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Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning
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 |
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. |
show all references
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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