattern FormaUon. Kybernetik 1972, 12:30-39.
20.
MEXNHAROTH, GIERER& Applications of a Theory of Biological Pattern Formation Based on Lateral Inhibition. J Cell Sci 1974, 15:321-346.
21. •
LACAmTC, HARRISONLG: From Gradients to Segments: MOdels for Pattern Formation in Early Drosophila embryogene.sis. Semin Dev Biol 1991, 2:107-117.
34. OSTERGF, MURRAYJD: Pattern Formation Models and De** velopmental Constraints. J E.xp Zool 1989, 251:186-202. This is the single most accessible brief description of lateral-inhibition models and their equivalence. It also serves ,as a short introducUon to how the mathematical properties of a model are explored.
EC Cox, 1-16 Moffett Laboratories, Deparmlent of Molecular Biology, Princeton University, Princeton, New Jersey 08544, USA.
Introduction "I hope 1 shall not shock ... too much if I add that it is also a good rule not to put overmuch cordidence in the observational results that are put forward until they have been confirmed by theory" [1].
Models in general The goal of mathematical modeling in the natural sciences is twofold; to summarize observations, and to make practical experimental predictions. Generally, the less obvious the predictions, the more successful the theory. Familiar examples of successful models in this sense are those used to explain allosteric enzyme behavior [2,3], and the Hodgkin-Huxley model for nerve transmission [4]. Both rationalize a great deal of puzzling data and are formally rigorous and detailed, making nontrivial experimental predictions that can and have been tested. Models with similar power for developmental biologists are only now beginning to emerge. Qualitative models permeate the developmental biolog,y literature. Ideas about gradients [5], positional information [6], and lateral inllibition [7] are now commonplace. Qualitative models often summarize experimental data, their purpose being not so much to explore what is feasible as to create a useful representation of the data. Despite their undoubted importance, both as thumbnail sketches and for heuristic purposes, they tend to fail when called upon to represent anything but the simplest phenomena. This is because realistic representations of most biological data inevitably lead to models that contain strongly non-linear terms, even for relatively simple cases, such as the behavior of allosteric enzymes [2,3]. Thus, in the long run quantitative models cannot be successful, as animals and plants are not the simple sum of their parts.
Quantitative models, on the other hand, attempt to take into account the full richness of behavior expected of multi-component interacting systems. They explore in a quantitative manner what is physically possible and realistic. Most are non-linear, and set constraints on the general properties of the behavior of the entire system, be they interactions between different cell types through receptors and ligands, or the mechanical forces that arise and are modulated during gastrulation. One consequence of their generality is that they cannot yet act as guides for the experimentalist who is interested in biochemical mechanisms. They can, however, distinguish between what is possible and what is not, a distinction that is notoriously difficult to make in complex systems (for a good example see [8.]). Quantitative models are now sufficiently advanced for us to confidently state, for example, that repeating spatial patterns are ultimately the products of short-range activating and long-range inhibiting forces, although we cannot confidently predict what those forces might be, or their biochemical basis. Just why this is so is explored in this review.
Lateral-inhibition models as examples Lateral-inhibition models establish spatial patterns by inhibiting the growth of nearby patterning elements. There is a large literature dealing with them, beginning with Turing's classic 1952 reaction-diffusion paper [9]. Turing realized that periodic patterns could emerge from a uniform distribution of two (or more) morphogens if they interacted in certain ways, usually one of them an activator and the other an inhibitor, and if their diffusion coefficients were appropriately chosen. One of his essential ideas was that the reacting and diffusing morphogens were everywhere, their synthesis and degradation coupled both kinetically and by diffusion. To illustrate his
Abbreviation cAMP--cyclic-AMP. (~) Current Biology Ltd ISSN 0959-437X
647
648
Pattern formation and developmental mechanisms ideas, he modeled the emergence of tentacles around the Hydra mouth, an exanlple he chose from Child's 1941 book [5]. Since Turing's paper, reaction-diffusion models have been used to explain a wide variety of spatial and temporal patterns, and the properties of Tufing-like models have been explored in great detail [10,11,12"]. These studies provide us with five quite general insights into the properties of patterning systems that are somewhat akin to conservation laws in physics. First, two or more morphogens (or their equivalent) are necessary for the establishment of stable patterns. Second, linear systems lead to unbounded growth of the pattern. Third, geometry and size are crucial deten'ninants of the final pattern. Fourth, the same basic fomlalism can be used to generate traveling waves of morphogens as well as stable and chaotic structures. Finally, self-regulating gradients and prepattems can be built into the pattern-generating mechanism. Models basecl on reacting and diffusing morphogens are not the only way to achieve lateral inl'tibition. For example, buckling forces generated by cell adhesion and movement can be used tO explain epithelial folding [13-15], and neural models have been created to explain patterns on sea shells [16] and optical dominance columns in the visual cortex [17]. It is not widely appreciated that these examples, and others like them, are mathematically similar. Thus, as Fig. 1 illustrates, models based on such different principles as diffusing morphogens, chemotaxing cells, haptotaxing fibroblasts, and neural activity are formally equivalent.
Reaction-Diffusion Hydra; Animal coat patterns; Shell patterns; Segmentation
Mechanical Tissue invagination
Neural Activity
-
-
Mthough this lesson might appear discouraging for the experimentalist - - and often provokes a response similar to Eigen's "With a fifth order polynomial you can fit an elephant" - - the detmled formal analysis of these models has provided important general insights that should not be ignored. First and foremost, the fact that lateralinhibition models can be realized in many different ways is a central result. It is not obvious how it could have been brought home without modeling. Second, mechanical models may work in the absence of time-dependent diffusing chemicals, yet also yield periodic spatial patterns. Unlike reaction-diffusion mechanisms, where a prepattern can precede morphological change, form emerges continuously; in other words, intermediate patterns, not just the final patterns, are visible [15,18]. Mechanical models also suggest that shape change can precede and perhaps even induce the regulation of new genes, and this is a provocative idea - - we usually think of things happening the other w W around. Third, most lateral-inhibition models are self-regulating [19]. Mutual interaction between activating and inhibiting elements ensures this feature, providing stability and robustness to a developing system [20]. Non-regulating mechanisms - - for exan~ple, the inteq)retation of positional values on linear gradients that are themselves unregulated - - invite error [21-].
Specific examples
Short-range activation and Lateral inhibition
Shell patterns; Cortical stripes
The equivalence follows directly from the mathematics used to model the separate systems, and can be reduced to the statement that short-range activation and long-range inl~ibition are necessary and sufficient conditions for the emergence of stable periodic structures in space, regardless of the physical mechanisms underlying the model. In fact, the statement is stronger than this: if one chooses biologically reasonable mechanisms receptors and ligands, neural circuits, and mechanical forces - - we are restricted to models that exhibit local activation and lateral inhibition.
Two ex,'unples, chosen from the slime-mold literature, illustrate the fruitful combination of theory and experiment in a relatively simple morphogenetic context.
Chemotaxis Slime molds
Fig. 1. Four different classes of models whose equilibrium behavior is governed by solutions to the same general eigenvalue problem. Throughout the text they are referred to as lateral-inhibition models, although in many contexts, particularly in the original Turing sense, this nomenclature is somewhat misleading. In all cases the models contain two major terms, one governing spatial information, the other kinetic or dynamic information, governing, for example, the synthesis and decay of morphogens. A detailed comparative discussion of these models may be found in [10,12.*1. Specific references for reaction-diffusion, mechanical, neural and chemotactic models are [11,15,17,22], respectively.
Dictyostelium discoideum When slime-mold anloebae starve they signal to each other, forming an aggregation territory. Each territory will become a coordinated and mobile mass of cells (the slug) with tile abili W to crawl vast distances (relative to their length). Both the signaling process [22] and slug motion [23] have been modeled in detail. Signaling is easier to model because the biochemistry of the cyclicAMP (cAMP) system is fairly well lmown, and an attractive model for cell-cell communication has been available for several years [24]. One of the observational results is that aggregating amoebae form concentric circles of signaling and responding cells [25,26]. This is what one would
Modeling and experimentin developmentalbiology Cox naively expect when founding cells send out a concentric pulse of cAMP. However, the same population of cells also contains spirals of signaling and responding cells. Both the circles and spirals collide with and extinguish each other. What mechanism might give rise to these behaviors, especiaUy the establishment of spirals? One might argue qualitatively that a spiral is simply a broken circle, and that once broken 'non-finear effects take over'; indeed, this is a good starting point. However, understanding why both circles and spirals arise naturally from a single biochemical mechanism that incorporates the biochemical details of the cAMP response, as well as diffusion of cAMP, requires taking into account both the experimental literature and the mathematical properties of a particular non-linear system (for a review see [22]). These and other theoretical insights make signaling in D. discoideum arguably the most thoroughly modeled and best understood system in the developmental literature. This 'standard model', by extension, might also be used to explain the apparent propagation of scroll waves from the anterior to the posterior of migrating slugs [27]. The unforced accommodation of experimental observations is the hallmark of good modeling. At the moment, the model is complex in detail. One challenge for the future is to reduce the complexity by identifying and emphasising the details that capture the essential features. For a related, yet still complex inorganic system, the Belousov-Zhabotinsky reaction, great progress along these lines continues to be made [28], and it is reasonable to assume that this will also happen in DicO,ostelium.
genes acting in cells arrayed on different surfaces yield different patterns. Lateral-inhibition models explain this in an unforced manner, and it is perhaps in this sense that Eddington was led to his dictum, as quoted at the head of this review.
Prospects Why, then, haven't quantitative models had a greater impact on experiment? Because, with few exceptions [8.], experimentalists do not interact closely with modelers, and ideas about lateral inhibition and the properties expected of models embodying them are often rediscovered empirically. Part of the explanation for this phenomenon must surely lie in the enormous power of modern experimental tools. Progress continues at an explosive rate and in such an environment there is no perceived need for theory. At the same time, most quantitative mathematical models are currendy too general and, in a sense, too powerful, although the wedding of experiment and theory is now highly advanced in Dictyostelium signaling. Realistic models, again of the kind devised to describe D. discoideum signaling, also require well defined biochemical data, and this is only beginning to become available in metazoans. In the long run, however, the growing complexity of the experimental data and the multidimensional and non-linear interactions between the various parts of the system, will require quantitative models, both to serve as useful representations of the experimental facts, and to discover unexpected behavior of the system.
Polysphondylium pallidum A second example of how morphogenetic models have been used in both the explanatory and predictive sense involves the work of McNally and Cox [29] on patterning in P. pallidum. Normally, the P. pallidum fruiting body appears similar to a small delicate pine tree, with whorls of spore-bearing arms arranged regularly about a central trunk. The radial arrangement of arms within whorls is symmetrical, arising from a radial prepattern on the surface of a spherical cell mass [30]. Both the qualitative and the quantitative spacing between adjacent arms, as well as the emergence of the prepattern on the mass, can be modeled by reaction-diff-usion theory [31]. Monster whorls, which form rarely from large eUipsoidal cell masses rather than spheres, are often preceded by striped, as opposed to spotted, prepatterns. This is at first sight surprising. However, striped patterns are also outcomes of reaction-diffusion models, and here theory is an invaluable guide, not only in explaining the range of patterns observed in this system, but also the kinetics of their appearance. The realization that reaction-diffusion models tend to yield unstable stripes has further stimulated the modeling of segment formation during Drosophila embryogenesis [32,33].
Acknowledgments
In both of the above examples, geometry plays a major role. Of surprise for the experimentalist is that the same
I thank members of my laboratory and Lionel Harrison for many useful comments on earlier versions of this essay.
References and recommended reading Papers of particularinterest, publishedwithin the annual period review, have been highlightedas: • of special interest °° of outstanding interest 1. EDDINGTON A: New Pathways in Science, 1st edn. New York: MacMillan; 1935. 2.
KOSHLANDDE, NEMETHY G, FILMERD: Comparison of Experimental Binding Data and Theoretical Models in Proteins Containing Subunits. Biochemistry 1966, 5:365-385.
3.
MONODJ, WYMANJ, CHANGEUXJ-P: On the Nature of AIIosteric Transitions: A Plausible Model. J Mol Biol 1965, 12:88--118.
4.
HODGKINAL, HUXLEYAF: A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. J Physiol 1952, 117:500-544.
5.
CHILD CM: Patterns and Problems in DevelopmenL Chicago: University of Chicago Press; 1941.
649
650
Pattern formation and developmental mechanisms 6.
WOLPERTL: Positional Information and the Spatial Pattern
of Cellular Differentiation. J Tbeor Biol 1969, 25:1--47. 7.
WIGGLESWORTHVB: Local and General Factors in the Develo p m e n t of 'Pattern' in Rhodnius prolixus (Hemiptera). J E.xp Biol 1940, 17:180-200.
8. •
EDGARBA, ODELLGM, SCHUBIGERG: A Genetic Switch, Based on Negative Regulation, Sharpens Stripes in Drosophila Embryos. Dev Genet 1989, 10:124-142. This paper illustrates how the coupling of several non-linear interacUons leads to complex spatial patterns in Drosophila. Moreover, modeling goes hand in hand with experiment. 9.
TURINGAM: The Chemical Basis of Morphogenesis. Philc~ Trans R Soc Lond [Biol] 1952, B237:37-32.
10.
HARRISONLG: Kinetic 77JeoO, of Living Pattertl. Cambridge: Cambridge University Press; 1992, in press.
11.
MEINHARDTH: Models of Biological Pattert, Formation. New York: Academic Press; 1982.
12. MURRAYJD: Mathennatical Biolog}~, Biomathematics Text& 1st •• edn. Edited by Levin SA. New York: Springer-Verlag; 1989, 19. An excellent survey of modeling strategy. ParUcularly strong on qualitative and quantitative reasoning, and unusual in the number of well described biological examples.
A good rex~ew of the arguments for regulated gradients. 22.
TYSONJJ, MURRAYJD: Cyclic AMP Waves During Aggregation of Dictyostelium Amoebae. Development 1989, 106:421-426.
23.
ODEU. G, BONNER J: HOW the Dicgyostelium discoideum grex crawls. Phil~ Trans R Soc Lond [Biol] 1986, B312:487.
24.
MARTIEI.J, GOLDBETERA: A Model Based on Receptor Desensitization for Cyclic AMP Signaling in Dictyostelium Cells. Biopbys J 1987, 52:807.
25.
AI.C,V,ar,~O,AF, MONK M: Signal Propagation During Aggregation in the Slime Mould Dictyostelium discoideum. J Gen Microbiol 1974, 85:321-334.
26.
DEVREOTESP, POTEI. M, MACKA','S: Quantitative Analysis of Cyclic AMP Waves Mediating Aggregation in DicO~ostelium discoideum. Dev Biol 1983, 96:405.
27.
SIEGERTF, WEIJERCJ: Analysis of Optical Density Wave Propagation and Cell Movement in the CeUular Slime Mold Dictyostelium discoideum. PIJFsica D 1991, 49:224-232.
28.
G','ORGY1L, FIELD RJ: A Three-variable Model of Deterministic Chaos in the Belousov-Zhabotinslo, Reaction. Nature 1992, 355:806--810.
29.
McNAI.I.',' JG, COX EC: Spots and Stripes: the Patterning Spectrum in the Cellular Slime Mold Polysphondylium pallidum. Development 1989, 105:323-333.
30.
BYRNEG, COX EC: Spatial Patterning in Polysphondylium: Monoclonal Antibodies Specific for Whorl Prepatterns. Det, Biol 1986, 117:442-455.
13.
GMITROJI, SCRIVEN LE: A Physicochemical Basis for Pattern and Rhythm. In lntracelhdar TralLg~ort. Edited by Warren KB. Academic Press: New York; 1966:221-255.
14.
GIERERA: Physical A s p e c t s of Tissue Evagination and Biological Form. Q Rev Bioplo,s 1977, 10:529--593.
15.
OSTER GF, MtlRRA'¢ JD, HARRIS AK: Mechanical aspects of mesenchymal morphogenesis. J Embool Ex~ Morphol 1983, 78:83-125.
31.
BYRNEG, Cox E: Genesis of a Spatial Pattern in the Cellular Slime Mold Polysphondylium pallidum. Proc Natl Acad Sci U S A 1987, 84:4140-4144.
16.
ERMENTROUTB, CAMPBELLJ, OSTER G: A Model for Shell Patterns Based on Neural Activity. The Veliger 1986, 28:369-388.
32.
17.
SWINDALENV: A Model for the Formation of Occular Dom. inance Stripes. Proc R Soc Lond [Biol] 1980, B208:243-264.
LACALLITC, WILKINSONDA, HARRISON LG: Theoretical Aspects of Stripe Formation in Relation to Drosophila Segmentation. Det,elopment 1988, 104:105-113.
33.
18.
ODEU.GM, OSTER G, ALBERCHP, BURNSIDEB: The Mechanical Basis of Morphogenesis I. Epithelial Folding and lnvaginadon. Dev Biol 1981, 85:446-462.
LYONSMJ, HARRISON LG: A Class of Reaction-diffusion Mechanisms w h i c h Preferentially Select Striped Patterns. Cbem PI~, Lett 1991, 183:158-160.
19.
GIERERA, MEINHARDTH: A Theory of Biological Pattern FormaUon. Kybernetik 1972, 12:30-39.
20.
MEXNHAROTH, GIERER& Applications of a Theory of Biological Pattern Formation Based on Lateral Inhibition. J Cell Sci 1974, 15:321-346.
21. •
LACAmTC, HARRISONLG: From Gradients to Segments: MOdels for Pattern Formation in Early Drosophila embryogene.sis. Semin Dev Biol 1991, 2:107-117.
34. OSTERGF, MURRAYJD: Pattern Formation Models and De** velopmental Constraints. J E.xp Zool 1989, 251:186-202. This is the single most accessible brief description of lateral-inhibition models and their equivalence. It also serves ,as a short introducUon to how the mathematical properties of a model are explored.
EC Cox, 1-16 Moffett Laboratories, Deparmlent of Molecular Biology, Princeton University, Princeton, New Jersey 08544, USA.
