## 3 _E. coli_\(\sigma\)[70] Promoter Recognition
### Kinetics and Mechanism of Open Complex Dissociation  
Dissociation of the stable open complex RP\({}_{\mathrm{O}}\) to free promoter DNA upon addition of a competitor (to make dissociation irreversible) exhibits first order, single-exponential kinetics with rate constant \(\mathrm{k_{d}}\) (Figure 4). For the two promoters investigated, \(\lambda\)Pr [65, 66] and _lac_UV5 [61], both of which form long-lived RP\({}_{\mathrm{O}}\) complexes at 37 \({}^{\circ}\)C, \(\mathrm{k_{d}}\) decreases strongly with increasing temperature. This negative activation energy of dissociation indicates that one or more intermediates are kinetically significant in dissociation, and (together with the single-exponential kinetics) indicates these intermediates are in rapid equilibrium with RP\({}_{\mathrm{O}}\) on the time scale of the rate-determining step in dissociation. These intermediates occur after the \"bottleneck\" step in the forward direction and do not contribute to the rate of RP\({}_{\mathrm{O}}\) formation. How many intermediates are there, and are they open or closed? A combination of kinetic and fast permanganate footprinting studies reveal that at least two such intermediates are kinetically significant at \(\lambda\)Pr (designated \(\mathrm{I_{2}}\) and \(\mathrm{I_{3}}\) in Figures 3 and 4); these are open complexes with the bubble extending from \(-11\) to \(+2\) (like RP\({}_{\mathrm{O}}\)) which are much less stable than RP\({}_{\mathrm{O}}\)[23, 24]. DNA closing occurs in the conversion of \(\mathrm{I_{2}}\) to \(\mathrm{I_{1,L}}\)[23]; this step is rate-determining in dissociation [24], as indicated in Figure 5. Though the barrier for conversion of RP\({}_{\mathrm{O}}\) to \(\mathrm{I_{2}}\) is much larger than that for conversion of \(\mathrm{I_{2}}\) to \(\mathrm{I_{1}}\), the rate constant \(\mathrm{k_{3}}\) for conversion of RP\({}_{\mathrm{O}}\) to \(\mathrm{I_{2}}\) does not determine the rate of dissociation because this step is rapidly reversible on the time scale of conversion of \(\mathrm{I_{2}}\) to \(\mathrm{I_{1}}\). Hence, the equilibrium constant \(\mathrm{K_{3}}\) for conversion of RP\({}_{\mathrm{O}}\) to \(\mathrm{I_{2}}\) and not the rate constant \(\mathrm{k_{3}}\) appears in the equation for \(\mathrm{k_{d}}\) in Figure 4.  
Even though more than one open intermediate is involved in dissociation of RP\({}_{\mathrm{O}}\) at \(\lambda\)Pr, kinetic data are well described by Mechanism 3, involving one intermediate open complex (\(\mathrm{I_{2}}\)):  
\[\frac{1}{\mathrm{K_{3}}}\,\mathrm{k_{-2}}\] \[\mathrm{RP_{O}}\rightleftarrows\mathrm{I_{2}}\rightarrow\mathrm{R +P}\]  
The observation of single exponential dissociation kinetics means that \(\mathrm{I_{2}}\) and any other intermediates like \(\mathrm{I_{3}}\) rapidly equilibrate with RP\({}_{\mathrm{O}}\) on the time scale of conversion of \(\mathrm{I_{2}}\) to closed complexes [139]. In other words, in dissociation of RP\({}_{\mathrm{O}}\) without a high salt upshift (see below), any \(\mathrm{I_{2}}\) formed usually reverts to RP\({}_{\mathrm{O}}\) but occasionally undergoes the rate-determining DNA closing step to form \(\mathrm{I_{1,L}}\), which rapidly dissociates. Analysis of Mechanism 3 relates the observed rate constant k\({}_{\rm d}\) for RP\({}_{\rm O}\) dissociation to the DNA closing rate constant k-2:  
\[\rm{k_{d}=f_{i2}k_{\rm{-2}}} \tag{8}\]  
where f\({}_{\rm 2}\) is the fraction of open complexes that are I\({}_{\rm 2}\):  
\[\rm{f_{i2}=\frac{1}{1+K_{3}}}\ =\ \frac{[I_{2}]}{[I_{2}]+[RP_{O}]} \tag{9}\]  
To date the intermediate open complex I\({}_{\rm 3}\) at \(\lambda\)P\({}_{\rm R}\) has been treated as part of the RP\({}_{\rm O}\) population, as shown in Figure 4, because neither the equilibrium constant nor rate constants for forming I\({}_{\rm 3}\) from RP\({}_{\rm O}\) have been determined [24]. Hence, the equilibrium constant K\({}_{\rm 3}\) in Equation (9) is a composite of those for the conversions of I\({}_{\rm 2}\) to I\({}_{\rm 3}\) and I\({}_{\rm 3}\) to RP\({}_{\rm O}\).