# User:J. C. Martinez-Garcia/Notebook/HMS Activities/2008/11/13

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## Semi white box model

The last days I worked in the analysis of the second-order model proposed by Eduardo Sontag (Eduardo's Sontag website). He is now in a sabbatical year at MIT (working in the Electrical Engineering Department). The model that he proposed to us, and that I implemented in the computer, is the following one:

• Equation one: dx(t)/dt = f(x(t))+a/(K+k*y(t))
• Second equation: e*dy(t)/dt = -y(t) + x(t)/(f+g*u(t))

where x corresponds to the variable which dynamics drives the bi-estable behavior and y corresponds to the variable involved directly with the expression of the AKT protein. For the function f(x) there exist several possibilities. The first one (implemented in the computer) is a cubic function:

f(x) = Vmax x(t)/(Km+xx(t)) - l x(t),

i.e a Michaelis-Menten function with degradation of x(t). I took, because of certain conditions involving the behavior of s(x):=a/(K+k*yx(t)), the following values for the parameters:

• Km=1;
• K = 1;
• Vmax = 1;
• f = 0.01;
• g = 1;
• e = 1/100; (to guarantee a two scales of time regime).

For the input u(t), this corresponds to the Zstk drug. As far as k is concerned, this is a variable associated to the cut of the feedback acting from the second equation to the first one. In the first version of the computer implementation I chose k to be a logical variable, being just equal to zero when no feedback is considering in this equation. It must be pointed out that the second equation ensures that:

• y(t)=x(t)/(f+g*u(t))) since e y(t)→0 when e is very small.

The question now is how to justify in biological terms the chosen functions. In particular, the choice of f(x(t)) must be justified. Eduardo Sontag is proposing a mechanism not based in cooperativity. I will consider this question now.

## Some words on robustness

I printed some papers by Eduardo SOntag considering robustness measures. These are the following ones:

1. A. Dayarian, M. Chavez, E. D. Sontag, A. M Sengupta (2008): Shape, Size and Robustness: Feasible Regions in the Parameter Space of Biochemical Networks (here is the paper: Media:daryian_chaves_sontag_sengupta_final_oct08.pdf).
2. M. Chavez, A. M. Sengupta, E. D. Sontag (2008): Geometry and topology of parameter space: investigating measures of robustness in regulatory networks (here is the paper: Media:chaves_sengupta_sontag_revised_JMB_23aug08.pdf).

I will read them later and I will cultivate the contact con Sontag (that is important to improve the impact of my research in systems biology from a theoretical point of view).