Spatial pattern formation is a key issue in early embroynic development.  Embroynic cells divide, migrate, and differentiate to form the various structures, markings and organs of the body.  Perhaps the most spectacular manifestation of this process is animal coat markings.  These patterns are formed by melanin-secreting cells which migrate from the neural crest to the epidermal and dermal layers of the skin

Maini, P. K.

Murray, J. D.

Myerscough, M. R.

Winters, K. H.

English

Mathematical Institute Eprints Archive

..:t--
C"'(
MathematicsApplied to
Biology and Medicine
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6.PatternFormationandRhythms 467
BifurcatingSpatialPatternsin a Cell-Chemotaxis
Model
I
I P.K. Mainil ,K. H.Winters2,M. R. Myerscough3,J. D. Murray4
1Centrefor MathematicalBiology, MathematicalInstitute,24-29St.Giles',Oxford
OXI3LB, UK
I 2TheoreticalStudiesDepartment,HarwellLaboratory,DidcotOXIl ORA,UK
3 Schoolof MathematicsandStatistics,F07,Universityof Sydney,NSW 2006,
Australia,
4AppliedMathematics,FS-20,UniversityofWashington,SeattleWA 98195,USA
1. Introduction
Spatialpatternformationisakeyissueinearlyembryonicdevelopment.Embryonic
cellsdivide,migrate,anddifferentiateoformthevarioustructures,markingsand
organsof thebody.Perhapsthemostspectacularmanifestationof thisprocessis
animalcoatmarkings.Thesepatternsareformedby melanin-secretingcellswhich
migratefromtheneuralcrestotheepidermalnddermallayersof theskin.Several
modelshavebeenproposedfor skinpatterning.For example,reaction-diffusion
models[I, 2,3]hypothesisetheexistenceofchemicals(morphogens)whichreactand
diffuseand,underappropriateconditions,generatespatiallyheterogeneouspatterns.
Cellsarethenassumedtobepre-programmedto ifferentiateaccordingtothelevelof
thechemicaltheyexperience.Muchof thejustificationfor suchmodelsis still
circumstantial.Analternativeclassofmodelsarethemechanochemicalmodels[3,4],
whichproposethatinitiallya spatialpatternincelldensityis setupandcellsthen
differentiateaccordingtotheirlocaldensity.Onesuchmodeltakesaccountof cell-
chemotaxis,theprocessby whichcellsmigrateupa chemicalgradient[5].Under
appropriateconditionsthismodelcanexhibitspatialpatternsin celldensity.The
hypothesisthatcellsinhighdensityaggregateswilldifferentiateandproducemelanin
thensuggeststhattheobservedpigmentpatternsimplyreflectpatternsin cell
density.Thismodelwasrecentlyproposedforthestripeandshadowpatternsonthe
alligator(AlligatorMississipiensis)[6].Experimentalresultsunequivocallyshow
thathigherdensitiesofmelaninformingcellsarefoundin thedarkstriperegionsof
468
BifurcatingSpatialPatterns
thealligator.
Here,weiIIusLratesomeof thespatialpatternsexhibitedby thecell-chemotaxis
model.We brieflydescribethemodelin Section2. In Section3 we presenta
selectionof spatiallypatternedsolutionsfoundusingthefiniteelementpackage
ENTWlFE. WealsoillusLratetheeffectonthepatternsofdomaingrowthsincein
manydevelopmentalsituationsspatialorganisationofpatterntakesplaceona time
scalecommensuratewithsignificantgrowthoftheembryo[6].
2. Model Equations
Chemotaxis a majorfactorin manydevelopmentalprocessesandthisis thekey
aggregationforcein themodel.Themodelinvolvestwodependentvariables,thecell
density,n(r,t),andthechemoattractantconcentration,e(r.t).whererand t arethe
spatialcoordinateandtimerespectively.Themodelconsistsof a pairof coupled
nonlinearpartialdifferentialequationswhichdescribesthemotionof thecellsandthe
production,diffusionanddegradationf thechemoattractanl.Thenondimensionalised
modelequationsare:
(1)
(2)
whereD, a,s,r, andN areallpositive,nondimensional,constants.
Weareinterestedinpatternformationonfinitetwo-dimensionaldomains- the
model for theskin of developingvertebrates- so we considertheseequationson a
finitedomaina withzerofluxboundaryconditions,namely
!!.Ve(d=!!.Vn(d=0 for!:Eaa (3)
wheren.is theunitoutwardnormaltotheboundaryaa. Themathematicalproblem
consistsofequations(I)and(2),withboundaryconditions(3).
Thesystemhastwouniformsteady-states;n=0,e=0andn=N , e=~ anda
1+N
linearanalysisaboutheseuniformstateshowsthathetrivialsteadystateisalways
unstablebutthatincertainparameterregimesthenontrivialspatiallyhomogeneous
solutioncanbedrivenunstablebyspatiallyheterogeneousperturbationsandevolve
toinhomogeneousspatialpatternsinnande.Theformof thepatternis determined
primarilybythenon-dimensionalp rameterset(D,a, r, N) andthesizeandshapeof
thedomain.Themodelandanalysisarediscussedindetailin [5,7].
3. Two-DimensionalSpatial Patterns
Weinvestigatepossiblesteadystatesof thefullnonlinearsystemusingthesoftware
packageENTWIFE,whichsolvesnonlinearproblemsbydiscretisationi thefir.ite-
6.PatternFormationandRhythms 469
elementapproximationusingastandardGalerkinformulation.Thepackagelocates
thecriticalvaluesofachosenparameterforwhichtheuniformsteadystatebifurcates
toanonuniformsteadystateandcontinuesthissolutionasthebifurcationparameter
changes[8].Herewepresentaselectionof theresultsofapplyingEN1WIFE tothe
steady-stateproblemwherethedomaina is takentobea I x4 rectangulardomain
(FiguresI -3).
4. Discussion
We havepresenteda celI-chemotacticmodelfor biologicalpatterngenerationandhave
ilIuSLratedsomeof thespatialpatternsobtainedby solvingthemodelequationsusing
ENTWIFE. A more detaileddiscussionof the numericalprocedureused and the
patternsobtainedcan be foundin [7, 8]. This simple modelexhibitsa surprisingly
complexanddiverserangeof patterns.In its applicationto theformationof snake
skin markings,the modelexhibitsseveralof the patternscommonlyobservedon
snakes[9].Testablepredictionsmadeby themodelarediscussedin [7].
S
0,8 G
10.3)
um
G H.F
nlEx
0'-
25 ex..
nlA nl J\ SJ
y
11AA n~A/
y y
Fig.I: Bifurcationdiagramfor fixedr, N, D ands andvaryinga ona I x 4
domain.This diagramonlyshowstheupperbranchesof primarybifurcations,
secondarybifurcationsfromthesebrancheswereobservedbutnotstudiedindetail.
The lettersA to H on thestatediagramdenotethepatternsin cell density(the
chemicalconcentrationsexhibitsimilar patterns).The linearisedsystemhas
eigenfunctionscos(nla)cos(mny/4) denotedbymode(n,m).Notethatmodes(0,4)
and (1,0) are degenerate.Shaded regions denote high cell density
[
i IN- nldxdY
]
Ilnll= 0 .
Jgwy
y
an/ at= DV2n - aV.(nVe) + srn(N-n)
diffusion chemotaxis cellgrowth
ae/ at= V2e + sn/(J+n) - se
diffusion production degradation
470
n(o,OJ
RO(.
Q
n~~
y
BifurcatingSpatialPatterns
nl P~
n~~y
nh R./
Y
Fig. 2:Magnifiedviewof thebifurcationof mode(0,1)inFigure1 withthevalue
ofnattheoriginplottedagainsta.ThelettersP toS denotethepatternsindicated.
(a)
L' 1;:,,--.)
increasingcell density
(b)
Fig. 3 : (a)A selectionof patternsin celldensity.(b)Theeffectof changingthewidthofthedomain.
6. PatternFormationandRhythms 471
Acknowledgement
Thiswork(JDM) wassupportedin partby GrantDMS-9003339fromtheU.S.
NationalScienceFoundation,and(KHW) by theUnderlyingProgramof the
U.K.A.E.A. MRM wouldliketo acknowledgeanOverseasResearchStudentship
(ORS)AwardfromtheBritishGovernmentandbothMRM andPKM wouldliketo
thankU.K.A.E.A.HarwelIfortheirhospitalityandsupport.
References
1.J. D. Murray.A prepatternformationmechanismfor animalcoatmarkings.J
Theor.Bioi.,88:161-199,1981.
2.J. B. L. Bard.A modelforgeneratingaspectsof zebraandothermammaliancoat
patterns.J. Theor.Bioi.,93:363-385,1981.
3.J. D. Murray.MathematicalBiology.Springer-Verlag,Heidelberg,1989.
4.J. D. Murray,P. K. Maini,andR. T. Tranquillo.Mechanochemicalmodelsfor
generatingbiologicalpatternandformindevelopment.PhysicsReports,171:59-84,
1988.
5.G. F. Osterand1. D. Murray.Patternformationmodelsanddevelopmental
constraints.J. Exp.Zool.,251:186-202,1989.
6. J. D. Murray,D. C. Deeming,and M. J. W. Ferguson.Size dependent
pigmentationpatternformationin embryosof alligatormississipiensis:timeof
initiationof patternformationmechanism.Proc.Roy.Soc.(London),B 239:279-
293,1990.
7.P. K. Maini,M. R. Myerscough,K. H. Winters,andJ. D. Murray.Bifurcating
spatiallyheterogeneoussolutionsin a chemotaxismodelfor biologicalpattern
generation.Bull.Math.Bioi., 1991.(inpress).
8.K. H. Winters,M. R. Myerscough,P. K. Maini,andJ. D. Murray.Tracking
bifurcatingsolutionsof a modelbiologicalpatterngenerator.ImpactComp.Sci.
Eng.,2:355-371,1990.
9. J. D. MurrayandM. R. Myerscough.Pigmentationpatternformationonsnakes.
J. Theor.Bioi.,149:339-360,1991.


Bard.A modelforgeneratingaspectsof zebraandothermammaliancoat

Ferguson.Size dependent pigmentationpatternformationin embryosof alligatormississipiensis:timeof initiationof

Murray.Bifurcating spatiallyheterogeneoussolutionsin a chemotaxismodelfor biologicalpattern generation.Bull.Math.Bioi.,

Murray.Tracking bifurcatingsolutionsof a modelbiologicalpatterngenerator.ImpactComp.Sci.

Bifurcating spatial patterns in a cell-chemotaxis model

Oxford University Research Archive (ORA)

Oxford University Research Archive

..:t--C"'(Mathematics Applied toBiology and MedicineJ Demongeot* & V Capasso*** TIMC-lMAG-CNRS UJF-INPGFaculte de Medecine38700 La Tranche, France**Dipartimento di MatematicaUniversita' di Milano IRMA(Istituto per Ricerche di Matematica Applicata)-CNRBari, Italy------Wuerz Publishing LtdWinnipeg, CanadaWuerz Publishing Ltd~@ Copyright 1993 by Wuerz Publishing LtdAll rights reseIVed. No part of this book may be reproduced or transmittedin any form or by any means, electronic or mechanical, includingphotocopying, recording, or any informationstorage and retrieval system,without permission in writing from the publisher.Mathematics Applied to Biology and MedicineJ Demongeot and V Capasso, editorsISBN 0-920063-63-2Printed in the United KingdomBiological SystemsA BookSeriesSeries EditorsPierre AugerBiometrie. Genetique.et Biologie des PopulationsUniversite Claude Bernard - Lyon I43, boul. du 11 novembre 191869622 Villeurbanne CedexFranceRoger V JeanOept of Math & Computer SciencesUniversite du Quebec300, avenue des UrsulinesRimouski QU G5L 3A1CanadaConsulting EditorsOvide ArinoI.P.RA. MathematiquesUniversite de Pau64000 PauFranceC LeguizamonComision Nacional deEnergia AtomicaBiomatematica -Oepta Radiobiologiaav del Ubertador 8250ArgentinaGeorge KlirOept of Systems ScienceT J Watson School of EngineeringSUNY -BinghamtonBinghamton NY 13901USARobert RosenOept of Physiology & BiophysicsOalhousie UniversityHalifax NS B3H 4H7CanadaV P SaxenaSchool of Math & Applied SciencesJiwaji UniversityGwaliorIndiaPublisherWuerz Publishing Lld895 McMillan AvenueWinnipeg MB R3M OT2Canadaphone (204) 453 7429fax (204) 453 65986. Pattern Formation and Rhythms 467Bifurcating Spatial Patterns in a Cell-ChemotaxisModelII P. K. Mainil ,K. H. Winters2 ,M. R. Myerscough3, J. D. Murray41 Centre for Mathematical Biology, Mathematical Institute, 24-29 St. Giles', OxfordOXI3LB, UKI 2 Theoretical Studies Department, Harwell Laboratory, Didcot OXIl ORA,UK3 School of Mathematics and Statistics, F07, University of Sydney, NSW 2006,Australia,4 Applied Mathematics, FS-20, University of Washington, Seattle WA 98195, USA1. IntroductionSpatial pattern formation is a key issue in early embryonic development. Embryoniccells divide, migrate, and differentiate to form the various structures, markings andorgans of the body. Perhaps the most spectacular manifestation of this process isanimal coat markings. These patterns are formed by melanin-secreting cells whichmigrate from the neural crest to the epidermal and dermal layers of the skin. Severalmodels have been proposed for skin patterning. For example, reaction-diffusionmodels [I, 2, 3] hypothesise the existence of chemicals (morphogens) which react anddiffuse and, under appropriate conditions, generate spatially heterogeneous patterns.Cells are then assumed to be pre-programmed to differentiate according to the level ofthe chemical they experience. Much of the justification for such models is stillcircumstantial. An alternative class of models are the mechanochemical models [3,4],which propose that initially a spatial pattern in cell density is set up and cells thendifferentiate according to their local density. One such model takes account of cell-chemotaxis, the process by which cells migrate up a chemical gradient [5]. Underappropriate conditions this model can exhibit spatial patterns in cell density. Thehypothesis that cells in high density aggregates will differentiate and produce melaninthen suggests that the observed pigment patterns simply reflect patterns in celldensity. This model was recently proposed for the stripe and shadow patterns on thealligator (Alligator Mississipiensis) [6]. Experimental results unequivocally showthat higher densities of melanin forming cells are found in the dark stripe regions of468Bifurcating Spatial Patternsthe alligator.Here, we iIIusLratesome of the spatial patterns exhibited by the cell-chemotaxismodel. We briefly describe the model in Section 2. In Section 3 we present aselection of spatially patterned solutions found using the finite element packageENTWlFE. We also illusLratethe effect on the patterns of domain growth since inmany developmental situations spatial organisation of pattern takes place on a timescale commensurate with significant growth of the embryo [6].2. Model EquationsChemotaxis is a major factor in many developmental processes and this is the keyaggregation force in the model. The model involves two dependent variables, the celldensity, n(r,t), and the chemoattractant concentration, e(r.t). where rand t are thespatial coordinate and time respectively. The model consists of a pair of couplednonlinear partial differential equations which describes the motion of the cells and theproduction, diffusion and degradation of the chemoattractanl. The nondimensionalisedmodel equationsare:(1)(2)where D, a, s, r, and N are all positive, non dimensional, constants.We are interested in pattern formation on finite two-dimensional domains - themodel for the skin of developing vertebrates - so we consider these equations on afinite domain a with zero flux boundary conditions, namely!!.Ve(d =!!.Vn(d = 0 for!: Eaa (3)where n. is the unit outward normal to the boundary aa. The mathematical problemconsists of equations (I) and (2), with boundary conditions (3).The system has two uniform steady-states; n =0, e =0 and n =N , e =~ and a1 + Nlinear analysis about these uniform states shows that the trivial steady state is alwaysunstable but that in certain parameter regimes the nontrivial spatially homogeneoussolution can be driven unstable by spatially heterogeneous perturbations and evolveto in homogeneous spatial patterns in n and e. The form of the pattern is determinedprimarily by the non-dimensionalparameter set (D, a, r, N) and the size and shape ofthe domain. The model and analysis are discussed in detail in [5,7].3. Two-Dimensional Spatial PatternsWe investigate possible steady states of the full nonlinear system using the softwarepackage ENTWIFE, which solves nonlinear problems by discretisation in the fir.ite-6. Pattern Formation and Rhythms 469element approximation using a standard Galerkin formulation. The package locatesthe critical values of a chosen parameter for which the uniform steady state bifurcatesto a nonuniform steady state and continues this solution as the bifurcation parameterchanges [8]. Here we present a selection of the results of applying EN1WIFE to thesteady-state problem where the domain a is taken to be a I x 4 rectangular domain(Figures I - 3).4. DiscussionWe have presented a celI-chemotactic model for biological pattern generation and haveilIuSLrated some of the spatial patterns obtained by solving the model equations usingENTWIFE. A more detailed discussion of the numerical procedure used and thepatterns obtained can be found in [7, 8]. This simple model exhibits a surprisinglycomplex and diverse range of patterns. In its application to the formation of snakeskin markings, the model exhibits several of the patterns commonly observed onsnakes [9]. Testable predictions made by the model are discussed in [7].S0,8 G10.3)umG H.FnlEx0'-25 ex..nlA nl J\ SJy11AA n~A/y yFig.I: Bifurcation diagram for fixed r, N, D and s and varying a on a I x 4domain. This diagram only shows the upper branches of primary bifurcations,secondary bifurcations from these branches were observed but not studied in detail.The letters A to H on the state diagram denote the patterns in cell density (thechemical concentrations exhibit similar patterns). The linearised system haseigenfunctions cos(nla) cos(mny /4) denoted by mode (n,m). Note that modes (0,4)and (1,0) are degenerate. Shaded regions denote high cell density[i IN- nldxdY]Ilnll= 0 .Jgwyyan/ at= DV2n - aV.(nVe) + srn(N-n)diffusion chemotaxis cellgrowthae/ at= V2e + sn/(J+n) - sediffusion production degradation470n(o,OJRO(.Qn~~yBifurcating Spatial Patternsnl P~n~~ynh R./YFig. 2: Magnified view of the bifurcation of mode (0,1) in Figure 1 with the valueof n at the origin plotted against a. The letters P to S denote the patterns indicated.(a)L' 1;:,,--.)increasing cell density(b)Fig. 3 : (a) A selection of patterns in cell density. (b) The effect of changing thewidth of the domain.6. Pattern Formation and Rhythms 471AcknowledgementThis work (JDM) was supported in part by Grant DMS-9003339 from the U.S.National Science Foundation, and (KHW) by the Underlying Program of theU.K.A.E.A. MRM would like to acknowledge an Overseas Research Studentship(ORS) Award from the British Government and both MRM and PKM would like tothank U.K.A.E.A. HarwelI for their hospitality and support.References1. J. D. Murray. A prepattern formation mechanism for animal coat markings. J.Theor. Bioi., 88:161-199, 1981.2. J. B. L. Bard. A model for generating aspects of zebra and other mammalian coatpatterns. J. Theor. Bioi., 93:363-385, 1981.3. J. D. Murray. Mathematical Biology. Springer-Verlag, Heidelberg, 1989.4. J. D. Murray, P. K. Maini, and R. T. Tranquillo. Mechanochemical models forgenerating biological pattern and form in development. Physics Reports, 171:59-84,1988.5. G. F. Oster and 1. D. Murray. Pattern formation models and developmentalconstraints. J. Exp. Zool., 251:186-202, 1989.6. J. D. Murray, D. C. Deeming, and M. J. W. Ferguson. Size dependentpigmentation pattern formation in embryos of alligator mississipiensis: time ofinitiation of pattern formation mechanism. Proc. Roy. Soc. (London), B 239:279-293, 1990.7. P. K. Maini, M. R. Myerscough, K. H. Winters, and J. D. Murray. Bifurcatingspatially heterogeneous solutions in a chemotaxis model for biological patterngeneration. Bull. Math. Bioi., 1991. (in press).8. K. H. Winters, M. R. Myerscough, P. K. Maini, and J. D. Murray. Trackingbifurcating solutions of a model biological pattern generator. Impact Comp. Sci.Eng., 2:355-371,1990.9. J. D. Murray and M. R. Myerscough. Pigmentation pattern formation on snakes.J. Theor. Bioi., 149:339-360, 1991.

Bifurcating spatial patterns in a cell-chemotaxis model

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