Finding the sensitivity of zero-order ultra sensitivity and its parameter dependence I am provided a one-dimensional reaction system showing zero-order ultra sensitivity.
$$\ce{X_1 -&gt;[f_1] X_2}, \quad \quad \quad \ce{X_2 -&gt;[f_2] X_1}$$
The reaction rate function is the reduced enzyme reaction according to Michaelis-Menten's law.
$$f_1(x) = E\left(\frac{x_1}{K_1+x_1}\right)$$
$$f_2(x) = \left(\frac{x_2}{K_2+x_2}\right)$$
$$K_1=K_2=K$$
$$x_1(t) + x_2(t) = x_0$$
where $x_0$ is a constant. Let the steady state of $x_2$ be denoted by $x_2^{\text{st}}$. I need to find the value of $E$ that satisfies
$$x_2^{\text{st}}(E) = \frac{x_0}{2}$$
and the logarithmic sensitivity. I tried to look into some published papers/textbooks but could not find similar problems. Can anyone give some suggestions on directions to look into?
I found the answer.
First, we need to find balanced equation for reaction. Then express the concentration of $xˆst$ in the steady state through E.
Then we can arrive to this kind of solution for the value of E.
$\displaystyle{E\frac{x_0 - \frac{x_o}{2}}{K + x_0 - \frac{x_0}{2}}} - {\frac{\frac{x_0}{2}}{K + \frac{x_0}{2}}} = 0.$
$\displaystyle{E\frac{\frac{x_0}{2}}{K + \frac{x_0}{2}}} - {\frac{\frac{x_0}{2}}{K + \frac{x_0}{2}}}= 0. $
E = 1