## 2 Mathematical Modeling of p53 Pathway  
To describe a dynamic system in the control system paradigm, mathematical modeling is carried out, where a dynamic system can be described in terms of differential equations. Such mathematical models for the p53 pathway have been presented in the literature using different methods including the time-delay-based method and stochastic-based method [21].  
Each modeling technique has its own limitations, where the time delay method has to do with real cell proteins and measure the time elapsed during translation and transcription directly. In the stochastic method, rather than taking the instant outcome, the impact of protein levels is quantized, making the system computationally rigorous.  
p53 activates _MDM_2 mRNA by triggering the transcription factor of MDM2. The rate of generation of MDM2 can be controlled by the weighted mean of previous p53 concentration levels. A three-module framework was presented in [38], which would measure the pulse generation of the produced p53, resulting in the breakdown of DNA strands, thus further prompting processes such as DNA reparation, ATM stimulation, and a feedback control loop between p53 and MDM2. Another method of observing the response of p53 on the pathway was presented in [39], which, by generating an input signal, revives p53 and where the subsequent behavior of several upstream processes including ATM, CHK2, and WIPI was observed.  
The system dynamics having negative feedback control loops in the p53 pathway was studied in [23], where it was found that the p53-MDM2 complex had sustained oscillations without damping. The research focused on the response of various stresses on the feedback control loop of the p53-MDM2 complex; moreover, the inhibitor was studied, where it was shown to have spiky oscillations. The model inclusively covers all details of the p53 pathway and is suitable enough to be implemented in control system design.  
The process of drug development is extremely expensive and time consuming. The drug development process can be enhanced to be more cost effective and accurate. The drug dosage design is also a major problem. For the proposed drug, the dynamics associated with p53 are equally as important as its PK/PD characteristics. Researchers have spent a great deal of effort in physiological-based kinetics (PBK) modeling of numerous drugs. A scheme could be deployed where the PBK characteristics of an anti-tumor drug are integrated with the p53 pathway model. Advances in computational power in recent years have enabled further advances of the in silico models and enhanced the development operation. The complete parametrization of the PBK mathematical model presented by [PERSON] et al. has been selected for this research.  
In the current research, the models presented in the literature by [PERSON] and [PERSON] are integrated as shown in Figure 2. The p53 pathway dynamics presented in the Hunziker model describe the positive and negative feedback of the mRNA of p53-MDM2. The [PERSON] model describes the PBK dynamics of Nutlin. In [37], a new term for the anti-tumor drug Nutlin 3a was added to the [PERSON] model and its response investigated.