Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning

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December 2011, 4(6): 1413-1428. doi: 10.3934/dcdss.2011.4.1413

Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning

Wenrui Hao ^1, , Jonathan D. Hauenstein ^1, , Bei Hu ^2, , Yuan Liu ^1, , Andrew J. Sommese ^1, and Yong-Tao Zhang ^3,

1.	Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States, United States, United States
2.	Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
3.	Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618

Received May 2009 Revised September 2009 Published December 2010

Abstract
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The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579--589] has multiple steady state solutions. In this paper, we describe the computation of seven steady state solutions found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numerical simulations via a time marching approach. The results of this paper show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size.

Keywords: reaction-diffusion, polynomial systems, dorsal-ventral patterning., Steady states.

Mathematics Subject Classification: Primary: 65H04, 65H20, 65M06, 65N06, 92B0.

Citation: Wenrui Hao, Jonathan D. Hauenstein, Bei Hu, Yuan Liu, Andrew J. Sommese, Yong-Tao Zhang. Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1413-1428. doi: 10.3934/dcdss.2011.4.1413

References:

[1]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, "Bertini: Software for Numerical Algebraic Geometry,", Available at \url{http://www.nd.edu/~sommese/bertini}., ().
[2]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, II, Software for numerical algebraic geometry: A paradigm and progress towards its implementation, in "Software for Algebraic Geometry," volume 148 of IMA Vol. Math. Appl., 1-14. Springer, New York, 2008.
[3]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Adaptive multiprecision path tracking, SIAM J. Numer. Anal., 46 (2008), 722-746. doi: 10.1137/060658862.
[4]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Stepsize control for adaptive multiprecision path tracking, in "Interactions of Classical and Numerical Algebraic Geometry" (eds. D. Bates, G. Besana, S. Di Rocco and C. Wampler), Contemporary Mathematics, 496 (2009), 21-31.
[5]	B. Gustafsson, H.-O. Kreiss and J. Oliger, "Time Dependent Problems and Difference Methods," Wiley, New York, 1995.
[6]	A. D. Lander, Morpheus unbound: Reimagining the Morphogen gradient, Cell, 128 (2007), 245-256.
[7]	L. Saude, K. Woolley, P. Martin, W. Driever and D. L. Stemple, Axis-inducing activities and cell fates of the zebrafish organizer, Development, 127 (2000), 3407-3417.
[8]	A. J. Sommese and C. W. Wampler, II, "The Numerical Solution of Systems of Polynomials Arising in Engineering and Science," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812567727.
[9]	J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag, New York, 1993.
[10]	A. A. Teleman, M. Strigini and S. M. Cohen, Shaping morphogen gradients, Cell, 105 (2001), 559-562.
[11]	J. Verschelde and R. Cools, Symbolic homotopy construction, Appl. Algebra Engrg. Comm. Comput., 4 (1993), 169-183. doi: 10.1007/BF01202036.
[12]	L. Wolpert, R. Beddington, J. Brockets, T. Jessel, P. Lawrence and E. Meyerowitz, "Principles of Development," Oxford University, 2002.
[13]	Y.-T. Zhang, A. Lander and Q. Nie, Computational analysis of BMP gradients in dorsal-ventral patterning of the zebrafish embryo, Journal of Theoretical Biology, 248 (2007), 579-589. doi: 10.1016/j.jtbi.2007.05.026.

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References:

[1]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, "Bertini: Software for Numerical Algebraic Geometry,", Available at \url{http://www.nd.edu/~sommese/bertini}., ().
[2]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, II, Software for numerical algebraic geometry: A paradigm and progress towards its implementation, in "Software for Algebraic Geometry," volume 148 of IMA Vol. Math. Appl., 1-14. Springer, New York, 2008.
[3]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Adaptive multiprecision path tracking, SIAM J. Numer. Anal., 46 (2008), 722-746. doi: 10.1137/060658862.
[4]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Stepsize control for adaptive multiprecision path tracking, in "Interactions of Classical and Numerical Algebraic Geometry" (eds. D. Bates, G. Besana, S. Di Rocco and C. Wampler), Contemporary Mathematics, 496 (2009), 21-31.
[5]	B. Gustafsson, H.-O. Kreiss and J. Oliger, "Time Dependent Problems and Difference Methods," Wiley, New York, 1995.
[6]	A. D. Lander, Morpheus unbound: Reimagining the Morphogen gradient, Cell, 128 (2007), 245-256.
[7]	L. Saude, K. Woolley, P. Martin, W. Driever and D. L. Stemple, Axis-inducing activities and cell fates of the zebrafish organizer, Development, 127 (2000), 3407-3417.
[8]	A. J. Sommese and C. W. Wampler, II, "The Numerical Solution of Systems of Polynomials Arising in Engineering and Science," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812567727.
[9]	J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag, New York, 1993.
[10]	A. A. Teleman, M. Strigini and S. M. Cohen, Shaping morphogen gradients, Cell, 105 (2001), 559-562.
[11]	J. Verschelde and R. Cools, Symbolic homotopy construction, Appl. Algebra Engrg. Comm. Comput., 4 (1993), 169-183. doi: 10.1007/BF01202036.
[12]	L. Wolpert, R. Beddington, J. Brockets, T. Jessel, P. Lawrence and E. Meyerowitz, "Principles of Development," Oxford University, 2002.
[13]	Y.-T. Zhang, A. Lander and Q. Nie, Computational analysis of BMP gradients in dorsal-ventral patterning of the zebrafish embryo, Journal of Theoretical Biology, 248 (2007), 579-589. doi: 10.1016/j.jtbi.2007.05.026.

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