American Institute of Mathematical Sciences

Advanced
2011, 8(4): 1035-1059. doi: 10.3934/mbe.2011.8.1035

Mathematical analysis and numerical simulation of a model of morphogenesis

Ana I. Muñoz 1, and  José Ignacio Tello 2,

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

Received  November 2009 Revised  January 2011 Published  August 2011

We consider a simple mathematical model of distribution of morphogens (signaling molecules responsible for the differentiation of cells and the creation of tissue patterns). The mathematical model is a particular case of the model proposed by Lander, Nie and Wan in 2006 and similar to the model presented in Lander, Nie, Vargas and Wan 2005. The model consists of a system of three equations: a PDE of parabolic type with dynamical boundary conditions modelling the distribution of free morphogens and two ODEs describing the evolution of bound and free receptors. Three biological processes are taken into account: diffusion, degradation and reversible binding. We study the stationary solutions and the evolution problem. Numerical simulations show the behavior of the solution depending on the values of the parameters.
Keywords: Steady states, Existence of solutions, Reaction diffusion equations, Morphogenesis, Numerical simulations..
Mathematics Subject Classification: Primary: 22E46, 53C35; Secondary: 57S2.
Citation: Ana I. Muñoz, José Ignacio Tello. Mathematical analysis and numerical simulation of a model of morphogenesis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1035-1059. doi: 10.3934/mbe.2011.8.1035
References:
[1]

F. W. Cummings, A model of morphogenesis, Phys. A, 3-4 (2004), 531-547. Google Scholar

[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.  Google Scholar

[3]

E. V. Entchev and M. González-Gaitán, Morphogen gradient formation and vesicular trafficking, Traffic, 3 (2002), 98-109. Google Scholar

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. E. Showalter, "Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, AMS, Providence, Rhode Island, 1997.  Google Scholar

[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.  Google Scholar

[18]

J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Series A, 25 (2009), 343-361. Google Scholar

[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. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. W. Cummings, A model of morphogenesis, Phys. A, 3-4 (2004), 531-547. Google Scholar

[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.  Google Scholar

[3]

E. V. Entchev and M. González-Gaitán, Morphogen gradient formation and vesicular trafficking, Traffic, 3 (2002), 98-109. Google Scholar

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. E. Showalter, "Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, AMS, Providence, Rhode Island, 1997.  Google Scholar

[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.  Google Scholar

[18]

J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Series A, 25 (2009), 343-361. Google Scholar

[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. Google Scholar

[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.  Google Scholar

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