= Solution
Treat $X\geq0$ as a prescribed input <protein> <concentration>, and $Y(t),Z(t)\geq0$ as the downstream gene-product <concentrations>. Define the <Hill activation function>
$$
h(u;K,n)=\frac{u^n}{K^n+u^n},\qquad K>0,\quad n>0.
$$
A minimal model with no basal expression is
$$
\boxed{\frac{dY}{dt}=a_Yh(X;K_{XY},n_{XY})-Y,\qquad
\frac{dZ}{dt}=a_Zh(X;K_{XZ},n_{XZ})-Z}.
$$
Here $a_Y,a_Z>0$ are maximal production rates, $K_{XY},K_{XZ}$ are <Hill constants> with the same units as $X$, and $n_{XY},n_{XZ}$ are dimensionless <Hill coefficients>. The negative terms describe a <first-order reaction> for loss of each gene product, with both rate constants set to one in the chosen time units. Thus $a_Y,a_Z$ have units of <concentration> per time and the unit loss rates have units of inverse time.
The equations model <regulation of gene expression> at the level of its effective <protein> output, without separate <messenger RNA> dynamics. The independent branches mean that neither $Y$ nor $Z$ regulates the other in this initial model. Additive basal production could be included, but is not required by the question and is set to zero throughout this example.
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