Table 3. A comparison of the pathway-reporter method and the dual-reporter method for constitutive expression under the model ${\mathbf{𝐌}}_2$.

Here PR (MP) gives the results of the mRNA-protien pathway reporters, while DR (Mat) gives the results of dual reporters calculated from the mature mRNA. We considered noise on the transcription rate ($K_m$), the protein synthesis rate ($K_p$), and the protein decay rate (${\delta }_p$). The mRNA decay rate is fixed at one. In each case, we varied $K_p$ according to a ${\displaystyle \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(5,0.4)}$ distribution and ${\delta }_p$ according to a ${\displaystyle \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(8,0.125)}$ distribution; the corresponding noise strengths are 0.20 and 0.125, respectively. We considered different noise distributions on $K_m$, which produce a range of noise strengths. The noise distribution parameters are selected to produce a mean mRNA of approximately 50 and a mean number of approximately 1000 proteins in each simulation. The values given are the average of 100 simulations, each calculated from 500 copy number samples, and the errors are ± one standard deviation. As our theory predicts, the mRNA-protein reporters identify the noise on the transcription rate parameter $K_m$ (${\eta }_{\mathrm{ext}}^2$).

Theory		Simulation
(r)1-2 ${\eta }_{\mathrm{ext}}^2$	Noise ($K_m$)	Pr (MP)	DR (Mat)
0.00	$K_m=50$	$0.00\pm 0.01$	0.00 ± 0.00
0.10	${\displaystyle {\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}}_{133.\dot{3}}(6,10.5)}$	$0.10\pm 0.01$	0.10 ± 0.01
0.20	${\displaystyle \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(5,10)}$	$0.20\pm 0.02$	$0.20\pm 0.01$
0.50	${\displaystyle {\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}}_{300}(1.5,7.5)}$	$0.51\pm 0.04$	$0.50\pm 0.03$