×

Apparition de motifs géométriques dans une membrane enzymatique. (French) Zbl 0572.92004

The paper presents an analysis of pattern formation for steady state solutions of a diffusion-reaction system. The study, motivated by morphogenesis in biology which shows similar features, is led using mathematical and numerical analysis. In particular the fundamental role played by sequential bifurcations is pointed out. A spatially uniform trivial steady state can lose its stability as the size of the geometrical domain varies.
The analysis enables to understand how an embryonic structure can undergo sequential cell differentiations. The bifurcated steady states are no more spatially uniform, so that the concentration field of a morphogen can induce cell differentiation in those regions where this concentration is above some level.
Reviewer: W.S.Barański

MSC:

92B05 General biology and biomathematics
35B32 Bifurcations in context of PDEs
35K55 Nonlinear parabolic equations
92Cxx Physiological, cellular and medical topics
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] A. M. TURING, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London,Vol. B 237 (1952), 37-72. · Zbl 1403.92034
[2] I. PRJGOGINE et G. NICOLIS, On symmetry breaking instabilities in dissipativeSystems, J. Chem. Phys., 46 (1967), 3542-3550.
[3] I. PRIGOGINE, R. LEFEVER, A. GOLDBETER et M. HERSCHKOWITZ-KAUFMAN, Symmetry breaking instabilities in biological Systems, Nature, 223 (1969), 913-916.
[4] H. G. OTHMER et L. E. SCRTVEN, Instability and dynamic pattern in cellular networks, J. Theor. Biol., 32 (1971), 507-537.
[5] A. GIERER et H. MEINHARDT, A theory of biological pattern formation, Kybernetika (Prague) 12 (1972), 30-39. · Zbl 0297.92007
[6] L. WOLPERT, Positional information and the developmenl of pattern and form: Cowan J. D. (éd.), Some mathematical questions in biology 5 (The American Mathematical Society, Providence, 1974).
[7] A. BABLOYANTZ et J. HIERNAUX, Modeis for cell differentiation and génération ofpolarity in diffusion-controlled morphogenetic fields, Bull. Math. Biol., 37 (1975), 637-657. Zbl0317.92016 · Zbl 0317.92016 · doi:10.1007/BF02459528
[8] B.C. GOODWIN, Analytical physiology of cells and deveioping organisms (Academic Press, New York, 1976).
[9] J. D. MURRAY, Lectures on nonlinear differential-equation models in biology Clarendon Press, Oxford, 1977). Zbl0379.92001 · Zbl 0379.92001
[10] G. NICOLIS et I. PRIGOGINE, Self-organization in nonequilibrium Systems, frontdissipative structures to order through fluctuations, fronmdissipative structures to order through fluctuations (Wiley-Interscience, New York, 1977). Zbl0363.93005 MR522141 · Zbl 0363.93005
[11] M. MIMURA et J. D. MURRAY, Spatial structures in a model substrate-inhibitiondiffusion System, Z.Naturforsch, 33 C (1978), 580-586.
[12] P. C. FIFE, Mathematical aspects of reacting and diffusing Systems, (Springer-Verlag, Berlin, 1979). Zbl0403.92004 MR527914 · Zbl 0403.92004
[13] J. HIERNAUX et T. ERNEUX, Chemical patterns in circular morphogenetic fields, Bull. Math. Biol., 41 (1979), 461-468. MR631874
[14] J. P. KERNEVEZ, G. JOLY, M. C. DUBAN, B. BUNOW and D. THOMAS, Hystérésis,oscillations andpattern formation inrealistic immobilized enzyme Systems,J. Math. Biology, 7 (1979), 41-56. Zbl0433.92014 MR648839 · Zbl 0433.92014 · doi:10.1007/BF00276413
[15] S. A. KAUFFMAN, R. M. SHYMKO et K. TRABERT, Control of sequential compartmentformation in Drosophila, a uniform mechanism may control the locations of successivebinary developmental commitments, Science, Vol. 199 (1978), 259-270.
[16] A. GARCIA-BELLIDO et J. P. MERRIAM, Parameters of the wing imaginal disc deve-lopment of Drosophila melanogaster, Develop. Biol., 24 (1971), 61-87.
[17] A. GARCIA-BELLIDO, P. RIPOLL et P. MORATA, Developmental compartmentaliza-tion ofthe wing disk of Drosophila, Nature NewBiol., 245(1973), 251-253.
[18] J. P. KERNEVEZ, Enzyme Mathematics : Studies in Mathematics and its applications, Vol. 10 (North-Holland, 1980). Zbl0446.92007 MR594596 · Zbl 0446.92007
[19] G. MEURAUT et J. C. SAUT, Bifurcation and stability in a chemical system, J. Math. Anal, and Appi. 59 (1977), 69-91. Zbl0355.35009 MR462242 · Zbl 0355.35009 · doi:10.1016/0022-247X(77)90093-2
[20] J. A. BOA et D. S. COHEN, Bifurcation of localized disturbances in a model bioche-mical reaction, Siam J. Appl. Math., Vol. 30, n^\circ 1(1976), 123-135. Zbl0328.76065 · Zbl 0328.76065 · doi:10.1137/0130015
[21] D. HENRY, Geometrie theory of semilinear parabolie équations, lecture notes in Vlathematics n^\circ 840, Springer-Verlag, NewYork, 1981. Zbl0456.35001 MR610244 · Zbl 0456.35001
[22] KATO T., Perturbation theory for linear operators (Springer-Verlag, New York, 1960). Zbl0435.47001 · Zbl 0435.47001
[23] G. LOSS, Bifurcation et stabilité. Publications mathématiques d’Orsay, N^\circ 31 (Université de Paris Sud, Orsay, 1972).
[24] H.P. KEENER et H. B. KELLER, Perturbed bifurcation theory, Arch. Rat. Mech. Anal., Vol. 50 (1973), 159-175. Zbl0254.47080 MR336479 · Zbl 0254.47080 · doi:10.1007/BF00703966
[25] D. W. DECKER, Topics in bifurcation theory, Ph. D. Thesis, California ïnstitute of Technology, Pasadena, California, 1978.
[26] H. B. KELLER, TWOnewbifurcation phenomena, IRIA Research Report n^\circ 369 (1979). Zbl0505.35009 · Zbl 0505.35009
[27] M. G. CRANDALL et P. H. RABINOWITZ, Bifurcation, perturbation of simple eigen values, and linearized stability, Arch. Rat. Mech. Anal. 52 (1973), 161-180. Zbl0275.47044 MR341212 · Zbl 0275.47044 · doi:10.1007/BF00282325
[28] M. KUBICEK, Dependence of solution of nonlinear Systems on a parameter, ACM Transactions on Mathematical Software, Vol 2, 1 (March 1976), 98-107. Zbl0317.65019 · Zbl 0317.65019 · doi:10.1145/355666.355675
[29] H.B. KELLER, Numerical solution of bifurcation andnon linear eigen value problems, 359-384 : Rabinowitz P.H. (éd.), Applications of bifurcation theory (Academic Press, New York, 1977). Zbl0581.65043 MR455353 · Zbl 0581.65043
[30] G. JOLY, J. P. KERNEVEZ, M. SHARAN, Calculation of the bifurcation branches inreaction-difjusion Systems (à paraître dans Acta Applicandae Mathematicae).
[31] J. P. KERNEVEZ, E. DOEDEL, M. C. DUBAN, J. F. HERVAGAULT, G. JOLY et D. THOMAS, Spatio-temporal organization in immobilized enzyme Systems, à paraître. Zbl0523.92008 · Zbl 0523.92008
[32] J. P. KERNEVEZ, J. D. MURRAY, G. JOLY, M. C. DUBAN et D. THOMAS, Propagationd’onde dans un système à enzyme immobilisée, CRAS 387, A (1978), 961-964. Zbl0391.65050 MR520780 · Zbl 0391.65050
[33] J. P. KERNEVEZ, G. JOLY et M. SHARAN, Control of Systems with multiple steadystates, pp. 635-649 in : Glowinski, R. and Lions, J. L. (éd.), Computing Methods in Applied Sciences and Engineering, North Holland, Amsterdam, 1982. Zbl0499.65041 MR784656 · Zbl 0499.65041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.