## 3 _E. coli_\(\sigma\)[70] Promoter Recognition
### Kinetics and Mechanism of Open Complex Formation
If the mechanism is more complicated than Mechanism 2, with additional on-pathway closed complexes in rapid equilibrium with one another on the time scale of DNA opening, Equations (5) and (7) increase in complexity, but the general principle still applies that the observed isomerization rate constant \(k_{2}\) is the product of \(k_{open}\) and the fraction of closed complexes that are \(I_{1,L}\). Only if all closed complexes are \(I_{1,L}\) (_i.e._, \(K_{1,L}>>1\)) does \(k_{2}=k_{open}\); in this case \(K_{1}=K_{1,E}K_{1,L}\) for Mechanism 2. Here we apply Mechanism 2 to interpret the large differences in \(K_{1}\) and especially in \(k_{2}\) for open complex formation by full length (FL) and upstream-truncated UT-47 \(\lambda P_{R}\) promoters at 37 \({}^{\circ}C\)[75]. Isomerization is much faster for FL \(\lambda P_{R}\) (\(k_{2}=0.66\) s\({}^{-1}\)) than for UT-47 \(\lambda P_{R}\) (\(k_{2}=0.03\) s\({}^{-1}\)), but closed complex binding is weaker (\(K_{1}=5.8\times 10^{6}\) M\({}^{-1}\) for FL; \(K_{1}=2.6\times 10^{7}\) M\({}^{-1}\) for UT-47.) For FL \(\lambda P_{R}\), low temperature equilibrium OH footprinting [59] and fast OH footprinting [140,141] reveal that the population of closed \(I_{1}\) intermediates is advanced with the DNA significantly protected to +20. Hence the observed isomerization rate constant \(k_{2}\) is probably not much less than the DNA opening rate constant \(k_{open}\). Here, for purposes of illustration, we estimate \(k_{open}\) to be 1 s\({}^{-1}\) at 37 \({}^{\circ}C\). (The maximum rate of transcription initiation at 37 \({}^{\circ}C\) is known to be \(\sim\) 1 s\({}^{-1}\)[61,62], indicating that \(k_{open}\geq 1\) s\({}^{-1}\).) From \(k_{2}/k_{open}=0.66\) and Equation (5), \(K_{1,L}=2\) so 2/3 of closed complexes are \(I_{1,L}\), consistent with the footprinting data on the \(I_{1}\) ensemble. From \(K_{1}=5.8\times 10^{6}\) M\({}^{-1}\) and the interpretation of \(k_{2}\) in Equation (5) and Mechanism 2, \(K_{1,E}=K_{1}/3=1.9\times 10^{6}\) M\({}^{-1}\). For FL \(\lambda P_{R}\) a plausible free energy _vs._ progress diagram for the steps of open complex formation is given in Figure 5A, which is drawn to scale for \([R]=30\) nM, corresponding to \(K_{1,E}[R]=0.06\) and a correspondingly small occupancy of \(I_{1,E}\) at 30 nM RNAP. The barrier height for the rate determining opening step (rate constant \(k_{open}=1\) s\({}^{-1}\)) is set for (\(I_{1,L}\)-\(I_{2}\))\({}^{\ddagger}\) transition state decomposition frequencies of \(10^{3}\) s\({}^{-1}\) in each direction [128]. For the UT-47 \(\lambda P_{R}\) variant, in which the distal UP element and far upstream DNA are missing, \(k_{2}\) is only 3% of the FL \(\lambda P_{R}\) value [75], even though there is no obvious reason for upstream truncation to affect the intrinsic opening rate constant kopen, and the closed complex binding constant K1 is 4.5 fold larger than for FL \(\lambda\)Pr, even though favorable distal UP element interactions are eliminated by truncation at -47. Should the large reduction in k2 for UT-47 be interpreted as a reduction in kopen or as a shift in the closed complex ensemble to less advanced species? Footprinting evidence supports the latter interpretation: in UT-47 I1 ensemble, the downstream boundary of DNase protection is at +2/+7, indicating that the downstream duplex is only partially bent into the cleft. Hence, the absence of far-upstream DNA results in a less advanced closed complex ensemble for UT-47 \(\lambda\)Pr. At 37 \({}^{\circ}\)C, interpretation of k2 = 0.02 s\({}^{-1}\) for UT-47 \(\lambda\)Pr[75] in terms of Mechanism 2 with kopen = 1 s\({}^{-1}\) indicates that K1,L \(\sim\) 0.02, indicating that only 2% of the population of closed complexes are I1,L and 98% are I1,E, very different from the 66% I1,L, 34% I1,E distribution for FL \(\lambda\)Pr. From K1 = 2.6 \(\times\) 10\({}^{7}\) M\({}^{-1}\)[75] and this interpretation of k2, K1,E = K1/1.02 = 2.5 \(\times\) 10\({}^{7}\) M\({}^{-1}\), 13-fold larger than that calculated for FL \(\lambda\)Pr.