Transcription of r

The scbR gene is transcribed into scbR mRNA (r).

Chemical equation

$O_{R} \rightarrow O_{R} + r$

Rate equation

$r= T_{R}\cdot [O_{R}]$ (for overlapping promoters) $r= \Omega_{R}\cdot [O_{R}]$ (for isolated promoters)

Parameters

The parameters of this reaction are the transcription rate of scbR ($T_{R}$ and the firing rate $k_{F}$). The transcription rate is derived by the strength of the promoter ($\Omega_{R}$) but also taking into account the transcriptional interference by the scbA promoter. In this model, we have assumed that the isolated promoter strength is equal to the number of transcripts produced per unit of time. Therefore, the parameter $\Omega_{R}$ is assumed to be equal to the transcription rate constant of the isolated promoter. These parameter values were derived from published data on E.coli mRNA transcription rate and calculations based on genomic properties of Streptomyces coelicolor A3(2). Additionally, the firing rate (elongation initiation rate) constant ($k_{F}$) is needed to test the strength of each promoter. This parameter is also derived from literature and is sampled from the same distribution for both scbR and scbA promoters, but is then multiplied by a heterogeneity factor $\chi$ to calculate the final $k_{F_R}$ for the scbR promoter. The heterogeneity factor for each promoter is sampled from a log-normal distribution.

Name	Value	Units	Value in previous GBL models	Remarks-Reference
$\Omega_{R}$	$0.16−8.24$	$min^{-1}$	$0.18 s^{-1}$ $(10.8 min^{-1})$ $0.15 s^{-1} (9 min^{-1})$ Range tested: $10^{-4}-10 s^{-1}$ ($0.006-600 min^{-1}$) Bistability range: $0.13-0.2 s^{-1}$ ($7.8-12 min^{-1}$) and $0.0075-0.3 s^{-1}$ ($0.45-18 min^{-1}$)	In a recent publication by R.A. Cox, genomic properties and macromolecular compositions of Streptomyces coelicolor A3(2) and E.coli were reported, along with equations that connect these properties. For S. coelicolor, the polypeptide elongation rate $ε_{aa}$ is reported to be in the range between 0.59 and 3.17 amino acids s-1, from which the mRNA elongation rate can be calculated according to Cox from the equation $ε_{mRNA}= 3ε_{aa}$ (factor 3 reflects the number of nucleotides per codon), therefore resulting in $1.77$ and $9.51$ nucleotides $\cdot s^{-1}$ as minimum and maximum values respectively. As ScbR has 648 pb, the transcription rate constant can be calculated as per $\Omega_{R}=\frac{648 bp/gene}{1.77 bp/s}=366.1 s/gene=6.1 min/gene$ and $\Omega_{R}=\frac{648 bp/gene}{9.51 bp/s}=68.14 s/gene=1.14 min/gene$, thus resulting in the final values of $0.16$ and $0.88 min^{-1}$. Cox et al. 2004 Cox et al. 2004 Additionally, Bremer et al. have reported an mRNA transcription rate of 55 noucleotides/s for E. coli, a value which is also shared by R.A. Cox, while Vogel et al. have published a range of mRNA transcription rates in the range of 28-89 noucleotides/s, depending on different growth rates of E. coli. By the same calculations, the corresponding transcription rate constants are $5.09 min^{-1}$ and $2.59-8.24 min^{-1}$. Bremer et al. 2004 Vogel et al. 2004
$k_{F}$	$18.2−33$	$min^{-1}$	N/A	Pai et al. reported a typical transcription initiation rate in QS systems to be $20 min^{-1}$. Pai et al. 2009 This value is also supported by Kennell et al. who calculated the transcription initiation rates from experimental data derived from in vitro experiments using E. coli. The results showed one initiation every 3.3 sec (therefore transcription rate $18.2 min^{-1}$). Kennell et al. 1977 Finally, Tadmor et al. reported a maximum transcription initiation rate of $33 min^{-1}$ in E. coli based on observational data. Tadmor et al. 2008

In order to include the strength of the promoter in the transcription rate, the following formula is used: $\frac{1}{k_{onR}}=\frac{1}{\Omega_{R}}-\frac{1}{k_{F_R}}$ . In this way, a distribution with the values for parameter $k_{onR}$ (rate of RNA polymerase binding the promoter) is obtained, which is necessary to link the firing rate with the total transcription rate according to the formulas suggested by Bendtsen et al. . From the $k_{F_{R}}$ and the $k_{onR}$, the maximal occupancy for the isolated scbR promoter can be calculated by using the formula: $\theta^o_{R}=\frac{k_{onR}}{k_{onR}+k_{F_{R}}}$. The promoter aspect ratio $\alpha_{R}$ is equal to $\frac{k_{onR}}{k_{F_{R}}}$, which leads to the equivalent equation $\alpha_{R}=\frac{\theta^o_{R}}{1-\theta^o_{R}}$. The total transcription rate in the case of the isolated (uncoupled) promoters is calculated through the formula: $\Omega_{R}=k_{F_{R}} \cdot \theta^o_{R}$. Similarly, the maximal occupancy of the isolated scbA promoter ($\theta^o_{A}$)is calculated as described in the section Transcription of a. In the case of the overlapping (coupled) promoters the maximal occupancy for the scbR promoter is $\theta_{R}=\frac{\alpha_{R}+1}{\alpha_{R}+\alpha_{A}+1}$, where $\alpha_{A}$ is the promoter aspect ratio for the isolated scbA promoter. Therefore, the final transcription rate constant is calculated as per $T_{R}=k_{F_{R}} \cdot \theta_{R} \cdot \theta^{o}_{R} =\Omega_{R} \cdot \theta_{R}$.

Parameters with uncertainty

When deciding how to describe the uncertainty for this parameter we must take into consideration that the reported values are either calculated or derived with approximation from experiments and from other macromolecular properties. Additionally, some of the values correspond to mRNA transcription rates of different bacteria species (E. coli). This means that there might be a notable difference between actual parameter values and the ones reported in literature. These facts influence the quantification of the parameter uncertainty and therefore the shape of the corresponding distributions. Therefore, by assigning the appropriate weights to the parameter values and using the method described here, the appropriate probability distributions were designed.

Although the weight of the distribution is put on the values calculated for S. coelicolor by setting $0.8346 min^{-1}$ as the mode of the log-normal distribution for the $\Omega_{R}$, we wish to explore the full range of reported values. Thus, the Spread is set to $3.55$ and the range where 68.27% of the values are found is between $0.235$ and $2.96$ $min^{-1}$.

With regards to the firing rate $k_F$, the reported values are within the range of $18-33 min^{-1}$ with the most probable values being $18-20 min^{-1}$. Since these values are reported as being the average rates (and $33 min^{-1}$ being the maximum), we will also sample lower values, so the final sampling range will be around the values $10-33 min^{-1}$. The mode of the distribution is set to $20.7 min^{-1}$ and the Spread is set to $1.34$. Therefore, the range where 68.27% of the values are found is between $15.4$ and $27.8$ $min^{-1}$.

From the literature values of $\Omega_{R}$ and $k_F$, the values for $k_{onR}$ are calculated. By assigning the same weights to each of the $k_{onR}$ values as of the $\Omega_{R}$ values, a distribution with a mode of $0.867 min^{-1}$ and a Spread of $4.1$ is generated. The range where 68.27% of the values are found is between $0.212$ and $3.546 min^{-1}$.

Finally, the heterogeneity factor can be set so that $k_{F_R}>k_{F_A}$ or $k_{F_R}<k_{F_A}$. In the first case, $\chi$ has values within the range 1-10 and in the second within the range 0.1-0.9. In order for the two promoters to have equal strength, $\chi$ is set to be equal to 1. Therefore, the mode of the heterogeneity factor $\chi$ is $3$ and the Spread is $1.2$ (68.27% of the values are found between $2.5$ and $3.6$), in the case where scbR promoter is stronger. In the opposite case the mode is $0.4$ and the Spread is $1.3$ (68.27% of the values are found between $0.3$ and $0.52$).

The probability distributions for the parameters, adjusted accordingly in order to reflect the above values, are the following:

The values retrieved from literature and their weights are indicated by the blue dashed lines, and the uncertainty for each value is indicated using the reported experimental error (green lines) or a default value of 10% error (orange lines).

The parameter information of the distributions is:

Parameter	Mode	Spread	μ	σ
$\Omega_{R}$	$0.8346$	$3.55$	$0.63056$	$0.90076$
$k_{onR}$	$0.867$	$4.09$	$0.78694$	$0.96428$
$k_{F}$	$20.7$	$1.34$	$3.1107$	$0.28276$
$\chi$ ($k_{F_R}>k_{F_A}$)	$3$	$1.2$	$1.1308$	$0.1794$