Stochastic models of the binding kinetics of VEGF-A to VEGFR1 and VEGFR2 in endothelial cells

Vascular endothelial growth factor receptors (VEGFRs) are receptor tyrosine kinases (RTKs) that regulate proliferation, migration, angiogenesis and vascular permeability of endothelial cells. Vascular endothelial growth factor receptor 1 (VEGFR1) and receptor 2 (VEGFR2) bind vascular endothelial growth factors (VEGFs), inducing receptor dimerisation and activation, characterised by phosphorylation of tyrosine residues in their cytoplasmic domain. Although recent experimental evidence suggests that RTK signalling occurs both on the plasma membrane and intra-cellularly, and reveals the role of endocytosis in RTK signal transduction, we still lack knowledge of VEGFR phosphorylation-site use and of the spatiotemporal regulation of VEGFR signalling. In this paper, we introduce four stochastic mathematical models to study the binding kinetics of vascular endothelial growth factor VEGF-A to VEGFR1 and VEGFR2, and RTK phosphorylation. The formation of phosphorylated (homoand hetero-) dimers on the cell surface is a two-step process: diffusive transport and binding. The first two of our models (Model 1 and Model 2) only consider VEGFR2. This simplification allows us to introduce new stochastic descriptors making use of a matrix-analytic approach. The two remaining models (Model 3.1 and Model 3.2) describe the competition of VEGFR1 and VEGFR2 for ligand availability, and are analysed making use of Gillespie (stochastic) simulations and the van Kampen approximation. Under the hypothesis that bound phosphorylated receptor dimers are the signalling units, we study the time to reach a threshold number of such complexes. Our results indicate that the presence of VEGFR1 does not only affect the timescale to reach a given signalling threshold, but it also affects the maximum attainable threshold. This result is consistent with the conjectured role of VEGFR1 as a decoy receptor, that prevents VEGF-A binding to VEGFR2, and thus, VEGFR2 attaining suitable phosphorylation levels. We identify an optimum range of ligand concentration for sustained dimer phosphorylation. Finally, sensitivity analysis identifies the rate of monomer formation as the parameter that our descriptors depend most strongly on.