Journal of Mathematical Biology

, Volume 67, Issue 5, pp 1171–1197 | Cite as

Spatial aspects in the SMAD signaling pathway

  • J. Claus
  • E. Friedmann
  • U. Klingmüller
  • R. Rannacher
  • T. Szekeres
Article

Abstract

Among other approaches, differential equations are used for a deterministic quantitative description of time-dependent biological processes. For intracellular systems, such as signaling pathways, most existing models are based on ordinary differential equations. These models describe temporal processes, while they neglect spatial aspects. We present a model for the SMAD signaling pathway, which gives a temporal and spatial description on the basis of reaction diffusion equations to answer the question whether cell geometry plays a role in signaling. In this article we simulate the ordinary differential equations as well as partial differential equations of parabolic type with suile numerical methods, the latter on different cell geometries. In addition to manual construction of idealized cells, we also construct meshes from microscopy images of real cells. The main focus of the paper is to compare the results of the model without and with spatial aspects to answer the addressed question. The results show that diffusion in the model can lead to significant intracellular gradients of signaling molecules and changes the level of response to the signal transduced by the signaling pathway. In particular, the extent of these observations depends on the geometry of the cell.

Keywords

Cell signaling Mixed differential equations Finite elements  Mesh generation 

Mathematics Subject Classification (2000)

34A34 34D05 34D20 35K57 65N30 65N50 

References

  1. ANSYS ICEM CFD (2010) Mesh generation. http://www.ansys.com/products/icemcfd.asp
  2. Ashall L, Horton CA, Nelson DE, Paszek P, Harper CV, Sillitoe K, Ryan S, Spiller DG, Unitt JF, Broomhead DS, Kell DK, Rand DA, Se V, White MRH (2009) Pulsatile stimulation determines timing and specificity of NF-kappaB-dependent transcription. Science 324(5924):242–246CrossRefGoogle Scholar
  3. Brown GC, Kholodenko BN (1999) Spatial gradients of cellular phospho-proteins. FEBS Lett 457:452–454CrossRefGoogle Scholar
  4. Caudron M, Bunt G, Bastiaens P, Karsenti E (2005) Spatial coordination of spindle assembly by chromosome-mediated signaling gradients. Science 309(5739):1373–1376CrossRefGoogle Scholar
  5. Chung S-W, Miles FL, Sikes RA, Cooper CR, Farach-Carson MC, Ogunnaike BA (2009) Quantitative modeling and analysis of the transforming growth factor signaling pathway. Biophys J 96(5):1733–1750CrossRefGoogle Scholar
  6. Ciarlet PG (1987) The finite element method for elliptic problems. North-Holland, DordrechtGoogle Scholar
  7. Clarke DC, Liu X (2008) Decoding the quantitative nature of TGF-\(\beta \)/Smad signaling. Trends Cell Biol 18(9):430–442CrossRefGoogle Scholar
  8. Claus J (2010) Spatial aspects in the simulation of the SMAD signal transduction pathway. Diploma thesis, Universität Heidelberg, Institut für Wissenschaftliches RechnenGoogle Scholar
  9. Deheuninck J, Luo K (2009) Ski and SnoN, potent negative regulators of TGF-beta signaling. Cell Res 19(1):47–57CrossRefGoogle Scholar
  10. GASCOIGNE (2002) High performance adaptive finite element toolkit. http://www.numerik.uni-kiel.de/mabr/gascoigne/
  11. Gonzlez-Prez V, Schmierer B, Hill CS, Sear RP (2011) Studying SMAD2 intranuclear diffusion dynamics by mathematical modelling of FRAP experiments. Integr Biol (Camb) 3(3):197–207CrossRefGoogle Scholar
  12. Heldin C-H, Miyazono K, ten Dijke P (1997) TGF-\(\beta \) signalling from cell membrane to nucleus through SMAD proteins. Nature 390:465–471CrossRefGoogle Scholar
  13. Hengl S, Kreutz C, Timmer J, Maiwald T (2007) Data-based identifiability analysis of non-linear dynamical models. Bioinformatics 23(19):2612–2618CrossRefGoogle Scholar
  14. Jungblut D (2010) Rekonstruktion von Oberflächenmorphologien und Merkmalskeletten aus dreidimensionalen Daten unter Verwendung hochparalleler Rechnerarchitekturen. PhD thesis, Goethe-Universität Frankfurt am MainGoogle Scholar
  15. Kajino T, Omori E, Ishii S, Ninomiya-Tsuji J (2007) TAK1 MAPK kinase kinase mediates transforming growth factor-beta signaling by targeting SnoN oncoprotein for degradation. J Biol Chem 282(13):9475–9481CrossRefGoogle Scholar
  16. Kholodenko BN, Brown GC, Hoek JB (2000) Diffusion control of protein phosphorylation in signal transduction pathways. Biochem J 350:901–907CrossRefGoogle Scholar
  17. Kholodenko BN (2006) Cell signalling dynamics in time and space. Nat Rev Mol Cell Biol 7(3):165–176CrossRefGoogle Scholar
  18. Klingmüller U, Bauer A, Bohl S, Nickel PJ, Breitkopf K, Dooley S, Zellmer S, Kern C, Merfort I, Sparna T, Donauer J, Walz G, Geyer M, Kreutz C, Hermes M, Götschel F, Hecht A, Walter D, Egger L, Neubert K, Borner C, Brulport M, Schormann W, Sauer C, Baumann F, Preiss R, MacNelly S, Godoy P, Wiercinska E, Ciuclan L, Edelmann J, Zeilinger K, Heinrich M, Zanger UM, Gebhardt R, Maiwald T, Heinrich R, Timmer J, von Weizscker F, Hengstler JG (2006) Primary mouse hepatocytes for systems biology approaches: a standardized in vitro system for modelling of signal transduction pathways. Syst Biol (Stevenage) 153:433–447CrossRefGoogle Scholar
  19. Legewie S (2009) Systems biological analyses of intracellular signal transduction. PhD thesis, Humboldt-Universität Berlin, Mathematisch-Naturwissenschaftliche Fakultät IGoogle Scholar
  20. Lev Bar-Or R, Maya R, Segel LA, Alon U, Levine AJ, Oren M (2000) Generation of oscillations by the p53-Mdm2 feedback loop: a theoretical and experimental study. PNASGoogle Scholar
  21. Melke P, Jönsson H, Pardali E, ten Dijke P, Peterson C (2006) A rate equation approach to elucidate the kinetics and robustness of the TGF-pathway. Biophys J 91(12):4368–4380CrossRefGoogle Scholar
  22. Meyers J, Craig J, Odde DJ (2006) Potential for control of signaling pathways via cell size and shape. Curr Biol 16(17):1685–1693CrossRefGoogle Scholar
  23. Neumann R (2009) Räumliche Aspekte der Signaltransduktion. Diploma thesis, Universität Heidelberg, Institut für wissenschaftliches RechnenGoogle Scholar
  24. NeuRA2 (2009) The neuron reconstruction algorithm. http://www.neura.org
  25. Neves SR, Tsokas P, Sarkar A, Grace EA, Rangamani P, Taubenfeld SM, Alberini CM, Schaff JC, Blitzer RD, Moraru II, Iyengar R (2008) Cell shape and negative links in regulatory motifs together control spatial information flow in signaling networks. Cell 133(4):666–680CrossRefGoogle Scholar
  26. Schmierer B, Tournier AL, Bates PA, Hill CS (2008) Mathematical modeling identifies Smad nucleocytoplasmic shuttling as a dynamic signal-interpreting system. Proc Natl Acad Sci USA 105(18):6608–6613CrossRefGoogle Scholar
  27. Straube R, Ward MJ (2009) An asymptotic analysis of intracellular signaling gradients arising from multiple small compartments. SIAM J Appl Math 70:248–269MathSciNetCrossRefMATHGoogle Scholar
  28. Tewari M, Rao A (2006) Systems biology approaches to TGF-/Smad signaling. In: Dijke P, ten Heldin C-H, Ridley A, Frampton J (eds) Proteins and cell regulation, Smad signal transduction, vol 5. Springer, Berlin, pp 361–378Google Scholar
  29. Wilkinson DS, Ogden SK, Stratton SA, Piechan JL, Nguyen TT, Smulian GA, Barton MC (2005) A direct intersection between p53 and transforming growth factor beta pathways targets chromatin modification and transcription repression of the alpha-fetoprotein gene. Mol Cell Biol 25(3):1200–1212CrossRefGoogle Scholar
  30. Yang J, Dai C, Liu Y (2005) A novel mechanism by which hepatocyte growth factor blocks tubular epithelial to mesenchymal transition. J Am Soc Nephrol 16(1):68–78CrossRefGoogle Scholar
  31. Zi Z, Klipp E (2007) Constraint-based modeling and kinetic analysis of the Smad dependent TGF-signaling pathway. PLoS ONE 2(9):e936CrossRefGoogle Scholar
  32. Zvaifler NJ (2006) Relevance of the stroma and epithelial-mesenchymal transition EMT for the rheumatic diseases. Arthrit Res Therapy 8:210CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • J. Claus
    • 1
  • E. Friedmann
    • 2
  • U. Klingmüller
    • 3
  • R. Rannacher
    • 2
  • T. Szekeres
    • 3
  1. 1.Center for Modeling and Simulation in the Biosciences (BIOMS)Universität HeidelbergHeidelbergGermany
  2. 2.Department of Applied MathematicsHeidelbergGermany
  3. 3.Systems Biology of Signal TransductionGerman Cancer Research CenterHeidelbergGermany

Personalised recommendations