# FNR Promotor

To let the bacteria sense low oxygen concentrations, a FNR promoter was used in this research. The promoter that was chosen is not a default one, since it is a so called tandem promoter. That is, it contains two FNR binding sites. The kinetics of this promoter are therefore different than the promoter previously described in the Decoy Sites model. Here the necessary adaptations of the Decoy Model to account for this special promoter will be described.

## Promoter Design

The promoter that was used in this research was taken from research by Barnard et al.BusbyFNRAnne M.L. Barnard, Jeffrey Green, and Stephen J.W. Busby, Transcription regulation by tandem-bound FNR at Escherichia coli promoters. Journal of bacteriology 185.20, 5993-6004 (2003). The research showed the promoter (FF(-91.5)FF(-41.5)) expresses an 1.8 times higher activity than the most commonly used single binding site FNR promoter (FF(-41.5)). From hereon the tandem FF(-91.5)FF(-41.5) promoter will be referred to as the FNR-tandem promoter and default FF(-41.5) will be called FNR-single promoter. The global design of the FNR-single and FNR-tandem promoter are displayed below:

                  -130      -120      -110      -100       -90       -80       -70       -60       -50       -40       -30       -20       -10        0
|         |         |         |         |         |         |         |         |         |         |         |         |         |
(FNR-single)                                                        **********************************TTGAT****ATCAA**********************CATAAT*******
(FNR-tandem)      **********************************TTGAT****ATCAA************************************TTGAT****ATCAA**********************CATAAT*******
(consensus sequences)                               ______________                                    ______________


The FNR consensus sequences contain two binding sites, each for one monomer of the FNR dimer.

## The Model

When converting the knowledge about the FNR-tandem promoter to a model, it had to be kept in mind that there is very few measurement data of the promoter available. The only data that can be acquired are the binding and dissociation ratios of FNR to one consensus site and the relative activity of the FNR-tandem promoter with respect to the FNR-single promoter. The relative activity of the FNR-tandem promoter suggests that there is minimal hindrance between the two bound FNR proteins, but there also is no significant synergy. Using this information the tandem characteristics of the promoter were incorporated into the model. Hereto the following reactions were added:

\eqalignno{\ce{T + P\, &<=>[\text{Kon}][\text{Koff}] TP} & (1) \\ \ce{P + T\, &<=>[\text{Kon}][\text{Koff}] PT} & (2) \\ \ce{TP + T\, &<=>[\text{Kon}][\text{Koff}] TPT} & (3) \\ \ce{T + PT\, &<=>[\text{Kon}][\text{Koff}] TPT} & (4) }
ComplexPromotorReactions Reactions of the FNR-tandem promoter
The promoter now has three different bound states: FNR bound to the first FNR consensus site of the promoter (TP), FNR bound to the second site (PT) and FNR bound to both sites (TPT). Since the relative activity of the promoter suggests that there is minimal hindrance occurring when a second FNR proteins is about to bind to the promoter, the ratios of all four reactions described in are assumed to be equal. This is a rough estimation, but the assumption is necessary due to the lack of research data.

## Parameter Estimation

To estimate the Kon and Koff rates of a FNR consensus site, and thus - as described before - of the individual consensus sites in the FNR-tandem promoter. The data used for the parameter estimation of the two rates was acquired from research by Lazazzera et al.LazazzeraFNRB.A. Lazazzera, H. Beinert, N. Khoroshilova, M.C. Kennedy and P.J. Kiley, DNA Binding and Dimerization of the Fe–S-containing FNR Protein from Escherichia coli Are Regulated by Oxygen. The Journal OF Biological Chemistry 271, 2762-2768 (1996) (Fig. 4 of the article). The model was adapted to include promoters with just one consensus site (i.e. no PT or TPT states), as used in the paper. Furthermore, the right promoter concentration was set to 5 nM, which was the concentration used in assay. Hereafter, the values of Kon and Koff were optimized to minimize the error of the model output with respect to the measurement data. The optimal values of Kon and Koff were determined to be 0.0993 $M^{-1} s^{-1}$ and 0.00985 $s^{-1}$ respectively.

The next step is to estimate the activity of the promoter. Usually the activity is assumed to be proportional to the amount of bound promoter normalized to the total amount of promoter. In this case however, there are multiple bound states of the promoter that have to be taken into account, as shown in the equations of .

\begin{aligned} \text{Activity} &\sim {\text{[Bound Promoter]}\over{\text{[Total Promoter]}}} \\ \text{Activity} &\sim {{f_{TP} * TP + f_{PT} * PT + f_{TPT} * TPT}\over{TP + PT + TPT + P}} &\bigwedge &f_{TP} + f_{PT} + f_{TPT} = 1\\ \end{aligned}
ExpressionEquations Equations describing the calculation of the activity of the FNR-tandem promoter
The FNR-tandem promoter has a higher activity than the FNR-single promoter because the upstream bound FNR facilitates the binding of RNA Polymerase to the DNA when the downstream site is also occupied by FNR. However when FNR is only bound to the upstream site there is no significant activity of the promoter. Therefore, the PT state is not taken into account when calculating the activity. The TPT state has a higher activity than the TP state, because of the contribution of the upstream bound FNR. Since all activity values are relative, the contribution of the TP state is set to 1 (same as when calculating the activity of the FNR-single promoter) and the contribution of the TPT state will be estimated using the data that the FNR-tandem promoter has an 1.8 times higher activity than the FNR-single promoter. Parameter estimation shows that to reach this activity, the TPT should be 1.67 times as active as the TP state. After normalization of the activity factor of TP becomes 0.375 and the activity factor of TPT 0.625, resulting in the following equations.
\begin{aligned} \text{Activity} &\sim {{0.375 * TP + 0.625 * TPT}\over{TP + PT + TPT + P}} \\ \end{aligned}
ExpressionEquationsFinal Equations describing the calculation of the activity of the FNR-tandem promoter

## References

The source code of all models can be found here.