## 3 _E. coli_\(\sigma\)[70] Promoter Recognition
### Kinetics and Mechanism of Open Complex Formation  
The kinetics of open complex formation in excess RNAP at concentration [R] are found to be single exponential with a rate constant (kobs), which is a hyperbolic function of [R]. These kinetic data are well described by Mechanism 1 (see also Figure 4) with a reversible initial step (with equilibrium constant K1) that includes promoter binding and forms a closed intermediate I1, followed by a rate determining \"isomerization\" step (with rate constant k2) to convert I1 to the stable open complex (RP0) [137,138]. \[ \begin{array}{cc}\text{K}_{1}&\text{k}_{2}\\ \text{R}+\text{P}\rightleftarrows\text{I}_{1}\rightarrow\text{RP}_{0}\end{array} \]  
\[ \begin{array}{cc}\text{Mechanism 1}\\ \text{Mechanism 1}\end{array} \]  
While evidence now exists for an ensemble of closed intermediates I1 and several different intermediate open complexes, Mechanism 1 remains useful for analysis of the kinetics of open complex formation, and the quantities K1 and k2 are interpretable in terms of the species in the I1 ensemble. The observation of single exponential kinetics means that the species in the I1 ensemble rapidly equilibrate with one another and with free P on the time scale of its conversion to RP0 [139]. Analysis of Mechanism 1 for this situation (see Figure 4, bottom) yields:  
\[\text{kobs}=\theta_{\text{I1}}\text{k}_{2} \tag{1}\]  
where \(\theta_{\text{I1}}\) is the fraction of closed promoter DNA present as I1 complexes:  
\[\theta_{\text{I1}}=\frac{[\text{I}_{1}]}{[\text{P}]+[\text{I}_{1}]}=\frac{ \text{K}_{1}[\text{R}]}{1+\text{K}_{1}[\text{R}]} \tag{2}\]  
The interpretation of K1 and k2 depends on the details of the I1 ensemble, as indicated for the particular case of two closed complexes below. Kinetics of Open Complex Formation for \(\dot{\lambda}P_{R}\) and \(\dot{\lambda}P_{R}\) Variants: Interpretation of \(k_{\text{obs}}\)  
As an example relevant for analysis of the single-exponential kinetics of open complex formation by full-length (FL) and upstream truncated (UT) \(\lambda\text{P}_{\text{R}}\) promoters, consider the situation where the I1 ensemble consists of one early closed complex (designated I1,E) and the most advanced closed complex (I1,L). Since the kinetics of RP0 formation are single-exponential, I1,E and I1,L are in rapid equilibrium with one another and with P on the time scale of conversion of I1,L to RP0. For this case Mechanism 1 is rewritten as:  
\[ \begin{array}{cc}\text{K}_{1,\text{E}}&\text{K}_{1,\text{L}}&\text{kopen}\\ \text{R}+\text{P}\rightleftarrows\text{I}_{1,\text{E}}\rightleftarrows\text{I}_ {1,\text{L}}\rightarrow\text{RP}_{0}\end{array} \]where \(k_{open}\) is the rate constant for the elementary, rate-determining DNA opening step involving \(I_{1,L}\). For Mechanism 2, \(k_{obs}\) in excess RNAP is given by Equation (3):  
\[k_{obs}=0_{11L}k_{open} \tag{3}\]  
where \(0_{11L}\) is the fraction of closed promoter DNA present as \(I_{1,L}\) complexes:  
\[0_{11L}=\frac{I_{1,L}}{P+I_{1,E}+I_{1,L}}=\frac{K_{1,E}K_{1,L}[R]}{1+K_{1,E}(1 +K_{1,L})[R]} \tag{4}\]  
Comparison of Equations (1)-(4) shows that the observed isomerization rate constant \(k_{2}\) of Mechanism 1 is interpreted using Mechanism 2 as:  
\[k_{2}=\frac{K_{1,L}}{1+K_{1,L}}k_{open}=f_{11,L}k_{open} \tag{5}\]  
where \(f_{11,L}\) is the fraction of \(I_{1}\) complexes that are \(I_{1,L}\):  
\[f_{11,L}=\frac{K_{1,L}}{1+K_{1,L}}=\frac{[I_{1,L}]}{[I_{1,E}]+[I_{1,L}]} \tag{6}\]  
The observed closed complex binding constant \(K_{1}\) of Mechanism 1 is interpreted using Mechanism 2 as:  
\[K_{1}=K_{1,E}\left(1+K_{1,L}\right)=\frac{[I_{1,E}]+[I_{1,L}]}{[P][R]} \tag{7}\]  
In words, \(K_{1}\) of Mechanism 1 is the overall \(I_{1}\) binding constant written in terms of total \(I_{1}\) concentration.