Mathematical models of the VEGF receptor and its role in cancer therapy

Abstract

We present an analysis of a stochastic model of the vascular endothelial growth factor (VEGF) receptor. This analysis addresses the contribution of ligand-binding-induced oligomerization, activation of src-homology 2 domain-carrying kinases and receptor internalization in the overall behaviour of the VEGF/VEGF receptor (VEGFR) system. The analysis is based upon a generalization of a Wentzel-Kramers-Brillouin (WKB) approximation of the solution of the corresponding master equation. 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 anti-angiogenic treatment.

Figure 1

Schematic of our receptor-binding model (table 1) including ligand-induced dimerization (receptor activation). The black rectangles represent the cytoplasmic kinase domains of the receptors. The asterisks denote that the dimerized receptors have undergone cross-phosphorylation and hence provide high-affinity docking sites for SH2 domain-carrying kinases. See text for a definition of $k_{\text{eff}}^x$.

Figure 2

Schematic of our receptor-binding model (table 3). The black rectangles represent the cytoplasmic kinase domains of the receptors. The asterisks denote that the dimerized receptors have undergone cross-phosphorylation and hence provide high-affinity docking sites for SH2 domain-carrying kinases. See text for a definition of $k_{\text{eff}i}^s$, i=1, … , 4.

Figure 3

Simulation results corresponding to Model 3 with $k_{\text{on}}^x=4.6\times {10}^3{\text{s}}^{-1}$ and $k_{\text{off}}^x={10}^{-3}{\text{s}}^{-1}$, respectively. This plot shows the time course of the proportion of surface dimers for different values of L. Key: solid line corresponds to L=10 nM, dashed line to L=1 nM, dot-dashed line to L=0.1 nM and dotted line to L=0.01 nM. Ligand is introduced at t=0.

Figure 4

(a) Simulation results corresponding to Model 3 with $k_{\text{on}}^x=4.6\times {10}^3{\text{s}}^{-1}$ and $k_{\text{off}}^x={10}^{-3}{\text{s}}^{-1}$. The squares in this plot show the maximum value achieved by the proportion of surface dimers (figure 3) as a function of ligand concentration. Solid line corresponds to a Hill curve $X=X_{\text{max}}{L}^n/(K^n+L^n)$ with n=1.2. (b) Experimental results by Park et al. (2003). Solid line corresponds to a Hill curve $X=X_{\text{max}}{L}^n/(K^n+L^n)$ with n=1.5. In both panels, dashed lines correspond to a Hill curve with n=1, which has been included for comparison. Ligand is introduced at t=0.

Figure 5

Simulation results corresponding to Model 3. These plots show the time course of the proportion of surface dimers bound to SH2 domains (x₁, x₂, x₃ and x₄) for L=10⁻⁸ M.

Figure 6

Simulation results corresponding to Model 3. (a) corresponds to results showing the proportion of surface dimers (upper plot) and the proportion of bound SH2 domains (lower plot) without anti-VEGF treatment for L=10 nM. (b) Results from a simulation in which anti-VEGF therapy was implemented by reducing the level of VEGF from L=10 nM to L=0.01 nM at time t=10⁻². (c): idem at t=10⁻³. Parameter values have been taken from table 2.

Figure 7

x_* as a function of the dimensionless quantity AL for different values of the dimensionless quantity $k_{\text{on}}^x$. Parameter values have been taken from table 2. Squares, $k_{\text{on}}^x=4.6\times {10}^3{\text{s}}^{-1}$; triangles, $k_{\text{on}}^x=4.6\times {10}^2{\text{s}}^{-1}$; and circles, $k_{\text{on}}^x=4.6\times {10}^1{\text{s}}^{-1}$.

Figure 8

Numerical solution for the first cumulant equations corresponding to Model 2 (table 3). (a) corresponds to log(AL)=−6, panel (b) to log(AL)=1 and (c) to log(AL)=6. Parameter values have been taken from table 2.

Figure 9

Simulation results corresponding to Model 2. (a) corresponds to results showing the proportion of surface dimers (upper plot) and the proportion of bound SH2 domains (lower plot) without anti-VEGF treatment for L=10 nM. (b) Results from a simulation in which anti-VEGF therapy was implemented by reducing the level of VEGF from L=10 to 0.01 nM at time t=10⁻³. (c): idem at t=10⁴. Parameter values have been taken from table 2.

Figure 10

Simulation results corresponding to Model 2 where the effect of combining anti-VEGF therapy and reduction of $k_{\text{on}}^x$ is demonstrated. Both therapies have been applied at t=10⁴. VEGF concentration has been reduced from L=10 to 0.01 nM. The dimerization rate has been reduced from $k_{\text{on}}^x=4.6\times {10}^3{\text{s}}^{-1}$ to $k_{\text{on}}^x=4.6\times {10}^1{\text{s}}^{-1}$. The rest of the parameter values have been taken from table 2.

Figure 11

Simulation results for Model 4 (equations (5.1)–(5.15)) for a physiological situation. This plot shows the proportion of receptors. Dashed line corresponds to L=10⁻⁸ M, solid line to L=10⁻⁹ M, dot-dashed line to L=10⁻¹⁰ M and dotted line to L=10⁻¹¹ M. Parameter values have been taken from table 2.

Figure 12

Simulation results for Model 4 (equations (5.1)–(5.15)) for a physiological situation. This plot shows the number of SH2 domains bound to receptor dimers on the surface of the cell. Solid line corresponds to L=10⁻⁸ M, dashed line to L=10⁻⁹ M, dot-dashed line to L=10⁻¹⁰ M and dotted line to L=10⁻¹⁸ M. Parameter values have been taken from table 2.

Figure 13

Simulation results for Model 4 (equations (5.1)–(5.15)). This plot shows the proportion of SH2 domains bound to surface receptors dimers. Solid line corresponds to L=10⁻⁸ M, dashed line to L=10⁻⁹ M, dot-dashed line to L=10⁻¹⁰ M and dotted line L=10⁻¹⁸ M. (a) corresponds to k_s=4.5×10⁻⁴ and (b) to k_s=9×10⁻⁴. Other parameter values have been taken from table 2.

Figure 14

Simulation results of the effect of anti-VEGF therapy on Model 4 (equations (5.1)–(5.15)) in a physiological situation. The levels of VEGF have been reduced from L=10 to 0.01 nM. (a) corresponds to drug administration at t=10⁻⁴ and (b) to administration time t=10⁻². Parameter values have been taken from table 2.

Figure 15

Simulation results of the effect of anti-VEGF therapy on Model 4 (equations (5.1)–(5.15)) in a pathological situation. The value of the rate of receptor synthesis has been increased to k_s=4.5×10⁻⁴, i.e. fivefold its physiological value. The levels of VEGF have been reduced from L=10 to 0.01 nM. (a) corresponds to an untreated case and (b) to drug administration at t=10⁻⁴. Other parameter values have been taken from table 2.

Figure 16

Simulation results of the effect of anti-VEGF therapy on Model 4 (equations (5.1)–(5.15)) in a physiological situation. The levels of VEGF have been reduced from L=10 to 0.01 nM. (a) corresponds to drug administration at t=10⁴ and (b) to a close-up of (a). Parameter values have been taken from table 2.

Figure 17

Variance of x for Model 1 (table 1) calculated at the steady state shown on a logarithmic scale. Circles correspond to $k_{on}^x=4.6\times {10}^2{\text{s}}^{-1}$ and triangles to $k_{on}^x=4.6\times {10}^2{\text{s}}^{-1}$. Parameter values have been taken from table 2.

Figure 18

(a) Simulation results corresponding to Models 3 and 4 (with k_s=9×10⁻⁵ s⁻¹). The squares (Model 3) and triangles (Model 4) in this plot show the maximum values achieved by the proportion of surface dimers (figure 3) as a function of ligand concentration when the models were simulated until t=1.2, i.e. 20 min in dimensional terms. (b) Experimental results by Cai et al. (2006).