DOI: 10.1126/science.1179047, 1616 (2010);329 Science

Shigeru Kondo and Takashi MiuraBiological Pattern FormationReaction-Diffusion Model as a Framework for Understanding

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Reaction-Diffusion Model as aFramework for UnderstandingBiological Pattern FormationShigeru Kondo1* and Takashi Miura2

The Turing, or reaction-diffusion (RD), model is one of the best-known theoretical modelsused to explain self-regulated pattern formation in the developing animal embryo. Although itsreal-world relevance was long debated, a number of compelling examples have gradually alleviatedmuch of the skepticism surrounding the model. The RD model can generate a wide variety ofspatial patterns, and mathematical studies have revealed the kinds of interactions required foreach, giving this model the potential for application as an experimental working hypothesis ina wide variety of morphological phenomena. In this review, we describe the essence of this theoryfor experimental biologists unfamiliar with the model, using examples from experimental studiesin which the RD model is effectively incorporated.

Over the past three decades, studies at themolecular level have revealed that a widerange of physiological phenomena are

regulated by complex networks of cellular or mo-lecular interactions (1). The complexity of suchnetworks gives rise to new problems, however,as the behavior of such systems often defies im-mediate or intuitive understanding. Mathematicalapproaches can help facilitate the understandingof complex systems, and to date, these approacheshave taken two primary forms. The first involvesanalyzing every element of a network quantita-tively and simulating all interactions by compu-tation (1). This strategy is effective in relativelysimple systems, such as the metabolic pathwayin a single cell, and is extensively explored in thefield of systems biology. However, for more com-plex systems in which spatiotemporal parameterstake on importance, it becomes almost impos-sible to make a meaningful prediction. A secondstrategy, one that includes simple mathematicalmodeling in which the details of the system areomitted, can be more effective in extracting thenature of the complex system (2). The reaction-diffusion (RD) model (3) proposed by AlanTuring is a masterpiece of this sort of mathe-matical modeling, one that can explain howspatial patterns develop autonomously.

In the RDmodel, Turing used a simple systemof “two diffusible substances interacting witheach other” to represent patterning mechanismsin the embryo and found that such systems cangenerate spatial patterns autonomously. The mostrevolutionary feature of the RD model is its in-troduction of a “reaction” that produces theligands (morphogens). If “diffusion” alone is at

work, local sources of morphogens are needed toform the gradient. In such cases, the positionalinformation made by the system is dependent onthe prepattern (Fig. 1, A and B). By introducingthe reaction, the system gains the ability to gen-erate various patterns independent of the pre-pattern (Fig. 1C). Unfortunately, Turing died soonafter publishing this seminal paper, but simula-

tion studies of the model have shown that thissystem can replicate most biological spatial pat-terns (4–6). Later, a number of mathematicalmodels (4) were proposed, but most followedTuring’s basic idea that “the mutual interactionof elements results in spontaneous pattern for-mation.” The RD model is now recognized as astandard among mathematical theories that dealwith biological pattern formation.

However, this model has yet to gain wideacceptance among experimental biologists. Onereason is the gap between the mathematical sim-plicity of themodel and the complexity of the realworld. The hypothetical molecules in the originalRD model have been so idealized for the pur-poses of mathematical analysis that it seemsnearly impossible to adapt the model directly tothe complexity of real biological systems. How-ever, this is a misunderstanding to which exper-imental researchers tend to succumb. The logic ofpattern formation can be understood with simplemodels, and by adapting this logic to complexbiological phenomena, it becomes easier to ex-tract the essence of the underlying mechanisms.Genomic data and new analytic technologieshave shifted the target of developmental researchfrom the identification of molecules to understand-ing the behavior of complex networks, makingthe RD model even more important as a tool fortheoretical analysis.

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1Graduate School of Frontier Biosciences, Osaka University,Suita, Osaka, 565-0871, Japan. 2Department of Anatomyand Developmental Biology, Kyoto University GraduateSchool of Medicine, Kyoto 606-8501, Japan.

*To whom correspondence should be addressed. E-mail:skondo@fbs.osaka-u.ac.jp

A

One morphogen

Two morphogens

Gradient 1D horizontal 1D vertical

B

C

Interactions "Wave" Spots and stripes Labyrinth

Gradients 2D pattern More complicated

Fig. 1. Schematic drawing showing the difference between the morphogen gradient model and Turingmodel. (A) A morphogen molecule produced at one end of an embryo forms a gradient by diffusion. Cells“know” their position from the concentration of the molecule. The gradient is totally dependent on theprepattern of the morphogen source (boundary condition). (B) Adding a second morphogen produces arelatively complex pattern; but with no interactions between the morphogens, the system is not self-regulating. (C) With addition of the interactions between the morphogens, the system becomes self-regulating and can form a variety of patterns independent of the prepattern. [Art work by S. Miyazawa]

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In this review, we describe the RDmodel andits experimental applications, addressing someof the points biologists tend to question.We hopemany biologists gain an appreciation for thisbeautiful theory and take advantage of it in theirexperimental work.

The Original Turing ModelAt the beginning of his original paper (3), Turingstated that his aim was to show that the com-bination of known physical elements is sufficientto explain biological pattern formation. The ele-ments selected by Turing were a theoretical pairof interacting molecules diffused in a contin-uous field. In his mathematical analysis, Turingrevealed that such a system yields six potentialsteady states, depending on the dynamics of re-action term and wavelength of the pattern [Fig.2A and Supporting Online Material (SOM) Text1-1]. In case I, the system converges to a stableand uniform state. In case II, a uniform-phase os-cillation of the morphogen concentration occurs.Such phase unification is seen in such systems ascircadian rhythms (7) and the contraction of heartmuscle cells (8). In case III, the system formssalt-and-pepper patterns, such as are made whendifferentiated cells inhibit the differentiation ofneighboring cells. This is seen, for example, withdifferentiated neuroprogenitor cells in the epithe-lium of Drosophila embryos (9). Case IV repre-sents an unusual state in which a pattern of thesort made in case III oscillates. No examples ofthis have been identified in a developmental sys-tem. In case V, a traveling wave is generated. Bio-logical traveling waves caused by this mechanisminclude the spiral patterns formed by the socialamoeba Dictyostelium discoideum on aggrega-tion (10), and the wave of calcium ions that tra-verses the egg of the frog Xenopus laevis onsperm entry (11).

In case VI, stationary patterns are made. Thefinding of this type of wave is the major achieve-ment of Turing’s analyses, and these are usuallyreferred to as Turing patterns. ATuring pattern isa kind of nonlinear wave that is maintained by thedynamic equilibrium of the system. Its wave-length is determined by interactions between mol-ecules and their rates of diffusion. Such patternsarise autonomously, independent of any preexist-ing positional information. [The supportingmate-rials contain an expanded explanation of the RDmodel for biologists who are unfamiliar with themathematical description (SOM Text S1-1), aswell as a mathematical explanation of how theperiodic pattern arises in the system (SOM TextS1-2).] This property makes it possible to explainhow patterns arise precisely in, for example, fer-tilized oocytes, which present little in the wayof positional information. The ability of Turingpatterns to regenerate autonomously, even afterexperimentally induced disturbances, is also im-portant and of great utility in explaining the au-tonomy shown by pattern-forming developmentalprocesses (4, 6). In addition, through the tuningof parameters and boundary conditions, the sys-

tem underlying Turing pattern formation can gen-erate a nearly limitless variety of spatial patterns.(Figure 2B shows representative two-dimensionalpatternsmade by simulationswith the RDmodel.)The intricate involutions of seashells (5), the ex-quisite patterning of feathers (12), and the breath-takingly diverse variety of vertebrate skin patterns(13) have all been modeled within the frameworkof the Turing model (Fig. 2C). The remarkablesimilarity between prediction and reality in thesesimulations points strongly to Turing mechanismsas the underlying principle in these and othermodes of biological patterning. (User-friendlysimulation software is available as supportingonline material; a manual of the software is inSOM Text 1-3.)

Compatibility of RD and Gradient ModelsExperimental biologists may recall the “gradientmodel,”which has been quite effective in explain-ing pattern-formation events (14). In the classicgradient model, the fixed source of the morpho-gen at a specific position provides positional in-formation (14) (Fig. 1). Although the assumptionof a morphogen source appears to be differentfrom the assumptions of the RD model, it can beintroduced into the RDmodel quite naturally as a“boundary condition.” In other words, the classicmorphogen gradient model can be thought ofas the specific case of the RD model in which thereaction term is removed. In many simulationstudies, such boundary conditions are used to makethe pattern more realistic (6). Recent experimentalstudies ofmorphogen gradients have shown that theprecision and the robustness of the gradient aresecured by the interactions ofmolecular elements(15). To model such situations, the authors used amathematical framework that is essentially iden-tical to that of the RD model. Both reaction andboundary conditions are essential to understand-ing complex real systems, and the RD model isuseful for modeling such cases.

Applying the “Simple” RD Model to aComplex RealityThe hypothetical molecules in the original RDmodel (3) are idealized for the purposes of math-ematical analysis. It is assumed that they controltheir own synthesis and that of their counterparts,and diffuse quickly across spaces that would bedivided by cellular membranes. Obviously, it isquite difficult to apply such a model directly tocomplex living systems.

Concerted efforts to align theoretical modelsto real-world systems, however, have begun tobear fruit, pointing to a much broader range ofsituations in which the general principles under-lying the Turing model might apply. Gierer andMeinhardt (16, 17) showed that a system needsonly to include a network that combines “a short-range positive feedback with a long-range nega-tive feedback” to generate a Turing pattern. Thisis now accepted as the basic requirement forTuring pattern formation (14, 16). This refinementleaves the types and numbers of reacting factors

unspecified, making it much simpler to envisionsystems that might fit the requirements.

The interacting elements need not be limitedto molecules, or even to discrete entities; a circuitof cellular signals will do just as well (18). Thereis also no need for the stimulus to be provided viadiffusion; othermodes of transmission can achievethe same end result. Theoretical modeling hasshown that a relayed series of direct cell-to-cellsignals can form awave having properties similarto one formed by diffusible factors (19). Otherforms of signaling, including chemotactic cell mi-gration (20), mechanochemical activity (21), andneuronal interactions [as in the Swindale model(22) of ocular dominance stripe formation], arealso capable of forming Turing-like patterns. Forall of these systems, a similar periodic pattern isformed when the condition of “short-range posi-tive feedback with long-range negative feedback”is satisfied.

Why systems represented by apparently differ-ent equations behave similarly, and howmuch thecapacity for pattern forming differs among them,are the important subjects from the mathematicsperspective. But if the dynamics of the systemsare nearly the same, experimental researchers canselect any of the models as their working hypoth-esis. In the case of fish skin patterning, althoughexperiments apparently suggest the involvementof a nondiffusing signal transduction mechanism,the simplest RDmodel can predict themovementof the pattern during fish growth (23) and theunusual patterns seen in hybrid fish (24).

Finding Turing Patterns in Real SystemsDuring embryogenesis, a great variety of periodicstructures develop from various nonperiodic cellor tissue sources, suggesting that waves of thesort generated by Turing or related mechanismsmay underlie a wide range of developmental pro-cesses. Using modern genetic and molecular tech-niques, it is possible to identify putative elementsof interactive networks that fulfill the criteria ofshort-range positive feedback and long-rangenegative feedback, but finding the network aloneis not enough. Skeptics rightly point out that justbecause there is water, it doesn’t mean there arewaves. Nomatter how vividly or faithfully a math-ematical simulation might replicate an actualbiological pattern, this alone does not constituteproof that the simulated state reflects the reality.This has been another major hurdle in identifyingcompelling examples of Turing patterns in livingsystems. The solution, however, is not so com-plicated; to show that a wave exists, we need toidentify the dynamic properties of the pattern thatis predicted by the computer simulation. Exper-imental demonstrations have focused on patternformation in the skin, because the specific char-acteristics of Turing patterns are more evident intwo dimensions than in one.

Turing Patterns in Vertebrate SkinObservation of the dynamic properties of Turingpatterns in nature was made by Kondo and Asai

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A

B

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Initial condition

Six stable states

Case VI (Turing pattern)Case V

Uniform, stationary

Oscillatory caseswith extremely short

wavelength

Oscillatory caseswith finite wavelength

Stationary waves withfinite wavelength(Turing pattern)

Uniform, oscillating Stationary waves withextremely short wavelength

Both morphogensdiffuse and reactwith each other

I II III

IV V VI

Fig. 2. Schematic drawing showing the mathematical analysis of the RDsystem and the patterns generated by the simulation. (A) Six stable statestoward which the two-factor RD system can converge. (B) Two-dimensionalpatterns generated by the Turing model. These patterns were made by anidentical equation with slightly different parameter values. These simulationswere calculated by the software provided as supporting online material. (C)

Reproduction of biological patterns created by modified RD mechanisms. Withmodification, the RD mechanism can generate more complex patterns such asthose seen in the real organism. Simulation images are courtesy of H. Meinhardt[sea shell pattern (5)] and A. R. Sandersen [fish pattern (13)]. Photos of actualseashells are from Bishougai-HP (http://shell.kwansei.ac.jp/~shell/). Images ofpopper fish are courtesy of Massimo Boyer (www.edge-of-reef.com).

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in a study of horizontal stripes in the tropical fish,Pomacanthus imperator (25). They have recentlyshown that this dynamic nature is shared bymany fish species, including the well-establishedmodel organism, zebrafish. Although zebrafishstripes may appear to be stationary, experimentalperturbation of the pattern triggers slow changes(23). Following laser ablation of pigment cells ina pair of black horizontal stripes, thelower line shifts upward before sta-bilizing in aBell-like curve (Fig. 3A)(23). As a result, the spatial intervalbetween the lines is maintained, evenwhen their direction changes. Thisstriking behavior is predicted bysimulation (Fig. 3B).

Fortunately, the zebrafish is ame-nable to a variety of experimentalapproaches that may lead to theidentification of the circuit of inter-actions that generates these patterns(26).Work to date has shown that theskin patterns of this fish are set up andmaintained by interactions betweenpigmented cells (26). Nakamasuworked out the interaction networkamong the pigment cells. Althoughthe shape of the network is differ-ent from that of the original Turingmodel, it fits the short-range posi-tive, long-range negative feedbackdescription (18). (The mutual inhi-bition between black and yellowcells behaves as a positive feedbackloop, as the expansion of black cellsweakens their counterpart.) Identi-fication of the genetic factors in-volved in the zebrafish pigmentationis under way (26), and it is hopedthat this will clarify the details of thesignaling network beneath thewavesin the skin of fish, and potentiallyall vertebrates. Many similar surfacepatterns are seen in invertebrates andplants. We suppose that in thesecases an essentially similar mecha-nism (RD mechanism) is involved,although their molecular basis maybe different.

Other well-studied examples in-clude the regular disposition of featherbuds in chick (27), and of hair fol-licles inmice (28). Jung et al. showedthat the spatially periodic pattern offeather buds regenerates even whenthe skin is recombined from disso-ciated cells (27). In the case ofmousehair follicles, alteration of the amountof putative key factors changes thepattern in a manner predicted bycomputer simulation. Sick et al. usedoverexpression and inhibition ofWntand Dkk in fetal mouse to studyhow such perturbations might affectthe patterning of follicle formation

(28). By simulation, they first predicted that aringed pattern of Wnt gene expression that doesnot occur in nature would result if ectopic pro-duction of a Wnt protein were to be controlledappropriately, and then confirmed this experi-mentally. They suggested that Wnt serves as ashort-range activator, and Dkk as a long-rangeinhibitor, in this system (28). This pair of factors

functions in various patterning processes as well,making this a critically important result for theintimations it provides of a wider role for Turingpatterns in development (Fig. 4). Interestingly,the growth of new hair relying on interactionsbetween neighboring follicles proceeds even inadult mice, and in one case a mutant was iden-tified in which a traveling wave of hair formation

gradually moved across the skinover the life of animals carrying thismutation (29). Plikus et al. (29) haveshown how the factors FGF (fibro-blast growth factor) and BMP (bonemorphogenetic protein) function togenerate the traveling wave (case Vin the Turing model).

Other Potential Turing-DrivenDevelopmental PhenomenaEstablishment of right-left asym-metry in vertebrates is triggered bythe unidirectional rotation of cilia atthe node, followed by the interac-tion of Nodal and Lefty that am-plify and stabilize faint differencesin gene expression (30, 31). Nodalenhances both its own expressionand that of Lefty, and Lefty inhibitsthe activity of Nodal. The fact thatLefty spreads further than Nodalsuggests that the inhibitory interac-tion propagates more quickly thandoes its activating counterpart, whichas we have seen, indicates that thissystem fulfills the fundamental re-quirements for Turing pattern for-mation (30) (Fig. 4).

In vertebrate limb development,precartilage condensation, which islater replaced by skeletal bone, oc-curs periodically along the anterior-posterior axis of the distal tip region.Because this patterning occurs with-out any periodic prepattern, theTuring model has long been sug-gested to describe the underlyingmechanism (32). In this system, trans-forming growth factor–b (TGF-b)has been invoked as a candidate forthe activator molecule (33). TGF-bcan stimulate its own productionand trigger precartilage conden-sation, and the sites of incipientcondensation exert a laterally actinginhibitory effect on chondrogenesis.Although no candidate inhibitor hasbeen identified, an interaction net-work comprising TGF-b functionand precartilage condensation maysatisfy the short-range activation andlong-range inhibition criteria (33).Miura and Shiota have shown thatnearly periodic spatial patterns ofchondrogenesis occur in the cultureof dissociated limb cells in vitro,

Day 13

Day 16

Day 20

Day 23

C

A B

Short-range positive feedback

Long-range negative feedback

Fig. 3. Movement of zebrafish stripes and the interaction network among thepigment cells. The pigment pattern of zebrafish is composed of black pigment cells(melanophores) and yellow pigment cells (xanthophores). The pattern is made by themutual interactionbetween these cells. (A)Melanophores in the twoblack stripeswereablated by laser, and the process of recovery was recorded. (B) Results of simulationby the Turingmodel. (C) Interaction network between the pigment cells deduced byexperiments. The red arrow represents a long-range positive (enhancing) effect,whereas the blue linewith the endbar represents a short-range negative (inhibitory)effect. A circuit of two negative interactions functions as a positive feedback.

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and that the addition of TGF-b changes the pat-tern in a manner consistent with the predictionsof Turing’s model (34).

Wnt and Dkk are also essential for lungbranching in vertebrates (35) and play a key rolein head regeneration in Hydra (36), in which in-volvement of the Turingmechanism has long beensuggested by theoretical and experimental studies.Involvement of the same molecules does not di-rectly prove that the same dynamic mechanism isat work. However, because they constitute theself-regulated system that is robust to artificialperturbations, it is reasonable to expect that theTuringmechanismmay underlie these phenomenaas it does those in the skin. Watanabe et al. haveshown that signal transduction via the gap junction(connexin41.8) plays a key role in pigment pat-tern formation in zebrafish (37). Loss of function

of the connexin gene (leopard)reduces the spatial periodicity andchanges the pattern from stripes tospots (37). Amutation in anothergap junction gene, connexin43,shortens each fin ray, resultingin shortened fins (38). Althoughthe detailed molecular mecha-nism underlying this phenome-non has yet to be elucidated, it ispossible that the same mecha-nism functions in other processesof animal development. The aboveexamples are by no means anexhaustive list of the candidatescurrently being examined as po-tential biological Turing patterns.More detailed discussions areprovided in (4) and (6).

Identification of the specificdynamics of the RD system iscritical to showing the applica-bility of the Turingmechanism tothe formation of a given pattern.In the case of pigment patternformation, it was possible to dis-turb the pattern and observe theprocess of regeneration. In mostother systems, such observationis complicated because experi-mental perturbations may belethal. This is one reason why itis difficult to demonstrate RD ac-tivity in some biological systems.(In SOMText 1-4, we have sum-marized the important points forresearchers to keep inmindwhenthey use the RD model as theworking hypothesis.) However,recent technological advances inimaging technologies are assist-

ing such studies.Moreover, the artificial generationof Turing patterns in cell culture should be possiblein the near future as the result of synthetic biology(39). Turing was born in 1912 and published hisRD model in 1952. Although he was unable towitness the impact of his hypothesis on the workof contemporary biologists, we are hopeful thatwith an increased acceptance among experimentalbiologists of the principles he first elucidated, wewill see Turing’s mechanism take its place as amodel for the understanding of spatial patternformation in living systems.

References and Notes1. U. Alon, An Introduction to Systems Biology (Chapman &

Hall/CRC, Boca Raton, FL, 2006)2. R. M. May, Nature 261, 459 (1976).3. A. M. Turing, Bull. Math. Biol. 52, 153, discussion 119

(1990).

4. J. Murray, Mathematical Biology (Springer-Verlag, Berlin,ed. 4, 2003).

5. H. Meinhardt, The Algorithmic Beauty of Seashells(Springer-Verlag, Berlin, 1995).

6. H. Meinhardt, Models of Biological Pattern Formation(Academic Press, London, 1982).

7. H. R. Ueda, Cold Spring Harb. Symp. Quant. Biol. 72,365 (2007).

8. L. S. Song et al., Ann. N. Y. Acad. Sci. 1047, 99 (2005).9. C. V. Cabrera, Development 110, 733 (1990).

10. D. Dormann, C. J. Weijer, Development 128, 4535(2001).

11. C. Sardet, F. Roegiers, R. Dumollard, C. Rouviere,A. McDougall, Biophys. Chem. 72, 131 (1998).

12. M. P. Harris, S. Williamson, J. F. Fallon, H. Meinhardt,R. O. Prum, Proc. Natl. Acad. Sci. U.S.A. 102, 11734(2005).

13. A. R. Sanderson, R. M. Kirby, C. R. Johnson, L. Yang,J. Graphics GPU Game Tools 11, 47 (2006).

14. L. Wolpert, Principles of Development (Oxford Univ.Press, New York, 2006).

15. T. Gregor, D. W. Tank, E. F. Wieschaus, W. Bialek, Cell130, 153 (2007).

16. H. Meinhardt, A. Gierer, Bioessays 22, 753 (2000).17. H. Meinhardt, A. Gierer, J. Cell Sci. 15, 321

(1974).18. A. Nakamasu, G. Takahashi, A. Kanbe, S. Kondo, Proc.

Natl. Acad. Sci. U.S.A. 106, 8429 (2009).19. E. M. Rauch, M. M. Millonas, J. Theor. Biol. 226, 401

(2004).20. P. K. Maini, M. R. Myerscough, K. H. Winters,

J. D. Murray, Bull. Math. Biol. 53, 701 (1991).21. J. D. Murray, G. F. Oster, A. K. Harris, J. Math. Biol. 17,

125 (1983).22. N. V. Swindale, Proc. R. Soc. Lond. B Biol. Sci. 208, 243

(1980).23. M. Yamaguchi, E. Yoshimoto, S. Kondo, Proc. Natl. Acad.

Sci. U.S.A. 104, 4790 (2007).24. R. Asai, E. Taguchi, Y. Kume, M. Saito, S. Kondo,

Mech. Dev. 89, 87 (1999).25. S. Kondo, R. Asai, Nature 376, 765 (1995).26. D. M. Parichy, Semin. Cell Dev. Biol. 20, 63 (2009).27. H. S. Jung et al., Dev. Biol. 196, 11 (1998).28. S. Sick, S. Reinker, J. Timmer, T. Schlake, Science 314,

1447 (2006).29. M. V. Plikus et al., Nature 451, 340 (2008).30. T. Nakamura et al., Dev. Cell 11, 495 (2006).31. B. Thisse, C. V. Wright, C. Thisse, Nature 403, 425

(2000).32. S. A. Newman, H. L. Frisch, Science 205, 662

(1979).33. S. A. Newman, Int. J. Dev. Biol. 53, 663 (2009).34. T. Miura, K. Shiota, Dev. Dyn. 217, 241 (2000).35. S. P. De Langhe et al., Dev. Biol. 277, 316 (2005).36. R. Augustin et al., Dev. Biol. 296, 62 (2006).37. M. Watanabe et al., EMBO Rep. 7, 893 (2006).38. M. K. Iovine, E. P. Higgins, A. Hindes, B. Coblitz,

S. L. Johnson, Dev. Biol. 278, 208 (2005).39. D. Sprinzak, M. B. Elowitz, Nature 438, 443

(2005).40. We thank D. Sipp for suggestions and help in preparation

of the manuscript. We also thank H. Meinhardt,A. R. Sandersen, and P. Przemyslaw for figures reproducedfrom their original works.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/329/5999/1616/DC1SOM TextFigs. S1 to S4ReferencesComputer Simulation

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Hair follicle spacing

Hydra regeneration

Lung branching

Left-right asymmetry

Wnt Dkk

Lefty

?BMP

Nodal

TGF-�or

FGF

Feather bud spacing in chick

Tooth pattern

Lung branching

Skeletal pattern in limb

Short-range positive

feedback

Condensationof cells

Long-rangenegativefeedback

Fig. 4. Possible networks of protein ligands may give rise to Turingpatterns in the embryo. Shown above are candidates for the RDmechanism proposed by molecular experiments. For a detailedexplanation of each network, refer to the text and the articles listedin the references. In all these cases, the network is identical to that ofthe activator-inhibitor model proposed by Gierer and Meinhardt(17). Condensation of cells by migration into a local region causessparse distribution of cells in a neighboring region. This can alsofunction as long-range inhibition. (Note that the involvement of theRD mechanism in some of the phenomena above has not been fullyaccepted by experimental researchers.)

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