## 2 Mathematical Modeling of p53 Pathway
### Modeling PBK Dynamics of Nutlin by [PERSON] et al. [25] modeled the response of p53 to the anti-tumor drug Nutlin and determined the extra-cellular concentration and dynamics of the anti-tumor drug with the help of the available pharmacokinetics data of Nutlin oral delivery in mice. It can be noted that due to the complex of Nutlin and plasma proteins, the amount of free Nutlin in plasma is significantly low. The data are incorporated in the following equation:  
\[N_{Qb}=B_{p}\frac{K_{B}N_{ext}}{1+K_{B}N_{ext}} \tag{5}\]  
where \(N_{Qb}\) represents the amount of bound Nutlin and \(B_{p}\) and \(K_{B}\) represent the binding site plasma protein and association binding constant having estimated values of \(2.86\times 10^{-4}M\) and \(0.085\times 10^{6}M^{-1}\), respectively. [PERSON] et al. took into consideration a uni-compartmental model associated with extra-cellular and intra-cellular parts of a cell, where the total concentration of Nutlin can be calculated as the sum of the concentration of free Nutlin, \(N_{ext}\), and concentration of bound Nutlin, \(N_{Qb}\).  
\[N_{conc}=N_{ext}+N_{Qb} \tag{6}\]  
The extra-cellular free Nutlin \(N_{ext}\) can be calculated in terms of \(N_{conc}\):  
\[N_{ext}=\frac{-(1+K_{B}B_{p}-K_{B}N_{conc})+\sqrt{\left(1+K_{B}B_{p}-K_{B}N_{ conc}\right)^{2}+4K_{B}N_{conc}}}{2K_{B}} \tag{7}\]  
The processes incorporating the intra-cellular free Nutlin can be written as:  
\[\frac{dn_{c}(t)}{dt} =g_{1}N_{ext}(t)+k_{diss3}(MDM_{i}(t)+MDM_{pi}(t)+MDM_{pi}(t)) \tag{8}\] \[-k_{B3}n_{c}(t)(MDM(t)+MDM_{p}(t)+MDM_{pn}(t))-e_{1}n_{c}(t)\]  
where \(n_{c}\) is the concentration of Nutlin in the cell.  
Bonding between proteins can also take place in the retina, where it is assumed to be similar to that in plasma. Therefore, the four assumptions are as follows:  
1. Drug dissemination takes place in a single compartment.
2. The elimination of only the free Nutlin takes place.
3. The elimination is linear.
4. The binding proteins are in a quasi-steady state.  
This results in the following pharmacokinetics equation:  
\[\frac{dN_{conc}}{dt}=p_{m}D_{r}(t)-\alpha_{2}N_{ext}(N_{conc}),N_{conc}(t_{0} )=0 \tag{9}\]  
where \(p_{m}\) is the conversion factor from \(mg\)\(Kg^{-1}\) to \(moles\), \(D_{r}\) is the rate of drug dosage, \(t_{0}\) is the initial time of delivery, and \(N_{conc}\) is according to Equation (6). When dealing with oral delivery, the gastro-enteric release is assumed to be exponential; hence, for a single dose of operation at \(t=t_{0}\), the equation can be re-written as the following equation:  
\[\frac{dN_{conc}}{dt}=p_{m}D_{dose}\alpha_{1}e^{-\alpha_{1}(t-t_{0})}-\alpha_{2}N _{ext}(N_{conc}),N_{conc}(t_{0})=0 \tag{10}\]  
where \(D_{dose}\) is the dose (in \(mg\)\(Kg^{-1}\)).  
The mathematical model presented in this manuscript considers the negative feedback loop involving p53 and its inhibitor, Mdm2, at the core of the p53 pathway, and uses it to examine the effect of Nutlin3-a to enhance the activity of the p53 protein. Therefore, this study is applicable to those cancer types where the cancer is caused by the inactivation of the p53 protein due to the overexpression of MDM2. The other cancer types that are caused by the mutation of the p53 gene or involve any other signal transduction networks will have completely different mathematical models involving proto-oncogenes as state variables and are outside of the scope of this study.