zbMATH — the first resource for mathematics

Well-posedness of a linear spatio-temporal model of the JAK2/STAT5 signaling pathway. (English) Zbl 1277.35210
Summary: Cellular geometries can vary significantly, how they influence signaling remains largely unknown. In this article, we describe a new model of the most extensively studied signal transduction pathways, the Janus kinase (JAK)/signal transducer and activator of transcription (STAT) pathway based on a mixed system of linear differential equations (PDEs + ODEs) coupled by Robin boundary conditions. This model was introduced to analyze the influence of the cell shape on the regulatory response to the activated pathway. In this article, we present an analysis of the wellposedness of the resulting system, i.e., the existence of a unique solution, its nonnegativity, boundedness and Lyapunov stability. As byproduct, we show the well-posedness and convergence of a suitable discretization of this model providing the basis for its reliable numerical simulation.

35K57 Reaction-diffusion equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
92C37 Cell biology
92C40 Biochemistry, molecular biology
35B35 Stability in context of PDEs
Full Text: Euclid
[1] Adams, R. A., Fournier, J. J. F., Sobolev spaces , Elsevier, 2003. · Zbl 1098.46001
[2] ANSYS ICEM CFD Mesh Generation , URL: http://www.ansys.com/-products/icemcfd.asp.
[3] Bachmann, J., Dynamic Modeling of the JAK2/STAT5 Signal Transduction Pathway to Dissect the Specific Roles of Negative Feedback Regulators, dissertation, University of Heidelberg, 2009.
[4] Bermon A., and Plemmons R. J., Nonnegative Matrices in the Mathematical Sciences , SIAM Publication, Philadelphia, 1994.
[5] Bramble J. H., and Hubbard B. E., Approximation of Solutions of Mixed Boundary Value Problems for Poisson’s Equation by Finite Differences , J. ACM 12, 114-123, DOI=10.1145/321250.321260 http://doi.acm.org/10.1145/321250.321260, 1965. · Zbl 0125.07305
[6] Brenner S. C., and Scott R. L., The Mathematical Theory of Finite Element Methods , Springer, Berlin-Heidelberg-New York, 1994. · Zbl 0804.65101
[7] Ciarlet P. G., Introduction to Numerical Linear Algebra and Optimization , Cambridge University Press, Cambridge, UK, 1988. · Zbl 0672.65001
[8] Claus J., Friedmann E., Klingmüller U., Rannacher R., and Szekeres T., Spatial aspects in the SMAD signaling pathway , J. Math. Biol., DOI 10.1007/s00285-012-0574-1, 2012. · Zbl 1277.92007
[9] Friedmann E., Pfeifer A.C., Neumann R., Klingmüller U. and Rannacher R., Interaction between experiment, modeling and simulation of spatial aspects in the JAK2/STAT5 Signaling pathway , in Model based parameter estimation: theory and applications, Springer Series Contributions in Mathematical and Computational Sciences, 2011.
[10] GASCOIGNE, High Performance Adaptive Finite Element Toolkit , URL: http://www.numerik.uni-kiel.de/ mabr/gascoigne/. URL:
[11] Forsythe G. E., and Wasow W. R., Finite-difference Methods for Partial Differential Equations , John Wiley, New York, 1960. · Zbl 0099.11103
[12] Hackbusch W., Elliptic Differential Equations: Theory and Numerical Treatment , Springer, Berlin, 1992. · Zbl 0755.35021
[13] Hairer E., Norsett S. P., and Wanner G., Solving Ordinary Differential Equations I: Nonstiff Problems , Springer, Berlin-Heidelkberg-New York, 1987. · Zbl 0638.65058
[14] Jost J., Partial Differential Equations , Springer, 2007.
[15] Lady\(\check{z}\)enskaja O. A., Solonnikov V. A., and Ural\(\acute{c}\)eva N. N., Linear and Quasilinears Equations of Parabolic Type , American Mathematical Society, 1968. · Zbl 0174.15403
[16] Liebermann G. M., Second Order Parabolic Differential Equations , World Scientific Publishing Co. Pte. Ltd, 1996. · Zbl 0884.35001
[17] Maiwald T., and Timmer J., Dynamical Modeling and Multi-experiment Fitting with Potters Wheel, Bioinformatics 24, 2037-43, 2008.
[18] Neumann R., Räumliche Aspekte in der Signaltransduktion , Diplomarbeit Ruprecht-Karls- Universität Heidelberg, 2009.
[19] Pfeifer A.C, Kaschek D., Bachmann J., Klingmüller U., and Timmer J., Model-based extension of high-throughput to high-content data , 2008.
[20] Schilling M, Maiwald T, Bohl S, Kollmann M, Kreutz C, Timmer J, and Klingmüller U., Computational processing and error reduction strategies for standardized quantitative data in biological networks , FEBS J. 272(24):6400-11, 2005.
[21] Schilling M., Pfeifer A.C., Bohl S., and Klingmüller U., Standardizing experimental protocols , Curr Opin Biotechnol., 2008.
[22] Shortley G. H., Weller R., Numerical Solution of Laplace’s Equation , J. Appl. Phys. 9, 334-348, 1938. · Zbl 0019.03801
[23] Swameye I., Müller T.G., Timmer J., Sandra O., and Klingmüller. U., Identification of nucleocytoplasmatic cycling as a remote sensor in cellular signaling by databased modeling , PNAS Proceedings of the National Academy of Sciences, 100(3):1028-1033, 2003.
[24] Timmer J., Müller T.G., Swameye I., Sandra O., and Klingmüller U., Modeling the nonlinear dynamics of cellular signal transduction , International Journal of Bifurcation and Chaos, 14, No. 6, 2069-2079, 2004. · Zbl 1065.92018
[25] Varga R. S., Matrix Iterative Analysis , Prentice-Hall, Englewood Cliffs, N. J., 1962. · Zbl 0133.08602
[26] Wloka J., Partial Differential Equations , Cambridge University Press, 1987. · Zbl 0623.35006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.