Mathematical Modeling of the VEGF Receptor

  • Tomás Alarcón
  • Karen M. Page


This chapter is devoted to the formulation and analysis of several models of the VEGF receptor and the initial steps in the signalling cascade following receptor activation. Our models take into consideration different factors and processes such as receptor cross-linking, endocytosis, recycling, degradation and synthesis. The effect of each one of these factors is studied. In particular, we present an analysis of a stochastic model of the vascular endothelial growth factor (VEGF) receptor, which accounts for ligand binding-induced oligomerisation, activation of SH2 domain-carrying kinases and receptor internalization. This is an analysis, based upon a generalisation of a WKB approximation of the solution of the corresponding Master Equation, of the role and contribution of each of these processes to the overall behaviour of the VEGF/VEGF receptor (VEGFR) system. The results of this analysis, in turn, allow us to formulate plausible mechanisms for tumour resistance to antiangiogenic therapy. We predict that tumour-mediated overexpression of VEGFRs in the endothelial cells (ECs) of tumour-engulfed vessels leads to an increased sensitivity of the ECs to low concentrations of VEGF, thus endowing the tumour with increased resistance to antiangiogenic treatment. We then show using a simplified version of the above model, that it exhibits different dynamical behaviours, which account for different cell responses to stimulation with growth factor, from perfect adaptation to sustained response thus providing a framework which attempts to understand how a single sensorial system can produce a variety of different responses.


Vascular Endothelial Growth Factor Vascular Endothelial Growth Factor Receptor Vascular Endothelial Growth Factor Concentration Receptor Endocytosis Receptor Tyrosine Kinase Activation 
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  1. .
    Alarcón, T. and Page, K.M. (2006). Stochastic models of receptor oligomerisation by bivalent ligand. J. R. Soc. Interface. 3, 545-559.PubMedCrossRefGoogle Scholar
  2. .
    Alarcón, T. and Page, K.M. (2007). Mathematical models of the VEGF receptor and its role in cancer therapy. J. R. Soc. Interface. 4, 283-304.PubMedCrossRefGoogle Scholar
  3. .
    Autiero, M., Waltenberg, J., Communi, D., Kranz, A., Moons, L., Lambrechts, D., Kroll, J., Plaisance, S., De Mol, M. Bono, F., Kilche, S., Fellbrich, G., Ballmer-Hofer, K., Maglione, D., Mayer-Beyrle, U., Dewerchin, M., Dombrowski, S., Stanimirovic, D., Van Hummelen, P., Heio, C., Hicklin, D.J., Persico, G., Herbert, J.M., Shibuya, M., Collen, D., Conway E.M., Carmeliet, P. (2003). Role of PIGF in the intra- and inter-molecular cross-talk between the VEGF receptor Flt1 and Flk1. Nat. Med. 9. 936-943.PubMedCrossRefGoogle Scholar
  4. .
    Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P. (2002). Molecular biology of the cell. Garland Publishing, New York, NY (USA).Google Scholar
  5. .
    Alon, U., M.G. Surette, N. Barkai, S. Leibler. (1999). Robustness in bacterial chemotaxis. Nature. 347, 168-171.Google Scholar
  6. .
    Amlal, H., Faroqui, S., Balasubramaniam, A., and Sheriff, S. (2006). Estrogen up-regulates neuropeptide YY1 receptor expression in a human breast cancer cell line. Cancer Res. 66, 3706-3714.Google Scholar
  7. .
    Berger, G., and Benjamin, L.E. (2003). Tumorigenesis and the angiogenic switch. Nature Rev. Cancer. 3, 401-410.Google Scholar
  8. .
    Cai, J., Jians, W.G., Ahmed, A. and Bulton, M. (2006). Vascular endothelial growth factor induced cells proliferation is regulated by interaction between VEGFR-s, SH-PTP1 and eNOS. Microvasc. Res. 71, 20-31.PubMedCrossRefGoogle Scholar
  9. .
    Cross, M.J., Dixelius, J., Matsumoto, T., and Claesson-Welsh, L. (2003). VEGF-receptor signal transduction. Trends Biochem. Sci. 28, 488-494.Google Scholar
  10. .
    Dixelius, J., Makinen, T., Wirzenius, M., Karkkainen, M.J., Wernstedt, C., Alitalo, K., Claesson-Welsh, L. (2003). Ligand-induced vascular endothelial growth factor receptor-3 (VEGFR-3) heterodimerisation with VEGFR-2 in primary lymphatic endothelial cells regulates tyrosine phosphorylation sites. J. Biol. Chem. 278, 40973-40979.PubMedCrossRefGoogle Scholar
  11. .
    Ebens, A., K. Brose, E.D. Leonardo, M.G. Hanson, F. Bladt, C. Birchmeier, B.A. Barres, M. TessierLavigne. (1996). Hepatocyte growth factor scatter factor is an axonal chemoattractant and a neurotrophic factor for spinal motor neurons. Neuron. 17, 1157-1172.PubMedCrossRefGoogle Scholar
  12. .
    Felder, S., Zhou, M., Hu, P., Urena, J., Ullrich, A., Chaudhuri, M., White, M., Shoelson, S.E., and Schlessinger, J. (1993). SH2 domains exhibit high affinity binding to tyrosine-phosphorylated peptides yet exhibit rapid dissociation and exchange. Mol. Cell. Biol. 13, 1449-1455.Google Scholar
  13. .
    Ferrara, N. (2002). Role of vascular endothelial growth factor in physiologic and pathologic angiogenesis: therapeutic implications. Semin. Oncol. 29 (6 Suppl. 16), 10-14.Google Scholar
  14. .
    Grotendorst, G.R., Chang, T., Seppa, H.E., Kleinman, H.K., and Martin, G.R. (1982). Platelet-derived growth factor is a chemoattractant for vascular smooth muscle cells. J. Cell Physiol. 113, 261-266.CrossRefGoogle Scholar
  15. .
    Hampton, T. (2005). Antiangiogenic therapy a two-trick pony? JAMA. 293, 1051.PubMedCrossRefGoogle Scholar
  16. .
    Helmreich, E.J.M. (2001). The biochemistry of cell signalling. Oxford University Press, New York, NY (USA).Google Scholar
  17. .
    Hicklin, D.J., and Ellis, L.M. (2005). Role of the vascular endothelial growth factor pathway in tumor growth and angiogenesis. J. Clin. Oncol. 23, 1011-1027.PubMedCrossRefGoogle Scholar
  18. .
    Holash, J., Wiegand, S.J., Yancopoulos, G.D. (1999). New model of tumor angiogenesis: dynamic balance between vessel regression and growth mediated by angiopoietins and VEGF. Oncogene. 18, 5356-5362.PubMedCrossRefGoogle Scholar
  19. .
    Jain, R. (2005). Normalization of tumour vasculature: an emerging concept in antiangiogenic therapy Science. 307, 58-62.Google Scholar
  20. .
    Kitahara, K. (1973). The Hamilton-Jacobi equation approach to fluctuation phenomena Adv. Chem. Phy. 29, 85-111.Google Scholar
  21. .
    Klominek, J., Baskin, B., and Hauzenberger, D. (1998). Platelet-derived growth factor (PDGF) BB acts as a chemoattractant for human malignant mesothelioma cells via PDGF receptor β-integrin α3β1 interaction. Clin. Exp. Metastasis. 16, 529-539.PubMedCrossRefGoogle Scholar
  22. .
    Knox, B.E., P.N. Devreotes, A. Goldbeter, L.A. Segel. (1986). A molecular mechanism for sensory adaptation based on ligand-induced receptor modification Proc. Nat. Acad. Sci. 83, 2345-2349.CrossRefGoogle Scholar
  23. .
    Kubo, R., Matsuo, K., and Kitahara, K. (1973). Fluctuation and relaxation of macrovariables J. Stat. Phys. 9, 51-96.CrossRefGoogle Scholar
  24. .
    Lauffenburger, D.A., and Linderman, J.J. (1993). Receptors: models for binding, trafficking, and signalling. Oxford University Press, New York, (USA).Google Scholar
  25. .
    Lash, G.E., A.Y. Warren, S. Underwood, P.N. Baker. (2003). Vascular endothelial growth factor is a chemoattractant for trophoblast cells. Placenta. 24, 549-556.PubMedCrossRefGoogle Scholar
  26. .
    Levchenko, A., P.A. Iglesias. (2002). Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils. Biophys. J. 82, 50-63.PubMedCrossRefGoogle Scholar
  27. .
    Mac Gabham, F., and Popel, A.S. (2004). Model of competitive binding of vascular endothelial growth factor and placental growth factor to VEGF receptors on endothelial cells. Am. J. Physiol. Heart Circ. Physiol. 286, H153-H164.Google Scholar
  28. .
    Mac Gabham, F., and Popel, A.S. (2005a). Differential binding of VEGF isoforms to VEGF receptor 2 in the presence of neuropilin-1: a computational model. Am. J. Physiol. Heart Circ. Physiol. 288, H2851-H2860.Google Scholar
  29. .
    Mac Gabham, F., and Popel, A.S. (2005b). Monte Carlo simulations of VEGF binding to cell surface receptors in vitro. Biochim. Biophys. Acta – Mol. Cell Res. 1746, 95-107.Google Scholar
  30. .
    Marshall, C.J. (1995). Specifity of receptor tyrosine kinase signaling: transient versus sustained extracellular signal-regulated kinase activation. Cell. 80, 179-185.PubMedCrossRefGoogle Scholar
  31. .
    Mitchell, H., A. Chowdhury, R.E. Pagano, E.B. Leof. (2004). Ligand-dependent and -independent tranforming growth factor-β receptor recycling regulated by clathrin-mediated endocytosis and Rab11. Mol. Biol. Cell. 15, 4166-4178.PubMedCrossRefGoogle Scholar
  32. .
    Ohshima, M., Y. Noguchi, Y. Ito, M. Maeno, K. Otsuka. (2001). Hepatocyte growth factor secreted by periodontal ligament and gingival fibroblasts is a major chemoattractant for gingival epithelial cells. J. Periodontal Res. 36, 377-383.CrossRefGoogle Scholar
  33. .
    Park, C.H, Schneider, I.C., and Maugh, J.M. (2003). Kinetic analysis of platelet-derived growth factor receptor/phosphoinositide 3-kinase/Akt signalling in fibroblast. J. Biol. Chem. 278, 37064-37072.PubMedCrossRefGoogle Scholar
  34. .
    Polo, S., Pece, S., and Di Fiore, P.P. (2004). Endocytosis and cancer. Curr. Opin. Cell Biol. 16, 156-161.Google Scholar
  35. .
    Posner, R.G., Wofsy, C., and Goldstein, B. (1995). The kinetics of bivalent ligand-bivalent receptor aggregation: Ring formation and the breakdown of equivalent site approximation Math. Biosciences. 126, 171-190.CrossRefGoogle Scholar
  36. .
    Reibman, J., S. Meixler, T.C. Lee, L.I. Gold, B.N. Cronstein, K.A. Haines, S.L. Kolasinski, G. Weissmann. (1991). Transforming growth factor β1, a potent chemoattractant for human neutrophils, bypasses classic signal-transduction pathways. Proc. Nat. Acad. Sci. 88, 6805-6809.PubMedCrossRefGoogle Scholar
  37. .
    Sasagawa, S., Y. Ozaki, K. Fujita, S. Kuroda. (2005). Prediction and validation of the distinct dynamics of transient and sustained ERK activation. Nature Cell Biol. 7, 365-373.Google Scholar
  38. .
    Sawada, J., A. Itakura, A. Tanaka, T. Furusaka, H. Matsuda. (2000). Nerve growth factor functions as a chemoattractant for mast cells through both mitogen-activated protein kinase and phosphatidylinositol 3-kinase signaling pathways. Blood. 95, 2052-2058.PubMedGoogle Scholar
  39. .
    Sawyer, T.K. (1998). Src homology-2 domains: Structure, mechanisms and drug discovery. Biopolymers. 47 243-261.PubMedCrossRefGoogle Scholar
  40. .
    Schlesinger, J., D. Bar-Sagi. (1995). Activation of Ras and other signalling pathways by tyrosine kinase receptors. In Symposia on Quantitative Biology: Molecular Genetics and Biology. (Cold Spring Harbor Laboratory Press, Cold Spring Harbor, USA).Google Scholar
  41. .
    Schneider, I.C., J.M. Haugh. (2005). Quantitative elucidation of a distinct spatial gradient-sensing mechanism in fibroblasts. J. Cell Biol. 171, 883-892.CrossRefGoogle Scholar
  42. .
    Shibuya, M., Claesson-Welsh, L. (2006). Signal transduction by VEGF receptors in regulation of angiogenesis and lymphoangiogenesis. Exp. Cell Res. 312, 549-560.CrossRefGoogle Scholar
  43. .
    Spiro, P.A., J.S. Parkinson, H.G. Othmer. (1997). A model for excitation and adaptation in bacterial chemotaxis. Proc. Nat. Acad. Sci. 94, 7263-7268.PubMedCrossRefGoogle Scholar
  44. .
    Sulzer, B., De Boer, R.J., and Perelson, A.S. (1996). Cross-linking reconsidered: binding and cross-linking fields and the cellular response. Biophys. J. 70, 1154-1168.PubMedCrossRefGoogle Scholar
  45. .
    Suzuma, I., Mandai, M., Takagi, H., Suzuma, K., Otani, A., Oh, H., Kobayashi, K., and Honda, Y. (1999). 17 β-estradiol increases VEGF receptor-2 and promotes DNA synthesis in retinal microvascular endothelial cells. Invest. Ophth. Vis. Sci. 40, 2122-2129.Google Scholar
  46. .
    Teis, D., and Huber, L.A. (2003). The odd couple: signal transduction and endocytosis. Cell. Mol. Life Sci. 60, 2020-2033.CrossRefGoogle Scholar
  47. .
    Terranova, V.P., DiFlorio, R., Lyall, R.M., Hic, S., Friesel, R., and Maciag, T. (1985). Human endothelial cells are chemotactic to endothelial cell growth factor and heparin. J. Cell Biol. 101, 2330-2334.CrossRefGoogle Scholar
  48. .
    Traverse, S., K. Seedorf, H. Patterson, C.J. Marshall, P. Cohen, A. Ullrich. (1994). EGF triggers neuronal differentiation of PC12 cells that overexpress EGF receptor. Curr. Biol. 4, 694-701.PubMedCrossRefGoogle Scholar
  49. .
    Tyson, J.J., Chen, K.C., and Novak, B. (2003). Sniffers, buzzers, toggles and blinkers: dynamics and signalling pathways in the cell. Curr. Opin. Cell Biol. 15, 221-231.Google Scholar
  50. .
    Van Kampen, N. (1992). Stochastic processes in physics and chemistry. North-Holland, Amsterdam.Google Scholar
  51. .
    Vaudry, D., P.J.S. Stork, P. Lazarovici, L.E. Eiden. (2002). Signalling pathways for PC12 cell differentiation: making the right connections. Science. 296, 1648-1649.PubMedCrossRefGoogle Scholar
  52. .
    Vilar, J.M.G., Jansen, R., and Sander, C. (2006). Signal processing in the TGF-β superfamily ligand receptor network. PLOS Comput. Biol. 2, 36-45.Google Scholar
  53. .
    Wiley, H.S., J.J. Herbst, B.J. Walsh, D.A. Lauffenburger, M.G. Rosenfeld. (1991). The role of receptor tyrosine kinase activity in endocytosis, compartmentation and down-regulation of the epidermal growth factor receptor. J. Biol. Chem. 266, 11083-11094.PubMedGoogle Scholar
  54. .
    Woolf, P.J., and Linderman, J.J. (2004). An algebra of dimerisation and its implications for G-protein coupled receptor. J. theor. Biol. 229, 157-168.PubMedCrossRefGoogle Scholar
  55. .
    Yaka, R., A. Gamliel, D. Gurwitz, R. Stein. (1998). NGF induces transient but not sustained activation of ERK in PC12 mutant cells incapable of differentiating. J. Cell. Biochem. 70, 425-432.PubMedCrossRefGoogle Scholar
  56. .
    Yi, T.-M., Y. Huang, M.I. Simon, J. Doyle. (2000). Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc. Nat. Acad. Sci. 97, 4649-4653.PubMedCrossRefGoogle Scholar
  57. .
    Ti Zhang, Hui-Chuan Sun, Yang Xu, Ke-Zhi Zhang, Lu Wang, Lun-Xiu Qin, Wei-Zhong Wu, Yin-Kun Liu, Sheng-Long Ye, and Zhao-You Tang. (2005). Overexpression of PDGF Receptor α in endothelial cells of hepatocellular carcinoma associated with metastatic potential. Clin Cancer Res. 11, 8557-8563.Google Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Centre de Recerca MatemàticaBarcelonaSpain
  2. 2.Department of MathematicsUniversity College LondonLondonUK

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