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Mathematical analysis and numerical simulation of a model of morphogenesis
1. | Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain |
2. | Departamento de Matemática Aplicada, E.U. Informática. Universidad Politécnica de Madrid, Ctra. de Valencia, Km. 7. 28031 - Madrid, Spain |
References:
[1] |
F. W. Cummings, A model of morphogenesis, Phys. A, 3-4 (2004), 531-547. |
[2] |
E. V. Entchev, A. Schwabedissen and M. González-Gaitán, Gradient formation of the TGF-beta homolog Dpp, Cell, 103 (2000), 981-991.
doi: 10.1016/S0092-8674(00)00200-2. |
[3] |
E. V. Entchev and M. González-Gaitán, Morphogen gradient formation and vesicular trafficking, Traffic, 3 (2002), 98-109. |
[4] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964. |
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1977. |
[6] |
B. Ka\'zmierczak and K. Piechór, Heteroclinic solutions for a model of skin morphogenesis. The case of strong attachment of the epidermis to the basal lamina, Application of Mathematics in Biology and Medicine, Univ. Comput. Eng. Telecommun. Kielce, (2004), 85-90. |
[7] |
M. Kerszberg and L. Wolpert, Mechanism for positional signalling by morphogen transport: A theoretical study, J. Theor. Biol., 191 (1998), 103-114.
doi: 10.1006/jtbi.1997.0575. |
[8] |
P. Krzyzanowski, P. Laurencot and D. Wrzosek, Well-posedness and convergence to the steady state for a model of morphogen transport, SIAM J. Math. Anal., 40 (2008), 1725-1749. |
[9] |
A. D. Lander, Q. Nie, B. Vargas and F. Y. M. Wan, Agregation of a distributed source morphogen gradient degradation, Studies in Appl. Math., 114 (2005), 343-374.
doi: 10.1111/j.0022-2526.2005.01556.x. |
[10] |
A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion?, Dev. Cell, 2 (2002), 785-796.
doi: 10.1016/S1534-5807(02)00179-X. |
[11] |
A. D. Lander, Q. Nie and F. Y. M. Wan, Internalization and end flux in morphogen gradient degradation, J. Comput. Appl. Math., 190 (2006), 232-251.
doi: 10.1016/j.cam.2004.11.054. |
[12] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. |
[13] |
Y. Lou, Q. Nie and F.Y.M. Wan, Effects of sog Dpp-receptor binding, SIAM J. Appl. Math., 65 (2005), 1748-1771.
doi: 10.1137/S0036139903433219. |
[14] |
J. H. Merking, D. J. Needham and B. D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity, Nonlinearity, 18 (2005), 2745-2773.
doi: 10.1088/0951-7715/18/6/018. |
[15] |
J. H. Merking and B. D. Sleeman, On the spread of morphogens, J. Math. Biol., 51 (2005), 1-17.
doi: 10.1007/s00285-004-0308-0. |
[16] |
R. E. Showalter, "Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, AMS, Providence, Rhode Island, 1997. |
[17] |
A. Teleman and S. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc, Cell, 103 (2000), 971-980.
doi: 10.1016/S0092-8674(00)00199-9. |
[18] |
J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Series A, 25 (2009), 343-361. |
[19] |
J. I. Tello, On the existence of solutions of a mathematical model of morphogens, in "Modern Mathematical Tools and Techniques in Capturing Complexity Series: Understanding Complex Systems" 495-509 (eds. L. Pardo, N. Balakrishnan and M. A. Gil), Springer, 2011. |
[20] |
L. Wolpert, Positional information and the spatial pattern of cellular differentiation, J. Theore. Biol., 25 (1969), 1-47.
doi: 10.1016/S0022-5193(69)80016-0. |
show all references
References:
[1] |
F. W. Cummings, A model of morphogenesis, Phys. A, 3-4 (2004), 531-547. |
[2] |
E. V. Entchev, A. Schwabedissen and M. González-Gaitán, Gradient formation of the TGF-beta homolog Dpp, Cell, 103 (2000), 981-991.
doi: 10.1016/S0092-8674(00)00200-2. |
[3] |
E. V. Entchev and M. González-Gaitán, Morphogen gradient formation and vesicular trafficking, Traffic, 3 (2002), 98-109. |
[4] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964. |
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1977. |
[6] |
B. Ka\'zmierczak and K. Piechór, Heteroclinic solutions for a model of skin morphogenesis. The case of strong attachment of the epidermis to the basal lamina, Application of Mathematics in Biology and Medicine, Univ. Comput. Eng. Telecommun. Kielce, (2004), 85-90. |
[7] |
M. Kerszberg and L. Wolpert, Mechanism for positional signalling by morphogen transport: A theoretical study, J. Theor. Biol., 191 (1998), 103-114.
doi: 10.1006/jtbi.1997.0575. |
[8] |
P. Krzyzanowski, P. Laurencot and D. Wrzosek, Well-posedness and convergence to the steady state for a model of morphogen transport, SIAM J. Math. Anal., 40 (2008), 1725-1749. |
[9] |
A. D. Lander, Q. Nie, B. Vargas and F. Y. M. Wan, Agregation of a distributed source morphogen gradient degradation, Studies in Appl. Math., 114 (2005), 343-374.
doi: 10.1111/j.0022-2526.2005.01556.x. |
[10] |
A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion?, Dev. Cell, 2 (2002), 785-796.
doi: 10.1016/S1534-5807(02)00179-X. |
[11] |
A. D. Lander, Q. Nie and F. Y. M. Wan, Internalization and end flux in morphogen gradient degradation, J. Comput. Appl. Math., 190 (2006), 232-251.
doi: 10.1016/j.cam.2004.11.054. |
[12] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. |
[13] |
Y. Lou, Q. Nie and F.Y.M. Wan, Effects of sog Dpp-receptor binding, SIAM J. Appl. Math., 65 (2005), 1748-1771.
doi: 10.1137/S0036139903433219. |
[14] |
J. H. Merking, D. J. Needham and B. D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity, Nonlinearity, 18 (2005), 2745-2773.
doi: 10.1088/0951-7715/18/6/018. |
[15] |
J. H. Merking and B. D. Sleeman, On the spread of morphogens, J. Math. Biol., 51 (2005), 1-17.
doi: 10.1007/s00285-004-0308-0. |
[16] |
R. E. Showalter, "Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, AMS, Providence, Rhode Island, 1997. |
[17] |
A. Teleman and S. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc, Cell, 103 (2000), 971-980.
doi: 10.1016/S0092-8674(00)00199-9. |
[18] |
J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Series A, 25 (2009), 343-361. |
[19] |
J. I. Tello, On the existence of solutions of a mathematical model of morphogens, in "Modern Mathematical Tools and Techniques in Capturing Complexity Series: Understanding Complex Systems" 495-509 (eds. L. Pardo, N. Balakrishnan and M. A. Gil), Springer, 2011. |
[20] |
L. Wolpert, Positional information and the spatial pattern of cellular differentiation, J. Theore. Biol., 25 (1969), 1-47.
doi: 10.1016/S0022-5193(69)80016-0. |
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