Draft:ATM-Wip1 (Cancer) Oscillator Model

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

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