Developmental Patterningof the Vertebrate LimbEdited by

J. Richard HinchliffeUniversity College of WalesAberystwyth, Wales, United Kingdom

Juan M. HurleUniversity of CantabriaSantander, Spain

and

Dennis SummerbellNational Institute for Medical ResearchLondon, United Kingdom

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A COMPARISON OF REACTION DIFFUSION AND MECHANOCHEMICAL

MODELS FOR LIMB DEVELOPMENT

Philip Maini

Centre for Mathematical BiologyMathematical Institute24-29 st. Giles'Oxford OX1 3LB

Several theoretical models have been proposed to attemptto elucidate the underlying mechanisms involved in the spatialpatterning of skeletal elements in the limb. Here, I brieflycomparetwo such models - reactiondiffusion (RD) and mechano-chemical (MC) - and highlight their properties and predictions.

Although similar mathematically, these models are basedon fundamentally different biological approaches. RD theoryasserts that a chemical prepattern is first set up in the limbby a system of reacting and diffusing chemicals (Turing, 1952).Cells then respond to this prepattern by differentiatingwherever the concentration of one of the chemicals lies abovea threshold value (Wolpert, 1969, 1981). Thus, the spatialpattern of chemical concentration is reflected by the spatialpattern of cell differentiation.

The MC theory (Oster et.al., 1983, Oster et.al., 1985)proposes that the mechanical and chemical interactions of cellswith their external environment- the extracellularmatrix -

leads to a spatial pattern in cell density. The cells in theaggregates thus formed differentiate. Thus, the spatialpattern of cell differentiation overlies the spatial patternof cell density.

The spatial patterns predicted by both RD and MC theory,from linear analysis, ar~ essentially .theeigenfunctions of theLaplacianoperator (Mainiand Solursh,1990). The more complextensor form of the MC model may lead to a richer set ofstructures than those exhibited by RD models. This possibilityhas yet to be analysed. Thus, at the moment, one cannotdistinguish between these models on a purely mathematicalbasis.

The similarity of the mathematical formulation of theabove hypotheses does, however, enable one to make some generalpredictions on the properties of the skeletal pattern in thelimb independent of the biological basis of the model. ForI

iDevelopmental Patterning of the Vertebrate Limb

~ited by J.R.Hinchliffe et al., Plenum Press, New York, 1991 161

example, both models suggest that there are only a limitednumber of ways in which elements may arise and bifurcate. Thishas lead to a more precise formulation of the idea ofdevelopmental constraints on vertebrate limb evolution (Osteret.al., 1988, Oster and Murray, 1989).

Both models predict that the complexity and form ofspatial pattern is dependent on scale, geometry and boundaryconditions. For example, they predict that decreasing 1imb budwidth will result in a loss of elements. This agrees withexperimental observation (Alberch and Gale, 1983).

Recently it has been shown that if two stage 19 anteriorchick limb halves are combined, the recombinant forms twohumeral elements (Wolpert and Hornbruch, 1990). As the recom-binant has the same size as a normal limb, both the abovemodels would predict a single humerus. The observationsuggests that the anterior stage 19 limb half contains cellsthat have already differentiated yet, at this stage, there isno visible aggregation of cells. This is inconsistent with theMC approach but may be consistent with the RD approach assumingthat the prepattern is laid down and stabilised before stage19. However, another interpretation of the result is thatduring normal development a prepattern may divide the earlylimb bud into domains in which cells are either competent orincompetent to differentiate into cartilage. Such a prepatternmay be set up by a gradient in homeobox gene expression or inretinoid distribution (Maini and Solursh, 1990). Therecombination experiment would then be seen as combining twosuch cell populations and, in effect, doubling the domainwidth. The eventual developmental fate of cells could then bedetermined by either of the above models. ~

This scenario suggests that pattern formation is ahierarchal process wherein each mechanism provides the initia '

condition for subsequent processes. This possibility warrantfurther experimental and theoretical investigation.

ACKNOWLEDGEMENT This work was supported in part by thNational Science Foundationunder Grant No. DMS 8901388. '

REFERENCES

P. Alberch and E. Gale, Size dependency during the developmeJof the amphibian foot. Colchicine induced digital loss a!reduction, J. Embryol. expo Morphol. 76, 177-197 (1983). '

P.K. Maini and M. Solursh, Cellular mechanisms of pat~

formation in the developing limb, Int. Rev. Cytology -j!appear). ~

G.F. Oster, J.D. Murray, and A.K. Harris, Mechanical asp~of mesenchymal morphogenesis, J. Embryol. expo Morphol. 78i.~'1

125 (1983). .

G.F. Oster, J.D. Murray, and P.K. Maini, A model ~i

chondrogenic condensations in the developing limb: the r~JIextracellular matrix and cell tractions, J. Embryol.i"'!~Morphol. 89, 93-112 (1985).j

162

G.F. Oster, N. Shubin, J.D. Murray, and P. Alberch, Evolutionand morphogenetic rules. The shape of the vertebrate limb inontogeny and phylogeny, Evolution 45, 862-884 (1988).

G.F. Oster and J.D. Murray, Pattern formation models anddevelopmental constraints, in J.P. Trinkaus Anniversary Volume1989 (in press).

A.M. Turing, The chemical basis of morphogenesis, Phil. Trans.Roy. Soc. Lond. B237, 37-72 (1952).

L. Wolpert, Position~l information and the spatial pattern ofcellular differentiation, J. Theor. BioI. 25, 1-47 (1969).

L. Wolpert, positional information and pattern formation, Phil.Trans. Roy. Soc. Lond. B295, 441-450 (1981).

L. Wolpert and A. Hornbruch, Double anterior chick limb budsand models for cartilage rudiment specification, Development109, 961-966 (1990).

163

Proceedings of a NATO Advanced Research Workshop onDevelopmental Patterning of the Vertebrate Limb,held September 23-26, 1990,in Santander, Spain

LIbrary of Congress Cataloglng-ln-Publ teat Ion Data

NATO Advanced Research Workshop on Develop.enta' Patternlng of theVertebrate Limb (1990 , Santander. Spatn)

Developmental patternlng of the vertebrate '-lmb I ed1ted by J.R1chard H1nchllffe, Juan M. Hurle, and Denn1s Sum.erbel1.

p. c.. -- (NATO ASI series. Series A, l1fe sciences; v.205)

'Publlshed In cooperation with NATO ScIentifIc AffaIrs DivIsion."'Proceed I ngs of a NATO Advanced Research Workshop on Deve lop.enta I

Pattern Ing of the Vertebra te Lj.b. he1d September 2:3-26, 1990. InSantander. Spa 1n "--CIP t. p. verso.

Inc I udes bIb 11ograph 1ca I references and Index.ISBN 0-:306-4:3927-11. Extre. I ties (Anatomy )--Congresses. 2. Vertebrates-

-Deve iopment--Congresses. 3. Extrem It ies (AnatOmy )--Evo Jut 10n--Congresses. 4. Morphogenes 1s--Congresses. 5. Pattern forut Ion(Blology)--Congresses. 1. Hlnchl1ffe. J. R. Il. Hurle. Juan M.Ill. Sumoerbe 11. Oenn1s. IV. North At Iant 1c Treaty Organ 1zat j on.ScIentifIc AffaIrs Dlvls10n. V. TItle. VI. SerIes.CL950. 7 .N:3B 1990596' .0:3' :32--dc20 91-22049

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