Using Stable Limit Cycles to Model p53-Mdm2 Protein Interactions in the Presence of DNA Damage
The protein p53 is constantly made and destroyed within a cell. Another protein, Mdm2, is a negative regulator of p53. In the presence of DNA damage, Mdm2 does not bind to p53, thus activating the p53 protein. This activation then promotes the mechanisms necessary to either fix or terminate the damaged cell. The concentration of both p53 and Mdm2 in a cell oscillate, referred to as a negative feedback loop. Blocking this loop, therefore preventing the binding of these two proteins, as a method to reactivate the p53 function is a promising cancer therapeutic strategy. To study the oscillatory behavior of p53 and Mdm2, several mathematical models have been developed. These models often only show oscillations for narrow ranges in parameter values and only oscillate for short periods of time before eventually reaching some steady state. We propose a new model of the p53-Mdm2 interaction which exhibits periodic solutions resulting from stable limit cycles. We establish conditions for existence and stability of these limit cycles, and we validate this model using four data sets of the p53-Mdm2 interaction. To improve model performance, we apply a two-step method where Fourier series are used to estimate frequency dependent variations in the existing data. The estimated parameter values indicate two distinct behaviors in the data sets – one with a closed negative feedback loop and the other with an open feedback look in the absence of the Hill function. In conclusion, the proposed model is capable of detecting stable limit cycles and underlying mechanisms governing p54-Mdm2 interactions.
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