Several theoretical models have been proposed to attempt to elucidate the underlying mechanisms involved in the spatial patterning of skeletal elements in the limb.  Here, I briefly compare two such models - reaction diffusion (RD) and mechanochemical (MC) - and highlight their properties and predictions

Maini, P. K.

English

Oxford University Research Archive

Developmental Patterningof the Vertebrate LimbEdited byJ. Richard HinchliffeUniversity College of WalesAberystwyth, Wales, United KingdomJuan M. HurleUniversity of CantabriaSantander, SpainandDennis SummerbellNational Institute for Medical ResearchLondon, United KingdomPlenum PressNew York and LondonPublished in cooperation with NATO Scientific Affairs DivisionNATO ASI SeriesAdvanced Science Institutes SeriesA series presenting the results of activities sponsored by the NA TO Science Committee,which aims at the dissemination of advanced scientific and technological knowledge,with a view to strengthening links between scientific communities.The series is published by an international board of publishers in conjunction with theNATO Scientific Affairs DivisionPlenum Publishing CorporationNew York and LondonKluwer Academic PublishersDordrecht, Boston, and LondonSpringer-VerlagBerlin, Heidelberg, New York, London,Paris, Tokyo, Hong Kong, and BarcelonaRecent Volumes in this SeriesVolume 202- Molecular Aspects of Monooxygenases and Bioactivationof Toxic Compoundsedited by Emel Aring, John B. Schenkman, and Ernest HodgsonVolume 203-From Pigments to Perception: Advances in UnderstandingVisual Processesedited by Arne Valberg and Barry B. LeeVolume 204- Role of Melatonin and Pineal Peptides in Neuroimmunomodulationedited by Franco Fraschini and Russel J. ReiterVolume 205-Developmental Patterning of the Vertebrate Limbedited by J. Richard Hinchliffe, Juan M. Hurle,and Dennis SummerbellVolume 206-Alcoholism: A Molecular Perspectiveedited by T. Norman PalmerVolume 207-Bioorganic Chemistry in Healthcare and Technologyedited by U. K. Pandit and F. C. AlderweireldtVolume 20B-Vascular Endothelium: Physiological Basis of Clinical Problemsedited by John D. Catravas, Allan D. Callow,C. Norman Gillis, and Una S. Ryan~Series A: Life SciencesA Life SciencesB PhysicsC Mathematical and Physical SciencesD Behavioral and Social SciencesE Applied SciencesF Computer and Systems SciencesG Ecological SciencesH Cell BiologyI Global Environmental ChangeA COMPARISON OF REACTION DIFFUSION AND MECHANOCHEMICALMODELS FOR LIMB DEVELOPMENTPhilip MainiCentre for Mathematical BiologyMathematical Institute24-29 st. Giles'Oxford OX1 3LBSeveral 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. ForIiDevelopmental Patterning of the Vertebrate Limb~ited by J.R.Hinchliffe et al., Plenum Press, New York, 1991 161example, 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. 'REFERENCESP. 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.~'1125 (1983). .G.F. Oster, J.D. Murray, and P.K. Maini, A model ~ichondrogenic condensations in the developing limb: the r~JIextracellular matrix and cell tractions, J. Embryol.i"'!~Morphol. 89, 93-112 (1985).j162G.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).163Proceedings of a NATO Advanced Research Workshop onDevelopmental Patterning of the Vertebrate Limb,held September 23-26, 1990,in Santander, SpainLIbrary of Congress Cataloglng-ln-Publ teat Ion DataNATO 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 IPattern 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 91-22049CIPISBN 0-306-43927-1@ 1991 Plenum Press, New YorkA Division of Plenum Publishing Corporation233 Spring Street, New York, N.Y. 10013All rights reservedNo part of this book may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming,recording, or otherwise, without written permission from the PublisherPrinted in the United States of America

A comparison of reaction diffusion and mechanochemical models for limb development

Mathematical Institute Eprints Archive

DevelopmentalPatterning
of the Vertebrate Limb
Edited by
J. Richard Hinchliffe
University College of Wales
Aberystwyth, Wales, United Kingdom
Juan M. Hurle
University of Cantabria
Santander, Spain
and
Dennis Summerbell
National Institute for Medical Research
London, United Kingdom
Plenum Press
New York and London
Published in cooperationwith NATO Scientific Affairs Division
NATO ASI Series
AdvancedScienceInstitutesSeries
A series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The series is publishedby an internationalboardof publishersin conjunctionwiththe
NATO ScientificAffairsDivision
PlenumPublishingCorporation
New York and London
Kluwer Academic Publishers
Dordrecht, Boston, and London
Springer-Verlag
Berlin,Heidelberg,New York, London,
Paris,Tokyo, Hong Kong, andBarcelona
Recent Volumesin this Series
Volume202- Molecular Aspects of Monooxygenases and Bioactivation
of Toxic Compounds
edited by Emel Aring, John B. Schenkman, and Ernest Hodgson
Volume203-From Pigments to Perception: Advances in Understanding
Visual Processes
edited byArne Valberg and Barry B. Lee
Volume204- Role of Melatonin and Pineal Peptides in Neuroimmunomodulation
edited by Franco Fraschini and Russel J. Reiter
Volume205-Developmental Patterning of the Vertebrate Limb
edited by J. Richard Hinchliffe, Juan M. Hurle,
and Dennis Summerbell
Volume206-Alcoholism: A Molecular Perspective
edited byT. Norman Palmer
Volume207-Bioorganic Chemistry in Healthcare and Technology
edited by U. K. Pandit and F. C. Alderweireldt
Volume20B-Vascular Endothelium: Physiological Basis of Clinical Problems
edited by John D.Catravas, Allan D. Callow,
C. Norman Gillis, and Una S. Ryan
~
Series A: Life Sciences
A Life Sciences
B Physics
C Mathematicaland PhysicalSciences
D Behavioraland Social Sciences
E Applied Sciences
F Computerand Systems Sciences
G Ecological Sciences
H Cell Biology
I Global EnvironmentalChange
A COMPARISONOF REACTION DIFFUSION AND MECHANOCHEMICAL
MODELSFOR LIMB DEVELOPMENT
Philip Maini
Centre for Mathematical Biology
Mathematical Institute
24-29 st. Giles'
Oxford OX1 3LB
Several theoretical models have been proposed to attempt
to elucidate the underlying mechanisms involved in the spatial
patterning of skeletal elements in the limb. Here, I briefly
comparetwo such models - reaction diffusion (RD) and mechano-
chemical (MC) - and highlight their properties and predictions.
Although similar mathematically, these models are based
on fundamentally different biological approaches. RD theory
asserts that a chemical prepattern is first set up in the limb
by a system of reacting and diffusing chemicals (Turing, 1952).
Cells then respond to this prepattern by differentiating
wherever the concentration of one of the chemicals lies above
a threshold value (Wolpert, 1969, 1981). Thus, the spatial
pattern of chemical concentration is reflected by the spatial
pattern of cell differentiation.
The MC theory (Oster et. al., 1983, Oster et. al., 1985)
proposes that the mechanical and chemical interactions of cells
with their external environment - the extracellular matrix -
leads to a spatial pattern in cell density. The cells in the
aggregates thus formed differentiate. Thus, the spatial
pattern of cell differentiation overlies the spatial pattern
of cell density.
The spatial patterns predicted by both RD and MC theory,
from linear analysis, ar~ essentially .theeigenfunctions of the
Laplacian operator (Mainiand Solursh, 1990). The more complex
tensor form of the MC model may lead to a richer set of
structures than those exhibited by RD models. This possibility
has yet to be analysed. Thus, at the moment, one cannot
distinguish between these models on a purely mathematical
basis.
The similarity of the mathematical formulation of the
above hypotheses does, however, enable one to make some general
predictions on the properties of the skeletal pattern in the
limb independent of the biological basis of the model. ForI
i
DevelopmentalPatterningof the VertebrateLimb
~itedby J.R. Hinchliffeetal.,PlenumPress,NewYork,1991 161
example, both models suggest that there are only a limited
number of ways in which elements may arise and bifurcate. This
has lead to a more precise formulation of the idea of
developmental constraints on vertebrate limb evolution (Oster
et. al., 1988, Oster and Murray, 1989).
Both models predict that the complexity and form of
spatial pattern is dependent on scale, geometry and boundary
condi tions. For example, they predict that decreasing 1imb bud
width will result in a loss of elements. This agrees with
experimental observation (Alberch and Gale, 1983).
Recently it has been shown that if two stage 19 anterior
chick limb halves are combined, the recombinant forms two
humeral elements (Wolpert and Hornbruch, 1990). As the recom-
binant has the same size as a normal limb, both the above
models would predict a single humerus. The observation
suggests that the anterior stage 19 limb half contains cells
that have already differentiated yet, at this stage, there is
no visible aggregation of cells. This is inconsistent with the
MC approach but may be consistent with the RD approach assuming
that the prepattern is laid down and stabilised before stage
19. However, another interpretation of the result is that
during normal development a prepattern may divide the early
limb bud into domains in which cells are either competent or
incompetent to differentiate into cartilage. Such a prepattern
may be set up by a gradient in homeobox gene expression or in
retinoid distribution (Maini and Solursh, 1990). The
recombination experiment would then be seen as combining two
such cell populations and, in effect, doubling the domain
width. The eventual developmental fate of cells could then be
determined by either of the above models. ~
This scenario suggests that pattern formation is a
hierarchal process wherein each mechanism provides the initia '
condition for subsequent processes. This possibility warrant
further experimental and theoretical investigation.
ACKNOWLEDGEMENT This work was supported in part by th
National Science Foundation under Grant No. DMS8901388. '
REFERENCES
P. Alberch and E. Gale, Size dependency during the developmeJ
of 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~JI
extracellular 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, Evolution
and morphogenetic rules. The shape of the vertebrate limb in
ontogeny and phylogeny, Evolution 45, 862-884 (1988).
G. F. Oster and J. D. Murray, Pattern formation models and
developmental constraints, in J.P. Trinkaus Anniversary Volume
1989 (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 of
cellular 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 buds
and models for cartilage rudiment specification, Development
109, 961-966 (1990).
163
Proceedingsof a NATO Advanced Research Workshop on
Developmental 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 the
Vertebrate 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 Develop.enta I
Pattern Ing of the Vertebrate Lj.b. he1d September2:3-26, 1990. In
Santander. Spa1n"--CIP t. p. verso.
Inc I udes bIb 11ograph 1caI references and Index.
ISBN 0-:306-4:3927-1
1. Extre. I ties (Anatomy)--Congresses. 2. Vertebrates-
-Deve iopment--Congresses. 3. Extrem It ies (AnatOmy)--Evo Jut 10n-
-Congresses. 4. Morphogenes1s--Congresses. 5. Pattern forut Ion
(Blology)--Congresses. 1. Hlnchl1ffe. J. R. Il. Hurle. Juan M.
Ill. Sumoerbe11. Oenn1s. IV. North At Iant 1c Treaty Organ 1zat j on.
ScIentifIc AffaIrs Dlvls10n. V. TItle. VI. SerIes.
CL950. 7.N:3B 1990
596' .0:3':32--dc20 91-22049
CIP
ISBN 0-306-43927-1
@ 1991 PlenumPress, NewYork
A Divisionof PlenumPublishingCorporation
233 SpringStreet,New York, N.Y. 10013
All rightsreserved
No partof thisbook maybe reproduced,storedin a retrievalsystem,or transmitted
in anyformor by anymeans,electronic,mechanical,photocopying,microfilming,
recording,or otherwise,withoutwrittenpermissionfromthe Publisher
Printedin theUnitedStatesof America


7.N:3B 1990 596' .0:3':32--dc20 91-22049 CIP ISBN 0-306-43927-1 @

A model ~i chondrogenic condensations in the developing limb: the r~JIextracellular matrix and cell tractions,

Double anterior chick limb buds and models for cartilage rudiment specification,

Pattern formation models and developmental constraints,

Position~l information and the spatial pattern of cellular differentiation,

Proceedingsof a NATO Advanced Research Workshop on Developmental Patterning of the Vertebrate Limb, held September 23-26,1990, in Santander,

sciences; v. 205) 'Publlshed In cooperation with NATO ScIentifIc AffaIrs DivIsion.&quot; 'Proceed I ngs of a NATO

Size dependency during the developmeJ of the amphibian foot. Colchicine induced digital loss a! reduction,

A comparison of reaction diffusion and mechanochemical models for limb development

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