Model for Osteosarcoma-9 as a potent factor in cell survival and resistance to apoptosis

In this paper we use a simple model to explore the function of the gene Osteosarcoma-9 (OS-9). We are particularly interested in understanding the role of this gene as a potent anti-apoptotic factor. The theoretical description is constrained by experimental data from induction of apoptosis in cells where OS-9 is overexpressed. The data available suggest that OS-9 promotes cell viability and confers resistance to apoptosis, potentially implicating OS-9 in the survival of cancer cells. Three different apoptosis-inducing mechanisms were tested and are modeled here. A more complex and realistic model is also discussed.

Figure 1

Number of viable cells after withdrawal of IL-3, in the presence and in the absence of OS-9, from FACS analysis. In (a) we reproduce the data from Fig. 2B in Ref. 6. In (b) we show the solutions for $c_v\left(t\right)$ with $\alpha =0.057$. The black curve depicts $c_v\left(t\right)$ when $\beta =0$, the gray curves depict $c_v\left(t\right)$, with the dark gray curve for ${\beta }_{17\mathrm{V}}=0.17$ and the light gray curve for ${\beta }_{17\mathrm{VIII}}=0.31$. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.056$, ${\beta }_{17\mathrm{V}}=0.19$ and ${\beta }_{17\mathrm{VIII}}=0.32$; the lower dashed curves represent the fittings to the minima data with ${\alpha }_0=0.058$, ${\beta }_{17\mathrm{V}}=0.15$ and ${\beta }_{17\mathrm{VIII}}=0.30$.)

Figure 2

Number of apoptotic cells after withdrawal of IL-3, in the presence and in the absence of OS-9, from FACS analysis. In (a) we reproduce the data from Fig. 2C in Ref. 6. In (b) we show the solutions for $c_a\left(t\right)$ with $\alpha =0.057$ and $\gamma =0.30$. The black curve depicts $c_a\left(t\right)$ when $\beta ,\delta =0$, the gray curves depict $c_a\left(t\right)$, with the dark gray curve for ${\beta }_{17\mathrm{V}}=0.17$ and ${\delta }_{17\mathrm{V}}=0.099$, and the light gray curve for ${\beta }_{17\mathrm{VIII}}=0.31$ and ${\delta }_{17\mathrm{VIII}}=0.086$. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.058$, $\gamma =0.26$, ${\beta }_{17\mathrm{V}}=0.15$ and ${\delta }_{17\mathrm{V}}=0.070$, and ${\beta }_{17\mathrm{VIII}}=0.30$ and ${\delta }_{17\mathrm{VIII}}=0.029$; the lower dashed curves represent the fittings to the minima data with $\alpha =0.056$, $\gamma =0.34$, ${\beta }_{17\mathrm{V}}=0.19$ and ${\delta }_{17\mathrm{V}}=0.14$, and ${\beta }_{17\mathrm{VIII}}=0.32$ and ${\delta }_{17\mathrm{VIII}}=0.16$.)

Figure 3

Number of viable cells after treatment with staurosporine $1\mu M$, in the presence and in the absence of OS9, from FACS analysis. In (a) we reproduce the data from Fig. 5B in Ref. 6. In (b) we show the solutions for $c_v\left(t\right)$ with $\alpha =0.20$. The black curve depicts $c_v\left(t\right)$ when $\beta =0$. The gray curves depict $c_v\left(t\right)$, with the dark gray curve for ${\beta }_{17\mathrm{V}}=0.18$ and the light gray curve for ${\beta }_{17\mathrm{VIII}}=0.24$. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.19$, ${\beta }_{17\mathrm{V}}=0.14$ and ${\beta }_{17\mathrm{VIII}}=0.23$; the lower dashed curves represent the fittings to the minima data with $\alpha =0.20$, ${\beta }_{17\mathrm{V}}=0.17$ and ${\beta }_{17\mathrm{VIII}}=0.20$.)

Figure 4

Number of apoptotic cells after treatment with staurosporine $1\mu M$, in the presence and in the absence of OS-9, from FACS analysis. In (a) we reproduce the data from Fig. 5C in Ref. 6. In (b) we show the solutions for $c_a\left(t\right)$ with $\alpha =0.20$ and $\gamma =0.72$. The black curve depicts $c_a\left(t\right)$ when $\beta ,\delta =0$; the gray curves depict $c_a\left(t\right)$, with the dark gray curve for ${\beta }_{17\mathrm{V}}=0.18$ and ${\delta }_{17\mathrm{V}}=0.19$, and the light gray curve for ${\beta }_{17\mathrm{VIII}}=0.24$ and ${\delta }_{17\mathrm{VIII}}=0.27$. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.20$, $\gamma =0.62$, ${\beta }_{17\mathrm{V}}=0.17$ and ${\delta }_{17\mathrm{V}}=0.09$, and ${\beta }_{17\mathrm{VIII}}=0.20$ and ${\delta }_{17\mathrm{VIII}}=0.20$; the lower dashed curves represent the fittings to the minima data with $\alpha =0.19$, $\gamma =0.90,{\beta }_{17\mathrm{V}}=0.14$ and ${\delta }_{17\mathrm{V}}=0.33$, and ${\beta }_{17\mathrm{VIII}}=0.23$ and ${\delta }_{17\mathrm{VIII}}=0.39$.)

Figure 5

Number of viable cells after treatment with staurosporine $0.1\mu M$, in the presence and in the absence of OS-9, from FACS analysis. In (a) we reproduce the data from Fig. 5B in Ref. 6. In (b) we show the solutions for $c_v\left(t\right)$ with $\alpha =0.15$. The black curve depicts $c_v\left(t\right)$ when $\beta =0$; the gray curves depict $c_v\left(t\right)$, with the dark gray curve giving ${\beta }_{17\mathrm{V}}=0.19$ and the light gray curve ${\beta }_{17\mathrm{VIII}}=0.35$. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.15$, ${\beta }_{17\mathrm{V}}=0.18$ and ${\beta }_{17\mathrm{VIII}}=0.35$; the lower dashed curves represent the fittings to the minima data with $\alpha =0.16$, ${\beta }_{17\mathrm{V}}=0.20$, and ${\beta }_{17\mathrm{VIII}}=0.34$.)

Figure 6

Number of apoptotic cells after treatment with staurosporine $0.1\mu M$, in the presence and in the absence of OS-9, from FACS analysis. In (a) we reproduce the data from Fig. 5C in Ref. 6. In (b) we show the solutions for $c_a\left(t\right)$ with $\alpha =0.15$ and $\gamma =0.20$. The black curve depicts $c_a\left(t\right)$ when $\beta ,\delta =0$; the gray curves depict $c_a\left(t\right)$, with ${\beta }_{17\mathrm{V}}=0.19$ and ${\delta }_{17\mathrm{V}}=0$ for the dark gray curve, with ${\beta }_{17\mathrm{VIII}}=0.35$ and ${\delta }_{17\mathrm{VIII}}=0$ for the light gray curve. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.16$, $\gamma =0.19$, ${\beta }_{17\mathrm{V}}=0.20$ and ${\delta }_{17\mathrm{V}}=0$, and ${\beta }_{17\mathrm{VIII}}=0.34$ and ${\delta }_{17\mathrm{VIII}}=0$; the lower dashed curves represent the fittings to the minima data with $\alpha =0.15$, $\gamma =0.20$, ${\beta }_{17\mathrm{V}}=0.18$ and ${\delta }_{17\mathrm{V}}=0$, and ${\beta }_{17\mathrm{VIII}}=0.35$ and ${\delta }_{17\mathrm{VIII}}=0$.)

Figure 7

Number of viable cells after treatment with $\mathrm{TNF}\alpha$, in the presence and in the absence of OS-9, from the FACS analysis. In (a) we reproduce the data from Fig. 6B in Ref. 6. In (b) we show the solutions for $c_v\left(t\right)$ with $\alpha =0.32$. The black curve depicts $c_v\left(t\right)$ when $\beta =0$; the gray curves depict $c_v\left(t\right)$, with the dark gray curve for ${\beta }_{17\mathrm{V}}=0.46$ and the light gray curve for ${\beta }_{17\mathrm{VIII}}=0.54$. (The upper dashed curves represent the fittings to the maxima data with ${\alpha }_0=0.31$, ${\beta }_{17\mathrm{V}}=0.47$, and ${\beta }_{17\mathrm{VIII}}=0.54$; the lower dashed curves represent the fittings to the minima data with ${\alpha }_0=0.33$, ${\beta }_{17\mathrm{V}}=0.46$, and ${\beta }_{17\mathrm{VIII}}=0.54$.)

Figure 8

Number of apoptotic cells after treatment with $\mathrm{TNF}\alpha$, in the presence and in the absence of OS-9, from the FACS analysis. In (a) we reproduce the data from Fig. 6C in Ref. 6. In (b) we show the solutions for $c_a\left(t\right)$ with $\alpha =0.32$ and $\gamma =0.045$. The black curve depicts $c_a\left(t\right)$ when $\beta$, $\delta =0$; the gray curves depict $c_a\left(t\right)$, with the dark gray curve for ${\beta }_{17\mathrm{V}}=0.46$ and ${\delta }_{17\mathrm{V}}=0$, and the light gray curve for ${\beta }_{17\mathrm{VIII}}=0.54$ and ${\delta }_{17\mathrm{VIII}}=0$. (The upper dashed curves represent the fittings to the maxima data with $\alpha =0.33$, $\gamma =0.036$, ${\beta }_{17\mathrm{V}}=0.46$, and ${\delta }_{17\mathrm{V}}=0$, and ${\beta }_{17\mathrm{VIII}}=0.54$ and ${\delta }_{17\mathrm{VIII}}=0$; the lower dashed curves represent the fittings to the minima data with $\alpha =0.31$, $\gamma =0.055$, ${\beta }_{17\mathrm{V}}=0.47$, and ${\delta }_{17\mathrm{V}}=0.33$, and ${\beta }_{17\mathrm{VIII}}=0.54$ and ${\delta }_{17\mathrm{VIII}}=0$.)