Table 3.

Definition of parameters and functions

$f(i,j)={\displaystyle \frac{{\alpha }^{i+j-1}}{{(2\alpha -1)}^i}}+{\displaystyle \frac{1}{2^{i+j-1}}}\left(\begin{array}{c}i+j-1\\ i\end{array}\right)[1-{\displaystyle \frac{2\alpha -1}{2\alpha }}{}_2{F}_1(1,i+j;j;\frac{1}{2\alpha })]$,
$g^{\mathit{aa}}(i,j)={\displaystyle \frac{\tilde{\alpha }{}^{i+j-1}}{{(2\tilde{\alpha }-1)}^i}}+{\displaystyle \frac{1}{2^{i+j-1}}}\left(\begin{array}{c}i+j-1\\ i\end{array}\right)[1-{\displaystyle \frac{2\tilde{\alpha }-1}{2\tilde{\alpha }}}{}_2{F}_1(1,i+j;j;\frac{1}{2\tilde{\alpha }})]$,
$\mathrm{\Xi }=2(T_g+T_m)+{\text{e}}^{T_g}[2(T_g-1)T_m+(T_g-2)T_g]$
$\eta =s_u/(s_u+s_b)$	Fraction of time the gene spends in the active state
${\rho }_k=r/k$	Mean number of bound RNAPs in the time 1/k
$\rho =r/d_m$	Mean number of bound RNAPs in the time $1/d_m$
$\mu =k/(k+d)$	Local RNAP survival probability (no-pausing case)
${\tau }_p=1/(d+k)$	Timescale of fluctuations of RNAP
${\tau }_g=1/(s_u+s_b)$	Timescale of fluctuations of gene
${\tau }_d=1/d$	Timescale of RNAP detachment
${\tau }_m=1/d_m$	Timescales of fluctuations of mature RNA
$\alpha =1/(1+{\tau }_p/{\tau }_g)$	Non-dimensional parameter
$\gamma =1/(1+{\tau }_p/{\tau }_m)$	Non-dimensional parameter
$\theta =1/(1+{\tau }_m/{\tau }_g)$	Non-dimensional parameter
$\beta =s_b/s_u$	Ratio of gene inactivation and activation rates
$b=r/s_b$	Mean burst size
${\upsilon }_k=s_u/k$	Ratio of gene activation and RNAP elongation rates
${\upsilon }_m=s_u/d_m$	Ratio of gene activation and mature RNA degradation rates
${\mathrm{\delta }}_g={\tau }_g/{\tau }_d$	Ratio of gene timescale and RNAP detachment timescale
$T_g=⟨T⟩/{\tau }_g$	Ratio of elongation timescale and gene timescale
$T_d=⟨T⟩/{\tau }_d$	Ratio of elongation timescale and RNAP detachment timescale
$T_r=r⟨T⟩$	Ratio of the mean elongation time to the timescale of initiation
$T_m=d_m⟨T⟩$	Ratio of the mean elongation time to the timescale of decay of mature RNA
$T_{s_b}=s_b⟨T⟩$	Ratio of the mean elongation time to the timescale of gene deactivation
$\sigma =r_p/r_a$	Ratio of the pausing and activation rates
${\pi }_{r_a}=r_a/(r_a+d_p)$	Probability of RNAP activation
${\pi }_{d_p}=d_p/(r_a+d_p)$	Probability of premature RNAP detachment from the paused state
$\lambda =\sigma {\pi }_{r_a}$	Probability of RNAP pausing from active state
$\tilde{\mu }=k/(k+d_a+r_p{\pi }_{d_p})$	Local RNAP survival probability (in pausing case)
${\tau }_{r_a}=1/r_a$	Timescale of RNAP activation from the paused state
${\tau }_{d_p}=1/d_p$	Timescale of premature termination of paused RNAP
${\tau }_p=1/(k+d_a)$	Typical time that an actively moving RNAP spends on a gene segment
${\tau }_{\mathit{pp}}=1/(r_a+d_p)$	Typical time spent in the paused state
${\lambda }_{r_p}={\pi }_{r_p}/(1-{\pi }_{r_p})$	Ratio of active RNAP timescale over RNAP pausing timescale
${\pi }_{r_p}=r_p/(r_p+k+d_a)$	Probability of the actively moving RNAP switching to the paused state
${\omega }_{r_a}={\pi }_{r_a}{\tau }_g/({\pi }_{r_a}{\tau }_{r_a}+{\tau }_g)$	Non-dimensional parameter
$\tilde{\alpha }=({\tau }_g+{\lambda }_{r_p}{\pi }_{d_p}{\tau }_g)/({\tau }_g+{\tau }_p+{\lambda }_{r_p}{\tau }_g(1-{\omega }_{r_a}))$	Non-dimensional parameter.