Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 3 / 3 / a / Solution

Treat $X\ge 0$ as a prescribed input protein concentration, and $Y(t),Z(t)\ge 0$ as the downstream gene-product concentrations. Define the Hill activation functionA minimal model with no basal expression isHere $a_Y,a_Z>0$ 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.