Draft:ATM-Wip1 (Cancer) Oscillator Model

Introduction

The ATM-(p53)-Wip1 oscillator is a parametric relaxation type of oscillator model with polynomial and biologically-interpretable terms introduced to model the topological structure constructed by bistable dynamics of ATM protein and a long negative feedback loop mediated by Wip1 (the product of PPM1D gene) protein. The ATM-Wip1 topological structure resembles the frustrated bistable unit that is known to produce oscillations as well as other modes of dynamics.

Model Equations

$dx/dt=-x(rx^{2}-r(a+b)x+ab+cy-rd)/\tau _{1}$

$dy/dt=(z+mx-ny)/\tau _{2}$

where

• $x$ represents ATM protein concentration levels in arbitrary units.
• $y$ represents Wip1 protein concentration levels in arbitrary units.

p53 protein (concentration) levels are not introduced as a dynamical variable; instead, it is assumed to follow ATM dynamics proportionally and algebraically under the quasy-steady state assumption for p53-Mdm2 interaction.

The constant parameters of the model are $a,b,c,d,n,z,\tau _{1},\tau _{2}$ and they take only positive values. The control parameters are $r$ and $m$. $r$ represents the damage severity and is restricted to $0<r<1$, with 0 representing no damage and 1 representing the most severe situation. $m$ represents inversely the duration of the DNA repair and is controlled by proapoptotic genes via mechanisms similar to the "death by integration". Normally, $m$ stays at a high value indicated as $m_{0}$ and decreased towards 0 as the repair continues .