J. theor. Biol. (1999) 197, 295–330Article No. jtbi.1998.0876, available online at http://www.idealibrary.com on

0022–5193/99/007295+36 $30.00/0 7 1999 Academic Press

A Mathematical Model for Outgrowth and Spatial Patterning of theVertebrate Limb Bud

R D* H G. O†

*Department of Mathematics, Washington State University, Pullman, WA 99163,U.S.A. and †Department of Mathematics, University of Utah, Salt Lake City,

UT 84108, U.S.A.

(Received on 1 June 1998, Accepted in revised form on 23 November 1998)

A new model for limb development which incorporates both outgrowth due to cell growthand division, and interactions between morphogens produced in the zone of polarizingactivity (ZPA) and the apical epidermal ridge (AER) is developed and analysed. Thenumerically-computed spatio-temporal distributions of these morphogens demonstrate theimportance of interaction between the organizing regions in establishing the morphogeneticterrain on which cells reside, and because growth is explicitly incorporated, it is found thatthe history of a cell’s exposure to the morphogens depends heavily on where the cell originatesin the early limb bud. Because the biochemical steps between morphogen(s) and geneactivation have not been elucidated, there is no biologically-based mechanism for translatingthe spatio-temporal distributions of morphogens into patterns of gene expression, but severaltheoretically plausible functions that bridge the gap are suggested. For example it is shownthat interpretation functions based on the history of a cell’s exposure to the morphogens canqualitatively account for observed patterns of gene expression. The mathematical model andthe associated computational algorithms are sufficiently flexible that other schemes for theinteractions between morphogens, and their effect on the spatio-temporal pattern of growthand gene expression, can easily be tested. Thus an additional result of this work is acomputational tool that can be used to explore the effects of various mutations andexperimental interventions on the growth of the limb and the pattern of gene expression. Infuture work we will extend the model to a three-dimensional representation of the limb andwill incorporate a more realistic description of the rheological properties of the tissue mass,which here is treated as a Newtonian fluid.

7 1999 Academic Press

1. Introduction

1.1.

Limb development is a model system for thestudy of tissue growth, pattern formation anddifferentiation, both from the experimental and

theoretical viewpoints. Limb development inbirds (primarily chick), mammals (primarilymouse) and amphibians has been studiedextensively for over 70 years, and in chick thereis a substantial base of experimental informationon which to build mathematical models [re-viewed in Maini & Solursh (1991), Tickle &Eichele (1994) and Duprez et al. (1996)]. Toindicate what a model of the type we develop

E-mail: dillon.theta.math.wsu.edu; othmer.math.utah.edu

DV

Flank AP

PD

Wing tip

(a)Humerus

Radius

Ulna

Digits

4

2

3

(b)

. . . 296

must eventually incorporate, we discuss chicklimb development in some detail in theremainder of the Introduction. However,the model we present does not incorporate allthese details, since much of what is knownabout spatio-temporal patterns of gene ex-pression in this system is still very qualitative.Thus the model we develop later should beviewed as the first step in an evolutionaryprocess, and one of our long-range objectives isto develop a computational model for growing,deforming tissues in which species are producedand diffuse about. Our aim is to provide acomputational tool that can be used to donumerical experiments both on chick limbpatterning and in other contexts where growthand patterning occur simultaneously. We beginwith a two-dimensional model in this paper,but will extend it to three space dimensions infuture work.

In chick, the wing site in the flank of theembryo is determined by Hamburger–Hamiltonstage 8 (26–29 hours of incubation*), theanterior–posterior orientation of the bud isdetermined by about stage 11 (Hornbruch &Wolpert, 1991), limb outgrowth is visible bystage 17, and the first skeletal element is visibleat stage 24. Pattern formation, by which wemean the establishment of spatial differences ingene expression and cell differentiation, isdescribed relative to three axes, the proximo-distal (PD) axis, which extends from somitesto wing tip, the anterior–posterior (AP) axis

and the dorsal–ventral (DV) axis [cf. Fig. 1(a)].The skeletal elements [humerus, radius andulna, wrist, and digits; cf. Fig. 1(b)] formin a proximal–distal and posterior–anteriorsequence. Fate maps show that the anterior halfof the limb bud gives rise to part of the humerus,the radius, and digit 2, while the posterior halfgives rise to part of the humerus, the ulna, anddigits 3 and 4 (Stark & Searls, 1973).

The avian limb bud stems from a thickening ofthe somatic layer of the lateral plate mesoderm,due to proliferation of mesenchymal cells andperhaps a localized decrease in the cellproliferation rate on either side of the site offuture outgrowth (Searls & Janners, 1971; Vogelet al., 1996). In the early stages mesenchymalcells provide the signals to initiate the process ofoutgrowth, and one or more members of thefibroblast growth factor family (collectively,FGFs, principally FGF-1, FGF-2, FGF-4, andFGF-8) may be a signaling molecule. Forexample it has been shown that beads soakedwith FGF-1, FGF-2 and FGF-4 can induceectopic limb bud outgrowth when implantedunder the ectoderm (Cohn et al., 1995), but morerecent evidence suggests that these are not theprimary inducers of limb outgrowth in vivo[reviewed in Vogel et al. (1996)]. Fgf-8transcripts are found in the prelimb field beforeoutgrowth in mouse (Crossley & Martin, 1995)and chick (Crossley et al., 1996), and ectopicapplication of FGF-8 to flank tissue inducesectopic limb formation (Vogel et al., 1996), butFgf-10 expression precedes this (Ohuchi et al.,1997). In addition, production of retinoic acid(RA) is high in prospective wing bud tissue(Helms et al., 1994), but whether RA is a

* The average stage length is 4 hours up to stage 23, andabout 6 hours thereafter.

F. 1. (a) The orientation of axes used to describe the limb; (b) a schematic of the adult wing skeleton in chick.

(a)

ZPA

AER

(b)

297

F. 2. A schematic of the limb bud, showing the AER and ZPA at (a) approximately stage 18 and (b) stage 25 in chick.At stage 24 the limb is approximately 1 mm long in both the PD and AP directions and 0.5 mm long in the DV direction.

primary inducing molecule or whether it has aneffect on cell–cell communication is not under-stood. Recent evidence (Ogura et al., 1996; Luet al., 1997) suggests that RA is required for theestablishment of the zone of polarizing activity(ZPA), a specialized group of cells that lies at theposterior margin of the limb bud (Zwilling, 1961;Hinchliffe & Griffiths, 1984).

Just as the morphological features can bedescribed relative to three axes, so also can thesignaling that is thought to control growth andto determine the spatio-temporal pattern of geneactivation. Continued proliferation and out-growth proximo-distally depends on interactionsbetween the mesenchymal cells and the overlyingectoderm. The ectoderm at the distal tip of thelimb bud forms the apical ectodermal ridge(AER), which is apparently determined by theboundary between ectodermal cells which ex-press the factor radical fringe and those that donot (Johnson & Tabin, 1997). However earlyoutgrowth of the limb bud and differentiation ofthe AER are independent, since mutations of thelimb deformity (ld) gene disrupt the latter but notthe former (Haramis et al., 1995), but this samemutant shows that localization of the AERis also dependent on dorsal–ventral polarity(Grieshammer et al., 1996; Kuhlman &Niswander, 1997); see also (Niswander, 1997).The dorsal–ventral polarity is itself determinedby unknown signals from the somites and the

lateral somatopluere at an earlier stage (Michaudet al., 1997).

In a normal limb the AER does not extendover the entire tip, but occupies only theposterior portion of the tip (cf. Fig. 2). Removalof the AER between stages 18 and 28stops outgrowth of the limb and leads to atruncated limb with distal skeletal deficiencies(Summerbell, 1974a). Non-AER ectoderm isnecessary for growth in pre-stage 16 wingmesenchyme, and ectoderm from either stage 16or 24 inhibits chondrogenesis in stage 24mesenchyme (Solursh & Jensen, 1988). Thestimulatory effect requires cell contact, whereasthe inhibitor is apparently a diffusible molecule.In some of the polydactylous mutants such astalpid3, the AER extends across the entire wingtip, and transplants of the mutant mesodermaltissue into a normal ectodermal sleeve induces ananterior extension of the AER and duplication ofdigits (MacCabe et al., 1975; Wolpert, 1976).The polydactylous mutants produce a limb budthat is wider in the AP direction than normallimb buds, perhaps as a result of the extendedAER.

The rapidly dividing mesodermal cells adja-cent to the AER form the so-called progress zone(PZ). Recent evidence suggests that a product ofthe homeobox gene Msx1 maintains cells in theprogress zone in an undifferentiated, rapidlyproliferating state, while more proximal cells

. . . 298

begin to differentiate (Robertson & Tickle,1997). Expression of Msx1 is regulated by asignal from the AER, probably one of the FGFfamily. In addition, the bone morphogeneticprotein Bmp-2 is also localized in the AER (atleast in mouse) and this may inhibit proliferation(Niswander & Martin, 1993). There is apparentlylittle influence of more proximal tissue on cells inthe progress zone, for if the tip of an early limbbud is grafted onto a late stage limb theproximo-distal sequence of skeletal elementsappropriate to the tip stage is produced by thegraft, while the converse graft produces deletionsof proximo-distal elements (Summerbell et al.,1973).

The AER is maintained by a factor producedeither in the progress zone or in a specializedgroup of cells, the zone of polarizing activity(ZPA), that lies at the posterior margin of theprogress zone (Zwilling, 1961; Hinchliffe &Griffiths, 1984). At the onset of outgrowth theZPA is located near the flank on the posteriormargin of the bud [Fig. 2(a)], but as outgrowthproceeds the region of maximal ZPA activitymoves progressively forward [Fig. 2(b)]. Thus the‘‘ZPA-ness’’ of tissue depends in part onproximity to the AER. Transplants of the ZPAto the anterior margin of the limb usually lead toduplication of skeletal elements, the pattern ofwhich depends on the location of the transplantrelative to the ZPA and the AER (Wolpert, 1987;Tickle et al., 1975). Only cells in the progresszone can respond to the polarizing action of theZPA (Summerbell, 1974b), and functional gapjunctions are required for communicationbetween ZPA cells and anterior mesenchyme,since blocking antibodies to gap junctionalproteins prevent ZPA-induced limb duplications(Allen et al., 1990). In light of the fact that ZPAcan induce extra digits, one might suppose thatthe polydactylous mutants contain additionalZPA tissue in the anterior part of the limb, butthis has been ruled out (Tickle, 1980). Instead itis the response of mesenchymal cells to thenormal ZPA signal, rather than the presence ofadditional ZPA tissue, that is altered in thesemutants (Tickle, 1980). In ld mutants the

polarizing activity of the ZPA is reduced, theAER cells fail to differentiate into their typicalcolumnar shape, and there are truncations in theautopod region (Haramis et al., 1995).

The polarizing activity of the ZPA is in turnmaintained by a factor, probably one of the FGFgrowth factors, that may be produced in theAER. Removal of posterior AER is followed bya decline in polarizing activity of the ZPA, butthe addition of FGF-4 soaked beads to posteriortissue in the absence of the ridge maintainspolarizing activity, and outgrowth of the limbbud continues under these conditions (Vogel &Tickle, 1993).

In addition to ectodermal tissue, there arethree major cell types present at later stages oflimb development: muscle cells, fibroblasts(which form the connective tissue, tendons, etc.)and pre-cartilage cells (chondrocytes). Themuscle cells are known to originate in somitictissue and to then migrate into the limb.However, the chondrocytes and fibroblasts botharise from determination of mesenchymal cells inthe progress zone. While the emphasis in theliterature is primarily on the spatial pattern ofchondrogenesis, it should be kept in mind thatthe spatial pattern formation problem has twoaspects: one is to produce the proper pattern ofcartilage anlage, which then lead to the bones,but the other is that the correct spatial pattern ofconnective tissue must also be produced. If allmesenchymal cells are destined to becomefibroblasts, and determination of cells aschondrocytes is merely a derailment of that fate,then the emphasis on the pattern of chondro-genesis is justified. However this has not beendemonstrated to date.

1.2.

Earlier work suggested that retinoic acid (RA)might be a morphogen* produced in the ZPA(Smith et al., 1989). An implanted bead whichreleases RA at the anterior margin of the limbproduces a digit pattern that is dose-dependent(Tickle et al., 1985): low concentrations lead toa normal digit pattern, higher concentrationsproduce supernumerary digits, but still higherconcentrations lead to wings in which only thehumerus and a knob of cartilage are formed.

* A diffusible substance that induces a concentration-de-pendent response at some step in the patterning process.

299

Tickle et al. (1985) showed that posteriorimplants give rise to anterior concentrations highenough to specify an additional digit 2, butdespite this the digit pattern is normal. Thaller &Eichele (1987) showed that the concentration ofRA in vivo is graded in the PA direction. It hasbeen found that there is also a gradient of acytoplasmic RA-binding protein (CRABP)(Maden et al., 1988) in the AP direction,opposite that of the RA gradient, the net effectof which is to steepen the RA gradient. Atpresent it is also not known whether RA affectsgap junctions in chick limb mesenchyme,although it is known to have a biphasic effect inother systems, with enhancement at levelscomparable to those found in chick limb (Mehtaet al., 1989; Allen et al., 1990). As we notedearlier, recent evidence suggests that RA isrequired for the establishment of the ZPA(Ogura et al., 1996; Lu et al., 1997).

Currently it is thought that FGF-4 or anothermember of this family is one of the morphogens.Another is Sonic hedgehog (Shh), a proteinsecreted by cells in the ZPA. Shh expression isfirst detected at stage 17, during initiation of limbbud formation (Riddle et al., 1993). Thereafter,Shh expression matches the location of the ZPAas determined by Honig & Summerbell (1985),both in position and in intensity of expression.Sonic is the probable mediator of polarizingactivity within the limb bud, because in additionto the fact that its domain of expressioncolocalizes with the ZPA, Shh is sufficient toconvey polarizing activity when misexpressed inthe limb bud. Shh may itself be localized near theZPA, but its range of influence is notunequivocally determined. The ‘‘active’’ N-terminal fragment is tethered to the cell, butother factors may also be involved. In Drosophilait has recently been shown that Hh binds topatched and a newly-discovered receptor tout-velu, which together determine the range of Hhsignaling (Bellaiche et al., 1998), and a similarsecondary factor may be found in chick limb aswell. When applied to the anterior portion of thelimb Sonic can induce mirror-image duplicationsof digits, and Sonic expression can be induced byRA (Riddle et al., 1993). Members of the Gligene family may be involved as downstreammediators of Shh effects, since Shh downregu-

lates Gli3 and upregulates Gli1 (Marigo et al.,1996), but little is known about the network ofinteractions at this level.

There is also strong evidence for interactionbetween Sonic and FGF-4, either (or both) director via intermediaries such as Bmp-2. Some basicobservations that suggest this interaction are asfollows.

, Sonic hedgehog alone is insufficient to induceexpression of Bmp or Hox genes, or mesoder-mal proliferation in the absence of the AER(Laufer et al., 1994). Sonic hedgehog virusinjected into the proximal–medial mesodermof stage 21 limb buds did not result in ectopicinduction of either Hoxd genes or Bmp-2,despite the fact that Sonic hedgehog ex-pression was comparable to that seen in distalinjections (Laufer et al., 1994).

, The influence of the AER on Sonic hedgehogactivity is further demonstrated by thefollowing experiment. The anterior half ofthe AER was surgically removed in stage20/21 limb buds, Sonic hedgehog virus wasinjected into the anterior marginal mesoderm,and the embryos were allowed to developfor an additional 36–48 hours. There wassufficient posterior AER remaining so thatembryos developed almost wild-type out-growth and patterning on the limb bud. Inthe absence of the AER, Sonic hedgehogdoes not induce mesodermal proliferation orthe expression of Hox or Bmp-2 genes. Thusa signal is required from the AER to sustainproliferation and gene induction inducedby Sonic hedgehog (Laufer et al., 1994). Inthe mutants ld FGF-4 is not expressed in theAER, and Shh expression is activated but notmaintained, which may account for thedecrease in polarizing activity (Haramis et al.,1995).

, FGFs promote mesodermal competence torespond to Sonic hedgehog. FGF-4 soakedbeads were stapled to AER-denuded anteriormesoderm infected with Sonic hedgehog virus.Hoxd-11, Hoxd-13 and Bmp-2 expressionwere induced at expression levels similar toendogenous expression and to expressionlevels seen in the presence of an AER (Lauferet al., 1994).

FGF–4(a)

Wnt–7a

Hoxd HoxdShhBmp–2

(late) (early)

++

Growth controlgene expression

AER(b)

FGF–4

ZPA

Shh

. . . 300

To summarize, Sonic hedgehog plays a role inpatterning of mesodermal tissue and mayregulate FGF-4 expression. FGF-4 inducesmesodermal proliferation and maintains Sonichedgehog expression. Both factors are necessary,because mesodermal tissue can only be patternedby Sonic hedgehog in association with compe-tency provided by FGF-4. Patterning andproliferation are always coincident, and thecurrent model is that FGF-4 from the AER andSonic hedgehog from the ZPA interact to controlthe continued outgrowth of the limb bud.However it is possible that exogenously-appliedFGF-4 mimics the activity of a different memberof the FGF family. In addition, the bonemorphogenetic protein Bmp-2 interacts withFGF-4, as is summarized in Fig. 3(a). However,we will not consider the role of Bmp-2 in the firstmodel, and thus we restrict ourselves to thescheme shown in Fig. 3(b).

This is not the complete story, since patterningalso occurs along the dorsal–ventral axis. Less isknown about this, but it appears that Wnt7a isthe primary factor that controls patterning alongthis axis (Johnson & Tabin, 1997). Thus patternformation along the three axes of the limb iscontrolled by a complex network of signalingmolecules that originate in the AER, the ZPAand the non-AER ectoderm, and the interactionsbetween these must be understood before adetailed understanding of patterning in thegrowing three-dimensional limb is possible.However this does not imply that a model basedon incomplete information about the gene andmorphogen control networks cannot contributeto our understanding of pattern formation.

1.3.

Wolpert (1969) postulated that the ZPAproduces a morphogen which diffuses through-out the tissue and is degraded in it. This wouldestablish a gradient in the PA (posterior–anterior) direction which could provide pos-itional information and could lead to a spatialpattern of differentiation. Tickle et al. (1975)estimated that such gradients have to beestablished within 10 hours across a distance of500–1000 mm, and concluded that transport bydiffusion is fast enough. If an impermeable

F. 3. (a) A model for the interactions of the three majorputative morphogens, FGF-4, Shh, and Bmp-2, asproposed by Tickle and co-workers (Duprez et al., 1996);(b) a schematic of the reduced kinetic interactions betweenZPA, where production of Shh is affected by FGF-4 whichdiffuses from the AER, and the reciprocal effect of Shh onproduction of FGF-4 in the AER.

barrier is placed along the PD axis, thenskeletal development occurs only on the pos-terior side of the barrier, which suggests that adiffusible morphogen is produced at the ZPA(Summerbell, 1979). The gradient model predictsthat transplants of ZPA at positions along theAP axis of a stage 16 wing bud should result ineither the elimination of the humerus, or itsduplication, or the formation of a mirror-imageduplicate of a single humerus, depending on theposition of the graft and on the thresholdconcentration. However, in no cases is thehumerus eliminated and rarely is it duplicated.Usually either a normal or a mirror-imageduplicate humerus forms (Wolpert, 1987). Thetheory also predicts that multiple ZPA graftsshould lead to fused or abnormally thick

301

digits, contrary to observation. Wolpert andHornbruch (Wolpert, 1987) conclude from thisthat there is another mechanism at work whichcontrols the thickness of the digits.

The progress zone model, in which differen-tiation is controlled by the number of divisionsa cell undergoes while in the progress zone, wasproposed to explain pattern formation anddifferentiation in the PD direction (Wolpertet al., 1975). It predicts that removal of the AERwill lead to distal truncation, as is observed,and it makes other predictions on the outcomeof grafting a donor wing tip onto a hoststump which agree closely with observation(Summerbell & Lewis, 1975). However, it doesnot have any regulative properties and thuscannot account for the immense capability of theearly limb bud to regulate. For instance, removalof slices of the early limb bud perpendicular tothe PD axis can lead to normal limbs(Summerbell, 1977), but according to the model,this should produce deletions along the PD axisof the final pattern. Oster et al. (1983) proposeda different model for pattern formation, one inwhich condensation of cells produces thepatterning. However, it has recently been shownthat if two anterior stage 20 limb halves arecombined, the recombinant frequently forms twohumeral elements (Wolpert & Hornbruch, 1990).This suggests that the anterior half of the limbcontains cells that are already determined at thisstage, yet visible aggregation of cells does notoccur until much later (Wolpert & Hornbruch,1990). Several other models, including somebased on formal rules of growth and patterning(Wilby & Ede, 1975), and others based onthe reaction and diffusion of morphogens(Meinhardt, 1982), have been proposed. Wilby &Ede (1975) show that formal rules for cellgrowth, division and movement can reproducethe shape of the growing limb bud, but theirmodel has no morphogens. Meinhardt (1982,1983) studies reaction–diffusion models ofactivator–inhibitor type for patterning along theAP and DV axes and shows how secondary fieldscan be generated by the interaction of theprimary patterns. However these models involveautocatalytic production of the morphogensthroughout the growing tissue, and there is nointeraction between the ZPA and the AER

morphogens in determining the positionalinformation along the AP axis. As we indicatedearlier, the preponderance of current experimen-tal information suggests that the primarymorphogens are produced in specialized regionson the boundary of the limb bud, and that thetwo regions are coupled by diffusion ofmorphogens between them.

1.4.

Thus there is currently no model that cansuccessfully explain the experimental obser-vations, and we believe that there are severalreasons for this. Firstly, existing models assumethat patterning occurs separately in the AP andPD directions, but transplant results show adependence on the distance between the ZPAand the AER (Wolpert, 1987), and on theposition along the PD axis at which the graft isimplanted in the host (Javois & Iten, 1981).Evidence cited previously gives the biochemicalbasis of the interaction between the AER and theZPA. Secondly, none of the existing modelsincorporate interactions between morphogensand growth, and thus none can adequatelyrepresent the effect of growth on the spatio-temporal patterns of the morphogens. Finally,none account for the role of the non-AERectoderm in patterning, nor do they incorporateany control of cell–cell communication in thepatterning process. As we stated earlier, ourlong-term aim is to develop a model that willenable us to test various hypotheses concerningpattern formation in the limb, and to develop acomputational model useful for understandinggrowth and patterning in other contexts. Aminimal limb model involves at least two spacedimensions, boundary conditions that vary withposition on the boundary, and a domain ofvariable shape. To assist readers in understand-ing the biological basis of the model, withoutnecessarily understanding the mathematicalimplementation of it, we first give a detailedverbal description of the model.

In the model we treat the growing limb as atwo-dimensional region which begins as atruncated disk, but whose shape is determined bythe forces exerted by the growing tissue. This canbe thought of as a section through the limb taken

. . . 302

at the centerline in the DV direction. Althoughthis precludes analysing transplants in which theDV polarity is altered (Javois & Iten, 1986), it isan essential first step, given the complexity of thesystem. It is essential that growth be included ina model, for the length increases from 00.25 to01.75 mm in the 36 hours between stage 18 andstage 25, and patterning occurs in this interval*.Differential or localized growth such as occursafter a ZPA transplant (Summerbell, 1981), isincluded in the model.

The model involves two diffusible morphogensthat are produced in specialized regions near theperiphery of the limb and diffuse throughout theinterior of it. The AER is modeled as a distinctregion on the distal boundary that serves both asthe source of a substance that maintains cells inthe undifferentiated state and as the source ofone of the morphogens. The first assumption willbe that the maintenance factor and themorphogen are identical, but that can bemodified in later versions.

We model the AER as a specialized region atthe distal end of the limb that serves as amorphogen source, the strength of whichdepends on the ZPA factor. The size of the AER,and hence the total amount of morphogenreleased, can be dynamically adjusted toconform with the observed changes of its sizefrom stage 17 onward. The existing one-dimensional models assume that the concen-

tration, rather than the flux, is specified at theboundary. However, our assumption seems moreappropriate because a constant concentrationrequires a mechanism by which the concen-tration is sensed and the production and/orrelease rates adjusted to maintain the concen-tration constant. In contrast, we simply prescribea certain capacity for producing the morphogenin the AER region.

The ZPA is the source of the secondmorphogen. It is located on the posteriorboundary near the distal end of the limb, and aswith the AER, we assume that it has a givencapacity to produce the morphogen. It is knownthat the ZPA originates at the flank but movesdistally as the limb grows, and we incorporatethis in the model by specifying that the ZPAremain within a fixed distance of the AER. Also,the ZPA is restricted to interior tissue near theboundary. All ectoderm is assumed to beimpermeable to both morphogens, and bothmorphogens are degraded in the interior of theregion. Initially we will assume that theconcentration of the morphogens satisfies a zeroflux condition at the proximal boundary, wherethe limb attaches to the flank in vivo, but othertypes of boundary conditions will be tested inlater work†.

There are numerous questions that can bestudied through use of the computational model,including the following.

, What is the spatial distribution of themorphogens, assuming that they are onlyproduced in a zone near the boundary, andhow does their distribution depend onparameters such as the production rates anddiffusion coefficients? Can this distribution beestablished in the available time for reasonablevalues of the diffusion coefficients, both undernormal conditions and after transplants of theZPA? Does the spatio-temporal history of cellscorrespond with results obtained from fatemaps?

, Can one define a threshold-based combinat-orial scheme of interpretation of the instan-taneous concentration landscape or thehistory of the landscape that will lead to theobserved spatial pattern of gene expression?The existence of two spatially-separated

* If we assume that the diffusion coefficient D is1×10−7 cm2 s−1, then the characteristic time t0L2/2D=055 hours for a length of 2 mm. Thus in the laterstages of patterning the characteristic diffusion time-scale iscertainly comparable to the time-scale for growth. If eventhe diffusion coefficient is 10 times this value the statementstill holds true.

† Specifying a zero concentration at the proximalboundary rather than a zero flux has a substantialconceptual attraction in that the concentration of themorphogen produced at the AER would rise in the progresszone as the limb elongates. In other words, labile cells thatleave the progress zone in the early stages of outgrowthexperience a lower concentration of this morphogen thando those that emerge later. In view of this, a sequence ofthresholds could produce the partitioning in the PDdirection into three levels that could correspond tohumerus, radius and ulna, and wrist and digits. Thishypothesis is consistent with the observations that exchangeof the AER does not change the fate of the underlyingtissue, for we would interpret this as just a replacement ofone source with another.

ZPA

AER Outgrowth

FGF–4production

Shhproduction

Growth and celldivision ( 1)

Diffusionand decay ofmorphogens

( 2)

303

sources of the morphogens and the two-dimensionality of the underlying spatialdomain make it plausible that such aninterpretation function can be devised.

, Can the model explain a significant fraction oftransplant results that are not explicable by theexisting one-dimensional models? In particu-lar, can one explain the effect of the distancebetween the transplanted ZPA and the existingAER, particularly for those transplants thatproduce splitting of the bud? Can one explainwhy removal of a rectangle of tissue withoutrejoining the cut edges leads to a partial lossof skeletal elements, whereas when theboundaries are sewn together, a normal wingresults (Javois & Iten, 1981)? Similarly, doesthe model exhibit the degree of regulationobserved following other types of surgicalintervention, and what is the role of cell–cellcommunication and growth in this regulation?

, Is it necessary to include some level ofself-organization in the model to account forthe fact that mesodermal cells that areseparated and allowed to reaggregate in anectodermal jacket without a ZPA formmoderately good digits (Pautou, 1973)? Theexperimental results on this point are notclear-cut, for the effect may be due to sortingof cells that are already differentiated, in whichcase the experiment has no bearing on patternformation.

The first two will be addressed here; the otherswill be studied in future publications.

The model presented in this paper consists ofa fluid-mechanical component that describeslimb bud outgrowth, a reaction–diffusion–advection component that determines the spatio-temporal distribution of the morphogens, and amoving boundary that represents the mechanicaland biochemical properties of the limb budectoderm. We do not incorporate any augmenta-tion of cell movement in response to the localenvironment, for example due to chemotaxis, butthis can be incorporated in the future if theresults indicate the need for it. In the followingsection we describe the mathematical formu-lation of the model and in Section 3 we presentsome analytical results for simplified geometriesand kinetics. In Section 4 we present numerous

simulations designed to illustrate the predictionsmade by the model. This section can be readindependently of the preceding two by readerswho wish to skip the mathematical details.

2. Mathematical Formulation of the Model for aGrowing Limb

The first step will be to develop thefluid-mechanical description for the growth andmovement of tissue in the limb. This componentof the model requires detailed elaboration,because to our knowledge, this type ofdescription has not been used before. We modelthe tissue as a viscous, incompressible fluidwhose volume increases by virtue of a distributedsource that stems from cell division and growth.The boundary of the limb is regarded as anelastic medium, and outgrowth results from the‘‘pressure’’ of the increasing volume against thisboundary. The processes involved are depicted inthe schematic shown in Fig. 4. The equations forthe fluid motion that drives outgrowth are givenin the following subsection. The reaction–diffusion equations that govern the evolution ofthe morphogen distributions in space and timeare given in Subsection 2.2, and a scaled versionof the complete set of governing equations isgiven in Subsection 2.3.

2.1.

The tissue in a growing limb is a complexmixture of cells, extracellular matrix, and other

F. 4. A schematic of the growing limb and the processesinvolved in the limb. The interior of the limb is denoted V,the AER region is denoted V1, the ZPA region is denotedV2, and the boundary of the limb is denoted G.

. . . 304

components, and to our knowledge its rheologi-cal properties have not been investigated.However the mechanical properties of otherembryonic chick cell aggregates have beenstudied by Philips & Steinberg (1969), Philipset al. (1977) and Philips & Steinberg (1978). Ina long-term culture vertebrate tissue massesexhibit liquid-like characteristics in response tostress, but the short-term response is more likethat of an elastic solid. Philips et al. (1977)describe this tissue as an ‘‘elasticoviscous’’ liquid,but a more common terminology is to describeit as a viscoelastic material (Fung, 1993). Sincewe are only interested in the slow motion due togrowth we neglect the elastic component of theresponse and model the tissue as a viscous fluid.The process of cell growth and cell divisionrequires transport of nutrients via diffusion andconvection across the limb bud ectoderm andthrough the extracellular matrix, and at a laterstage, through the capillary system. We idealizethis complex process as a distributed sourceS(c, x, t) of volume within the limb bud*. Thelocal source strength may depend upon chemicalspecies such as growth factors contained in c, thelocation x of the tissue within the limb bud, andthe age of the limb.

We assume that the fluid density is constant,but since there is growth, the continuity equationtakes the form

9 · u=S(c, x, t). (1)

wherein u is the local fluid velocity. Later we willindicate how this growth term depends on themorphogen concentrations and other variables.We further assume that the fluid motion isdescribed by the Navier–Stokes equations, whichprovide the simplest description of a viscousfluid. These are given by (Batchelor, 1973)

r1u1t

+ r(u · 9)u=−9p

+ m092u+139S1+ rF. (2)

Here r is the fluid density, p is the pressure, andm is the fluid viscosity. The term F is the forcedensity (force per unit area in two dimensions)that limb bud ectoderm exerts on the fluidsurrounding it. As will be seen below, F isnon-vanishing only in thin layers surroundingthe limb bud boundary.

Equations (1) and (2) describe the tissuedynamics in the interior of the limb. In addition,the growth model must include a movingboundary G that represents the limb budboundary. The instantaneous configuration ofthe boundary in two dimensions is given by thefunction X(s, t), where s is a Lagrangian labelfor a point on the boundary. We specify thatX(s, t) moves at the local fluid velocity, andtherefore

1X1t

= u(X(s, t), t). (3)

Since the limb boundary is treated as an elasticmaterial, the force per unit length f(s, t) ateach point on the boundary is a function ofthe instantaneous configuration. In a three-dimensional model the limb bud boundarycould be modeled entirely by tangentialelastic spring forces. In two dimensions, weinclude elastic links between the anterior andposterior edges to represent the circumferentialforces in the three-dimensional ectodermthat prevent the limb bud from ballooningoutward. The boundary is taken to beneutrally buoyant and thus the limb budboundary forces are transmitted directly tothe fluid via the force density F, which is givenby

F(x, t)=gG

f(s, t)d(x−X(s, t)) ds. (4)

In this equation the integration is over thepoints of the boundary G and d is thetwo-dimensional Dirac delta function. The limbbud grows out from the flank of the embryo, andfor simplicity we regard the flank as animmovable boundary. This is accomplished bytethering the points on the proximal boundary inFig. 4 to fixed points in space with stiff elasticspring forces.

* S is the volumetric growth per unit volume per unittime.

305

2.2. ––

In the interior V of the limb the evolution ofthe morphogens c=(c1, c2), where c1 representsthe AER morphogen and c2 represents the ZPAmorphogen, is described by a system ofreaction–diffusion–convection equations of thefollowing form.

1c1t

+9 · (uc)=D92c+R(c) (5)

The diffusion matrix D is a diagonal matrixwhose entries are the diffusion coefficients of thetwo morphogens. We have assumed here that thediffusion coefficients are constants, but wecould easily incorporate dependence on themorphogens to describe control of cell–cellcommunication by the morphogens.

As we indicated previously, the AER morpho-gen is only produced in the AER (V1) and theZPA morphogen is only produced in the ZPA(V2). Thus R=(R1, R2) has the form

Rk =6rk (c)− kkck

−kkck

x $ Vk

otherwise,(6)

where rk (c)q 0 except possibly at c= 0.The convective term 9 · (uc) in eqn (5) can be

written as c9 · u+u · 9c, and using the growth asgiven by eqn (1), eqn (5) becomes

1c1t

+Sc+ u · 9c=D92c+R(c). (7)

On G we specify the homogeneous Neumann orno-flux boundary conditions

n · D9c=0, (8)

where n is the outward normal to G.

2.3.

In order to cast the governing equations intodimensionless form, we introduce characteristiclength and velocity scales and a characteristicchemical concentration, which we denote by L,U and C, respectively. We then define thefollowing scaled variables: t= t/T, x= x/L,X� =X/L, u= u/U, c= c/C and p= p/P, S� =S/S0. We further set T=L/U, P= rU2 and

S0 =1/T, and then obtain the following dimen-sionless equations.

1u1t

+(u · 9)u=−9p+Re−1092u+13

9S� 1+F�

1c1t

+S� c+ u · 9c=D� 92c+G� (c)

1X�1t

= u(X� (s, t), t)

9 · u=S� (c)

F� (x, t)=g f� (s, t)d(x−X� (s, t)) ds. (9)

Here the Reynolds number Re is defined asRe=LU/n, where n= m/r is the kinematicviscosity. The remaining quantities are defined asF� =LF/U 2, f� =Lf/U2, G� =LR/(CU), S� =LS/U, D� =D/(LU), s= s/L. To simplify thenotation the overbars are dropped hereafter, butall variables remain dimensionless.

A characteristic length-scale for vertebratelimb development is L=0.1 cm, the approxi-mate width of the early chick limb bud. Betweenstages 18 and 25 the PD length of the wing budincreases from approximately 0.023 to 0.174 cmover a time span of approximately 36 hours.Thus the average rate of limb bud outgrowth isapproximately 0.0042 cm hr−1, which yields acharacteristic velocity U=1.2×10−6 cm s−1.For these characteristic length- and velocity-scales, a unit of dimensionless time t correspondsto approximately 23.15 hr. The Reynolds num-ber for these values of U and L for a fluid witha kinematic viscosity of water (n1 0.01 cm2 s−1)is Re0 10−5. Since reasonable values of n formesodermal tissue are likely to be several ordersof magnitude larger, Re may be several orders ofmagnitude smaller. Initially, we shall take thediffusion rate for growth factors in limb tissue tobe approximately 10−7 cm2 s−1. This gives us adimensionless diffusion coefficient D1 1.

2.4.

The numerical algorithm for solving thecomplete model equations given by the system at(9) is based on the immersed boundary method,first used by Peskin (1977) to model blood flow

. . . 306

in the heart. This method has since beendeveloped into a general method that can be usedto study flows interacting with moving elasticstructures. Details of the implementation of thenumerical algorithm are described in theAppendix. A crucial feature of this method in ourapplication is that the limb bud is not the entirecomputational domain for the solution of thefluid dynamical equations. The limb is embeddedwithin a larger rectangular domain andwithin thislarger domain, the limb bud boundary contrib-utes a singular force field in the fluid equations.The volumetric sources that model growth in thelimb are balanced by sinks distributed in theregion exterior to the limb. Because the fluidequations are solved on a fixed regular domain,we can impose periodic boundary conditions andcan solve the discretized Navier–Stokes equationsusing a Fast Fourier Transform algorithm. Theadvection–diffusion–reaction equations for themorphogens are solved using a finite differencemethod with an upwind scheme for the advection.The method for approximating the Neumannboundary conditions for the morphogens at themoving limb bud boundary [eqn (8)] is discussedin the Appendix.

3. Simplified One-dimensional Models

3.1. -

To gain some insight into the effect the spatialseparation between the AER and the ZPA hason the magnitude and spatial distribution of themorphogen concentrations, we consider severalone-dimensional model problems in which thegrowth, and hence the fluid velocity, is zero. Thefirst problem, which will show the effect ofdiffusion coefficients and decay constants on thedistributions that result from coupled, spatially-separated, positive feedback mechanisms, is onein which the enzymes are localized at oppositeends of an interval. Thus the terms ri in (6) arelocalized in space, but the degradation of themorphogens occurs throughout the domain.Since the equations have been non-dimensional-ized, the interval is [0, 1], and we suppose that theAER is at x=1 and the ZPA at x=0. In thissituation the governing equations for themorphogen concentrations reduce to

1c1

1t=D1

12c1

1x2 − k1c1

1c2

1t=D2

12c2

1x2 − k2c1

(10)

for x $ (0, 1), with the boundary conditions

−D21c2

1x(0, t)= r2(c1(0, t)) (11)

D11c1

1x(1, t)= r1(c2(1, t)) (12)

−D11c1

1x(0, t)=D2

1c2

1x(1, t)=0. (13)

These equations reflect the assumptions that themorphogens diffuse throughout the domain andare degraded by first-order reactions, that theproduction of ZPA morphogen at x=0 iscontrolled by the amount of AER morphogenpresent, and that the production of AERmorphogen at x=1 is controlled by the amountof ZPA morphogen present at x=1. It should benoted that each morphogen only controls theproduction rate of the other; it is itself notconsumed in the process.

We first show how to obtain time-independentsolutions of these equations, which must satisfythe system

D1d2c1

dx2 − k1c1 =0 (14)

D2d2c2

dx2 − k2c2 =0 (15)

−D2dc2

dx(0)= r2(c1(0)) (16)

D1dc1

dx(1)= r1(c2(1)) (17)

D1dc1

dx(0)=D2

dc2

dx(1)=0. (18)

The solution of eqns (14) and (15) which satisfiesthe boundary conditions in eqn (18) is given by

c1(x)=A1 coshX k1

D1x

c2(x)=A2 coshX k2

D2(1− x).

(19)

307

The amplitudes A1 and A2, which must bepositive, are determined from the remainingboundary conditions given in eqns (16) and (17).Thus the AER morphogen c1 has a maximum atx=1, and the ZPA morphogen has a maximumat x=0, as expected. One finds that theamplitudes of these distributions are solutions ofthe nonlinear system

D2A2X k2

D2sinhX k2

D2= r2(A1)

D1A1X k1

D1sinhX k1

D1= r1(A2).

(20)

We assume that r1(0)= r2(0)=0, whichsimply means that there is no basal productionof morphogen in the absence of the othermorphogen. This could easily be changedwithout altering the overall conclusion signifi-cantly. Under this assumption the system (20)always has the solution (A1, A2)= (0, 0) for allvalues of the Di s and ki s but the question iswhether it has a non-zero solution. One seesfrom (20) that if one of the Ai s is zero then sois the other, so there is no possibility of anon-zero concentration of one morphogen and azero concentration of the other.

We can write (20) in the form

A2 =V2 · r2(A1) (21)

A1 =V1 · r1(A2) (22)

where

V−1i 0zkiDi sinhX ki

Di,

and then there is a positive solution for A1 andA2 if and only if the curves defined by (21) and(22) intersect in the interior of the positivequadrant of the A1 −A2 plane. The ratefunctions ri are typically monotone increasingfunctions, at least for small c, and they shouldsaturate at large c. Under these two conditionsone can show that there is at least one positiveintersection of the curves if

V20 dr2

dA11(21)

q 1V1 0 dr1

dA21−1

(22)

,

when these are evaluated at (0, 0). The subscripts(21) and (22) denote quantities computed fromthe equation with that number.

This cannot be determined in general withoutknowledge of the rate functions, and wetherefore suppose that

r1(c2)=V1c2

K2 + c2r2(c1)=V2

c1

K1 + c1.

(23)

That is, we assume Michaelis–Menten kineticsfor the production of both morphogens. We thenhave

A2 =V2V2A1

K1 +A1(24)

A1 =V1V1A2

K2 +A2. (25)

These can be solved explicitly and one finds thatthe non-zero solution is given by

A2 =V1V2V1V2 −K1K2

K1 +V2V2(26)

A1 =V1V1A2

K2 +A2. (27)

Therefore (A1, A2)q (0, 0) if and only if

V1V2

K1K2qV−1

1 V−12 , (28)

or

V1

K2

V2

K1qzk1k2zD1D2 sinhX k1

D1sinhX k2

D2.

(29)

On the left-hand side the term V1/K2 (resp. V2/K1)is the slope of the corresponding productionterm r1 (resp. r2) at zero concentration, while theright-hand side is determined by the diffusioncoefficients and the decay rates.

If ki /Di is small, either because the decay ratesare small or the diffusion coefficients are large,then

sinhX ki

Di0X ki

Di

. . . 308

and (29) reduces to

V1V2

K2K1q k1k2. (30)

This is purely kinetic criterion which simplystates that the product of the maximal slopes ofthe production terms must exceed the product ofthe degradation rates.

At the other extreme, if the decay rates are toolarge there will certainly be no positive solution.Furthermore if all the kinetic parameters arefixed and the diffusion constants are decreased,then (29) will certainly not be satisfied forsufficiently small Di s, and again there will be nopositive solution. Since the length of the domainis used to make the diffusion coefficientsdimensionless, one can always guarantee thatthere is no positive solution for sufficiently largeL*.

It is clear from the foregoing analysis that thecondition

V1V2

K2K1−zk1k2zD1D2 sinhX k1

D1sinhX k2

D1=0

(31)

represents a transverse bifurcation point atwhich a solution (A1, A2) passes from the thirdquadrant to the first quadrant as this quantityincreases through 0. Further analysis shows thatthe positive solution for (A1, A2) is stable, at leastwhen the difference in (31) is positive andsufficiently small.

3.2.

The foregoing leads to an analytical criterionthat guarantees a non-zero solution, and itindicates the interplay between production rates,decay rates, and the diffusion coefficients.However in reality both the AER and the ZPAare distributed over a region of the growing limbbud, and next we consider a one-dimensional

model of this. We assume that the AER lies inthe interval x $ (x3, 1) and that the ZPA lies inthe interval x $ (x1, x2), where x1 Q x2 E x3 Q 1.Thus we assume here that the ZPA occupies afixed interval, but later we adopt a morefunctional definition of ZPA-ness.

In this case the reaction terms R1 and R2 in eqn(6) may be expressed in the form

R1(x, c)=H(x− x3)r1(c)− k1c1

R2(x, c)=H(x− x1)H(x2 − x)r2(c)− k2c2

(32)

where H is the Heaviside function (H(x)=0 forxE 0 and H(x)=1 for xq 0) and the ri are asgiven in (23). The governing equations are

1c1

1t=D1

12c1

1x2 +R1(x, c)

1c2

1t=D2

12c2

1x2 +R2(x, c)(33)

for x $ (0, 1), with the boundary conditions

D11c1

1x(0, t)=D2

1c2

1x(0, t)=0 (34)

D11c1

1x(1, t)=D2

1c2

1x(1, t)=0. (35)

In general one cannot obtain analyticalsolutions of these equations, but if bothproduction rates are constant the equations arelinear and uncoupled. This arises formally ifboth c1 and c2 are large relative to thecorresponding Michaelis constant in the rateexpressions, in which case the production ratesare saturated, and we use this formal connectionlater. In this special case the steady-statedistributions of the morphogens are given by thefollowing equations

c1(x)=6 a10(el1x +e−l1x)

a13(el1x +el1(2− x))+ a1

xE x3

x3 Q xE 1,

(36)

c2(x)=gF

f

a20(el2x +e−l2x)a21el2x + b21e−l2x + a2

a22(el2x +e−l2(2− x))

xE x1

x1 Q xE x2

x2 Q xE 1

(37)

* This has the interesting implication that the morpho-genetic interactions between two spatially-separated orga-nizing regions will be turned off when the size of the systemreaches a critical value. This may be a useful mechanism inother contexts, such as in anterio-posterior patterning inearly development of the vertebrate neural plate, wheresignals from each end of a planar domain are thought tocontrol patterning (Ruiz i Altaba, 1994).

(a) (b)

0.1

1.0

10.0

D1 = 0.01

(c)

0.1

1.0

10.0

D2 = 0.01

309

Here ai =Vi /ki , li =zki /Di , i=1, 2 and thecoefficients are given as follows.

a10 = a10el1x3 − el1(2− x3)

2−2e2l1 1,

a13 = a10el1x3 − el1(2− x3)

2−2e2l1−

e−l1x3

2 1,

a20 = a20−e−l2(−2+ x2) + e−l2(−2+ x1) + el2x2 − el2x1

2e2l2 −2 1,

a21 = a20−el2x1 + el2x2 + e−l2x1 − e−l2(−2+ x2)

2e2l2 −2 1,

a22 = a20−el2x1 − e−l2x2 + el2x2 + e−l2x1

2e2l2 −2 1,

b21 =

a20−el2(2+ x1) − e−l2(−2+ x2) + e−l2(−2+ x1) + el2x2

2e2l2 −2 1.

(38)

In Fig. 5 we show the effect of changes in thediffusion coefficients of the species for fixedvalues of the kinetic parameters and the spatialdomains in which the AER and ZPA arelocalized. In Fig. 5(a) we show a base case inwhich both diffusion coefficients are 1, and in theother panels we show the effect of varying thediffusion coefficients. Increasing Di leads toflatter profiles, whereas decreasing Di leads to

sharper profiles and higher concentration levelswithin the AER [Fig. 5(b)] and ZPA regions[Fig. 5(c)]. As Di : 0 the solution approximatesa step function with c1 =V1/k1 in the AER andc2 =V1/k2 in the ZPA.

3.3.

When the production rates are not constantone must resort to numerical computation of thesolutions to eqns (33–35). Figure 6 shows theresults for the kinetic parameters used inFig. 5(a–c) and several values of the diffusioncoefficients. One sees in Fig. 6(a) that at fixed D2,increasing D1 (the diffusion coefficient of theAER morphogen) increases the level of the AERmorphogen in the ZPA region and decreases it inthe AER region. Because production of the ZPAmorphogen depends on the level of the AERmorphogen when the kinetics are not saturated,this leads to an increase in the level of the ZPAmorphogen throughout the domain. Figure 6(b)shows that at fixed D1, increasing D2 decreasesthe ZPA morphogen in the ZPA region andincreases it elsewhere, thereby producing anincrease of the AER morphogen. These resultsare qualitatively similar to what is shown inFig. 5 because the kinetic terms are essentiallysaturated for the values of Di shown, but here achange in either diffusion coefficient affects thedistribution of both species. Furthermore, thereis a significant difference between the resultsusing constant production rates and concen-tration-dependent rates when both diffusioncoefficients are very low. In the present casesmall diffusion rates (for example D1 =1,

F. 5. Steady-state solutions with saturated kinetics. x1 =0.5, x2 =0.8, x3 =0.9. V1 =V2 =2000, k1 =50, k2 =50. Thehorizontal axis is x $ [0, 1], the vertical axis shows c1 and/or c2 $ [0, 40] in dimensionless units. The values of the parametersVk and kk are those used in the full simulations in a later section. (a) c1 (peak on right) c2 (central peak) with D1 =1 andD2 =1; (b) c1 with D1 =0.01, 0.1, 1, 10; (c) c2 for D2 =0.01, 0.1, 1, 10. In (b) and (c) smaller diffusivities produce largerpeak values in the morphogen concentration.

(a)

1.0

0.1

0.11.0

10.0

D2 = 1.0

D1 = 10.0

(b)

0.1

1.0

10.0

10.0

1.0

D1 = 1.0

D2 = 0.1

. . . 310

F. 6. Steady-state solutions to the evolution equations (33)–(35), computed using the same kinetic parameters andproduction regions as in Fig. 5. The horizontal axis is x $ [0, 1], the vertical axis (range [0, 40]) shows c1 (—) and c2 (. . .)in dimensionless units. (a) D2 =1 and variable D1; (b) D1 =1 and variable D2.

D2 =0.01) leads to solutions that converge to thesteady state c=(0, 0), because there is insuffi-cient transport of the complementary morpho-gen between the production sites to offset thedegradation that occurs throughout the domain.By contrast, when production of a morphogen isindependent of the level of the complementarymorphogen, as in Fig. 5, the spatial distributionsof the morphogens approach step functionssupported on the production regions as thediffusion coefficients approach zero.

It is known that the maximum expression ofZPA activity is not immediately adjacent to theAER, but rather, lies in the region a few hundredmicrons proximal to the AER. The precedingresults are consistent with this observation butthey reflect the fact that the ZPA was specifiedgeometrically rather than functionally. Themechanisms controlling the graded distributionof ZPA activity are not known at present, but onepossibility is that it stems from spatially varyingdifferences in the mesodermal tissue itself. Analternative is that the ZPA morphogen pro-duction exhibits a biphasic response to the AERmorphogen. This could arise, for instance fromactivation of an enzyme at low AER morphogenlevels and inhibition at high levels. Computationsin which the production rate V2 depends on theconcentration of the AER substance c1 as follows

V2(c1)=61.0

0

if c1 Q cs1

otherwise(39)

produce profiles of the ZPA substance that hasa maximum at some distance from the AER, asin Fig. 6.

4. Numerical Simulations Incorporating Growthand Cell Movement

4.1. -

In this section we describe numerical resultsfrom simulations of the full model, includinggrowth and cell movement. Initially, the curvedpart of the limb bud boundary is an arc of acircle chosen to approximate a stage 19–20 limbbud. The initial configuration of the limb buddomain is the same in each of the foursimulations described later.

The estimates of growth rates given earliershow that diffusion, and hence establishment ofthe morphogen distributions, is rapid comparedwith growth when the distances are small, but thetime-scales of these processes become morecomparable as outgrowth proceeds. To eliminateartifacts due to the initial morphogen distri-butions, we first set the growth to zero andcompute an approximate steady state for themorphogen distributions. We then use thismorphogen distribution as the initial conditionfor simulations that include growth and cellmovement. The quasi-steady distributions forthe morphogens are obtained by integrating theevolution eqn (5), with S0 0 and u=0, forward

311

in time, using the methods described in theAppendix, until the concentration distributionsare approximately constant in time. Thus, theinitial morphogen distributions are approximatesolutions to the equation

D92c+R(c)=0 (40)

with boundary conditions given by eqn (8). Oncethe initial distributions for S=0 are determined,we turn on the growth rate throughout the limb.The dependence of the volumetric growth rate onthe morphogens must be postulated, and wesuppose that it depends only on c1, themorphogen or growth factor produced in theAER, as follows.

S= s1c1 + s2. (41)

The growth rate comprises a constant com-ponent s2 =0.1 that represents a basal growthrate, and a component proportional to theconcentration of the growth factor c1. We set thefirst order rate constant s1 equal to 10/3. Aninterpretation of (41) is that growth depends onbinding of a growth factor to surface receptors,and that the concentration of growth factor ismuch less than the Km for binding.

In the four simulations that follow the AER isa region in the limb bud of roughly constantdimensions localized along the distal edge of thelimb bud. The ZPA-competent tissue is a regionof constant width on the posterior edge of thelimb bud, beginning at the proximal boundaryand ending just short of the AER region.Although the width of ZPA-competent region isfixed, the ZPA-competent length elongates withlimb bud outgrowth. The algorithms fordetermining the ZPA and AER regions aredescribed in detail in the Appendix. As will bediscussed below, the production rates of theAER and ZPA morphogens within these regionsmay vary both spatially and temporally. Since

the growth factor c1 is produced in the AER,the growth rates are highest in the distal region.This is consistent with experimental observationswhich show that the mitotic rate is highest in thePZ and about one-fourth that rate in theproximal limb bud.

Simulation 1: Uncoupled AER and ZPAproduction. As was discussed in the Introduction,the biochemical reactions that produce the AERand ZPA morphogens are believed to becoupled. However, to establish a base case forlater comparison purposes we first suppose thatthe production rates of the morphogens areuncoupled. As a result, in this simulation activeZPA extends along the entire posterior margin ofthe domain. In later simulations, it is theinteraction with the AER factor that limits theeffective extent of the ZPA. Here the morphogenproduction rates Rk that appear in eqn (5) havethe form

Rk (c)=6gk (1− ck )

−kkck

x $ Vk

otherwise(42)

for k=1, 2. In all examples V1 is identified withthe AER and V2 with the ZPA, as in Fig. 2. Thedimensionless kinetic parameters are set atgk =10000 in this example in order to ensurethat ck will be approximately 1 in Vk . Thedimensionless diffusion constants are set atDk =1 for k=1, 2 and the morphogen degra-dation rates are set at kk =50. The outline of thegrowing limb bud domain and contours ofthe morphogen concentrations are shown atseveral time steps in Row I of Fig. 7. Since theproduction of c1 is independent of c2 and theAER is initially symmetric about the PD axis,the profiles of c1 are also symmetric about the PDaxis initially, and because the local growth ratesdepend only on the concentration of c1, limb budoutgrowth is also symmetric initially. One sees inthe figure that the shapes of the limb bud andconcentration profiles of the AER growth factorremain essentially symmetric about the PD axisthroughout the entire period. The location of theimmersed boundary points are obscured by thesuperposition of morphogen contours in thisfigure, but the reader can look at Fig. 8, to bedescribed below, for a better picture of theboundary*.

* We remark that the algorithm itself contains a slightasymmetry that arises as follows. A single boundary pointis added whenever the boundary stretches by a prescribedamount. The new boundary point is inserted between thetwo adjacent immersed boundary points that have movedfurthest apart. The addition of new points to the boundarygenerally does not occur symmetrically with respect to thePD axis.

(a)I

c2

c1

0.5

c2

c10.5

(a)II

(b)I (c)I (d)I

(b)II (c)II (d)II

c2

c1

0.5

(a)III (b)III (c)III (d)III

c2

c1

0.5

(a)IV (b)IV (c)IV (d)IV

. . . 312

F. 7. Contour lines of the morphogen concentrations c1 and c2. Row I: simulation 1 at dimensionless times (a) 0; (b)0.5625; (c) 1.125; (d) 1.875 [real times of (a) 0; (b) 13; (c) 26; and (d) 43 hours]. Rows II–IV: simulations 2–4, showing theconcentration contours at dimensionless times (a) 0; (b) 0.375; (c) 0.75; (d) 1.3125 [real times of (a) 0; (b) 8.7; (c) 17.4; (d)30.4 hours]. If we identify panel (a) in each row with stage 19, then these times correspond to stages 19, 21, 23, and 25,respectively, in normal chick limb development. Here and hereafter, the bounding square in each panel delimits thecomputational domain, each side of which is 2.0 mm in length (see Appendix). The contours shown represent equally-spacedlevels of c1, which is highest near the AER and decreases monotonically along the PD axis, and c2, which is highest alongthe posterior boundary and decreases monotonically along the posterior–anterior axis. In Row I the contour levels shownbegin at 0.1 and increase in increments of 0.1 for each species. In Rows II–IV the contour levels begin at 0.5 and increasein increments of 0.5 for each species.

Simulation 2: Coupled feedback interactionbetween AER and ZPA morphogen production. Inthis simulation we use the Michaelis–Menten

kinetics given at (23) for the production of themorphogens and couple these with the standardlinear decay terms. Thus the kinetic functions are

(a) (b) (c) (d)

313

F. 8. The locations of fluid markers at selected times within the growing limb bud. The four rows correspond to thefour rows shown in Fig. 7, and the time snapshots within a row correspond to those shown in that figure.

as follows.

R1 =V1c2

K2 + c2− k1c1, for x $ V1

and R1 =−k1c1, otherwise

R2 =V2c1

K1 + c1− k2c2, for x $ V2

and R2 =−k2c2, otherwise. (43)

The dimensionless parameters are identical inboth species and are set at Vk =2000, kk =50,Kk =1 and Dk =1, k=1, 2. As in Simulation 1,the AER substance c1 is only produced in theAER, but it decays throughout the limb bud withtime constant k−1

1 . Similarly, the ZPA factor c2 isproduced only in the ZPA and decays through-out the limb bud.

Row II in Fig. 7 shows the dramatic effectthat spatial localization and coupling of the

. . . 314

production rates has on the spatio-temporaldistributions of the morphogens. When com-pared with Row I, the high point of the AERmorphogen distribution, which is slightly greaterthan 2 at the final time (panel II-d), shifts towardthe ZPA and there is a gradient of AER factorwithin the AER, with c1 higher in the posteriorand lower in the anterior portion of the AER.(Since the average level of c1 is higher in Row IIthan in Row I, the simulation in Row I wasextended in time to produce the same overallgrowth.) The asymmetry in the production of thegrowth factor, which is higher in the posterior–distal region of the limb than in the anteriorregion, leads to a pronounced asymmetry in theshape of the growing limb. Note also that themaximum levels of both morphogens decrease asoutgrowth proceeds, because both morphogensare degraded throughout the growing volume oftissue, but only produced in domains of constantsize*. For example the maximum of the ZPAfactor exceeds 6 in panel a of Row II, whereasthe maximum is only about 3.5 in panel d of thatrow.

Simulation 3. In this and the followingsimulation the reaction kinetics and kineticparameters are identical to those in Simulation 2,but the diffusion coefficients are varied. In thissimulation we set D1 =1 and D2 =0.5, whichmeans that the ZPA species diffuses at half therate used in Simulation 2. One sees in Row IIIof Fig. 7 that the maximum concentration levelsof c2 are higher at later times than in Simulation2. In particular, the maximum value of c2 at thefinal time (panel d) is approximately 4.5 here, ascompared with 3.5 at the same time in Row II.Furthermore, just as in the results for theone-dimensional simulations shown in Fig. 5, theconcentration levels of c2 fall off more rapidlywhen the diffusion coefficient is reduced, andhence the contour levels are more compressed inspace. Because the ZPA factor is more localizedin space when its diffusion coefficient is reduced,this leads to a lower overall production of thegrowth factor and hence a reduction in the

outgrowth of the limb. Indeed, the final lengthshown in panel III-d is only 93% of the lengthat the same time shown in panel II-d. Since theonly difference in the parameters between RowII and Row III is the change in the diffusioncoefficient of the ZPA factor, this shows that thecoupling between production of the factors canhave unanticipated effects on the growth of thelimb. We will say more about this phenomenonin the discussion of fluid markers below.

Simulation 4. In this simulation the diffusioncoefficients are set at D1 =2, D2 =0.5, andotherwise the parameters are the same aspreviously. The contours in c1 and c2 are shownin Row IV of Fig. 7. There are two significanteffects of the increase in the diffusion of the AERfactor. Firstly, the maximum levels of the AERfactor are reduced significantly (the maximum ofc1 is approximately 1.0 in panel d of Row IV,compared with just under 2.5 in panel d of RowII), because when the AER factor is moreuniformly distributed in space the total degra-dation rate is increased substantially. As a result,the rate of limb bud outgrowth is lower inSimulation 4 than in Simulation 3. Secondly, thefaster diffusion of the AER factor means theZPA morphogen is produced in significantamounts over a larger portion of ZPA-competent tissue, and thus the concentration ofthe ZPA factor rises throughout much of theposterior margin of the limb. Thus the ZPAsubstance is more proximally expressed inSimulation 4 than in Simulation 3. However, themaximum concentration level of the ZPA speciesis lower in Simulation 4 (approximately 4) thanin Simulation 3 (approximately 4.5).

4.2.

Since growth in the limb is asymmetric whenthe ZPA and AER are localized in space andinteract via diffusible substances, it is necessaryto track cells in order to determine the temporalpattern of morphogen concentrations to whichcells are exposed during outgrowth. Without thisinformation it is difficult to propose mechanismsby which the morphogen concentrations can betranslated into levels of gene expression. Inaddition to tracking the overall shape of the limband the concentration distributions within it, wetrack individual fluid markers in each of the

* The AER-competent region is of fixed size, but theZPA-competent region elongates. Because of morphogencoupling, the active region of the ZPA species is localizeddistally and the size of the active region is roughly constant.

315

simulations described in the preceding section.These markers, which may be regarded asproxies for individual limb bud cells or theirprogeny, travel at the local fluid velocity, andthus accurately reflect the relative displacementsbetween selected points in the limb as outgrowthproceeds.

Fluid markers are placed on each interior gridpoint at t=0, and subsequently each marker isconvected along at the local fluid velocity. Thelocations of these fluid markers within thegrowing domain for the four simulationsdescribed previously are shown in Fig. 8. Initiallythe markers are regularly spaced, since theycoincide with the grid points, but because theymove with the local velocity, which is determinedby the local growth rate, the distance betweenadjacent pairs of markers changes with time.Evidence of high growth rates in the region ofhighest concentration of c1, the AER growthfactor, can be seen in the wider spacing ofmarkers in the distal region, but there is a basallevel of growth throughout the limb. The axiallysymmetric distribution of c1 in Simulation 1(Row I Fig. 7) leads to axially symmetric growthrates and an axially symmetric fluid markerdistribution (Row I of Fig. 8). The asymmetricdistribution of c1 due to coupling with the ZPAmorphogen c2 (see Rows II–IV of Fig. 7) resultsin asymmetric local growth rates and to amarked asymmetry in the fluid marker distri-bution, as shown in Rows II–IV of Fig. 8. Inparticular, the spacing between adjacent columnsof fluid markers is much greater in theposterior–distal region than in the anterior–distal region. As noted above, the reduction ofthe diffusion coefficient D2 for the ZPA species inSimulation 3 results in a posterior shift in theproduction of the AER species. This change inthe distribution of c1 results in more pronouncedposterior–distal growth rates, as evidenced bythe increased posterior spacing between adjacentcolumns of the distal fluid markers (cf. Row III

of Fig. 8). In Simulation 4 the diffusion constantD2 remains the same as in Simulation 3, but D1

is increased. This leads to a more uniformdistribution of the AER factor c1 and moreuniform growth throughout the limb, as can beseen in the fluid marker distribution shown inRow IV of Fig. 8.

Further insight into the pattern of cellmovement during growth can be gained byplotting the trajectories of selected fluid markers.The results for the four simulations are shown inFig. 9(a–d). Initially nine pairs of fluid markers,located along the posterior–distal edge of thelimb bud are chosen from those shown in Fig. 8.The pairs are designated (a, A) . . . (i, I) fromposterior to anterior. In each pair, the moreproximal marker is designated by a lower-caseletter, the distal marker by a capital letter. Inpanel 1 of Fig. 9 we show the tracks of the fluidmarkers for Simulation 1*. The pair (e, E) isinitially on the centerline in the PD direction andremains there throughout the simulation—areflection of the axially-symmetric distribution ofc1 and concomitant axially symmetric growthrates. The other pairs are displaced posteriorly oranteriorly, depending on their initial location.The trajectories for Simulation 2 are shown inpanel 2 of Fig. 9, where it is seen that the pairs(d, D) and (e, E) are displaced anteriorly as wellas distally. This anterior displacement resultsfrom higher growth rates in the posterior–distalregion. As in Simulation 1, (a, A), (b, B), and(c, C) are displaced posteriorly and distally. Thefact that growth occurs primarily near the AERis reflected in the fact that cell markers thatinitially lie near the AER are far removed fromit at the final time.

Panel 3 of Fig. 9 shows the trajectories forSimulation 3, where, in contrast to Simulation 2,the fluid marker pair (c, C) follows a more axialpath and (d, D) and (e, E) are displaced moreanteriorally. As was noted in the previoussection, the AER growth factor is moreconcentrated in the posterior–distal region of thelimb bud in the third simulation. The highergrowth rates in the posterior–distal region pushthe fluid marker pairs (c, C), (d, D), and (e, E)anteriorally. There is however less effect on thetrajectories of cells that begin on the anterior sideof the mid-line.

* Since the computational domain is of fixed size and thefluid is incompressible, the volumetric source that arisesfrom growth must be ‘‘absorbed’’ in sinks located outsidethe limb bud. These are placed at four grid points near thecorners of the computational domain, and the results inPanel 1 show that the location of these sources preserves thesymmetry of the results.

I

1

A

E

I

A

E

2

I

A

E

3

I

A

E

4

. . . 316

F. 9. Paths of selected fluid markers (a, A), . . . , (i, I) for simulations 1–4. Labels on the curves mark the trajectoriesof selected fluid markers.

The anterior displacement of these pairs is lesspronounced in Simulation 4, as is shown in panel4 of Fig. 9. This is consistent with the higherdiffusivity of the AER factor, which leads tomore uniform, albeit lower, growth ratesthroughout the limb bud (cf. also Rows III andIV of Fig. 8). In this simulation there isproportionately less growth near the AER, andcells that begin near the AER remain closer to itthroughout outgrowth. An interpretation of thisin the context of the progress-zone modeldescribed in the Introduction is that cells thatbegin in the progress zone never stray far fromit; they are held more or less in their relativeplace by the growth that occurs proximally.

Since the trajectories of fluid markers can beinterpreted as the trajectories of individual cellsor their progeny, the preceding results show thatthe location of individual cells within the limbbud as a function of time depends on the detailsof the model, and in particular, is strongly

influenced by the diffusivities of the AER andZPA factors. As can be anticipated and will beshown below, the time-dependent location of thecells relative to the AER and ZPA leads to atime-dependent micro-environment of morpho-gen levels to which the cells are exposed.

An alternate representation of the differentialgrowth throughout the limb is obtained byfollowing the motion of the trapezoidal blocks oftissue formed by connecting four adjacentmarkers, two in the proximal row and two in thedistal row. The eight blocks that result arelabeled as follows: Block 1: (a, A, b, B), Block 2:(b, B, c, C), . . . , Block 8: (h, H, i, I) (see Fig. 10).Each block undergoes growth, deformation anddisplacement during outgrowth, and the differ-ence between the final and initial area of a blockrepresents the cumulative growth in that block.The spatial location of a block at any time T canbe interpreted as a fate map, from t=0 to t=T,for cells in that block. The details of this

8

1

7

2

6

3

5

4

1

12

6

3

5

4

2

1

2

6

3

54

3

1

2

6

3

5

4

4

317

transformation depend primarily on the localgrowth rate, which is governed by the spatio-temporal distribution of the morphogens. Forexample one sees that in Simulations 2–4, Block3 is the largest and has a significant area in thepost- and pre-axial regions. The boundarybetween Blocks 4 and 5 represents the midline att=0, and one sees that it suffers a substantialdisplacement in the anterior direction inSimulations 2–4.

4.3.

The local patterns of growth as a function oftime, which are manifested in the separationbetween markers in Figs 8 and 9 and in thegrowth of area in Fig. 10, reflect the history ofexposure to the growth-controlling morphogenat the fluid markers. On the other hand, thecontour lines shown in Fig. 7 reflect an Eulerianor ‘‘fixed-in-space’’ viewpoint that reflects whata cell is exposed to at a certain point inspace–time. At present it is not known whether

it is the instantaneous concentration of morpho-gens that determines gene activation, or whetherit is the history of exposure (or at least aminimum exposure time) that is paramount. Forthis reason we next display the morphogenconcentrations in the Lagrangian framework, inwhich concentrations are tracked along celltrajectories.

The morphogen concentration ck (t) at thelocation of a fluid marker can be expressed as

ck (t)= ck (Xi (t), t) (44)

where Xi (t) gives the path of the i-th marker. Theconcentrations ck of the AER and ZPAsubstances at the location of the markers whosetrajectories are given in Fig. 9 are shown as afunction of time in Fig. 11. One sees there thatin simulations 2 and 3 the concentration of theAER morphogen is a strictly decreasing functionof time for all cells but the one that originates atthe marker (C). This decrease is to be expected,since the growth rate is highest near the tip,

F. 10. Blocks 1–8 for simulations 1–4. Each frame shows the initial and final block outlines and limb bud configurations.

Simulation 2

Simulation 3

Simulation 4

a b c d e f

A B C D E F

a b c d e f

A B DC E F

a b c d e f

A B C D E F

. . . 318

F. 11. Concentrations of c1 (—) and c2 (. . .) at the location of the individual fluid markers. The hash marks on thevertical axis of each graph represent equal increments of dimensionless concentration from (0, 6). The hash marks on thehorizontal axis represent equal increments in dimensionless time from (0, 1.3125). For each of the simulations shown, themost posterior pair (a, A) is shown at the left of the figure.

and thus the apex grows away from the markedcells. There are small increases at intermediatetimes for the cell marked C in all simulationsbecause (a) it remains quite close to the tipthroughout outgrowth, and (b) its traject-ory is displaced posteriorly, and hence towardhigher AER factor, especially in Simulation 2.

The temporal profile of the concentration ofZPA factor is quite different for cells thatoriginate in the anterior half as compared withthose that originate in the posterior half. In theformer the ZPA factor is decreasing in time, asis the AER factor, but cells in the posterior halfexperience a well-defined temporal maximum of

319

ZPA factor, and this occurs earliest nearest theZPA and later toward the midline.

5. Interpretation Functionals Relating GeneExpression to Morphogen Distributions

The patterns of gene expression in the growinglimb bud, such as those discussed later for Hox,may be directly or indirectly related to thespatio-temporal profiles of substances producedin the AER and ZPA. Transduction of anextracellular signal into gene expression usuallyinvolves complex signal transduction mechan-isms based on intracellular second messengers,and at present the molecular basis of Hox genecontrol has not been elucidated. Thus it is notknown if there is a direct connection, but theindirect evidence for a connection is strongenough to justify some theoretical analysis. Inthis section we suggest several possible interpret-ation mechanisms by which the morphogendistributions could be translated into patterns ofgene expression.

The simplest map between morphogen con-centration and gene expression is obtained bymaking the latter directly dependent on theformer. In the previous section we showed thatthe concentration of the AER factor is a strictlydecreasing function of time along most celltrajectories. This fact can be used to define a genecontrol mechanism as follows. Let us definethe progress zone (PZ) as the region in which theconcentrations of the AER morphogen and theZPA factor are above a given threshold, andthen cells leave the PZ when either concentrationlevel drops below the respective threshold.Suppose we set the thresholds at c1 =1 andc2 =1 in Simulation 2 (cf. Fig. 11). Then one seesfrom that figure that only the cells corres-ponding to the three posterior-most markers[(a, A)–(c, C)] are ever in the PZ, and these cellsleave the PZ in a posterior to anterior sequencein time. This is consistent with the observationthat posterior differentiation typically precedesanterior differentiation. Other maps baseddirectly on the morphogen distributions arepossible, but we turn next to maps based on arudimentary form of signal transduction.

In the simplest model we assume that theactivation and transcription of the genes in

question depends on the concentration of c1 andc2 at the cell site. The gene product wk , which isthe indicator of gene expression, is produced ineach cell according to the differential equation

dwk

dt= f(ck (t))− bkwk (45)

where as before, ck (t) is the k-th morphogenconcentration in the Lagrangian description, i.e.at the cell site at time t. We suppose thatproduction of wk depends only on the concen-tration of ck and that it is degraded according tofirst-order kinetics with rate constant bk . Theform of the response function f is describedbelow. In this model, the production rate of aparticular gene product wk depends only on thesingle morphogen ck , but a more complex modelcould be constructed in which the productionrates depend on both morphogens.

The solution to (45) is

wk (t)=wk (0)e−bk t +gt

0

f(ck (s))e−bk (t− s) ds.

(46)

Thus the current concentration of the geneproduct, wk (t), depends on the entire history ofthe morphogen concentration ck at the cell site.The decay rate of wk determines how rapidly the‘‘memory’’ of past concentrations fades, and thefunction f determines how the morphogenconcentration is filtered via the signal transduc-tion scheme. The following four characteristicforms for the response function f will beconsidered here.

f1 =ck

1+ ck

f2 =ck

1+ c2k

f3 =c6

k

1+ c6k

f4 =c6

k

100+ c6k.

The graphs of these functions are shown inFig. 12, where one sees that f1, f3 and f4 areincreasing functions of ck , k=1, 2, and each of

1.0

0.5

0.010 2 3 4 5 6

f3

f4

f1

f2

. . . 320

F. 12. Graphs of the four functions used in theinterpretation schemes.

varies significantly with the spatial marker whenusing f2 as the filter. It is also evident that, asexpected, the switch-like functions f3 and f4

enhance the spatial gradients of the geneproducts along the anterior–posterior axis ascompared with those for the morphogenconcentrations. In particular, one sees that theanterior–posterior gradient of w2 is significantlysharper than that of c2 when using either f3 or f4

as the response function.To better compare our results with experimen-

tally-observed spatial patterns of gene ex-pression, we have also computed the contours ofwk over the entire limb domain at selected times*.The contours for wk that result from applying theresponse functions f1 and f2 are shown in Figs 15and 16. The contours in Fig. 15 are suggestive ofthe Hoxa and Hoxd expression patterns seen atHH stage 22 (see the Discussion). However, itshould be noted that the pattern seen here isqualitatively similar throughout the simulation,whereas the patterns of Hox expression in chicklimb apparently change significantly duringdevelopment. Since f1 is monotone increasingone expects that the concentrations of the wk swill be monotone decreasing in the distal toproximal and posterior to anterior directions,and one sees that this is generally the case. Incontrast, because f2 increases rapidly to itsmaximum at wk =1 and falls off gradually atlarger concentrations, the region of maximumexpression of wk does not necessarily coincidewith the region of maximum expression of ck . InFig. 16(a), which corresponds to stage 21, themaximum expression of w1 is found in thesubridge proximal to the limb bud apex. In thelater stages the high point is found at the apex.

To emphasize the effects of changes in theresponse functions, we show the w2 distributionfor all four response functions in Fig. 17. Toprovide a basis for comparison the sameconcentration levels are shown for all responsefunctions. The functions f1, f3, and f4 are eachmonotone in wk and produce expression patternswith the highest concentrations in w2 in the ZPA.The concentration levels decrease monotonicallyin a posterior–distal to anterior–proximal se-quence, but this monotonicity is lost in thefunctional f2. Moreover, as can be anticipatedfrom earlier discussion, the expression of w2 is

these saturates at 1. The function f2 has a singlemaximum and decays to zero for large ck , f1 is ofMichaelis–Menten form, and f3 and f4 aresigmoidal and differ only in the half-maximalconcentration. The primary difference betweenthe Michaelis–Menten and sigmoidal functions isin the shape of the function at the toe nearck =0. The sigmoidal functions f3 and f4 areintended to approximate ‘‘switches’’ of differentsharpness under which the production of wk

switches rapidly from low to high as ck increases.In the following we apply these functionals to theconcentration histories in Simulation 2. Theconclusions for the other simulations are similar.

We set b=10 and w(0)=0 and compute thetemporal evolution of wk for all pairs of fluidmarkers in the posterior half of the limb. Theresults for f1 and f2 are shown in Fig. 13 and thosefor f3 and f4 are shown in Fig. 14. These figuresshould be compared with the graphs of ck for thesame fluid markers shown in Fig. 11. Clearly thegraphs of wk are much smoother than are thosefor ck because the concentration is smoothed viathe integration in eqn (46). Moreover, becausethe ‘‘gene product’’ wk is zero initially, the wk shave a pronounced maximum and the timeabove half-maximal concentration dependsstrongly on the response function. For examplef1, f2 and f3 produce relatively broad peaks, andin addition, the location of the peak in time

* In order to generate these contours, we solve eqn (46)for each of the fluid markers in the limb domain. Theconcentration data at the irregularly spaced fluid markersis interpolated to the regular grid using the subroutineBIVAR supplied with the NCAR graphics package.

e E

f1

e

f2

E

d D d d

c C c C

b B b B

a A a A

321

much more localized in space for f3 and f4. Acomparison of Figs 16 and 17 shows that thequalitative picture of the spatial distributiondepends heavily on the decay parameter.

To further emphasize the effect of the decayrate on the spatial distribution of gene productwe show the distribution of w2 for the responsefunction f2 and a lower decay rate in Fig. 18. One

sees in Fig. 18 that decreasing the decay rateincreases the expression domain of w2, asexpected, but the value of the decay rate can haveunexpected effects. One sees that there is aY-shaped expression domain for w2, which issuggestive of what might be used for determiningthe bifurcation into the radius and ulna, orperhaps later, the digits.

F. 13. The temporal profiles of wk for simulation 2 at the fluid markers (a, A), (b, B), . . . , (e, E), using the functionsf1 and f2. The vertical axis represents the concentration of wk $ [0, 0.1]; the horizontal axis represents dimensionless timefor t $ [0, 1.3125].

e

f3

E e E

f4

d D d D

cC

c C

b B b B

a A a A

. . . 322

F. 14. The temporal profiles of wk for simulation 2 at the fluid markers (aA), (bB), . . . , (eE) (ordered top to bottom),using the functions f3 and f4. The vertical axis represents the concentration of wk $ [0, 0.12]; the horizontal axis representsdimensionless time for t $ [0, 1.3125].

6. Discussion

6.1. -

Important features of our model whencompared with previous models are the incor-poration of outgrowth of the limb bud, themutual feedback interactions between the AER

and ZPA factors, and the ability to determine themorphogen distributions in two space dimen-sions as a function of time, both at fixedlocations in space, and along cell trajectories inthe growing limb. As a result, we can determinethe spatial and temporal history of individualcells and cell lines within the changing environ-ment of morphogen concentrations. That the

(a) (b) (c) (d) (e)

(a) (b) (c) (d) (e)

323

F. 15. Concentration contours for w1 (upper) and w2 (lower) produced with the functional f1 at dimensionless times(a) 0.3375; (b) 0.50625; (c) 0.675; (d) 0.9375; (e) 1.2; [real times of (a) 7.8; (b) 11.7; (c) 15.6; (d) 21.7; (e) 27.8 hours andstages (a) 21; (b) 22; (c) 23; (d) 24; (e) 25] in simulation 2. Higher concentration levels are indicated by closer spaced crosshatching. Three dimensionless concentration levels are shown: 0.02 Qwk Q 0.04, 0.04Qwk Q 0.06 and 0.06Qwk .

computational model can give insights into celland morphogen dynamics in vivo is suggested bynoting that the fluid marker maps shown inFig. 8 are qualitatively similar to the fate mapsin Bowen et al. (1989) in the distal half of thelimb bud, and that the block maps shown inFig. 10 are qualitatively similar to the fate mapsshown in Vargesson et al. (1997) Fig. 1(B) and1(G). Because the computations provide infor-mation about cell lineage via the fate maps oftissue blocks, as well as the temporal history ofmorphogen concentration levels within these

tissue blocks (Fig. 11), we can investigate therelationship between cell lineage and geneexpression.

In the classical version of the progress zonemodel discussed in the Introduction, the PZ isregarded as a specialized region within the distalsubridge, and the amount of time cells spend inthe PZ influences the course of proximal–distaldifferentiation. In our model we could define thePZ as a region in which the AER growth factoris above a threshold level, and as can be seen inFig. 7, the contour lines in the AER morphogen

F. 16. Concentration levels for w1 (upper) and w2 (lower) in simulation 2 produced with the function f2 at the stagesshown in Fig. 15. Four dimensionless concentrations levels are shown: 0.015 Qwk Q 0.020, 0.020Qwk Q 0.025,0.025Qwk Q 0.030, and 0.030Qwk .

(a) (b) (c) (d)

. . . 324

F. 17. Concentration levels for w2 using the four response functions. (a) f1; (b) 2f2; (c) f3; (d) f4. The contour plots showthe solution for simulation 2 at dimensionless time 0.9375 (real time 21.7 hours) corresponding to HH stage 24. Fivedimensionless concentrations levels are shown: 0.04 Qw2 Q 0.08, 0.08Qw2 Q 0.12, 0.12Qw2 Q 0.16, 0.16Qw2 Q 0.2,0.2Qw2. The parameters for the functions are the same as those used above, except b=2.5.

c1 define a stable PZ in the distal subridge. Usingthis definition of the progress zone, one candetermine when cells enter and/or leave theprogress zone by determining the local concen-tration of c1 at the cell site, as shown in Fig. 11.Of course the PZ, and hence the residence timein it, may be determined by a downstreamproduct of the morphogens, and the filteredhistories of the morphogen concentrationsshown in Figs 13 and 14 illustrate how the timespent in the PZ changes as the definition of thePZ changes.

One of the most important open problems inlimb development concerns the relationshipbetween the spatio-temporal pattern of primarysignaling molecules such as Sonic and thevarious growth factors, the patterns of geneexpression, and the translation of these patternsinto patterns of cell differentiation to cartilageand connective tissue types. Given the complex-ity of the networks involved in the control ofgene expression, the possible redundancy be-tween the effects of homologous genes, and the

paucity of information about the control of celldifferentiation, it is not possible to synthesize anintegrated scheme at present. However one canaddress simpler issues, such as the possiblerelationship between the pattern of signalingmolecules and patterns of gene expression. Atpresent most is known about the spatio-temporalpattern of expression of the Hox genes, which arethought to be the primary genes involved in thepatterning necessary for the proper spatiallocalization of the cartilage condensations, andtherefore we briefly review this information. Alisting of many other genes that are known to beexpressed in various stages of limb developmentis given in Tickle & Eichele (1994).

The vertebrate Hox genes comprise fourcomplexes of gene clusters, Hoxa, Hoxb, Hoxcand Hoxd, and genes in each complex havehomologs in each of the other clusters. Thepatterns of expression of Hoxa and Hoxd havebeen most completely mapped, and a summaryof the pattern at HH stage 22 is shown in Fig. 19.The five Hoxd genes, labeled 9–13 from 3' to 5'on the chromosome, are expressed in aspatially-nested pattern that is centered roughlyat the ZPA [cf. Fig. 19(b)] (Izpisua-Belmonteet al., 1991). This spatial pattern of geneexpression reflects the temporal order in whichgenes are read from 3' to 5'. Hoxd-9 andHoxd-10 are expressed throughout the prospec-tive limb field at stage 16, Hoxd-11 appears atstage 18, and expression of Hoxd-12 andHoxd-13 begin thereafter (Nelson et al., 1996).

Our computational results demonstrate thatthe interaction between two spatially-localizedsources of FGF and SHH can produce a stabletwo-dimensional distribution of each of themorphogens and of secondary gene products as

F. 18. Concentration levels for w2 produced with thefunctional 2f2 at stage 23, but with a decay rate of b=1.Seven contours levels are shown: (0.04, 0.08), (0.08, 0.12),(0.12, 0.16), (0.16, 0.20), (0.20, 0.24), (0.24, 0.28), (0.28,0.32).

(a)Hox a9–a11

Hox a9–a13

Hox a9

(b)

Hox d9–d13

Hox d9–d11Hox d9

325

F. 19. A schematic of the spatial pattern of Hoxa (a)and Hoxd (b). After Robertson & Tickle (1997).

stage 28, and high concentration levels of FGFmay be required for the continued expression ofthis gene. The results of our computationalmodel are in qualitative agreement with thisconclusion, in the sense that cells of the samelineage will nonetheless experience differenttemporal histories of morphogen exposure.

6.2.

As we stated earlier, our objective here is notonly to study the effects of spatial localization ofthe ZPA and the AER, and of the couplingbetween them, but also to develop a compu-tational tool that can be used to investigate theeffects of experimental interventions during limbdevelopment, and to explore various hypotheti-cal maps between morphogen distributions andgene expression. In future work we plan toextend the model to three space dimensions andto explore some of the grafting and blockingexperiments that have been done. We also planto investigate the interactions between the majorknown signaling pathways, to develop a bettermodel for the viscoelastic properties of cells, andto include chemotactic motion of cells.

Supported in part by an NSF PostdoctoralResearch Fellowship and NSF grant DMS 9805501 toR. Dillon and by NIH Grant (GM21923 to H.Othmer.

REFERENCES

A, F., T, C. & W, A. (1990). The role of gapjunctions in patterning of the chick limb bud.Development 108, 623–634.

B, G. K. (1973). An Introduction to FluidMechanics. Cambridge: Cambridge University Press.

B, Y., T, I. & P, N. (1998). Tout-velu isa Drosophila homologue of the putative tumoursuppressor EXT-1 and is needed for Hh diffusion. Nature394, 85–88.

B, J., H, J. R., H, T. J. & R,A. M. F. (1989). The fate map of the chick forelimb-budand its bearing on hypothesized developmental controlmechanisms. Anat. Embryol. 179, 269–283.

C, M. J., I-B, J. C., A, H. & H,J. K. (1995). Fibroblast growth factors induce additionallimb development from the flank of chick embryos. Cell80(5), 739–746.

C, P. H. & M, G. R. (1995). The mouse geneFgf-8 encodes a family of polypeptides and is expressedin regions that direct outgrowth and patterning in thedeveloping embryo. Development 121, 439–451.

well. This morphogenetic landscape could be thebasis of a stable positional information systemavailable to the cells over stages 19–25. Althoughthe level sets of the morphogen distributions maynot be identified directly with Hoxa or Hoxdexpression, their existence does allow thepossibility of a stable positional informationsystem based on interdependent morphogengradients in two spatial directions. The level setsof the morphogen distributions are also qualitat-ively similar to the distribution of Hoxa andHoxd expression, at least at some stages.

Vargesson et al. (1997) have demonstratedthat gene expression may depend on both celllineage and morphogen concentration levels. Inthat study a lipophilic dye was injected into stage20 wing buds under the AER and examined laterat stage 28. At stage 20 the expression ofHoxd-13 is localized in the posterior–distalmesenchyme of the limb bud, but by stage 28 theexpression domain has expanded considerably inthe anterior direction. Conceivably the expan-sion of this expression domain could be dueentirely to cell growth, as would be predicted bythe fate maps of tissue blocks. However, it wasfound that labeled cells on the proximalperiphery at stage 20 do not express the gene at

. . . 326

C, P. H., M, G., MA, C. A. &M, G. R. (1996). Roles for FGF8 in the induction,initiation, and maintenance of chick limb development.Cell 84(1), 127–136.

D, D. M., K, K., F-W, P. H.,T, C. & B, P. M. (1996). Activation of Fgf-4and HoxD gene expression by BMP-2 expressing cells inthe developing chick limb. Development 122, 1821–1828.

F, Y. C. (1993). Biomechanics: Mechanical Properties ofLiving Tissues. Berlin: Springer-Verlag.

G, U., M, G., P, J. M., A,U. K. & M, G. R. (1996). The chick limblessmutation causes abnormalities in limb bud dorsal–ventralpatterning: implications for the mechanism of apical ridgeformation. Development 122, 3851–3861.

H, A. G., B, J. M. & Z, R. (1995). Thelimb deformity mutation disrupts the SHH/FGF-4feedback loop and regulation of 5' HoxD genes duringlimb pattern formation. Development 121, 4237–4245.

H, J., T, C. & E, G. (1994). Relationshipbetween retinoic acid and sonic hedgehog, two polarizingsignals in the chick wing bud. Development 120,3267–3274.

H, J. R. & G, P. J. (1984). Experimentalanalysis of the control of cell death in chick limb buddevelopment. In: Society for Experimental BiologySeminar Series, Vol. 25. Cell Ageing and Cell Death;Meeting (Davies I. & Sigee, D. C., eds) pp. 223–242.Cambridge, England, New York: Cambridge UniversityPress.

H, L. S. & S, D. (1985). Maps of strength ofpositional signaling activity in the developing chick wingbud. J. Embryol. Exp. Morphol. 87, 163–174.

H, A. & W, L. (1991). The spatial andtemporal distribution of polarizing activity in the flank ofthe pre-limb-bud stages in the chick embryo. Development111, 725–731.

I-B, J. C., T, C., D, P., W,L. & D, D. (1991). Expression of the homeoboxHox-4 genes and specification of position in chick wingdevelopment. Nature 350, 585–589.

J, L. C. & I, L. E. (1981). Position of origin ofdonor posterior chick wing bud tissue transplanted to ananterior host site determines the extra structures formed.Dev. Biol. 82(2), 329–342.

J, L. C. & I, L. E. (1986). The handedness andorigin of supernumerary limb structures following 180degree rotation of the chick wing bud on its stump. J.Embryol. Exp. Morphol. 91, 135–152.

J, R. L. & T, C. J. (1997). Molecular models forvertebrate limb development. Cell 90, 979–990.

K, J. & N, L. (1997). Limb deformityproteins: role in mesodermal induction of the apicalectodermal ridge. Development 124(1), 133–139.

L, E., N, C. E., J, R. L., M, B. A.& T, C. (1994). Sonic hedgehog and Fgf-4 act througha signaling cascade and feedback loop to integrate growthand patterning in the developing limb bud. Cell 79,993–1003.

LV, R. J. & L, Z. (1994). The immersed interfacemethod for elliptic equations with discontinuous co-efficients and singular sources. SIAM J. Numer. Anal. 31,1019–1944.

L, H.-C., R, J.-P., G, L., T, C. &E, G. (1997). Retinoid signaling is required for the

establishment of a ZPA and for the expression of Hoxb-8,a mediator of ZPA formation. Development 124(9),1643–1651.

MC, J. A., MC, A. B., A, U. K. &MC, J. R. (1975). Limb development inDiplopodia4: a polydactylous mutation in the chicken.J. Exp. Zool. 191, 383–392.

M, M., O, D. E., S, D. & C, F.(1988). Spatial distribution of cellular protein binding toretinoic acid in the chick limb bud. Nature 335, 733–735.

M, P.K. & S,M. (1991). Cellular mechanisms ofpattern formation in the developing limb. Int. Rev. Cytol.129, 91–133.

M, V., J, R. L., V, A. & T, C. J.(1996). Sonic hedgehog differentially regulates expressionof GL1 and GLI3 during limb development. Dev. Biol.180(1), 273–283.

M, P., B, J. S. & L, W. R. (1989).The actions of retinoids on cellular growth correlate withtheir actions on gap junctional communication. J. CellBiol. 108, 1053–1065.

M, H. (1982). Models of Biological PatternFormation. London: Academic Press.

M, H. (1983). A boundary model for patternformation in vertebrate limbs. J. Embryol. Exp. Morphol.76, 115–137.

M, J. L., L, F. & L D, N. M. (1997).The dorsoventral polarity of the presumptive limb isdetermined by signals produced by the somites and by thelateral somatopleure. Development 124(8), 1453–1463.

M, K. W. & M, D. F. (1994). NumericalSolution of Partial Differential Equations. Cambridge:Cambridge University Press.

N, C. E., M, B. A., B, A. C., L, E.,DM, E., M, L. C., G, E.,T, L., P, L. F. & T, C. (1996).Analysis of Hox gene expression in the chick limb bud.Development 122, 1449–1466.

N, L. (1997). Limb mutants: what can they tell usabout normal limb development? Curr. Opin. Gen. Dev.7, 530–536.

N, L. & M,G.R. (1993). Mixed signals fromthe AER: FGF-4 and BMP-2 have opposite effects onlimb outgrowth. In: Limb Development and Regenerationpp. 625–633. New York: Wiley–Liss.

O, T., A, I. S., V, A., R, C.,E, R. M. & B, J. C. I. (1996). Evidence thatShh cooperates with a retinoic acid inducible co-factor toestablish ZPA-like activity. Development 122, 537–542.

O, H., N, T., Y, A., A, A.,O, T., I, Y., Y, H., K, T.,N, T.,Y,M., I,N. & N, S. (1997). Themesenchymal factor, FGF10, initiates and maintains theoutgrowth of the chick limb bud through interaction withFGF8, an apical ectodermal factor. Development 124(11),2235–2244.

O, G. F., M, J. D. & H, A. K. (1983).Mechanical aspects of mesenchymal morphogenesis.J. Embryol. Exp. Morph. 78, 83–125.

P, M. P. (1973). Analyse de la morphogenese du pieddes Oiseaux a l’aide de melange cellulaires interspe-cifiques. I. Etude morphologique. J. Embryol. Exp.Morphol. 29, 175–196.

P, C. S. (1977). Numerical analysis of blood flow in theheart. J. Comp. Phys., 25, 220–252.

327

P, C. S. & MQ, D. M. (1995). A general methodfor the computer simulation of biological systemsinteracting with fluids. In: Biological Fluid Dynamics(Ellington, C. P. & Pedley, T. J., eds). Company ofBiologists, Ltd.

P, H. M. & S, M. S. (1969). Equilibriummeasurements of embryonic chick cell adhesiveness. I.Shape equilibrium in centrifugal fields. Proc. Nat. Acad.Sci. U.S.A., 64, 121–127.

P, H. M. & S, M. S. (1978). Embryonictissues as elasticoviscous liquids. I. Rapid and slow shapechanges in centrifuged cell aggregates. J. Cell Sci. 30,1–20.

P, H. M., S, M. S. & L, B. H. (1977).Embryonic tissues as elasticoviscous liquids. II. Directevidence for cell slippage in centrifuged aggregates. Dev.Biol. 59, 124–134.

R, R. D., J, R. L., L, E. & T, C.(1993). Sonic hedgehog mediates polarizing activity of theZPA. Cell 75, 1401–1416.

R, K. E. & T, C. (1997). Recent molecularadvances in understanding vertebrate limb development.Br. J. Plastic Surgery 50(2), 109–115.

R A, A. (1994). Later. Trends Neurosci. 17,233–243.

S, R. L. & J, M. Y. (1971). The initiation oflimb bud outgrowth in the embryonic chick. Dev. Biol.24, 198–213.

S, S. M., P, K., S, O., W, S. E.,T, C. & E, G. (1989). Molecular approachesto vertebrate limb morphogenesis. Development (Suppl),121–131.

S, M. & J, K. L. (1988). The accumulation ofbasement membrane components during the onset ofchondrogenesis and myogenesis in the chick wing bud.Development 104(1), 41–50.

S, R. J. & S, R. L. (1973). A description of chickwing bud development and a model of limb morphogen-esis. Dev. Biol. 33, 138–153.

S, D. (1974a). Interaction between the proximo-distal and antero-posterior coordinates of positionalvalue during the specification of positional information inthe early development of the chick limb-bud. J. Embryol.Exp. Morphol. 32, 227–237.

S, D. (1974b). A quantitative analysis of theeffect of excision of the AER from the chick limb-bud.J. Embryol. Exp. Morphol. 32, 651–660.

S,D. (1977). Regulation of deficiencies along theproximal distal axis of the chick wing-bud: a quantitativeanalysis. J. Embryol. Exp. Morphol. 41, 137–159.

S, D. (1979). The zone of polarizing activity:evidence for a role in normal chick morphogenesis.J. Embryol. Exp. Morph. 50, 217–233.

S,D. (1981). Control of growth and the develop-ment of pattern across the anteroposterior axis of thechick limb bud. J. Embryol. Exp. Morphol. 63, 161–180.

S, D. & L, J. H. (1975). Time, place andpositional value in the chick limb-bud. J. Embryol. Exp.Morph. 33, 621–643.

S, D., L, J. H. & W, L. (1973).Positional information in chick limb morphogenesis.Nature 244, 492–496.

T, C. & E, G. (1987). Identification andspatial distribution of retinoids in the developing chicklimb bud. Nature 327, 625–628.

T, C. (1980). The polarizing region and limbdevelopment. In: Development in Mammals (Johnson, M.H., ed.), vol. 4, pp. 101–136. Amsterdam: Elsevier/North-Holland Biomedical Press.

T, C. & E, G. (1994). Vertebrate limbdevelopment. In: Annual Review of Cell Biologypp. 121–152. Annual Reviews, Inc.

T, C., L, J. & E, G. (1985). A quantitativeanalysis of the effect of all-trans-retinoic acid on thepattern of chick wing development. Dev. Biol. 109, 82–95.

T, C., S, D. & W, L. (1975).Positional signalling and specification of digits in chicklimb morphogenesis. Nature 254, 199–202.

V, N., C, J. D., V, K., C, C.,W, L. & T, C. (1997). Cell fate in the chicklimb bud and relationship to gene expression. Develop-ment 124(10), 1909–1918.

V, A. & T, C. (1993). FGF-4 maintains polarizingactivity of posterior limb bud cells in vivo and in vitro.Development 119, 199–206.

V, A., R, C. & I-B, J.-C.(1996). Involvement of FGF-8 in initiation, outgrowthand patterning of the vertebrate limb. Development 122,1737–1750.

W, O. K. & E, D. A. (1975). A model for generatingthe pattern of cartilage skeletal elements in the embryonicchick limb. J. theor. Biol. 52, 199–217.

W, L. (1969). Positional information and the spatialpattern of cellular differentiation. J. theor. Biol. 25, 1–47.

W, L. (1976). Mechanisms of limb development andmalformation. Br. Med. J. 32, 65–70.

W, L. (1987). An introduction to cells. Br. Med. J.295(6604), 987–988.

W, L. & H, A. (1990). Double anteriorchick limb buds and models for cartilage rudimentspecification. Development 109, 961–966.

W, L., L, J. & S, D. (1975). Morpho-genesis of the vertebrate limb. Ciba Symp. 29, 95–130.

Z, E. (1961). Limb morphogenesis. Adv. Morpho-gen. 1, 301–330.

APPENDIX

The rectangular fluid domain is discretizedusing a uniform rectangular grid. The fluidvariables (u, p, F, S) are discretized and definedat the grid points. The limb bud boundary is alsodiscretized and represented by a finite number ofpoints Xp , p=1, . . . , N, but in general theimmersed boundary points Xp do not coincidewith points of the fluid grid. The boundary forcesfp are defined at the points Xp , and interpolationbetween the immersed boundary points and thefluid grid is effected by means of a discreted-function given below. The concentration ofchemical species c is defined on the same grid asthe fluid variables. Since the computational fluiddomain is rectangular we are able to use Fast

. . . 328

Fourier Transform methods for time-steppingthe fluid equations.

The algorithm for numerical solution of thecoupled fluid-chemical system can be summar-ized as follows. At the beginning of each timestep n we have the fluid velocity field un, thelocations Xn

p of the immersed boundary points,and the chemical concentration field cn. In orderto update these values during this time step wedo the following.

1. Calculate the force density fnp at each

immersed boundary point.2. Interpolate the forces to the grid to

determine the force density F [eqn (4)].3. Calculate the local growth rate Sn =S(cn).4. Solve the Navier–Stokes eqns (1–2) for

un+1.5. Interpolate the fluid velocity field to each

immersed boundary point and move thatpoint at its local fluid velocity to obtainXn+1

p [eqn (3)].6. Solve the advection–diffusion–reaction eqn

(5) for cn+1.

A.1. Elastic Boundary Forces

The ectodermal boundary of the two-dimensional limb bud is modeled by a discreteset of immersed boundary points Xp forp=1, . . . , N(t). The location of each point istime dependent. Each immersed boundary pointXp is connected to its neighbors Xp−1 and Xp+1 byelastic forces of the form

f1k = s

k+1

p=k−1

p$ k

S1(>Xk −Xp>−L0)(Xk −Xp )>Xk −Xp>

.

(A.1)

where S1 is a spring stiffness constant and L0 isthe resting length. The initial spacing betweenadjacent pairs of immersed boundary points isalso given by the resting length L0 and is one-halfthe mesh width of the two-dimensional fluiddomain. The immersed boundary points alongthe proximal boundary are also tethered to fixedpoints in space via elastic spring, and thus these

points are subject to an additional force of theform

f2k =S2(Xk −X0

k ), (A.2)

where S2 is a spring stiffness constant and X0k is

the initial location of a proximal boundary pointXk . The tethering force f2

k =0 at all immersedboundary points away from the proximalboundary.

Each immersed boundary point along theposterior boundary is also connected via anelastic spring force to an immersed boundarypoint on the anterior boundary. If we label theset of non-proximal boundary points(X1, . . . , XM ), from the lower left corner of thelimb bud counterclockwise around the limb tothe upper left corner, we form an elastic linkbetween pairs (X1, XM ), (X2, XM−1), . . . , (XM/

2−1, XM/2+1). The transverse force at aposterior immersed boundary point Xk is derivedfrom an equation of the form

f3k =S3(>Xk −Xp>−Dw )(Xk −Xp )/>Xk −Xp>.

(A.3)

where p=M− k+1. The spring constants S3

are taken to be zero unless the distance betweenthe pair of points is greater than DW , the nominallimb bud width. These transverse forces con-strain the limb so that it elongates along theproximal distal axis rather than ballooning as thelimb bud grows.

The total elastic force at each immersedboundary point is the sum fk =S3

j=1fjk . These

forces f are interpolated to the two-dimensionalnumerical fluid grid points by a discrete versionof eqn (4). The interpolation is based on anapproximate delta function of the formdh (x)=d(x) d(y) where h is the mesh width and

d(r)=gG

G

F

f

14h 01+cos

pr2h1

0

=r=Q 2h

=r=e 2h.(A.4)

We refer the reader to Peskin (1977) for furtherdetails.

As the limb bud expands, the immersedboundary points become more widely spaced.During the simulation, we introduce a new

329

boundary point between an adjacent pair if theelastic link exceeds a prescribed length.

A.2. Numerical Solution of the Navier–StokesEquations

The numerical method for solving theNavier–Stokes eqns (1)–(2) is similar to themethod described in Peskin & McQueen (1995).Our model includes an additional term in eqn (1)due to the distributed source S in the continuityeqn (2). Equations (1)–(2) are discretized on aregular rectangular grid of with mesh width h

r0un+1 − un

Dt+ s

2

s=1

uns D2

s un1=−D0pn+1

+ m0s2

s=1

D+s D−

s un+1 +1/3D0Sn1+ rF n

D0 · un+1 =Sn (A.5)

where Sn =S(cn), D+s , D−

s are the forward andbackward divided difference operators fors= x, y, D0 is the centered divided difference,D0 · is the center divided difference divergenceoperator and D2

s is the upwind divided difference(Peskin & McQueen, 1995). Equation (A.5) canbe rewritten as:

0I−mDtr

s2

s=1

D+s D−

s 1un+1 +Dtr

D0pn+1 = vn

D0 · un+1 =Sn (A.6)

with

vn =0I−Dt s2

s=1

uns D2

s 1un +mDt3r

D0Sn

+DtF n. (A.7)

Equation (A.6) is a constant-coefficient systemof difference equations in the unknowns(un+1, pn+1). We can define the discrete Fourier

transformation of the grid function f on thespatial domain (0, L)2 by

f k1, k2 = sN−1

j1, j2 =0

exp(−i(2p/N)(j1k1 + j2k2))fj1 j2

(A.8)

for 0E k1, k2 EN−1 and L=Nh. With thisdefinition, the discrete Fourier transform of eqn(A.6) is given by:

01+4mDtrh2 s

2

s=1

sin2(pks /N)1(uxs )n+1k1k2

+iDtrh

sin(2pks /N)pxn+1k1k2

= (vxs )nk1k2

i

hs2

s=1

sin02pks

N 1(uxs )k1k2 =S nk1k2

. (A.9)

We can now multiply eqn (A.9) by

i

hsin02pks

N 1 (A.10)

and sum over s=1, 2, to obtain

pxn+1k1k2

=$01+4mDtrh2 s

2

s=1

sin2(pks /N)1S nk1k2

−i

hs2

s=1

× sin(2pks /N)(vxs )k1k2%>Dtrh2 s

2

s=1

sin2(2pks /N).

(A.11)

With px known, we can solve eqn (A.9) for ux toobtain:

(uxs )n+1k1k2

=(vxs )n

k1k2−

iDtrh

sin(2pks /N)pxn+1k1k2

1+4mDtrh2 s

2

s=1

sin2(pks /N)

.

(A.12)

Note that the denominator in eqn (A.11) is zerofor (k1, k2)= (0, 0), (0, N/2), (N/2, 0), or (N/2,N/2). For these values of (k1, k2) we can solveeqn (A.9) directly for (uxs )k1k1 since the terminvolving pxk1k2 is zero. This is equivalent tosetting pxk1k2 =0 in eqn (A.12). Once uxn+1 is

. . . 330

computed, we make no further use of the pxn+1

and obtain the fluid velocity field un+1 using theinverse discrete Fourier transform.

This model might be formulated for the Stokesequations [obtained from eqns (1)–(2) by settingthe left-hand side of eqn (2) to zero]. However,for a solution to exist on a periodic domain, theintegral of F over the domain must be zero. Thiscondition will not necessarily hold if a portion ofthe limb bud immersed boundary is tethered asit is in our model. Thus a time dependentequation must be employed. In the method usedhere, the computational cost of including theadvective term (u · 9)u is minimal since this termis evaluated at the previous time step.

A.3. Solution of theAdvection–Reaction–Diffusion Equations

The irregularly shaped-moving boundary ofthe growing limb bud creates special difficultiesfor the numerical solution of the advection–reaction–diffusion eqns (5). The model requiresNeumann (zero-flux) boundary conditions forthe chemical species at this interface. We solvethese equations on the same grid as the fluidequations. At regular grid points within the limbbud domain, we use a standard five-point stencilfor the Laplacian 92c. At irregular grid points atleast one of the grid points in the standardfive-point stencil lies outside of the limb buddomain. As a result, the irregular grid pointsrequire special treatment. Step (6) of thealgorithm can be further subdivided into severalsteps:

6a. determine the (x, y) coordinates of thepoints where the limb bud boundaryintersects a vertical or horizontal grid line;

6b. determine the grid points that lie insidethe limb bud domain;

6c. solve the advection–reaction–diffusionsystem.

Since the time steps for the fluid solver aresmall, we can use an explicit method in step (6c)for solving the advection–reaction–diffusion

system. Since we are only interested in thechemical concentrations within the limb budboundary, we can ignore the solution at gridpoints outside of the limb bud. At regular gridpoints, we use the standard five point stencil ina central difference approximation to theLaplacian. Near the boundary, one or more ofthe grid points in the standard stencil will lieoutside of the limb bud domain. At theseirregular grid points, we make use of theNeumann boundary condition to develop anequation for the Laplacian approximation. Ifonly one point is missing from the five-pointstencil (Type 4), we use an approximationdescribed in Morton & Mayers (1994). Atirregular points with two (Type 3) or three pointsmissing (Type 2), we assume that the limb budboundary for the morphogen equations con-forms to the grid lines. This gives us an O(h)approximation to the Laplacian on the per-turbed boundary. The Type 4 solutions can bepoor if the immersed boundary is very close toa grid point. Since our limb bud is growing, thereare points that come arbitrarily close to theboundary. We circumvent the computationaldifficulty by assuming that the boundary isfurther away than it really is. While we lose someaccuracy, the method is stable and inexpensive tocompute. A more elegant approach based on the‘‘immersed interface method’’ of LeVeque & Li(1994) could be incorporated into our numericalsoftware.

A.4. Determination of the Source Strength

At each time step, we must evaluate the localsource strength Sn

i,j at each mesh point. Si,j maydepend on the local concentration of one ormore growth factors cn

i,j . For the simulationsshown in Section 4 and from eqn (41), the localsource strength is given by Sij = s1(c1)ij + s2.Mass balance over the entire fluid domainrequires Si,jSn

i,j =0. The source terms within thefluid domain are balanced by four sinks locatedoutside of the limb domain. We assume thatSij =0 outside of the limb bud domain except atthe sinks.