A Computer Assisted Approximation for the Generation of Patterns of Superficial Coarseness of the Pericarp and of the Seedhead of Some Plants: Coincidences in the Numerical Results
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Jun 7, 2012
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Abstract
This article proposes a phenomenological model that describes the pattern formation of the seedcoat (seedhead) of plant seeds through reaction-diffusion equations with parameters within Turing’s space. With the purpose of studying pattern formation, several numerical examples concerning simplified geometries of a seed are solved. The finite element method is used for the numerical solution along with the Newton-Raphson method for the approximation of partial non-linear differential equations. The numerical examples show that the model may represent the formation of different types of plant seedcoats
Keywords
Botanics, progeny testing, seeds development, finite element method Turing testBotánica, pruebas de descendencia, desarrollo de las semillas, método de elementos finitos, prueba de Turing
References
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BARTHLOTT, W. Scanning electron microscopy of the epidermal surface in plants. En Claugher, D (ed.). Scanning electron microscopy in taxonomy and functional morphology. Oxford: Clarendon Press, 1990.
BELYTSCHKO, T.; LIU, W. K. y MORAN, B. Nonlinear Finite Elements for Continua and Structures. New York: John Wiley and Sons, 2000.
BOLEK, Y.; OGLAKCI, M. y OZDIN, K. Genetic variation among cotton (G. hirsutum L.) cultivars for motes, seed-coat fragments and loading force. Field Crops Research. 2007, vol. 101, núm 2, pp. 155-159.
CARTWRIGHT, J. Labyrinthine turing pattern formation in the cerebral cortex. Journal of Theoretical Biology. 2002, vol. 217, pp. 97-103.
CHAPLAIN, M.; GANESH, A. J. y GRAHAM, I. G. Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumor growth. Journal of Mathematical Biology. 2001, vol. 42, pp. 387-423.
CRAMPIN, E. J. y MAINI, P. K. Reaction–diffusion models for biological pattern formation. Methods and Applications of Analysis. 2001, vol. 8, núm. 3, pp. 415-428.
DE WIT, A. Spatial patterns and spatiotemporal dynamics in chemical systems. Advances in Chemical Physics. 1999, vol. 109, pp. 435-513.
EDDE, P. y AMATOBI, C. I. Seed coat has no value in protecting cowpea seed against attack by Callosobruchus maculatus (F.). Journal of Stored Products Research. 2003, vol. 39, núm. 1, pp. 1-10.
FONT QUER, P. Diccionario de botánica. 8ª reimpresión. Barcelona: Labor, 1982.
GARZÓN, D. Simulación de procesos de reacción-difusión: Aplicación a la morfogénesis del tejido óseo. PhD tesis. Universidad de Zaragoza, España, 2007.
GARZÓN-ALVARADO, G. A.; GARCÍA-AZNAR, J. M. y DOBLARÉ, M. Appearance and location of secondary ossification centres may be explained by a reaction–diffusion mechanism. Computers in Biology and Medicine. 2009, vol. 39, pp. 554-561.
GEIRER, A. y MEINHARDT, H. A theory of biological pattern formation. Kybernetik. 1972, vol. 12, pp. 30-39.
GRAVEN, P.; DE KOSTER, C.; BOON, J. y BOUMAN, F. Structure and macromolecular composition of the seed coat of the Musaceae. Annals of Botany. 1996, vol. 77, pp. 105-122.
HARRISON, L.; WEHNER, S. y HOLLOWAY, D. Complex morphogenesis of surfaces:theory and experiment on coupling of reaction-diffusion patterning to growth. Faraday Discussions. 2002, vol. 120, pp. 277-293.
HEATHERLY, L.; KENTY, M. y KILEN, T. Effects of storage environment and duration on impermeable seed coat in soybean. Field Crops Research. 1995, vol. 40, núm. 1, pp. 57-62.
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KAPRAL, R. y SHOWALTER, K. Chemical waves and patterns. New York: Kluwer, 1995.
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MADZVAMUSE, A. Time-stepping schemes for moving grid finite elements applied to reactiondiffusion systems on fixed and growing domains. Journal of Computational Physics. 2005, vol. 24, núm. 1, pp. 239-263.
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MADZVAMUSE, A.; MAINI, P. K. y WATHEN, A. J. A moving grid finite element method applied to a model biological pattern generator. Journal of Computational Physics. 2003, vol. 190, pp. 478-500.
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MADZVAMUSE, A.; THOMAS, R. D. K.; MAINI, P. K. y WATHEN, A. J. A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves. Bulletin of Mathematical Biology. 2002, vol. 64, pp. 501-530.
MAINI, P. K.; PAINTER, K. J. y CHAU, H. N. P. Spatial pattern formation in chemical and biological systems. Journal of Chemical Society., Faraday Trans. 1997, vol. 93, pp. 3601-3610.
MEI, Z. Numerical bifurcation analysis for reaction-diffusion equations. Berlín: Springer-Verlag, 2000.
MEINHARDT, H. Models of biological pattern formation. New York: Academic Press, 1982.
PARODI, L. R. Gramíneas. En Enciclopedia argentina de agricultura y jardinería. Tomo I, segundo volumen: Descripción de plantas cultivadas. Buenos Aires: ACME SACI, 1987, p. 1112.
SICK, S.; REINKER, S.; TIMMER J. y SCHLAKE, T. WNT and DKK determine hair follicle spacing through a reaction–diffusion mechanism. Science. 2006, vol. 314, pp. 1447-1450.
STRASSBURGER, E. Tratado de botánica. 8a ed. Barcelona: Omega, 1994.
TURING, A. The chemical basis of morphogenesis. Philosophical Transactions of th Royal Society. Lond. B. 1952, vol. 237, pp. 37-72.
BARTHLOTT, W. Scanning electron microscopy of the epidermal surface in plants. En Claugher, D (ed.). Scanning electron microscopy in taxonomy and functional morphology. Oxford: Clarendon Press, 1990.
BELYTSCHKO, T.; LIU, W. K. y MORAN, B. Nonlinear Finite Elements for Continua and Structures. New York: John Wiley and Sons, 2000.
BOLEK, Y.; OGLAKCI, M. y OZDIN, K. Genetic variation among cotton (G. hirsutum L.) cultivars for motes, seed-coat fragments and loading force. Field Crops Research. 2007, vol. 101, núm 2, pp. 155-159.
CARTWRIGHT, J. Labyrinthine turing pattern formation in the cerebral cortex. Journal of Theoretical Biology. 2002, vol. 217, pp. 97-103.
CHAPLAIN, M.; GANESH, A. J. y GRAHAM, I. G. Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumor growth. Journal of Mathematical Biology. 2001, vol. 42, pp. 387-423.
CRAMPIN, E. J. y MAINI, P. K. Reaction–diffusion models for biological pattern formation. Methods and Applications of Analysis. 2001, vol. 8, núm. 3, pp. 415-428.
DE WIT, A. Spatial patterns and spatiotemporal dynamics in chemical systems. Advances in Chemical Physics. 1999, vol. 109, pp. 435-513.
EDDE, P. y AMATOBI, C. I. Seed coat has no value in protecting cowpea seed against attack by Callosobruchus maculatus (F.). Journal of Stored Products Research. 2003, vol. 39, núm. 1, pp. 1-10.
FONT QUER, P. Diccionario de botánica. 8ª reimpresión. Barcelona: Labor, 1982.
GARZÓN, D. Simulación de procesos de reacción-difusión: Aplicación a la morfogénesis del tejido óseo. PhD tesis. Universidad de Zaragoza, España, 2007.
GARZÓN-ALVARADO, G. A.; GARCÍA-AZNAR, J. M. y DOBLARÉ, M. Appearance and location of secondary ossification centres may be explained by a reaction–diffusion mechanism. Computers in Biology and Medicine. 2009, vol. 39, pp. 554-561.
GEIRER, A. y MEINHARDT, H. A theory of biological pattern formation. Kybernetik. 1972, vol. 12, pp. 30-39.
GRAVEN, P.; DE KOSTER, C.; BOON, J. y BOUMAN, F. Structure and macromolecular composition of the seed coat of the Musaceae. Annals of Botany. 1996, vol. 77, pp. 105-122.
HARRISON, L.; WEHNER, S. y HOLLOWAY, D. Complex morphogenesis of surfaces:theory and experiment on coupling of reaction-diffusion patterning to growth. Faraday Discussions. 2002, vol. 120, pp. 277-293.
HEATHERLY, L.; KENTY, M. y KILEN, T. Effects of storage environment and duration on impermeable seed coat in soybean. Field Crops Research. 1995, vol. 40, núm. 1, pp. 57-62.
HOFFMAN, J. Numerical methods for engineers and scientists. New York: McGraw Hill, 1992.
HOLLOWAY, D. y HARRISON, L. Pattern selection in plants: Coupling chemical dynamics to surface growth in three dimensions. Annals of Botany. 2008, vol. 101, p. 361.
HOLZAPFEL, G. A. Nonlinear solid mechanics. New York: John Wiley, 2000.
HUGHES, T. J. R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. New York: Courier Dover Publications, 2003.
KAPRAL, R. y SHOWALTER, K. Chemical waves and patterns. New York: Kluwer, 1995.
KARCZ, J.; KSIAZCZYK, T. y MALUSZYNSKA, J. Seed Coat Patterns in Rapid-Cycling Brassica Forms. Acta Biologica Cracoviensia, Series Botanica. 2005, vol. 47, núm. 1, pp. 159-165.
LEFEVRE, J. y MANGIN, J. A reaction-diffusion model of human brain development. PLOS Computational Biology. 2010, vol. 6, núm. 6, pp. 1-10.
MADZVAMUSE, A. A numerical approach to the study of spatial pattern formation. D Phil thesis, University of Oxford, Inglaterra, 2000.
MADZVAMUSE, A. Time-stepping schemes for moving grid finite elements applied to reactiondiffusion systems on fixed and growing domains. Journal of Computational Physics. 2005, vol. 24, núm. 1, pp. 239-263.
MADZVAMUSE, A. y MAINI, P. K. Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains. Journal of Computational Physics. 2007, vol. 225, núm. 1, pp. 100-119.
MADZVAMUSE, A.; MAINI, P. K. y WATHEN, A. J. A moving grid finite element method applied to a model biological pattern generator. Journal of Computational Physics. 2003, vol. 190, pp. 478-500.
MADZVAMUSE, A.; SEKIMURA, T.; THOMAS, R. D. K.; WATHEN A. J. y MAINI, P. K. A moving grid finite element method for the study of spatial pattern formation in Biological problems. En: Sekimura, T.; Noji, S.; Nueno, N. y Maini, P. K. (eds). Morphogenesis and Pattern Formation in Biological Systems – Experiments and Models. Tokyo: Springer-Verlag, 2003, pp. 59-65.
MADZVAMUSE, A.; THOMAS, R. D. K.; MAINI, P. K. y WATHEN, A. J. A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves. Bulletin of Mathematical Biology. 2002, vol. 64, pp. 501-530.
MAINI, P. K.; PAINTER, K. J. y CHAU, H. N. P. Spatial pattern formation in chemical and biological systems. Journal of Chemical Society., Faraday Trans. 1997, vol. 93, pp. 3601-3610.
MEI, Z. Numerical bifurcation analysis for reaction-diffusion equations. Berlín: Springer-Verlag, 2000.
MEINHARDT, H. Models of biological pattern formation. New York: Academic Press, 1982.
PARODI, L. R. Gramíneas. En Enciclopedia argentina de agricultura y jardinería. Tomo I, segundo volumen: Descripción de plantas cultivadas. Buenos Aires: ACME SACI, 1987, p. 1112.
SICK, S.; REINKER, S.; TIMMER J. y SCHLAKE, T. WNT and DKK determine hair follicle spacing through a reaction–diffusion mechanism. Science. 2006, vol. 314, pp. 1447-1450.
STRASSBURGER, E. Tratado de botánica. 8a ed. Barcelona: Omega, 1994.
TURING, A. The chemical basis of morphogenesis. Philosophical Transactions of th Royal Society. Lond. B. 1952, vol. 237, pp. 37-72.
How to Cite
Garzón Alvarado, D. A., & Ramírez Martínez, A. M. (2012). A Computer Assisted Approximation for the Generation of Patterns of Superficial Coarseness of the Pericarp and of the Seedhead of Some Plants: Coincidences in the Numerical Results. Ingenieria Y Universidad, 16(1), 27. https://doi.org/10.11144/Javeriana.iyu16-1.acpg
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