## Communications in Mathematical Analysis

### Well-Posedness of a Linear Spatio-Temporal Model of the JAK2/STAT5 Signaling Pathway

#### Abstract

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.

#### Article information

Source
Commun. Math. Anal. Volume 15, Number 2 (2013), 76-102.

Dates
First available in Project Euclid: 9 August 2013

https://projecteuclid.org/euclid.cma/1376053392

Mathematical Reviews number (MathSciNet)
MR3093582

Zentralblatt MATH identifier
1277.35210

#### Citation

Friedmann, E.; Neumann, R.; Rannacher, R. Well-Posedness of a Linear Spatio-Temporal Model of the JAK2/STAT5 Signaling Pathway. Commun. Math. Anal. 15 (2013), no. 2, 76--102. https://projecteuclid.org/euclid.cma/1376053392

#### References

• Adams, R. A., Fournier, J. J. F., Sobolev spaces, Elsevier, 2003.
• ANSYS ICEM CFD Mesh Generation, URL: http://www.ansys.com/-products/icemcfd.asp.
• 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.
• Bermon A., and Plemmons R. J., Nonnegative Matrices in the Mathematical Sciences, SIAM Publication, Philadelphia, 1994.
• 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.
• Brenner S. C., and Scott R. L., The Mathematical Theory of Finite Element Methods, Springer, Berlin-Heidelberg-New York, 1994.
• Ciarlet P. G., Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press, Cambridge, UK, 1988.
• 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.
• 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.
• GASCOIGNE, High Performance Adaptive Finite Element Toolkit, URL: http://www.numerik.uni-kiel.de/~mabr/gascoigne/.
• Forsythe G. E., and Wasow W. R., Finite-difference Methods for Partial Differential Equations, John Wiley, New York, 1960.
• Hackbusch W., Elliptic Differential Equations: Theory and Numerical Treatment, Springer, Berlin, 1992.
• Hairer E., Norsett S. P., and Wanner G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin-Heidelkberg-New York, 1987.
• Jost J., Partial Differential Equations, Springer, 2007.
• 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.
• Liebermann G. M., Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd, 1996.
• Maiwald T., and Timmer J., Dynamical Modeling and Multi-experiment Fitting with PottersWheel, Bioinformatics 24, 2037-43, 2008.
• Neumann R., Räumliche Aspekte in der Signaltransduktion, Diplomarbeit Ruprecht-Karls- Universität Heidelberg, 2009.
• Pfeifer A.C, Kaschek D., Bachmann J., Klingmüller U., and Timmer J., Model-based extension of high-throughput to high-content data, 2008.
• 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.
• Schilling M., Pfeifer A.C., Bohl S., and Klingmüller U., Standardizing experimental protocols, Curr Opin Biotechnol., 2008.
• Shortley G. H., Weller R., Numerical Solution of Laplace's Equation, J. Appl. Phys. 9, 334–348, 1938.
• 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.
• 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.
• Varga R. S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962.
• Wloka J., Partial Differential Equations, Cambridge University Press, 1987.