# 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.

$$\ce{X_1 ->[f_1] X_2}, \quad \quad \quad \ce{X_2 ->[f_2] X_1}$$

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?

• Why not write $\ce{X_1 <=>[f_1][f_2] X_2}$ instead? – Rodrigo de Azevedo Jun 28 '20 at 16:36

First, we need to find balanced equation for reaction. Then express the concentration of $$xˆst$$ in the steady state through E.
$$\displaystyle{E\frac{x_0 - \frac{x_o}{2}}{K + x_0 - \frac{x_0}{2}}} - {\frac{\frac{x_0}{2}}{K + \frac{x_0}{2}}} = 0.$$
$$\displaystyle{E\frac{\frac{x_0}{2}}{K + \frac{x_0}{2}}} - {\frac{\frac{x_0}{2}}{K + \frac{x_0}{2}}}= 0.$$