a cellular oscillator model for periodic pattern formation

*AuthE-mail: y

-CurrpartmenUniversiU.S.A.

J. theor. Biol. (2001) 213, 171}181doi:10.1006/jtbi.2001.2414, available online at http://www.idealibrary.com on

0022}51

A Cellular Oscillator Model for Periodic Pattern Formation

JOHANNES JAEGER*- AND BRIAN C. GOODWIN

Schumacher College, ¹he Old Postern, Dartington, ¹otnes, Devon ¹Q9 6EA, ;.K.

(Received on 30 March 2001, Accepted in revised form on 30 July 2001)

In this paper, we present a model for pattern formation in developing organisms that is basedon cellular oscillators (CO). An oscillatory process within cells serves as a developmental clockwhose period is tightly regulated by cell autonomous or non-autonomous mechanisms.A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillatorsfreezing into spatial order as the clocks slow down and stop at di!erent times or phases in theircycles. We apply a CO model to vertebrate somitogenesis and show that we can reproduce thedynamics of periodic gene expression patterns observed in the pre-somitic mesoderm. We alsoshow how varying somite lengths can be generated with the CO model. We then discuss themodel in view of experimental evidence and its relevance to other instances of biologicalpattern formation, showing its versatility as a pattern generator.

( 2001 Academic Press

1. Introduction

Repeated or periodic patterns are a very commontheme in development and evolution. A plant'sbasic body plan, for instance, consists of repeat-ing shoot and root elements. Many animals aresegmented or show striped or concentric patternson their skin. Up to the present, reaction}di!u-sion (RD) models are the most commonly usedtheoretical models to explain the biologicalformation of periodic patterns. RD modelsassume di!erential di!usion of two or more inter-acting morphogens in a growing tissue (see Tur-ing, 1952; Meinhardt, 1982, 1986, 1999).

There are several problems concerning theapplicability of RD models to biological patternformation. An interaction of di!using morpho-gens, as required by an RD mechanism, remains

or to whom correspondence should be [email protected] address: Graduate Program in Genetics, De-t of Molecular Genetics and Microbiology, Statety of New York at Stony Brook, NY 11794-5222,

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to be discovered in biological systems. Moreover,it is an open question if long-range di!usionof morphogens is a common phenomenon inbiological tissues. There is some evidence thatdi!usion of morphogens does indeed occur (seefor example McDowell et al., 1997). However,other cases have been reported where the mor-phogen is distributed through a complex andprobably energy-intensive transport mechanism(Entchev et al., 2000) or where the gradient isformed through cell proliferation and concomi-tant dilution of the morphogen (Pfei!er et al.,2000). Considering these complications, we seea strong need for alternative models of patternformation in cellularized tissues. In this paper, wediscuss such a theoretical mechanism based oncellular oscillators (COs), which is able to pro-duce various repeated or periodic patterns incellularized growing tissues.

The basic idea behind CO-type models is thatbiological pattern formation is based on preciselytimed developmental changes within cells in agrowing tissue. Developmental timing requires

( 2001 Academic Press

172 J. JAEGER AND B. C. GOODWIN

an accurate cellular clock. Such a clock mecha-nism is most likely based on cell-intrinsic bio-chemical oscillations. The cell cycle itself, with itsperiodic expression of cyclin genes, is a goodexample of such a cellular oscillation (Rosenthalet al., 1980). Tight control of the cell cycle iscrucial for normal development (see for exampleEdgar & Lehner, 1996) and a lack thereof canlead to severe malformations and cancer.

The cell cycle is by no means the only knowncellular oscillation. Goldbeter (1996) reviewedstudies on biochemical oscillations in nucleotidesynthesis and glycolytic pathways. Moreover,a number of genetic ultradian and circadian cel-lular oscillations have been described (Lloyd& Stupfel, 1991; Dunlap, 1999). Such cellularoscillations have periods that range from a fewseconds up to days. Therefore, cells show periodicbehaviour on multiple time-scales as all theseoscillations occur simultaneously in a living cell.Ways of using this intrinsic temporal order incells to generate spatial patterns in developingorganisms were explored some time ago byGoodwin & Cohen (1969) and Goodwin (1976).Now we examine the recent evidence for linkingcellular oscillations to pattern formation.

A connection between cellular oscillations andpattern formation has been most thoroughlyestablished for vertebrate somitogenesis. Theprocess of somitogenesis is highly conservedamong all vertebrates (McGrew et al., 1998).Somites are patches of cells that form on eachside of the developing neural tube in vertebrateembryos and go on to develop into vertebrae,axial muscles, blood vessels and dermal tissue(Gossler & Hrabey de Angelis, 1998). During gas-trulation in the chick embryo, primitive streakcells get incorporated into the pre-somiticmesoderm (PSM) through the movement of anorganizing region called Hensen's node alongthe embryonic axis. Somites form at the anteriorend of the PSM at a "xed distance from theprogressing Hensen's node. In the chicken, theformation of each somite takes about 90 min(Gossler & Hrabey de Angelis, 1998).

Electron microscopy and molecular studieshave revealed that the PSM shows a pre-patternbefore overt somite formation occurs (reviewed inGossler & Hrabey de Angelis, 1998). Interestingly,some of the genes involved in somitogenesis show

cycles of oscillatory gene expression in the PSM.In the chicken, c-hairy1 mRNA is initially presentat high concentrations in a broad domain in theposterior PSM (Palmeirim et al., 1997). Within90 min, this domain of c-hairy1 expression movesanteriorly through the tissue, becoming narroweras it progresses until it becomes stationary in theposterior half of the newly forming somite. Thiscycle of c-hairy1 expression is repeated for eachnew somite. It is not dependent on signallingfrom neighbouring tissues since it can still beobserved in cultured explants of PSM tissue(Palmeirim et al., 1997). Very similar oscillatorygene expression patterns have been discoveredfor the lunatic fringe homologues in chicken andmouse (McGrew et al., 1998; Forsberg et al.,1998) and the basic helix}loop}helix (bHLH)transcription factors c-hey2 in chicken (Leimeis-ter et al., 2000), HES1 in mouse (Jouve et al.,2000) and Her1 in zebra"sh (Sawada et al., 2000).

These oscillatory gene expression patternsstrongly suggest the presence of a molecularsegmentation clock in the PSM (Stern &Vasiliauskas, 1998; Dale & PourquieH , 2000). Sucha segmentation clock has been proposed byCooke & Zeeman (1976) in their Clock andWavefront Model. In this model, tissue cellsundergo a sudden change in their adhesive prop-erties at a certain stage of their development.Since more anterior cells are older than the moreposterior ones, a &wavefront' of di!erentiationwill sweep through the tissue in anterior to poste-rior direction. The periodicity of the segmenta-tion process is created by an oscillatory cellular&&clock'' that switches between a permissive andan inhibitory state. Only cells in the permissivestate are able to undergo the physiologicalchange. The wavefront therefore advances in dis-crete steps de"ned by the oscillation, which formssomite boundaries upon switching back to theinhibitory state (Cooke & Zeeman, 1976).

In a recent paper, Kerszberg & Wolpert (2000)have further developed the idea of an oscillatorysegmentation clock in their Clock and TrailModel. The authors assume a progress zoneposterior to the PSM. A progress zone*"rstproposed for a model of limb development(Summerbell et al., 1973)*is an undi!erentiatedregion of tissue growth that gives rise to all thecells in the newly formed tissue. In the Clock

PERIODIC PATTERN FORMATION 173

and Trail Model, all cells in the progress zoneoscillate in phase, acting as a segmentation clock.As cells leave the progress zone and enter thePSM, their oscillation stops and the phase atwhich the oscillation stops becomes "xed intoa spatial wave pattern, laying down a &trail' ofperiodic positional information (Kerszberg &Wolpert, 2000).

However, neither of the above models ad-dresses the dynamical waves of gene expressionobserved in the PSM. These gene expressionpatterns strongly resemble waves that can beobserved in oscillatory chemical #ow systems.Such #ow systems have been modelled using#ow-distributed oscillator (FDO) models (Kaern& Menzinger, 1999; Kaern et al., 2000a, b). FDOswere initially de"ned as in"nitesimal volumes ofan oscillating chemical reaction in a #ow. Bymodifying the oscillation period of the oscillatorsin the #ow as well as the initial oscillation at theupstream #ow boundary, di!erent kinds oftravelling or stationary waves can be created inthe #ow tube. These waves can actually beobserved in experiments using the oscillatoryBelousov}Zhabotinsky (BZ) reaction in a plug#ow tube (Kaern & Menzinger, 1999). Kaernet al. (2000a, b) suggest that such an FDOmechanism underlies vertebrate somitogenesis.In biological terms, the upstream #ow boundarycan be interpreted as the boundary betweena growth zone and the tissue itself and the #owcan be seen as a growing and di!erentiating tis-sue, in which cells &#ow' in at one boundary andleave the tissue again at the other end (Kaernet al., 2000b). The FDOs therefore correspond tooscillating cells in the tissue. In this way, theFDO model can accurately explain the observedperiodic gene expression patterns in the PSM.Moreover, its conditions (#ow, boundary andoscillation) readily apply to a wide range of bio-logical processes. However, the FDO model isformulated as a model of #uid dynamics (Kaern& Menzinger, 1999). It is somewhat di$cult totranslate into biological terms and cannot easilybe modi"ed or extended to suitably model vari-ous phenomena of biological pattern formationother than somitogenesis or segmentation. In ourview, this justi"es an attempt to simplify andreformulate the FDO mechanism in biologicalterms.

In this paper, we suggest a cellular oscillator(CO) model of periodic pattern formation thatattempts to combine the versatility of RD mod-els, the clarity of the &Clock and Trail'Model andthe ability to explain dynamic gene expressionpatterns of the FDO model. After a general de-scription of the model in Section 2, we discuss itsapplication to vertebrate somitogenesis (Sec-tion 3) and its concurrence with experimentalresults (Section 4). Finally, we brie#y discuss itsapplication to other biological pattern formingprocesses, such as segmentation in arthropods orannelids (Section 5) and suggest that CO-typemodels can be applied to a wide range of bio-logical pattern forming processes (Section 6).

2. Model: Cellular Oscillators

CO-type models are based on cellular oscil-lators in a growing tissue. By de"nition, cellularoscillators are cells that show intrinsic oscilla-tions in some physiological and/or structuralproperty. Such oscillations can occur at variousfrequencies simultaneously. For simplicity, wewill call the oscillating property relevant to theproblem at hand the cell state. The exact molecu-lar nature of the cell state is not crucial to themodel. It can be thought of as the concentrationof a metabolite, second messenger, protein orRNA gene product or the state of the post-trans-lational modi"cation of a protein. The cell statecan also be any combination of several or all ofthe above possibilities.

In a CO-type model, the cellular oscillationplays two important roles. First, the oscillationserves as a biochemical clock. We assume that thenumber of oscillation cycles the cell has under-gone can be &stored' and &read' by the cell, therebyindicating its physiological age. There are severalpossible biochemical mechanisms for this, includ-ing accumulation of a long-lived gene product(Schnell & Maini, 2000) or context-dependentactivation of downstream genes by a short-livedtranscription factor. However, the clock mecha-nism does not have to involve regulation of geneexpression. A post-translational biochemicalclock could be based on the accumulation ofa modi"ed form of a protein as observed forcyclin-dependent kinases during the cell cycle(reviewed in Johnson & Walker, 1999) or the

FIG. 1. The CO model of somitogenesis: The PSM isrepresented as a single row of cells with anterior (A) to theleft and posterior (P) to the right. Each row in the "gurerepresents the progress zone (discs) and the cells of the PSM(squares) at a speci"c moment in time. Tissue growth onlyoccurs at the posterior-most end of the tissue where theprogress zone is located (represented by a single disc). Cellsenter the PSM from the progress zone at a given rate (onecell per time step in this "gure). For simplicity, cells whichget incorporated into the PSM will only undergo two moreoscillation cycles before the oscillation stops abruptly. Seethe main text for details.


cyclic destruction of an inhibitor of gene expres-sion, similar to the mechanism of Hox gene regu-lation proposed by Kondo et al. (1998).

The second role of the oscillation in the tissueis to lay down a spatial periodic pattern of di!er-ential cell states as the tissue grows. The oscilla-tion of the cell state slows down and comes toa halt as cells grow older and di!erentiate. Inthis way, the oscillation of cell states can become"xed as a periodic spatial pattern in the tissue ifneighbouring cells are at di!erent states ofthe cycle when the oscillation stops. This canthen lead to morphological di!erentiation andboundary formation in the tissue (see Kerszberg& Wolpert, 2000; Kaern et al., 2000b for furtherdiscussion).

In the way described above, cellular oscilla-tions can lead to pattern formation in any kind ofcellularized growing tissue. Tissue growth canoccur either through cell divisions within thetissue or through incorporation of cells froma growth or progress zone (Summerbell et al.,1973). Therefore, the resulting periodic patterncan take many di!erent forms depending on themode and rate of tissue growth and the period ofthe cellular oscillation. For example, stripes andsegments can be formed if a tissue of cellularoscillators grows apically along a single axis.More complex patterns, such as concentric ringsor curved stripes, can be produced if we assumegrowth within the tissue at varying rates in twoor three dimensions. Evidently, the shape of theresulting patterns also depends on the oscillationperiod. If the oscillation is fast compared to thetissue growth rate, narrow stripes will result,whereas wide bands or gradients of di!erent cellstates can be produced with a slow oscillation.

Cellular oscillations can be triggered orin#uenced by events outside the cell such as inter-actions with the extracellular matrix or neigh-bouring cells. Such extracellular in#uences on theclock can easily be incorporated into a CO modelas escapement mechanisms that reset the internalclock. Furthermore, the CO model allows easyintegration of other developmental mechanismssuch as cell migration or tissue rearrangements.Therefore, CO-type models are biologicallyplausible and easily adaptable, which makesthem perfectly suited for modelling a wide rangeof biological pattern formation processes.

3. Model: Somitogenesis

As an example for pattern formation based oncellular oscillations, we introduce a CO model forvertebrate somitogenesis. Figure 1 shows anoverview of our model. The PSM is representedby a row of cells that grows at its posterior end,where Hensen's node is located. Kerszberg& Wolpert (2000) have suggested that Hensen'snode can be seen as a progress zone, in which allcells oscillate in phase. Following their proposal,we assume a progress zone at the posterior end of

FIG. 2. CO computer simulations: The left panel showsa simulation in which the tissue growth rate remains con-stant. In the right panel, the growth rate decreases in time sothat the width of the somites becomes narrower, as observedin mammalian tail somites. Anterior is to the left, posteriorto the right, while the vertical axis is time. Single lines ofthe graphs represent the PSM at each moment in time. Thetissue expands posteriorly (to the right) as it grows. Theprogress zone is located at the posterior end of the tissue.The grey zone at the bottom of the graph represents space inwhich no cells have grown yet.


the tissue, which is represented as a single cellularoscillator (discs in Fig. 1).

Cells enter the PSM from the progress zone ata certain growth rate, where they become tissuecells (squares in Fig. 1). Upon crossing the borderfrom Hensen's node into the PSM, the cells oscil-late initially with the same period as the progresszone. After a certain time span, the cellular oscil-lations become slower and stop. For simplicity,the tissue cells in Fig. 1 undergo two oscillationsafter entering the tissue before they stop abruptly.In this way, the temporal periodicity of the pro-gress zone's oscillation becomes translated intothe spatial periodicity of the somites. This peri-odic pattern of cell states can then lead to somiteformation supposing that each cell can di!eren-tiate further according to its cell state (see alsoKerszberg & Wolpert, 2000).

We have carried out computer simulations tostudy stripe formation in the CO model in moredetail (see Appendix A for mathematical formula-tion; also compare to similar simulations inKaern et al., 2000b). The output generated bythese simulations roughly corresponds to the de-piction of the model in Fig. 1. The tissue startsgrowing as a single cell at the bottom of eachgraph and expands posteriorly as we move up-wards along the time axis (Fig. 2). The progresszone is located at the posterior end of the tissue.Cells that enter the PSM undergo a number ofslowing oscillation cycles, i.e. the period of thetissue cells follows an exponential function (seealso Appendix A, Fig. A2). In this way, initiallyuniform oscillations in the tissue become spatialstripes in the tissue after a certain time delay.

The width of the resulting somites is propor-tional to the oscillation period in the progresszone and/or the growth rate of the tissue (see alsoAppendix A). If we assume a constant growthrate, the resulting somites will be evenly spaced ascan be seen in the left panel of Fig. 2. On the otherhand, a variable growth rate will produce somitesof varying size. The right panel of Fig. 2 showsa pattern similar to the one observed in manymammals, where a few wide lumbar vertebraestand in contrast to many narrow tail vertebrae.This can be achieved in CO-type models bysimply reducing the growth rate over time (seealso Kerzberg & Wolpert, 2000; Kaern et al.,2000b).

The slowing down of the oscillation produceswaves of varying cell states that move throughthe PSM tissue. If we trace the developing PSMin Fig. 2 by following single rows of cells frombottom to top in the graph, we observe broadinitiation of the waves at the posterior end of thetissue. After a while, the wave narrows as it movesanteriorly and slows down exponentially. Notethat the wave never actually stops (Fig. 2), due tothe recursive interdependence between oscilla-tion period and the cell's physiological age (seeAppendix A for details). However, we do notthink that this poses a serious di$culty to themodel. The wave's velocity becomes su$cientlylow to maintain the spatial pattern within thediscrete cellularized tissue for a signi"cantamount of time. In fact, the amount of timeduring which the pattern stays stable increases


exponentially as the wave slows down. We arguethat this &&quasi-stability'' is su$cient for thelimited time spans and precision required fordi!erentiation and boundary formation in arelatively small cellularized tissue as encounteredin an embryo.

4. Model: Comparison with Experiment

The time between the initiation of one wave ofgene expression and another corresponds pre-cisely to the time required for one spatial stripe toform in the simulated tissue. This dynamical pat-tern shows a very strong resemblance to thewaves of c-hairy1 and lunatic fringe expressionobserved by Palmeirim et al. (1997) and McGrewet al. (1998) in the chick embryo. Moreover, a cellintrinsic oscillation is consistent with the fact thatoscillatory c-hairy1 expression continues in cul-tured explants of PSM tissue (Palmeirim et al.,1997).

This CO model of somitogenesis assumes thatthere is an oscillating progress zone, which is inthis case Hensen's node. Recent evidence seemsto support such an assumption, since somite pre-cursor cells that show periodic mRNA expressionof several cycling genes have been discoveredin Hensen's node (Dale & PourquieH , 2000).However, it is not clear if a sharp boundarybetween progress zone and PSM exists. There-fore, it might be possible that cells enter thePSM at slightly di!erent physiological ages.This does not pose a serious problem to theCO model, as long as these age di!erences aresmall compared to the oscillation period of thecells.

A CO mechanism for somitogenesis is consis-tent with the size regulation observed in embryosof the clawed frog Xenopus laevis. Xenopus em-bryos that are reduced in size retain the normalnumber of somites (Cooke, 1975). In reducedembryos, the progress zone is proportionallysmaller than in a normal-sized one, meaning thatless cell divisions occur and the growth rate of thePSM becomes smaller. As we have mentionedabove, the width of the resulting stripes is directlyproportional to the growth rate. As the PSMgrows to only half the size of a normal PSM, andits growth rate is also halved compared to nor-mal development, the number of somites formed

is not altered. However, they are narrower thansomites in normal-sized embryos.

A linkage between the cellular oscillation andthe cell cycle (see Keynes & Stern, 1988; Collieret al., 2000) is supported by heat shock experi-ments performed by Primmett et al. (1989).A single heat shock on the second day of chickembryonic development results in periodicanomalies in every 6}7 somites that are formedlater in development. The 10 hr or so that it takesto form six somites corresponds roughly to theperiod of the cell cycle in the PSM (Primmettet al., 1989). Such a connection between segmen-tal clock and cell cycle is completely consistentwith the CO model, as long as the period of theclock is di!erent from the period of the cell cycle.The recurring defects observed in the heat shockexperiments can be explained by a lastingdamage to a speci"c phase of the cell cycle, whichalso a!ects the oscillation that underlies somiteformation. For example, there could be heat inac-tivation of a stable component common to cellcycle and segmental clock.

5. Model: Arthropod and Annelid Segmentation

We suggest that a CO mechanism might alsoapply to segmentation in invertebrates. However,the experimental evidence for a CO mechanismof segmentation in invertebrates is rather scarce.Short-germ insects, crustaceans, spiders, myria-pods and annelids form their segments in a man-ner very similar to vertebrate somitogenesis.New stripes of engrailed expression and latermorphological segments are formed at a "xeddistance from an apical growth zone in the abdo-men in various species of these taxa (Patel et al.,1989; Wedeen & Weisblat, 1991; Gilbert, 1997;Shankland & Savage, 1997). This is consistentwith a CO mechanism as described for vertebratesomitogenesis. Moreover, some centipedes showalternating small and large segments along theirbody axis (Barnes et al., 1993). Such a pattern caneasily be generated using the CO model, by intro-ducing an oscillation in the tissue's growthrate or the period of the progress zone oscillator(Fig. 3).

Fragmentation and heat-shock experimentsin the locust Schistocerca gregaria have been per-formed that are consistent with a CO mechanism

FIG. 3. Invertebrate segmentation: The left panel showsa simulation, in which the progress zone's oscillation periodalternated between two values. This produces a pattern ofalternating small and large segments, as observed in manycentipedes (see Barnes et al., 1993). The right panel shows thee!ect of heat shock (hs) on Schistocerca segmentation.During the heat shock, cell division ceases, but the cellularoscillation continues, causing a gap in the segmentationpattern.


of segmentation. Fragmentation experimentshave shown the presence of a growth or progresszone in the apical abdomen that is required forthe segmentation process (Mee, 1986). Moreover,heat shocks applied to Schistocerca embryos dur-ing segment formation cause deletions about 2}3segments posterior to the last visible segment atthe time of the heat shock (Mee & French,1986a, b). Mee & French (1986a) mention thatheat shocks can temporarily inhibit cell divisionin Schistocerca embryos. If cell division is sud-denly stopped in our CO model without inter-rupting the cellular oscillation, gaps will occur inthe resulting pattern (Fig. 3). This would accountfor the observed deletions in the abdominalsegments.

Although the morphological similarities be-tween the mode of segmentation in vertebratesand short-germ insects are striking, there is nomolecular evidence for a CO mechanism in Schis-tocerca, as of yet. No oscillatory gene expressionpatterns have been detected in the locust's

abdomen so far. Detection of such oscillationswould be possible through in situ hybridizationor antibody staining. However, up to the present,molecular analyses of segmentation in invert-ebrates or insects other than the fruit #yDrosophila melanogaster have been rather scarceand in most cases their temporal resolution re-mains too limited to detect cellular oscillations.

6. Conclusions

In this paper, we propose that cellular oscil-lators might underlie a wide variety of develop-mental processes. By applying a CO-type modelto vertebrate somitogenesis, we show that it isconsistent with morphological and molecular ex-perimental evidence and can explain a variety ofobserved phenomena such as size regulation inXenopus embryos and varying somite sizes alongthe mammalian embryonic axis. Since we con-sider the dynamics of developmental processes ascrucial to the understanding of biological mecha-nisms of pattern formation, we emphasize themodel's capacity to faithfully reproduce periodicgene expression in the PSM.

The versatility of CO models means that theyare able to produce many of the patterns thathave thus far been explained using reaction}dif-fusion models (see Meinhardt, 1982, 1986, 1999).Oscillatory behaviour is known to occur in reac-tion}di!usion systems (see for example Muratov,1997), so there is a formal equivalence betweensystems based on di!erent molecular processes.However, CO-type models are based on simplerbiological assumptions than reaction-di!usion.This allows for easier biological interpretation ofthe model and more accurate predictions forguiding experimental research.

In contrast to the Clock and Wavefront Model(Cooke & Zeeman, 1976) and the Clock and TrailModel (Kerszberg & Wolpert, 2000), CO-typemodels are able to correctly reproduce thedynamics of gene expression in the pre-somiticmesoderm. We argue that this is an importantdi!erence, since we consider the temporal dy-namics of developmental processes to be crucialfor pattern formation.

Basically, the CO-type model proposed here isa simpli"cation and reformulation of the FDOmodel proposed by Kaern & Menzinger (1999;


see also Kaern et al., 2000a, b). One major ad-vantage of our reformulation is the transforma-tion of the frame of reference from a #ow of cellsto a stationary growing tissue. A second advant-age is that the CO-type model proposed hereonly requires one single oscillatory process thatvaries over time, whereas the FDO model as-sumes two independent oscillations, one at the#ow boundary and the second one within the#ow (Kaern & Menzinger, 1999). Since cells arecontinuous dynamical systems, in which the cellstate strongly depends on the previous states ofthe cell, it is not realistic to assume the indepen-dence of the two oscillations. Therefore, onlya subset of possible FDO mechanisms apply togrowing tissues, (see also Kaern et al., 2000b).This subset corresponds to the CO-type modelpresented here. A third advantage of our formu-lation of the model is that cellular age sensing isa cell-intrinsic physiological process based on thenumber of oscillations a cell has undergone. Tous, this seems to be a more realistic assumptionthan cells measuring &&objective'' or &&physical''time independent of their own state.

We are aware of the fact that by keeping themodel as general and simple as possible we haveneglected potential e!ects of cell signalling andadhesion. Furthermore, our model does not ad-dress underlying molecular mechanisms nor laterstages of somitogenesis, where somite polaritybecomes established (see Kerszberg & Wolpert,2000, for further discussion). Our aim here was topropose a conceptual sca!old for modelling peri-odic processes during morphogenesis, which isphrased in simple mathematical terms to makeCO models easily understandable and modi"ablefor a wide range of biological pattern formingprocesses.

CO mechanisms are easily detectable by expe-riment. If a CO mechanism does indeed apply toa developmental process, characteristic periodicwaves of gene expression, protein modi"cation ormetabolite concentration, which are dependenton the tissue's growth rate, will be observed in thegrowing tissue. Such expression patterns have asyet only been described for genes involved invertebrate somitogenesis. However, we suggestthat the assumptions underlying CO-type modelsare likely to apply to many other developingtissues as well. There is growing evidence that

cellular oscillations play a crucial role in variousdevelopmental processes and might underlie theprecise timing that is required for normal growthand development. Therefore, we hope thatCO-type models will stimulate not only noveltheoretical models of pattern formation but alsonew experimental approaches to the analysis ofperiodic cellular processes.

We would like to thank Nick Monk, Hilde Jans-sens, Michael Menzinger, Peter Dearden and MichaelAkam for discussion, criticism and/or sharing resultsprior to publication.

This research (JJ) was funded by a Roche ResearchFoundation fellowship. The algorithm of the COmodel of somitogenesis was implemented in C ona Macintosh G3 PowerBook using the CodeWarriorintegrated programming environment. Graphicaloutput was produced with Mathematica 4.0.

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APPENDIX A

A CO Model of Somitogenesis

The Cellular Oscillator (CO) model we pro-pose here is a discrete model of pattern formationin a one-dimensional cellularized tissue. Thecomputer simulations described in this paperwere implemented as follows. Each tissue cell isa cellular oscillator with its own phase / and

FIG. A1. Behaviour of the oscillation period in tissuecells. Each tissue cell's period increases exponentially aftera given delay upon entering the tissue. This causes theoscillation to slow down and eventually stop.


its current oscillation period ¹, which areinterdependent. At one end of the tissue thereis a progress zone, from which new cells getincorporated into the tissue. Each new tissuecell inherits phase / and current period ¹

from the progress zone. Since all cells in theprogress zone oscillate in phase, it is representedas one single cellular oscillator. The tissuecells and the progress zone show distinct time-dependencies of their oscillation periods. Ateach discrete time step t the following threeactions are carried out for each cell and theprogress zone:

1. Each cell's physiological age a is de"ned asthe increase of the cell-internal phase D/, i.e. thenumber of oscillation cycles a cell has undergonesince it has left the progress zone:

a(t)"/(t)!/0

Note that a is not identical with the cell's &&objec-tive'' age Dt"t!t

0, which cannot be measured

by the cell itself, but only by an observer outsidethe cell.

2. Each tissue cell's period ¹tis determined as

a function of its physiological age a:

¹t(a)"¹

t0#e(A(a~B))

where ¹t0

is the initial period at the timewhen the cell left the progress zone. A and Bare constants that determine the pattern ofslowing down and stopping of the oscillation.Thus, each ¹

tincreases with time as shown in

Fig. A1.The oscillation period of the progress zone¹

pcan be any function of the general form

¹p(t)"¹

p0#f (t)

where ¹p0

is the initial period of the progresszone at the beginning of the simulation runand f (t) is any linear, exponential or periodicfunction that appropriately describes the physio-logical oscillatory behaviour of the progresszone. Thus, ¹

pcan vary over time as shown in

Fig. A2.

3. In our CO model, the oscillator's phase / isde"ned as

/(t)"/0#

t+i/t0

1¹

i

where /0

is the cell's initial phase at time t0

whenit left the progress zone. Thus, / is updated atevery time step t according to

/(t)"/(t!1)#1

¹(a (t))

By repeating the above three steps, we generatea two-dimensional array of phases / for each cellat each time step t. The cell state z of each cell isnow a periodic function of these cellular phases/. The exact nature of the periodic functiondescribing the cell state is not crucial to the

FIG. A2. Possible oscillatory behaviour of the progress zone. In the most simple case (top left) the period remains constant.However, the oscillation period of the progress zone can vary linearly, exponentially (see Fig. A1) or periodically over time.


model. We have chosen a simple harmonic cosinefunction that produces values between !10and #10:

z(/)"10 cos(2n/)

The use of a sinusoidal function may be unreal-istic for biological oscillators with a complexdynamical behaviour. However, the use of morerealistic nonlinear periodic functions, like sawtooth or square waves, would not fundamentallyalter the results of this CO model.

As the tissue grows with a growth rate of r, itinitially shows uniform oscillations in all cell

states. Over time, as the older cells slowdown their oscillations, stable spatial wavepatterns will form in the tissue. Their wavelengthdepends on the tissue's growth rate r and theprogress zone's oscillation period ¹

pat the time

when the tissue cells were "rst incorporated intothe tissue:

j"r¹p

Thus, patterns of variable wavelength caneasily be created by varying the oscillation periodof the progress zone over time.

a cellular oscillator model for periodic pattern formation

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