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Mathematical Modeling of the VEGF Receptor

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Abstract

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.

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Notes

  1. 1.

    These three types of VEGFR are surface receptors. There is also a soluble form of VEGFR-1.

  2. 2.

    Throughout the paper, we use the same covention: an upper-case letter represents numbers of molecules of a given type, whereas the corresponding lower-case letter represents the proportion of molecules of that particular kind with respect to the total number of molecules. An exception to this rule is L, whose meaning is explained in the text.

  3. 3.

    Teis and Huber (2003) distinguish between active and inactive RTKs. We will assume that “active” refers to dimerised receptors, which seems to be pretty clear from the context, and that “inactive” refers to both unbound receptors and non-dimerised ligand/receptor complexes.

  4. 4.

    Within the statistical physics community, this approximation is often referred to as the eikonal approximation.

  5. 5.

    Kubo et al. (1973) states the multidimensional result without a proof.

  6. 6.

    The system (1.13)–(1.26) has 14 equations but the conservation law \(u + {u}^{i} + b + {b}^{i} + 2{ \sum \nolimits }_{j}({x}_{j} + {x}_{j}^{i}) = {n}_{R}\) allows us to reduce the dimensionality of the system by one unit.

  7. 7.

    In fact, this method produces a hierarchy of “kinetic” equations where the cumulants of order n depend on the all the cumulants of order up to n − 1.

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Alarcón, T., Page, K.M. (2012). Mathematical Modeling of the VEGF Receptor. In: Jackson, T.L. (eds) Modeling Tumor Vasculature. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0052-3_1

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