# p53 Model

We study the model of cellular p53 regulation given in [1]. We adapt the model by replacing $[p53]^{1.8}$ by $[p53]^{2}$, and obtain the following dynamics.

$\left\{ \begin{array}{lcl} [p53]' & = & (0.5 - 9.963e-6*[p53]*[MDM2_{nuclear}] - 1.925e-5*[p53])*3600 \\ {[\text{RNA}_{nuclear}]}' & = & (1.5e-3 + 1.5e-2*([p53]^2/(547600 + [p53]^2)) - 8e-4*[\text{RNA}_{nuclear}])*3600 \\ {[\text{RNA}_{cytoplasmic}]}' & = & (8e-4*[\text{RNA}_{nuclear}] - 1.444e-4*[\text{RNA}_{cytoplasmic}])*3600 \\ {[\text{MDM2}_{cytoplasmic}]}' & = & (1.66e-2*[\text{RNA}_{cytoplasmic}] - 9e-4*[\text{MDM2}_{cytoplasmic}])*3600 \\ {[\text{MDM2}_{nuclear}]}' & = & (9e-4*[\text{MDM2}_{cytoplasmic}] - 1.66e-7*[\text{MDM2}_{cytoplasmic}]^2 - 9.963e-6*[\text{MDM2}_{nuclear}]*[ARF])*3600 \\ {[ARF]}' & = & (0.5 - 3.209e-5*[ARF] - 9.963e-6*[\text{MDM2}_{cytoplasmic}]*[ARF])*3600 \end{array} \right.$

We firstly consider a small initial set which is given by
$[p53] \in [19.9,20.1], [\text{RNA}_{nuclear}] \in [19.9,20.1], [\text{RNA}_{cytoplasmic}] \in [19.9,20.1],$
$[\text{MDM2}_{nuclear}] \in [19.9,20.1], [\text{MDM2}_{cytoplasmic}] \in [19.9,20.1], [ARF] \in [19.9,20.1]$
The following figure shows the interval overapproximations for the flowpipes computed by Flow* 2.1.0 over the time horizon [0,10]. The total time cost is around 110 seconds.

For the following larger initial set with the same time horizon, Flow* 2.1.0 spent 200 seconds to compute all the flowpipes.
$[p53] \in [19.8,20.2], [\text{RNA}_{nuclear}] \in [19.8,20.2], [\text{RNA}_{cytoplasmic}] \in [19.8,20.2],$
$[\text{MDM2}_{nuclear}] \in [19.8,20.2], [\text{MDM2}_{cytoplasmic}] \in [19.8,20.2], [ARF] \in [19.8,20.2]$

### References

[1] G. Leenders and J. A.Tuszynski.
Stochastic and deterministic models of cellular p53 regulation.
Frontiers in Oncology. 2013;3:64.