Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 4 / a

For constant positive reaction rates, the integrating factor for the RNA equation givesFor completeness, the protein transient when ${\gamma }_P\ne {\gamma }_R$ isAt equal degradation rates, the last term becomes $k_P[R(0)-R_{\ast }]t{e}^{-{\gamma }_{P}t}$, its continuous limit.

One transcript per minute means $k_R=1{\min }^{-1}$ for a single active gene. A mean RNA lifetime of ten minutes means ${\gamma }_R=0.1{\min }^{-1}$, not a ten-minute half-life. The steady mean is therefore $\boxed{⟨R⟩=10}$ RNA molecules per cell.

For the Fano factors below, $R,P$ denote molecule numbers in a fixed cell. If they instead denote dimensional concentrations, the same deterministic equations hold after rescaling, but the conversion to molecule-number variance introduces the cell-volume factor; variance divided by concentration is not intrinsically a dimensionless count Fano factor.