Solute Probes of Conformational Changes in Open Complex Formation by E. coli RNA Polymerase at the λP_R Promoter: Evidence for Unmasking of the Active Site in the Isomerization Step and for Large-Scale Coupled Folding in the Subsequent Conversion to RP_o

Abstract

Transcription initiation is a multi-step process involving a series of requisite conformational changes in RNA polymerase (R) and promoter DNA (P) that create the open complex (RP_o). Here we use the small solutes urea and glycine betaine (GB) to probe the extent and type of surface area changes in the formation of RP_o between Eσ⁷⁰ RNA polymerase and λP_R promoter DNA. Effects of urea quantitatively reflect changes in amide surface and are particularly well suited to detect coupled protein folding events. GB provides a qualitative probe for the exposure or burial of anionic surface. Kinetics of formation and dissociation of RP_o reveal strikingly large effects of the solutes on the final steps of RP_o formation: urea dramatically increases the dissociation rate constant k_d, whereas GB decreases the rate of dissociation. Formation of the first kinetically significant intermediate I₁ is disfavored in urea, and moderately favored by GB. GB slows the rate-determining step that converts I₁ to the second kinetically significant intermediate I₂; urea has no effect on this step. The most direct interpretation of these data is that recognition of promoter DNA in I₁ involves only limited conformational changes. Notably the data support the following hypotheses: 1) the negatively charged N-terminal domain of σ⁷⁰ remains bound in the “jaws” of polymerase in I₁; 2) the subsequent rate-determining isomerization step involves ejecting this domain from the jaws, thereby unmasking the active site; and 3) final conversion to RP_o involves coupled folding of the mobile downstream clamp of polymerase.

Kinetic studies of the process of open complex (RP_o) formation by E. coli RNA polymerase holoenzyme (R: catalytically competent core enzyme, α₂ββ′ω bound to promoter DNA-recognizing σ⁷⁰ subunit, also abbreviated Eσ⁷⁰) at the λP_R promoter (P) have demonstrated that a minimum of three steps, involving two kinetically-significant intermediates (I₁ and I₂), are required to describe the mechanism (1-4):

$$\begin{array}{ccccccc}\text{R}+\text{P}& \begin{array}{c}\begin{array}{c}\text{rapid}\\ \text{equilibrium}\\ {\text{k}}_1\end{array}\\ \rightleftarrows \\ {\text{k}}_{-1}\end{array}& {\text{I}}_1& \begin{array}{c}\begin{array}{c}\text{slow}\\ {\text{k}}_2\end{array}\\ \rightleftarrows \\ \begin{array}{c}{\text{k}}_{-2}\\ \text{slow}\end{array}\end{array}& {\text{I}}_2& \begin{array}{c}{\text{k}}_3\\ \rightleftarrows \\ \begin{array}{c}{\text{k}}_{-3}\\ \text{rapid}\\ \text{equilibrium}\end{array}\end{array}& {\text{RP}}_{\text{o}}\end{array}$$ (Mechanism I)

Evidence for at least two kinetically significant intermediates has also been found for open complex formation at the lacUV5 (5), T7A1 (6-9) and λP_RM up 1 promoters (10). Although the structures of these intermediates (I₁, I₂) may be promoter-dependent (e.g. the extent to which DNA from -5 to +25 (where +1 is the start site) is protected by the “jaws” of RNAP in I₁, cf. (9, 11-13)), all proposed mechanisms for RP_o formation at σ⁷⁰ promoters invoke a critical slow (rate-determining) isomerization step after formation of a first kinetically significant intermediate, I₁. For λP_R, this slow step follows a sequence of rapidly equilibrating steps that form I₁ from free polymerase and promoter DNA. I₁ and all previous complexes are unstable to a 10 s challenge with competitor, whereas I₂ and RP_o (competitor-resistant (CR) complexes) survive this challenge.

For λP_R, the interconversion between intermediates I₁ and I₂ is the rate-limiting step in both the forward and back directions (1, 3, 4). In the forward direction, k₂ increases strongly with increasing temperature. The dependence of k₂ on temperature corresponds to a large, temperature-independent activation enthalpy (34 ± 2 kcal mol^-1) proposed to reflect protein rearrangements that nucleate DNA melting (2, 3, 11). In the back direction, k_-2 is the slow step following the RP_o ⇄ I₂ rapid equilibrium, yielding the composite dissociation rate constant k_d = k_-2/(1+K₃). The observed reduction in k_d with increasing temperature demonstrates the existence of the second intermediate I₂. Both the magnitude of activation enthalpy (∼30 kcal mol^-1 at ∼15 °C) and the heat capacity change of dissociation indicate that large conformational changes also occur in the final steps, presumably including the propagation of DNA melting (3).

What is the correspondence between these data and movements in the polymerase machinery that occur in DNA opening? The combination of kinetic, biochemical, and genetic studies of Eσ⁷⁰ and high resolution structural descriptions of the homologous thermophilic enzymes (14, 15) has led to distinct hypotheses for the sequence of conformational changes that form RP_o (including DNA bending, kinking, unwinding and/or unstacking; protein folding, unfolding and/or hinge bending) (2, 14, 16). To date, direct evidence exists for only one large-scale protein rearrangement in the process of bacterial RP_o formation: movement of the single-stranded DNA mimic region 1.1 of σ⁷⁰ out of the DNA binding channel (17). Smaller-scale conformational changes are deduced from a comparison of crystal structures of the free T. aquaticus holoenzyme and bound to a “forked” promoter DNA fragment (double-stranded from -41 to -12; single-stranded overhang of the nontemplate strand from -11 to −7). On binding, the β′ “pincer” or “clamp” moves ∼3 Å and the position of the β flap domain (bound to σ⁷⁰ region 4) shifts ∼6 Å (15, 18). These relatively small shifts contrast with the large flexibility inherent in the multidomain structures of free σ⁷⁰ and core. For example, conserved regions 2 and 4 of σ⁷⁰ move some 15 Å apart on binding core (19) and a comparison of EM and crystal structures suggests that the β/β ′ jaws of core polymerase can “flex” from ∼25 to 50 Å apart (20). Is this flexibility exploited during open complex formation, and when?

Mapping structural transitions onto individual steps of RP_o formation is challenging due to the intrinsic instability of kinetic intermediates, and the difficulty in obtaining a relatively homogenous population of a given intermediate for study. In favorable cases, homogeneous populations of intermediates have been obtained by forming complexes at (or downshifting to) low temperature (cf. (5, 11, 21-23)). For example, early complexes (I₁ or earlier species, presumably on the pathway to forming RP_o) have been characterized at binding equilibrium at low temperature, which slows or prevents their conversion to later intermediates (cf. (11, 13, 21, 24, 25)), and at short times of reaction (9, 10, 22). Attempts to characterize I₂ at λP_R by KMnO₄ footprinting after temperature-downshifts (11, 23, 26) appear to be complicated by the fact that the intrinsic KMnO₄ reactivity of the thymine bases in the open region of RP_o may be strongly temperature dependent (27); M. Capp, unpublished). To date neither the amount nor the positions of DNA opening in I₂ have been unambiguously determined by downshift experiments. Outside of demonstrations of its existence, I₂ (and other possible intermediates subsequent to it in rapid equilibrium on the time scale of I₂ → (I₁-I₂)^‡) remain virtually uncharacterized.

We have previously used temperature effects as a signature of conformational changes in the individual steps of open-complex formation, finding that both the equilibrium constant (K₁) for formation of the first intermediate I₁ (R + P ⇄ I₁) and the rate constant (k_d) for dissociation of RP_o (RP_o ⇄ I₂ → I₁) exhibit large temperature dependences characterized by large-magnitude heat capacity changes (ΔC_p^o) (2, 3, 28). A universal contribution to ΔC_p^o in protein processes arises from changes in the amounts of water-accessible nonpolar and polar surface area (ΔASA) (29-32). For protein folding, protein-protein interactions with coupled folding, and other processes where the surface buried is largely nonpolar, changes in ASA are quantitatively predicted by the observed negative ΔC_p^o (cf. (29)).

For formation of protein-nucleic acid interfaces, changes in polar and charged surface are typically as large or larger than changes in nonpolar ASA. As a result, the net contribution to ΔC_p^o from formation of the interface is predicted to be small in magnitude as a result of compensation between contributions from the burial of nonpolar and polar ASA. However, large negative ΔC_p^o characterize most protein-nucleic acid interactions studied to date. In some cases, this thermodynamic behavior signifies protein folding coupled to binding (29). However, other coupled processes characterized by temperature-dependent enthalpies are also found to significantly contribute to a negative ΔC_p^o. In particular, large negative ΔC_p^o accompany protein-nucleic acid interactions involving DNA base unstacking (33), protonation (34), or protein surface salt bridge disruption (35). These ΔC_p^o are not proportional to the corresponding changes in ASA. Since all of these coupled processes may be involved in the steps of RP_o formation, other probes of large-scale coupled folding and other coupled processes are needed.

In this study, we exploit the small biochemical solutes urea and glycine betaine (N,N,N-trimethyl glycine; GB) as probes of changes in polar amide (nitrogen, oxygen in amide groups) and anionic (carboxylate, phosphate) biopolymer surface, respectively. Combining quantitative data for the effects of urea on the unfolding of globular proteins (32) and α-helices (36) and for the interaction of urea with DNA and various native protein surfaces we deduced that urea interacts principally with polar amide surface of proteins and nucleic acid bases (37) and that, to a good approximation, the urea dependence of an equilibrium or rate constant of a process can be directly interpreted in terms of the change in polar amide surface area (ΔASA_amide; Ų) involved (38)

$${\left(\frac{\partial ln{K}_{\text{obs}}}{\partial m_{\text{urea}}}\right)}_{a_{\text{salt}}}^{m_{\text{urea}}\to 0}=(1.4(\pm 0.3)\times {10}^{-3})\Delta {\text{ASA}}_{\text{amide}}$$ (1)

The use of urea as a quantitative probe of changes in exposure of polar amide surface in a protein-DNA interaction has been calibrated for the specific 1:1 binding of lac repressor (LacI) to SymL operator DNA (38). The amount of polar amide surface area buried in forming a 1:1 tetramer-operator complex, predicted from the initial slope of the urea dependence of the binding constant at constant salt activity (∂lnK_obs/∂m_urea = -2.1±0.2 m^-1) is 1500±300 Ų. This agrees quantitatively with the amount of polar amide surface predicted to be buried from structural data (39, 40), 1526 Ų (of which 491 Ų is buried in the LacI-DNA binding interface, 515 Ų in coupled folding of the hinge helices, and 520 Ų in the protein-protein interface formed by docking of the folded DBD on the core domain of repressor (38)). Notably, two-thirds of the magnitude of the initial slope arises from the burial of amide ASA in coupled conformational changes, even though these changes only contribute one-half of the total amount of surface buried. Thus we conclude that urea is particularly suitable for detecting protein folding events and creation of protein-protein interfaces in protein-nucleic acid interactions (38).

The use of GB as a quantitative probe of anionic surface was also tested in the study of LacI binding to SymL (38). GB increases the LacI- SymL binding constant as expected for a complex that buries 632 Ų of DNA phosphate surface. However, the initial slope of the binding constant at constant salt activity (∂lnK_obs/∂m_GB = 1.8±0.2 m^-1) predicts that 450±100 Ų of anionic surface is buried. We (38) proposed a possible explanation for this quantitative discrepancy: a weak favorable preferential interaction of GB with multiple exposed aromatic residues on the DBD of LacI (Tyr 7 and Tyr 17), by analogy with the aromatic binding site observed in the crystal structure of GB bound to the ProX porter (41). If the cation-π interactions seen in the GB-ProX crystal structure are generalizable, GB would oppose formation of any interface that buries previously exposed aromatic residues, while favoring burial of DNA phosphates. Such an interaction with the native state of the LacI DBD would explain its unexpectedly large stabilization by GB (42) as well as the unexpectedly-small stabilization of the LacI-SymL complex by GB. In addition, effects of GB on processes that wrap DNA nonspecifically on a protein surface (e.g. integration host factor, nucleosome assemblies) may be complicated (see Discussion). Although interactions of GB with different types of biopolymer surface appear to be substantially more complicated than effects of urea, systems investigated to date exhibit a strong correlation between large effects of GB and large-scale burial or exposure of anionic surface (37, 38, 42).

Here we determine the effects of urea and GB on the rate and equilibrium constants of steps or portions of the RNAP-promoter association mechanism (I), and use these to discuss extant proposals for the mechanism of open complex formation. This approach has the distinct advantage of probing structural changes as they occur on the pathway. A rate constant can be analyzed using eq 1 in the same manner as that used for an equilibrium constant. According to transition state theory, the rate constant for a process (k) is proportional to the equilibrium constant for conversion of the reactant(s) of that process to the subsequent transition state (K_obs^‡). Assuming that the transition state decomposition frequencies in the coefficient of proportionality are independent of urea concentration, the urea dependence of the logarithm of the rate constant is equal to the urea dependence of the log of the equilibrium constant for the formation of the transition state: ∂lnk/∂m_urea = ∂lnK_obs^‡/∂m_urea. Thus, ∂lnk/∂m_urea is proportional to the amount of polar amide surface area exposed or buried in the formation of the transition state.

Experimental Procedures

Buffers

Different buffers and reaction conditions were employed in different series of experiments to optimize the quality and range of data that could be collected for each. This optimization was based on preliminary data collected under various reaction conditions.

For the study of the effect of urea concentration on the kinetics of irreversible association of RNAP and λP_R promoter DNA, binding buffer ( ${\text{BB}}_{\text{urea}}^{\text{assoc}}$) contained 0.086 molal (m) KCl, 0.036 m Tris-HCl (pH 8.0 at 25 °C), 0.014 m NaCl, 0.0032 m MgCl₂, 0.947 m glycerol, 0.001 m DTT, 100 μg/mL BSA, and 0, 0.33, 0.66, or 1.01 m urea.

For the study of the effect of GB concentration on the kinetics of association of RNAP and λP_R promoter DNA, binding buffer ( ${\text{BB}}_{\text{GB}}^{\text{assoc}}$) contained 0.107 m KCl, 0.044 m Tris-HCl (pH 8.0 at 28 °C), 0.014 m NaCl, 0.0086 m MgCl₂, 0.948 m glycerol, 0.001 m DTT, 100 μg/mL BSA, and 0, 0.69, or 1.47 m GB.

Effects of urea on the kinetics of dissociation of competitor-resistant (CR) complexes were investigated at 17.1 °C in ${\text{BB}}_{\text{urea}}^{\text{dissoc}}$ (0.123 m KCl, 0.010 m MgCl₂, 0.041 m Tris-HCl (pH 8.0 at 17.1 °C), 0.003 m NaCl, 0.001 m DTT, 100 μg/mL BSA, 0.20 M glycerol, and 0 to 0.63 m urea). Effects of GB on the kinetics of dissociation were investigated at 17.1 °C in ${\text{BB}}_{\text{GB}}^{\text{dissoc}}$ (0.126 m KCl, 0.010 m MgCl₂, 0.041 m Tris-HCl (pH 8.0 at 17.1 °C), 0.003 m NaCl, 0.001 m DTT, 100 μg/mL BSA, 0.52 m glycerol, and 0 to 0.65 m GB).

To keep the concentrations of all solution components whose effects are not being studied constant on the molal scale, the stock solutions of both urea and GB were made by dissolving solid urea or GB monohydrate into previously prepared BB. As GB was used in its monohydrate form, the additional water introduced with the GB reduced the molalities of the other components in ${\text{BB}}_{\text{GB}}^{\text{assoc}}$ and ${\text{BB}}_{\text{GB}}^{\text{dissoc}}$ slightly. At the highest GB concentration studied (1.47 m), these reductions were < 1.5% for all solution components; the effects of these changes in molal concentration were calculated to be negligible. In the case of ${\text{BB}}_{\text{urea}}^{\text{dissoc}}$, for convenience, solid urea was dissolved into solvent containing final concentrations of every component except for glycerol, so that a constant molarity of glycerol was obtained in these experiments (increasing the molality of glycerol slightly (insignificantly) from 0.24 to 0.25 m with increasing urea concentration).

Storage Buffer (SB) for RNAP holoenzyme, core deletion mutants (ΔSI1 and ΔSI3), and σ⁷⁰ subunit consists of 6.8 M glycerol (50% v/v), 10 mM Tris-HCl (pH 7.5 at 4 °C), 100 mM NaCl, 0.1 mM dithiothreitol (DTT), and 0.1 mM Na₂EDTA.

Wash Buffer (WB) for nitrocellulose filter binding assays consists of 0.1 M NaCl, 10 mM Tris-HCl (pH 8.0 at room temperature), and 0.1 mM Na₂EDTA.

Wild type Eσ⁷⁰ RNA polymerase holoenzyme

E. coli K12 wild type RNA polymerase holoenzyme was purified and stored in individual 500 μL samples as described (2). Promoter binding activities of RNAP samples were determined at the time of use by analysis of forward titrations of promoter DNA with RNAP performed in BB in the complete binding of limiting reagent regime at 37 °C as described (1) and used to calculate the concentrations of active holoenzyme reported here. Wild type Eσ⁷⁰ RNAP preparations used in this study were 40-60% active. No systematic differences in promoter association kinetics were found between RNAP preparations when corrected for activity.

Core deletion mutants, ΔSI1 and ΔSI3

Mutant RNAP core enzymes (ΔSI1, deletion of residues 226 to 350 of β and ΔSI3, deletion of residues 943 to 1130 of β′) were generous gifts of I. Artsimovitch in the form of BL21 E. coli cells containing plasmids pIA329 and pIA331 (43). The published method of preparation (43) was used to purify both ΔSI1 and ΔSI3, with slight modifications. During this procedure, RNA contamination was detected in the sample using an agarose gel stained with Gelstar (Cambrex, Rockland, ME). Therefore, 0.22 mg/ml RNase was pumped through the chitin column after the sample had been loaded. While this eliminated the RNA, a small amount of RNase was present in the final sample.

Alternatively, ΔSI3 was prepared by overexpressing α, β, and β′ΔSI3 [containing a hexa-histidine tag at the C-terminus] from plasmid pIA321 (which carries an ampicillin resistant marker; also a gift from I. Artsimovich) using the autoinduction method of (44) and purification protocol described in (43). β′ΔSI3 was extensively dialyzed against SB and stored at -70 °C. Working stocks were thawed and stored at -20 °C and reconstituted with σ⁷⁰ as described below. No difference in dissociation kinetics was observed between chitin-tagged and his-tagged RNAP.

Wild type E. coli σ⁷⁰

Wild type E. coli σ⁷⁰ was purified as described (45), with the following modifications. Larger buffer volumes were generally used to resuspend cell and protein pellets. The resuspended protein pellet was diluted two-fold with reconstitution buffer, instead of denaturing buffer. It was dialyzed against buffer E, instead of reconstitution buffer. The sample was first run through a Q-Sepharose anion-exchange column, then a 6 mL S-Sepharose cation-exchange column, instead of just a DEAE-Toyopearl TSK 650M column. The protein was found to be >90% pure σ⁷⁰.

λP_R promoter DNA

A ³²P-DNA fragment containing the λP_R promoter was 3′ end-labelled and purified as described in (46). BssH II and Sma I cleavage of pBR81 centrally position the λP_R wild type sequence (-60 to +20) in a fragment which extends from −115 to +76 relative to the transcription start site (+1) of the promoter. The specific activity of the fragment was generally ∼10⁸ cpm/mole.

Dissociation Kinetics

Effects of urea and GB on the kinetics of dissociation of pre-formed CR complexes were determined for wild type RNAP, ΔSI1, and ΔSI3 in 0-0.63 m urea, and for wild type RNAP in 0-0.65 m GB. Dissociation was made irreversible by the addition of an excess of heparin, and the kinetics of decay were monitored at 17.1 °C in the appropriate BB^dissoc using nitrocellulose filter binding. Polyanionic heparin is found to be an inert (i.e. non-displacing) competitor, both in the presence and absence of urea and GB; rates of dissociation at 0 and 0.63 m urea are independent of heparin concentration in the range 50-400 μg/mL (data not shown). Heparin was also shown to be an effective competitor for free RNAP in experiments in which free RNAP, initially incubated with heparin at 100 μg/mL, subsequently did not bind promoter DNA (data not shown). For experiments with ΔSI1 and ΔSI3, holoenzyme was reconstituted by incubating wild type σ⁷⁰ subunit with mutant core in at least a 1.5:1 molar ratio in SB on ice for at least 45 minutes before commencement of the experiment.

λP_R DNA (0.08-0.09 nM) and RNAP (10 – 40 nM for wild type, 5 or 5.8 nM for ΔSI1, 1.7 or 16 nM for ΔSI3) were combined in the appropriate BB^dissoc (+/- urea or GB) and incubated for at least 1 hour at 17.1 °C in a water bath to allow association to proceed to equilibrium. At time t=0, 50 μL of heparin in the appropriate BB^dissoc (at 17.1 °C; final concentration: 100 μg/mL) was combined with the 1450 μL reaction. At recorded time points, 100 μL aliquots were removed and filtered at room temperature. Two independent experiments were performed for each unique combination of holoenzyme construct (wild type, ΔSI1, or ΔSI3) and urea or GB concentration studied. Before heparin was added, 100 μL was removed from the reaction and transferred to 37 °C, where it was incubated for at least 30 minutes, then combined with 3.45 μL of heparin (at 37 °C; final concentration: 100 μg/mL). After ∼10 s, 100 μL of this sample was filtered to determine the total amount of DNA capable of binding to RNAP, cpm₃₇.

Association Kinetics

The irreversible kinetics of formation of CR complexes stable to a brief (∼10 s) challenge by heparin (100 μg/mL final concentration) were measured as a function of urea (in ${\text{BB}}_{\text{urea}}^{\text{assoc}}$ at 25 °C) or GB (in ${\text{BB}}_{\text{GB}}^{\text{assoc}}$ at 28 °C) concentration using rapid quench mixing and nitrocellulose filter binding as detailed previously (47). In most cases, the DNA and RNAP solutions contained the same concentrations of the appropriate BB^assoc before being mixed. In some cases where the final desired concentration of RNAP was high, the RNAP sample contained a concentration of SB that was higher than that present in the final BB. Because the RNAP was stored in SB, increasing the concentration of RNAP required adding higher amounts of SB to the RNAP sample. In such cases, the DNA sample contained a concentration of SB that was correspondingly lower than that in the final BB, so that, after mixing the RNAP and DNA, the solution contained the final BB concentrations of all components. As a control, both procedures (using either matched or unmatched buffers) were compared at lower [RNAP]; no difference was seen between the two methods.

Final concentrations of λP_R DNA were 0.08-0.24 nM and of RNAP were 1.2-75.2 nM. All experiments contained at least a 5.4-fold excess of RNAP over λP_R DNA to ensure that the kinetics were pseudo-first order. At time t, the reaction was quenched and expelled from the loop into a total volume of ∼270 mL (including reaction sample, heparin, and push buffer), resulting in a final concentration of ∼86-100 μg/mL heparin. No effect of heparin concentration was seen over this range.

Nitrocellulose filter binding assays

Nitrocellulose filter binding assays were performed as described (2). For dissociation reactions, the total counts per minute filtered (cpm_TOT, generally ∼1000-3500 cpm) was determined by spotting 20 μL from the reaction mixture onto a dried nitrocellulose filter. For association reactions, cpm_TOT was determined by performing a reaction with BB in the RNAP sample port and applying the entire sample onto three dried nitrocellulose filters. Filter efficiency (∼0.70-0.99) was determined by dividing cpm₃₇ (for dissociation reactions) or the plateau cpm_t→∞ (for rapid mix association reactions) by cpm_TOT in a reaction. Cerenkov radiation (cpm) from dried filters was measured in a Hewlett Packard Tri-Carb 2100 TR scintillation counter.

Data Analysis

All data were analyzed using SigmaPlot 6.0 (SPSS Inc., Chicago, IL) on a Dell Optiplex GX270 running Windows XP Professional v. 2002.

Determination of dissociation rate constants (k_d)

Where kinetics of dissociation of CR complexes were single-exponential (in the absence of solute and in the presence of urea; see Results), observed rate constants k_d (= k_-2/(1+K₃), with K₃ = k₃/k_-3 (2, 48)) were determined by fitting data for ${\theta }_t^{\text{CR}}$ (where ${\theta }_t^{\text{CR}}={\text{cpm}}_{\text{t}}/({\text{cpm}}_{37}-{\text{cpm}}_{\text{bkgd}})$ and cpm_t = cpm_obs-cpm_bkgd) to:

$${\theta }_t^{\text{CR}}={\theta }_o^{\text{CR}}{\text{e}}^{{-\text{k}}_{\text{d}}\text{t}}$$ (2)

where ${\theta }_o^{\text{CR}}$ is the value of ${\theta }_t^{\text{CR}}$ at time t = 0 (i.e. the initial fraction of DNA in CR complexes). For dissociation experiments in GB, where the decay is not well described by eq 2, the kinetics were empirically characterized by the half-time t_1/2 for complete dissociation. This was determined for each GB concentration (except for the 0 m control, which was well fit by a single exponential) by smoothing the data in the midrange of ${\theta }_t^{\text{CR}}(0.7>{\theta }_t^{\text{CR}}/{\theta }_o^{\text{CR}}>0.3)$ by fitting it to a quadratic equation and determining the time at which ${\theta }_t^{\text{CR}}={0.5\theta }_o^{\text{CR}}$, using the initial time point (taken within 25 s of mixing with heparin) as ${\theta }_o^{\text{CR}}$.

Determination of composite forward rate constants (α_CR)

In general, for each combination of RNAP and urea or GB concentration studied, association reactions were performed in duplicate (the only exception being the reactions at 0.66 m urea, each of which was only performed once). For each set of duplicates, the forward rate constant (α_CR) was determined by fitting the data ( ${\theta }_t^{\text{CR}}={\text{cpm}}_{\text{t}}/{\text{cpm}}_{\text{t}\to \infty }$) to a single exponential rise to a plateau:

$${\theta }_t^{\text{CR}}=1-{\text{e}}^{{-\alpha }_{\text{CR}}\text{t}}$$ (4)

For a preliminary set of data performed under the conditions used for the dissociation reactions, the association reaction was reversible (wherein an equilibrium distribution of free RNAP and DNA, I₁, and CR complexes is reached); these data required a relaxation analysis involving subtraction of k_d from relaxation rate constant β_CR to calculate α_CR (see Supporting Information).

Determination of K₁ and k₂ from the RNAP concentration dependence of α_CR

For single-exponential kinetics in excess RNAP ([RNAP]_T ≫ [promoter DNA]_T, where [RNAP]_T represents the total active concentration of RNAP in a reaction), α_CR is a hyperbolic function of RNAP concentration (2):

$${\alpha }_{\text{CR}}=\frac{{\text{K}}_1{\text{k}}_2{[\text{RNAP}]}_{\text{T}}}{1+{\text{K}}_1{[\text{RNAP}]}_{\text{T}}}=\frac{{\text{k}}_{\text{a}}{\text{k}}_2{[\text{RNAP}]}_{\text{T}}}{{\text{k}}_2+{\text{k}}_{\text{a}}{[\text{RNAP}]}_{\text{T}}}$$ (5)

where k_a ≡ K₁k₂ is the composite overall second order association rate constant, K₁ (= k₁/k_-1) is the equilibrium constant for formation of the first kinetically-significant intermediate (I₁, in rapid equilibrium with free RNAP and DNA), and k₂ is the microscopic rate constant for the subsequent (rate-determining; k₂ < k_-1) conversion of I₁ to I₂. (For convenience, the subscript T is dropped from [RNAP]_T below.) Values of α_CR (weighted by 1/σ², where σ is the standard deviation) were plotted versus [RNAP] and fit to eq 5 to determine K₁ and k₂ at any given set of conditions.

Urea and GB concentration dependences of equilibrium and kinetic constants

Plots of lnK₁ versus solute (GB, urea) molality exhibit curvature; solute dependences of lnK₁ were therefore fit as a quadratic function of molality. The logarithms of other kinetic constants (k₂, k_d, t_1/2) were well fit by a linear dependence on solute molality. Fits were weighted by 1/σ² where σ is the standard deviation of lnX.

Correction of observed solute dependences for the interactions of solute with KCl and MgCl₂

Both urea and GB have slightly favorable interactions with KCl and MgCl₂ in solution (49 and M. Capp, unpublished). Therefore, addition of either solute will reduce the thermodynamic activities of KCl and MgCl₂, and thereby favors steps in open-complex formation that are driven by a reduction in salt activity. Since our kinetics experiments were performed at constant molal concentrations of KCl and MgCl₂, and not at constant activities, a correction for the effects of urea and GB concentrations to constant KCl and MgCl₂ activities is necessary. In the Appendix, experimental osmotic and isopiestic data for the effects of urea and GB on the non-ideality of KCl and MgCl₂ solutions are used to obtain these corrections, which are significant because of the strong effect of changes in salt activity on protein-nucleic acid interactions.

PONDR predictions and ASA calculations

The default predictor VL-XT of PONDR (Predictors of Natural Disordered Regions) was used to predict the degree of disorder in T. th., T. aq. and E. coli β and β′ subunits from their amino acid sequences (50, 51). Sequences were submitted using the web interface at www.pondr.com. Surface areas were calculated as previously described (30).

Results

Irreversible Dissociation Kinetics: Large Effects of Urea and GB on the Second Half of the Mechanism of Open Complex Formation

The effects of urea and GB on the kinetics of dissociation of competitor-resistant (CR) complexes (I₂, RP_o) provide information about the net burial or exposure of polar amide and anionic surfaces in the second half of the mechanism of open complex formation (subsequent to formation of the (I₁-I₂)^‡ transition state between I₁ and I₂ in Mechanism I). Since the half–life of CR complexes at λP_R at moderate [salt] at T ≥ 25 °C is inconveniently large (≥ 8 hours), studies were performed at 17.1 °C, where k_d is more accurately determined. Kinetic data for irreversible dissociation of CR complexes in the presence of the competitor heparin were collected for a range of concentrations of urea (up to 0.63 m) and GB (up to 0.65 m) under otherwise identical reaction conditions. These data are plotted in Figure 1A and B as the fraction of promoter DNA in CR complexes, ${\theta }_t^{\text{CR}}$, versus time. Urea and GB have large effects in opposite directions on the rate of dissociation of CR complexes; urea increases and GB decreases it.

Figure 1.

Effects of urea and glycine betaine (GB) concentration on the dissociation of competitor-resistant (CR) RNAP-λP_R complexes (I₂, RP_o). Decay of CR complexes at 17.1 °C in ${\text{BB}}_{\text{urea}}^{\text{dissoc}}$ (at 0, 0.21, 0.42, 0.63 m urea; Panel A) or ${\text{BB}}_{\text{GB}}^{\text{dissoc}}$ (at 0, 0.21, 0.43, 0.65 m GB; Panel B) is shown. Data are plotted as the fraction of promoter DNA bound to RNAP in CR complexes ( ${\theta }_t^{\text{CR}}$) versus time after addition of heparin to a final concentration of 100 μg/mL. Curves represent fits of the data to single-exponential decays, assuming complete dissociation of CR complexes (eq 2).

For all urea concentrations investigated, Figure 1A demonstrates that dissociation of CR complexes fits well to a single-exponential time course (eq 2), decaying to zero at long times. Values of k_d (Table 1) increase strongly with urea concentration: k_d is approximately seven-fold faster in 0.63 m urea than in the absence of urea. At the urea concentrations in Figure 1A, k_d is smaller at 28 °C than at 17 °C (data not shown). These results indicate that the negative activation energy of dissociation characteristic of CR complexes in the absence of this solute persists in urea. We conclude from this data that the mechanism of dissociation of CR complexes in urea is still described by a rapid equilibrium between RP_o and I₂ on the time scale of I₂ converting back to (I₁-I₂)^‡ (48).

Table 1.

Urea concentration dependence of the dissociation rate constant k_d^a

	kd (s-1)
murea	Wild type	ΔSI1	ΔSI3
0 m	(6.2 ± 0.3) × 10-5	(9.5 ± 0.4) × 10-5	(1.9 ± 0.1) × 10-4
0.21 m	(1.29 ± 0.04) × 10-4	(1.76 ± 0.06) × 10-4	(3.4 ± 0.1) × 10-4
0.42 m	(2.17 ± 0.06) × 10-4	(3.4 ± 0.1) × 10-4	(6.4 ± 0.3) × 10-4
0.63 m	(4.3 ± 0.1) × 10-4	(5.1 ± 0.3) × 10-4	(1.00 ± 0.06) × 10-3
${\left(\frac{\partial {\text{lnk}}_{\text{d}}}{\partial {\text{m}}_{\text{urea}}}\right)}_{{\text{m}}_4,{\text{m}}_5}$	3.0 ± 0.1 m-1	2.7 ± 0.2 m-1	2.7 ± 0.1 m-1
${\left(\frac{\partial {\text{lnk}}_{\text{d}}}{\partial {\text{m}}_{\text{urea}}}\right)}_{{\text{a}}_4,{\text{a}}_5}$ b	3.2 ± 0.1 m-1	2.9 ± 0.2 c m-1	2.9 ± 0.1 c m-1

^aDetermined at 17.1 °C in ${\text{BB}}_{\text{urea}}^{\text{dissoc}}$ (see Methods).

^bDetermined from eq A6 (see Appendix).

^cDetermined assuming that the values of Sk_d for ΔSI1 and ΔSI3 are the same as that of wild type RNAP.

As shown in Figure 1B, addition of GB slows the entire time course of dissociation of CR complexes greatly. In addition, in the range 0.21 – 0.65 m GB, only the first 60-70% of the dissociation reaction fits the exponential form of eq 2; the final 30-40% of the decay is slower than expected. Given the complexity of the decay of CR complexes in GB, we describe the effect of GB on dissociation in terms of an empirical half-time t_1/2 (see Methods). Effects of GB on dissociation are large: the half-time t_1/2 is approximately 6-fold larger in 0.65 m GB than in the absence of GB. Evidence for conversion of open complexes formed at high temperature between T4 gp55 containing RNA polymerase and the T4 late promoter to a thermostable state at lower temperature has been obtained by KMnO₄ footprinting (52). To test if a corresponding process is induced by addition of GB, KMnO₄ footprints of CR complexes in the presence and absence of 0.6 M GB were compared (C. A. Davis, in preparation); no significant differences in either relative or absolute reactivities of thymines in the open region are detected.

The large effect of urea on k_d is symptomatic of large-scale coupled unfolding in the kinetically significant steps of dissociation, equivalent to large-scale coupled folding in the second half of the mechanism of open complex formation, as discussed below. To test whether either of two large insertions in the β (SI1, residues 226 to 350) or β′ (SI3, residues 943 to 1130) subunits of E. coli RNAP contribute to the effect of urea on k_d, we characterized the effects of urea on the kinetics of dissociation on RNAP mutants lacking these insertions. As observed for wild type polymerase, the kinetics of decay of CR complexes in urea are single exponential (Figure 2A, B); dissociation rate constants are listed in Table 1. The initial fraction of DNA in CR complexes in these experiments, ${\theta }_o^{\text{CR}}$, is a function of both [RNAP] and urea concentration. For ΔSI1, ${\theta }_o^{\text{CR}}$ (determined from a combination of the two data sets at 5 and 5.8 nM RNAP in Figure 2A) decreases from ∼1 at 0 m urea to 0.3 at 0.63 m urea. For ΔSI3, ${\theta }_o^{\text{CR}}$ (from the representative data set at 16 nM RNAP in Figure 2B) decreases from ∼1 at 0 m urea to 0.8 at 0.63 m urea.

Figure 2.

Effects of urea concentration on the dissociation of competitor-resistant (CR) complexes formed with the RNAP deletion mutants ΔSI1 and ΔSI3. Nitrocellulose filter binding data for the irreversible dissociation of CR complexes formed between ΔSI1 RNAP (Panel A; residues 226-350 deleted from the β subunit of RNAP) or ΔSI3 RNAP (Panel B; residues 943-1130 deleted from the β′ subunit) and λP_R promoter DNA at 17.1 °C in ${\text{BB}}_{\text{urea}}^{\text{dissoc}}$ are shown for various urea concentrations (0, 0.21, 0.42, 0.63 m). Data are plotted as the fraction of DNA bound to RNAP in CR complexes ( ${\theta }_t^{\text{CR}}$) versus time. (Because the two data sets for ΔSI3 were taken at two significantly different [RNAP], Panel B shows a representative data set at 16 nM RNAP.) Curves represent fits of the data to single-exponential decays (eq 2); values of the rate constant k_d obtained from these fits are listed in Table 1. (For ΔSI3, values in Table 1 are averages of k_d from fits to individual data sets.) Panel C compares the effect of urea on the rate of dissociation of ΔSI1 (open triangles), ΔSI3 (filled squares), and wild type (filled circles) RNAP-λP_R complexes. Solid lines are weighted linear fits of the data. Slopes of these fits are listed in Table 1.

As observed previously on the T7 A1 promoter (43), both deletion mutants dissociate faster than wild type polymerase from the λP_R promoter in the absence of urea (k_d increases 1.5-fold for ΔSI1 and 3-fold for ΔSI3 at 0 m urea under these solution conditions). However, Figure 2C (a semi-logarithmic plot of k_d versus urea concentration for wt and the two mutants) reveals that this effect is not a function of urea concentration. A comparison of the slopes in Figure 2C shows that the urea dependences of k_d for wild type RNAP, ΔSI1, and ΔSI3 are the same within error (Table 1).

Irreversible Association Kinetics as a Function of RNA Polymerase and Solute (Urea or Glycine Betaine) Concentrations

Initial experiments performed under the same solution conditions used to investigate dissociation indicated that effects of urea on K₁ and k₂ were relatively small in comparison to effects of urea on k_d. Under these conditions, association is reversible at low [RNAP] and/or higher [urea]. Therefore a relaxation analysis is required to analyze the data (in which k_d is subtracted from the relaxation rate constant β_CR to get the composite association rate constant α_CR). Since this analysis increases the uncertainty, we were unable to quantify these relatively small effects of urea with sufficient accuracy (see Supporting Information). To quantify these effects, we found conditions where formation of CR complexes is irreversible, even in the presence of 1.01 m urea, and where K₁ is large enough so that k₂ can be accurately determined. These requirements were met by raising the temperature to 25 °C and by reducing the concentrations of KCl and MgCl₂ to those of ${\text{BB}}_{\text{urea}}^{\text{assoc}}$. The effects of GB on the association kinetics were also investigated under irreversible conditions (in ${\text{BB}}_{\text{urea}}^{\text{assoc}}$ at 28 °C). At these temperatures, the kinetics are sufficiently fast that manual mixing cannot accurately determine k₂, necessitating the use of a rapid-quench mixer (2).

Representative data obtained at low (1.1 – 5 nM) and high (37 – 72.5 nM) [RNAP] at three concentrations of urea (0, 0.66, 1.01 m) and GB (0, 0.69, 1.47 m) are shown in Figures 3 and 4. In all cases, the association kinetics are single-exponential. Exponential fits of the kinetic data (to eq 4) are shown as the solid curves in Figures 3 and 4; these yield composite association rate constants α_CR for each RNAP and solute concentration investigated. Since the association mechanism is multi-step, the observation of single-exponential kinetics indicates that the steps of initial binding and bending of promoter DNA to form the competitor-sensitive intermediate I₁ rapidly equilibrate on the time scale of the subsequent slow conversion of I₁ to I₂ (2, 48).

Figure 3.

Effects of urea on the irreversible kinetics of formation of competitor-resistant (CR) complexes between RNAP holoenzyme and λP_R promoter DNA. Examples of rapid quench flow association kinetic data are shown for the highest and lowest RNAP concentrations studied at 0 (A), 0.66 (B), and 1.01 m (C) urea at 25 °C in ${\text{BB}}_{\text{urea}}^{\text{assoc}}$. Data are plotted as the fraction of promoter DNA in the form of CR complexes, ${\theta }_t^{\text{CR}}$, as a function of time. Solid lines are fits of the data to irreversible single-exponential kinetics (eq 4, with ${\theta }_{\text{eq}}^{\text{CR}}=1$).

Figure 4.

Effects of glycine betaine (GB) on the irreversible kinetics of formation of competitor-resistant (CR) complexes between RNAP holoenzyme and λP_R promoter DNA. Examples of rapid quench flow association kinetic data are shown for the highest and lowest RNAP concentrations studied at 0 (A), 0.69 (B), and 1.47 m (C) GB at 28 °C in ${\text{BB}}_{\text{GB}}^{\text{assoc}}$. Data are plotted as the fraction of promoter DNA in the form of CR complexes, ${\theta }_t^{\text{CR}}$ as a function of time. Solid lines are fits of the data to irreversible single-exponential kinetics (eq 4, with ${\theta }_{\text{eq}}^{\text{CR}}=1$).

To visualize the effects of urea concentration on the binding constant K₁ for formation of the intermediate I₁ and on the isomerization rate constant k₂, values of the association rate constant α_CR for each RNAP and urea concentration investigated are plotted in Figure 5. The behavior of α_CR as a function of [RNAP] is clearly hyperbolic at each urea concentration, consistent with eq 5. A well-defined plateau value of α_CR in the accessible range of [RNAP] is observed at each urea concentration. These plateau values of α_CR (attained at [RNAP] ≥ 30 nM; Figure 5A) are the same at all urea concentrations, revealing that the isomerization rate constant k₂ for I₁ → I₂ does not depend significantly on urea concentration.

Figure 5.

Effects of RNAP and urea concentrations on the composite association rate constant α_CR. Panel A: Values of α_CR obtained from all irreversible association kinetic experiments (see Figure 4 for examples) are plotted as a function of [RNAP] at 0, 0.33, 0.66, and 1.01 m urea. The RNAP concentration scale of Panel A emphasizes the lack of effect of urea on α_CR at high [RNAP]. Solid lines are fits of the data to eq 6; values of K₁ and k₂ determined from these fits are listed in Table 2. The expanded concentration scale plot in Panel B shows the significant effect of urea on α_CR at low [RNAP].

At low [RNAP], the modest but significant reduction in α_CR at the higher urea concentrations investigated (observed more clearly in the expanded-scale Figure 5B) indicates that the equilibrium constant K₁ decreases at higher urea concentration. Significantly, the behavior of α_CR as a function of RNAP concentration at 0.33 m urea is the same within the uncertainty as in the absence of urea (Figure 5B); the effect of urea on α_CR at any low RNAP concentration clearly increases with increasing urea concentration. Fits of all α_CR to the hyperbolic eq 5 for the R + P ⇄ I₁ → products mechanism are shown in both panels of Figure 5. These fits yield the values of K₁ and k₂ listed in Table 2.

Table 2.

Urea and GB concentration dependences of the equilibrium constant for formation of I₁ (K₁) and the forward rate constant for conversion of I₁ to I₂ (k₂) in Mechanism I

25 °C ( ${\text{BB}}_{\text{urea}}^{\text{assoc}}$)	K1 (M-1)	k2 (s-1)
0 m urea	(4.5 ± 0.4) × 108	(2.34 ± 0.06) × 10-1
0.33 m urea	(4.0 ± 0.3) × 108	(2.31 ± 0.03) × 10-1
0.66 m urea	(2.8 ± 0.2) × 108	(2.38 ± 0.05) × 10-1
1.01 m urea	(1.9 ± 0.3) × 108	(2.43 ± 0.12) × 10-1
${\left(\frac{\partial \text{ln}\left(\text{K}\text{or}\text{k}\right)}{\partial {\text{m}}_{\text{urea}}}\right)}_{{\text{m}}_4,{\text{m}}_5}$	-0.3 ± 0.3 m-1	0.03 ± 0.03 m-1
${\left(\frac{\partial \text{ln}\left(\text{K}\text{or}\text{k}\right)}{\partial {\text{m}}_{\text{urea}}}\right)}_{{\text{a}}_4,{\text{a}}_5}$ a	-0.7 ± 0.3 m-1	0.03 ± 0.03 m-1
28 °C ( ${\text{BB}}_{\text{GB}}^{\text{assoc}}$)	K1 (M-1)	k2 (s-1)
0 m GB	(6.5 ± 0.8) × 107	(2.0 ± 0.1) × 10-1
0.69 m GB	(1.5 ± 0.4) × 108	(1.4 ± 0.1) × 10-1
1.47 m GB	(2.9 ± 0.5) × 108	(8.5 ± 0.6) × 10-2
${\left(\frac{\partial \text{ln}\left(\text{K}\text{or}\text{k}\right)}{\partial {\text{m}}_{\text{GB}}}\right)}_{{\text{m}}_4,{\text{m}}_5}$	1.4 ± 0.4 m-1	-0.6 ± 0.1 m-1
${\left(\frac{\partial \text{ln}\left(\text{K}\text{or}\text{k}\right)}{\partial {\text{m}}_{\text{GB}}}\right)}_{{\text{a}}_4,{\text{a}}_5}$ a	0.7 ± 0.4 m-1	-0.6 ± 0.1 m-1

^aDetermined from eq A6 (see Appendix).

Values of the association rate constant α_CR for each RNAP and GB concentration investigated are plotted in Figure 6. As observed with urea, α_CR is a hyperbolic function of RNAP concentration at each GB concentration; however, for GB the well-defined plateau value of α_CR at high [RNAP] decreases strongly with increasing GB concentration, revealing that k₂ decreases strongly with increasing GB concentration. At low [RNAP], on the other hand, α_CR is essentially independent of GB concentration, indicating that the equilibrium constant K₁ for formation of the intermediate I₁ must increase sufficiently with increasing GB concentration to compensate for the effect of GB on k₂, so that the low-[RNAP] limit of α_CR (α_CR → K₁k₂[RNAP]_T) is independent of GB concentration. Fits of the [RNAP] dependences of α_CR to hyperbolic eq 5 at each GB concentration are shown in Figure 6, and values of the fitted quantities K₁ and k₂ are listed in Table 2.

Figure 6.

Effects of RNAP and glycine betaine (GB) concentrations on the composite association rate constant α_CR. Values of α_CR obtained from all irreversible association kinetic experiments (see Figure 5 for examples) are plotted as a function of [RNAP] at 0, 0.69, and 1.47 m GB. Solid lines are fits of the data to eq 6; values of K₁ and k₂ determined from these fits are listed in Table 2.

Analysis of Effects of Urea and GB on the Rate and Equilibrium Constants for the Steps of Open Complex Formation

Figure 7 summarizes the results of irreversible association and dissociation kinetic studies (Figures 1, 5, 6). Figure 7A plots the logarithm of the normalized equilibrium constant K₁ for formation of I₁ as a function of solute molality. Curvature is detected in the urea concentration dependence, as is observed for binding of LacI to SymL operator DNA in the range 0 – 3 m urea (38). This curvature may reflect a destabilization of some aspect of the free RNAP structure or of the polymerase assembly at the higher urea concentrations used here (possibly destabilization of the sigma-core binding interaction. For this reason, the initial slope of the fit was used in subsequent calculations. The initial slope of the urea concentration dependence is negative, as expected for burial of polar amide surface in binding, but is small in magnitude (∂lnK₁/∂m_urea° = -0.3 ± 0.3 m^-1); correction to constant salt (KCl and MgCl₂) activity does not increase the magnitude beyond that expected for the burial of polar amide surface of RNAP in the interface with promoter DNA (see Discussion), and provides no evidence for any large-scale coupled folding. The initial GB slope is positive (1.4 ± 0.4 m^-1), the direction expected for burial of anionic phosphate surface of DNA in forming I₁ (see Discussion).

Figure 7.

Summary of effects of urea and GB on the steps of the mechanism. Natural logarithms of K₁ (A), k₂ (B), and 1/k_d or t_1/2 (C) (all normalized to the appropriate values in the absence of solute) are plotted versus urea or GB concentration. In (A), solid lines represent quadratic fits to the data. In (B) and (C), solid lines represent linear fits of the lnk₂ and lnk_d data, respectively; dashed line in (C) is for lnt_1/2. Values of the slopes (for Panel A, the initial slope) of the fits are listed in Tables 1 and 2, along with the corresponding quantities after correction to constant salt activity.

For the subsequent, rate-determining conformational change, effects of urea and GB on the logarithm of the normalized rate constant k₂ are shown in Figure 7B. Remarkably, this step is essentially independent of urea concentration (∂lnk₂/∂m_urea = 0.03 ± 0.03 m^-1; values of k₂ at all urea concentrations are the same within error), indicating that there is no detectable net change in exposure of amide surface in converting I₁ to the subsequent transition state (I₁-I₂)^‡. GB significantly reduces the rate of this step; the GB derivative of the logarithm of k₂ is −0.6 ± 0.1 m^-1, consistent with significant exposure of anionic surface in this step. The correction of the data to constant salt activity does not change this result detectibly because the salt dependence of this step is small in magnitude (Sk₂ ≈ -1; WSK, in preparation). No curvature is detected in plots of lnk₂ versus m_solute over the ranges examined.

Figure 7C plots the logarithm of the normalized inverse dissociation rate constant (for urea) or half-time (for GB) as a function of solute molality. Analyzed in this manner, the urea data are interpretable as effects on the conversion of the (I₁-I₂)^‡ transition state to the equilibrium mixture of RP_o and I₂ (presumably mostly RP_o) for the conditions investigated. Clearly the effect of urea on these steps is dramatically greater than its effect on the steps of the first half of the mechanism (-∂lnk_d/∂m_urea = 3.0 ± 0.1); most of the polar amide surface buried in forming the open complex is buried in the second half of the association mechanism in conformational changes that occur subsequent to initial binding and formation of I₁. This situation is analogous to that observed for the effect of urea on the LacI-SymL interaction, where the majority (65-70%) of the urea effect results from conformational changes and only 30-35% is due to formation of the protein-DNA interface. Proposals for the late conformational changes in open complex formation detected by urea are given below. For GB, the situation is complicated by the fact that the final stages of the dissociation kinetics (Figure 1B) are not well fit by a single-exponential decay. Even with this complication, it is clear that the latter half of the mechanism is more sensitive to GB concentration than either the initial binding step or the subsequent isomerization step.

For the overall process of CR complex formation from free RNAP and promoter DNA, the equilibrium constant ${\text{K}}_{\text{obs}}^{\text{CR}}$ is the ratio of the composite forward rate constant k_a (=K₁k₂) to the dissociation constant k_d ( ${\text{K}}_{\text{obs}}^{\text{CR}}={\text{k}}_{\text{a}}/{\text{k}}_{\text{d}}$). Thus, the effect of urea on the overall process can be found from the individual effects on the forward and reverse processes: ${(\partial \text{ln}{\text{K}}_{\text{obs}}^{\text{CR}}/\partial {\text{m}}_3)}_{{\text{a}}_4,{\text{a}}_5}={(\partial \text{ln}{\text{K}}_1/\partial {\text{m}}_3)}_{{\text{a}}_4,{\text{a}}_5}+{(\partial {\text{lnk}}_2/\partial {\text{m}}_3)}_{{\text{a}}_4,{\text{a}}_5}-{(\partial {\text{lnk}}_{\text{d}}/\partial {\text{m}}_3)}_{{\text{a}}_4,{\text{a}}_5}$. The urea dependence of the overall process of CR complex formation at constant salt activity is -3.9±0.3 m^-1 (corresponding to a net burial of (2.8±0.6)×10³ Ų of polar amide surface).

Discussion

Open complex formation by Eσ⁷⁰ RNAP involves a sequence of conformational changes in both polymerase and promoter DNA that create the transcription bubble and position the template strand in the active site. Here we have exploited low (nondenaturing) concentrations of urea to probe protein conformational changes along the pathway. Similarly we have used low concentrations of GB to detect changes in the water-accessibility of anionic oxygens on the DNA backbone and polymerase (Asp, Glu sidechains, C-termini). Strikingly we find the largest effects of both solutes on the final steps in the mechanism, indicating that large-scale motions in the polymerase machinery occur after the formation of an extensive interface with duplex promoter DNA (∼-82 to +25; CAD, in preparation), as part of the descent from the high free energy (I₁-I₂)^‡ transition state to the final open complex.

We have recently proposed that the effects of urea concentration on an equilibrium or rate constant can be directly interpreted in terms of the amount of amide surface area exposed or buried in a process using the relationship ΔASA_amide (Ų) = (710±150)(∂lnκ/∂m_urea) where the derivative is taken at constant salt activity and at m_urea→ 0, and where κ is either an equilibrium or rate constant (eq 1; (38)). The dependence of ln(1/k_d) on m_urea reported here predicts that ∼2300±500 Ų of ASA_amide are buried in events that convert the (I₁-I₂)^‡ complex to RP_o (see Table 1). This value of ΔASA_amide is 1.3 times greater than that found for the interaction between LacI and its high affinity symmetric operator site SymL (38), and similar to that buried when a small protein such as CheY or apomyoglobin forms its highly ordered native state from an extended, denatured state (see (38) and references therein). Notably, for LacI binding, two-thirds of the total of amide surface buried occurs outside of the LacI-SymL interface in protein interfaces created by folding of the hinge helices and docking of the DNA binding domain on the core domain of repressor. By analogy with repressor binding and protein folding data, the urea dependence of k_d provides strong evidence that a region or regions on polymerase involving on the order of ∼100 residues folds late in the mechanism, and likely creates new protein-protein interfaces in doing so.

Numerous studies support the hypothesis that DNA melting occurs during the latter half of the mechanism (see, for example, 2, 3, 5, 13, 53, 54). DNA opening should expose amide-like functional groups on the bases (37), a process which should be favored by increasing [urea]. Since the anticipated contribution of DNA opening to the urea dependence of ln(1/k_d) is in the opposite direction to what is observed, the amount of folding may be greater than predicted above.

What regions of Eσ⁷⁰ could be disordered in the free enzyme, not fold in forming I₁ or in converting I₁ to the subsequent transition state, but then fold in the latter half of the mechanism of open complex formation? We tested whether either of two E. coli sequence insertions in the conserved polymerase architecture are responsible. Lack of density for residues in the downstream lobe of β (SI1: 226-350) and in the downstream β′ clamp (SI3: 943-1130) in the EM structure of the E. coli core enzyme indicates that these regions are disordered, or folded but flexibly tethered to ordered domains (20). Additionally, their points of insertion in the β jaw and β′ downstream clamp (conserved regions G and G′; see Figure 9) make them likely candidates to participate in clamping DNA downstream of +10 (20, 43, 55-57). Consistent with their proposed role, deletion of either insertion destabilizes competitor-resistant complexes at λP_R relative to wild type (Figure 2). However, the urea dependence of k_d is independent of either insertion within experimental uncertainty (Figure 2C), eliminating them as the origin of the effect. Indeed SI3 has recently been shown to be folded, forming a beta sandwich-hybrid domain that is connected to the trigger loop by two long, flexible tethers of ∼13 amino acids each (58).

Figure 9.

Schematic representation of the proposed conformational changes that convert I₁ to the transition state between and I₁ and I₂ (or I₂) and the subsequent changes that convert this species to RP_o at the λP_R promoter. Orientation of polymerase is the same as that shown in the right half of Figure 8 where the β subunit is closest to the viewer. The orientation here emphasizes events occurring in the active site channel, and does not include DNA upstream of −12. Dimensions of holoenzyme and DNA are drawn approximately to scale; holoenzyme outline is based on PDB file 1IW7 (14) and the shape of E. coli β′ SI3 is based on PDB file 2AUK (58). Coloring is: core subunits (α, β, β′; ω cannot be seen in this view), purple; σ, orange; nontemplate (nt) strand, black; template (t) strand, pink; and Mg²⁺ bound in the active site, red. In I₁, regions G and G′ of β′ and SI1 of β are represented as being partially disordered. Downstream DNA is placed in the channel by a sharp bend at −11/-12 (2, 12), but remains relatively base-paired downstream of this distortion. The negatively charged NTD domain of σ (region 1.1) is shown as lacking secondary structure and bound in the channel near the active site and the downstream lobe of β. Wrapping of upstream DNA in I₁ (not shown; (12)) positions β′ G, G′ and SI3 in such a way to facilitate the release of 1.1. Isomerization of I₁ to I₂ is proposed to involve the ejection of 1.1 from the jaws, unmasking the active site. The release of 1.1 allows the DNA to descend further into the channel, permitting the downstream lobe of β to close and SI1 to fold. As the DNA descends, aromatic residues of sigma capture bases in the −10 region of the nt strand, leading to DNA untwisting over ∼1 turn of the double helix. Repositioning and untwisting of the DNA in the channel triggers the conversion of (I₁ -I₂) ^‡ (or I₂) to RP_o; we hypothesize that basic residues in channel bind DNA backbones of the nt and t strands to complete the formation of the transcription bubble in RP_o. DNA downstream of +5 is clamped by β′ G, G′ and SI3. Position of SI3 in RP_o is based on (58).

Prediction of Regions of Disorder in E. coli RNA Polymerase

The E. coli RNA polymerase has proved remarkably refractory to forming well-ordered crystals, suggesting that it is inherently more flexible than its thermophilic homologs, and/or that the sequence insertions prevent highly ordered lattices from forming. To identify alternative candidates for regions that may be disordered in the E. coli enzyme, we applied the computer algorithm PONDR (see Methods; 50, 51). PONDR has successfully predicted such regions from the primary amino acid sequence of other proteins (cf. 59, 60, 61). From the set of sequences predicted to be disordered from the β and β′ sequences of T. aq., T. th. and E. coli, we considered sequences 15 amino acids in length or longer (the longer the region, the higher the probability that it is unfolded (50)). This set includes domains involved in creating subunit-subunit interfaces. Very possibly these regions are unfolded before assembly, and fold as part of interface formation during the assembly of core. Indeed such folding coupled to assembly appears to be required for formation of the intertwined structure of core polymerase (62). Perhaps not surprisingly, the remaining predicted unfolded (dynamic or flexible) regions cluster around the perimeter of the polymerase jaws (shown in Figure 9 in crimson red). The most extensive regions of disorder occur in conserved regions G and G′ of the β′ subunit. Notably, this region participates in interfaces formed in crystallization of the thermophilic polymerases; possibly an ordered conformation of this flexible domain is selected and ordering is coupled to crystallization. Crystallization of the E. coli enzyme may be disfavored relative to the thermophilic enzymes in part due to the insertion and flexibility of SI3 in this critical region (see above).

Proposal: Coupled Folding of β′ Conserved Regions G and G′ occurs in I₂ → RP_o

Regions G and G′ of (β′ form the downstream mobile clamp; this clamp is proposed to bind promoter DNA downstream of the initiation bubble (55, 57). Strikingly, deletion of conserved residues in this domain (β′Δ1149-1190) or a single amino acid substitution (β′G1161R) greatly destabilizes open complexes at the lacUV5, λP_R, RNA II and rrnB P1 promoters (63). Under the experimental conditions of that study, the half-time for dissociation of open complexes at lacUV5, RNA II and λP_R decreased from several hours to several minutes, and a comparable degree of destabilization was observed for the intrinsically unstable rrnb P1 promoter–polymerase complex (63). The magnitude of this effect not only supports the hypothesis that this domain is critically involved in creating the open complex but that it also undergoes significant rearrangements to do so. The number of residues predicted from the urea dependence of k_d (∼120-140) is quantitatively consistent with the number of residues predicted by PONDR to be unfolded in the β′ downstream clamp (∼120 residues). Thus we propose that this region folds and binds late in open complex formation. Based on its position in the structure, it is likely that these rearrangements create new protein-protein interfaces between G and G′ and regions near the active site. The increase in t_1/2 of dissociation with increasing GB concentration is consistent with a net burial of anionic ASA in the formation of RP_o, possibly reflecting in part the folding and binding of the mobile clamp to the downstream DNA. A schematic representation of this and other proposed conformational changes (see below) deduced from these solute studies is shown in Figure 9.

Studies to date do not determine whether the proposed remodeling of regions G and G′ would occur in forming I₂, or in events that convert I₂ into RP_o. Possible clues are provided by recent advances in defining the mechanism of action of the small regulatory protein DksA. DksA critically modulates ribosomal promoter activity in response to changes in in vivo conditions, including amino acid starvation (64). Binding of DksA to RNA polymerase in vitro increases the dissociation rate of CR complexes at all promoters studied including λP_R (65), and effects of DksA and the alarmone ppGpp on dissociation are synergistic (64-66). Based on structural homology to GreA (67) and GreB (68), biochemical studies (65) and the EM structure of E. coli GreB and core polymerase (56), DksA has been proposed to bind in the secondary channel of polymerase (65). How does placement of DksA in the secondary channel destabilize open complexes?

In order for DksA to accelerate the decay of competitor-resistant complexes at λP_R, it must bind I₂ and/or the I₁-I₂ transition state more tightly than RP_o. No precedent exists for a protein to solely bind the transition state. For example, in the mechanism for catalysis by catalytic antibodies (generated using transition state analogs in order to achieve a high affinity for the transition state) the antibody is also observed to bind the substrate and product, albeit with reduced affinity relative to the transition state (69)). Thus it is likely that DksA binds to I₂. Why would positioning of DksA in the secondary channel disfavor RP_o formation relative to I₂?

As illustrated in Figure 8, the secondary channel (62) is defined entirely by β′: two helices form one side of the channel, (called the rim helices in (56)), the floor of the active site channel at the bottom, and by the F-helix on top. Importantly, the other side of the channel is created by β′ region G. If this region is disordered, the secondary channel is not completely formed until it folds, and SI3 presumably occupies multiple positions. The proposed orientation of DksA in the secondary channel via its interface with the rim helices and Mg²⁺ in the active site (65) suggests to us that the C-terminal globular domain of DksA would sterically inhibit the rearrangement of the downstream clamp. In particular the placement of SI3 in the position modeled by (57) may overlap the C-terminal domain of DksA. Thus we hypothesize that the dynamic (unfolded) state of the downstream clamp pre-exists in free core and holoenzyme (DksA binds free core and holoenzyme in vitro (64, 65)), persists in I₁ and I₂, and that this conformational state (see Figure 9) favors DksA binding. While speculative, this line of reasoning leads to testable hypotheses: the downstream clamp of β′ folds in the transition of I₂ to RP_o (see Figure 9) and DksA somehow blocks or disfavors this repositioning, thereby destabilizing RP_o relative to I₂.

Figure 8.

Regions of disorder in RNA polymerase holoenzyme predicted by PONDR (50, 51). Cartoon based on PDB file 1IW7 (14). Regions (≥15 residues) predicted to lack a fixed tertiary structure in solution are highlighted in red. This data set excludes predicted disordered sequences that are found in subunit – subunit interfaces. PONDR regions cluster around the exterior of the jaws defined by β and β′, and also include flexible regions in the active site channel (the “rudder” of β′ (T. th. residues 587-607, E. coli residues 311-331) and residues in two “fork” loops of β (T. th. 399-417, 441-454, E. coli residues 511-537, 561-574). Regions predicted to have the most extensive degree of disorder are conserved regions G and G′ of β′. Subunit color scheme: α_I, green; α_II gold; β, light blue; β′, pink; σ^A, bisque; ω, teal. Figure 8 was created using MolScript (81) and Raster3D (82).

Effects of urea and GB on the binding and isomerization steps in the mechanism of RP_o formation

Formation of the first kinetically significant intermediate at the λP_R promoter creates an interface involving 100 bp of promoter DNA: •OH footprinting studies reveal that the DNA backbone is protected from at least −82 to +25 (CAD, in preparation). We proposed that DNase I protection to +25 in I₁ requires DNA binding in the (β/β′ jaws or pincers of polymerase (2, 11), and we recently demonstrated that entry of DNA into the jaws is facilitated by the presence of DNA upstream of −47 (12). While footprinting reveals important information about DNA structure and interactions, information about possible conformational changes in RNAP in I₁ at λP_R were first inferred from the temperature and [salt]-dependence of k_a (3, 4) and then from their effects on K₁ ((2), WSK, in preparation). Here the dependence of K₁ on [urea] and [GB] yield new insights into conformational changes occurring in I₁.

Unlike k_d, the dependence of K₁ on urea (Figure 7, Table 2) provides no evidence for extensive protein folding or unfolding or DNA opening in forming I₁. In particular, the initial slope (∂lnK₁/∂m_urea)_m4,m5 at constant salt molality is small (-0.3±0.3 m^-1); an approximate correction to constant activities of both KCl and MgCl₂ (see Methods) yields a constant-salt-activity initial slope (∂lnK₁/∂m_urea)_a4,a5 = -0.7±0.3 m^-1. Using eq 1, this slope corresponds to burial of approximately 500±200 Ų of polar amide surface in formation of I₁. What is region 1.1 of sigma doing in this step? Can the negatively charged region 1.1 and downstream DNA co-occupy the jaws in I₁ or is 1.1 required to leave the active site channel in order for DNA to enter?

To address this critical question, we compared the experimentally predicted value of ΔASA_amide with an estimate based available structural data and reasonable models for the downstream and upstream interfaces (Table 3). For wrapping of upstream DNA, we used the nucleosome and IHF as models. For downstream DNA interactions in the β/β′ jaws (-5 to +20), we utilized our model of I₁ (2) which is based on DNaseI, •OH, and KMnO₄ footprinting of I₁ at λP_R (11, 12) and the high resolution T. aq. core structure (70). The relatively open jaw width (25-30 Å) in the core structure allows duplex DNA to enter “high” in the channel with little steric clash. The X-ray crystal structures of the −35 DNA-σ^A complex (71), and the αCTD-CAP-DNA co-crystal (72) allow a more accurate assessment of surface involved in contacts from −55 to ∼-26. In sum, modeling indicates that ∼440 Ų of polar amide surface is buried in contacts in I₁ (Table 3). Although we do not have a model for the sigma region 2 – -10 DNA complex, values in Table 3 suggest that the amount of amide ASA buried in this interface cannot exceed 100 Ų.

Table 3.

Estimates of amounts of polar amide and anionic surface buried in the first kinetically significant intermedate (I₁) in open complex formation at the λP_R promoter.

Elements of the I1 interface	DNA contacts (bp)	−ΔASAamide (Å2)	−ΔASAanionic (Å2)	Model/comments
Wrapping from ∼-56 to −80	25	100a	762a	Based on hydroxyl radical footprint (CAD, in preparation)
Binding of two alpha CTD to ∼-41 to −55	15	197	520	PDB file 1LB2 (72)
σ region 4 – -35 DNA interactions	10	77	318	PDB file 1KU7 (71)
σ region 2 – -10 DNA interactions	∼5	ND	ND
DNA bound in the “open” jaws from ∼-5 to +20	25	70	200	Based on the model of I1 from (2): DNA duplex “high” in jaws, jaw width about 25 – 30 Å in T. aq. core structure 1I6V (70)
Total	∼80	444	1800	Missing an estimate for σ region 2 – -10 DNA interactions, as well as any interactions with DNA between −35 -10 regions
Predicted based on dlnK1/dmurea at constant salt activity		500±100

^aEstimates for amide and anionic surface (phosphate oxygens) buried in wrapping upstream DNA calculated using the average amount of −ΔASA_amide/bp (4.1 Ų) and of −ΔASA_anionic/bp (30.5 Ų) buried in the IHF-H′DNA (1IHF (79)) complex and in the nucleosome (1KX3 (80)). ND, not determined.

Since the structural estimates presented in Table 3 explain most if not all of the effect of urea on K₁, the most straightforward hypothesis is that region 1.1 remains bound in the jaws in I₁. However, we cannot rule out the possibility that 1.1 exits but rebinds elsewhere, with no net amide exposure. If this region (residues 50-75 based on •OH footprinting (73)) is entirely buried and then released into solution in an extended state (17), ∼750 Ų of polar amide surface would be exposed, yielding a value of ∂lnK₁/∂m_urea of ∼ +1 m^-1. Reconciliation of this prediction with the experimental observation requires burial of an additional ∼750 Ų of polar amide ASA (eq 1). The absence of regions on polymerase likely to fold in this step, the relatively nonspecific interface formed in the β/β′ jaws with DNA from −5 to +25, and the effect of GB on k₂ (below) suggest to us that 1.1 remains in the jaws in I₁.

The initial GB dependence of K₁ for formation of I₁ is modest; at constant KCl and MgCl₂ activity, the initial slope (∂lnK₁/∂m_GB)_a4,a5 is approximately 0.7±0.4 m^-1. However, this value is only 40% of the GB dependence determined for the formation of the complex between LacI – SymL operator (∂lnK₁/∂m_GB= 1.8±0.2 m^-1 (38)). PDB file 1EFA (39) predicts ∼ 630 Ų of anionic ASA are buried in the LacI dimer-SymL 20 bp oligomer complex, and Table 3 estimates ∼1800 Ų of anionic ASA is buried in I₁. As in the case of LacI, qualitative but not quantitative agreement is found between the predicted slope of the GB dependence of K₁ for formation of I₁ and what is expected from structural models.

This disagreement might result in part from a compensating effect of weak GB-aromatic interactions, as discussed in the Introduction. Potential GB binding sites defined by multiple aromatic residues exist in region 2.3-2.4 of sigma; interaction of these residues with GB might reduce their tendency to interact with the −10 region of the promoter in I₁. In addition, wrapping of upstream DNA in I₁ coupled to salt bridge disruption (35, 74) may also complicate direct interpretation of GB effects on K₁. If the burial of phosphate oxygens is coupled to the rehydration of carboxylate surface, the opposing changes in anionic ASA are predicted to reduce the GB effect due to compensation. Owing to the lack of appropriate models, calculation of ΔASA_anionic in Table 3 requires an uncertain estimate for the burial of phosphate oxygen ASA.

Evidence for Release of region 1.1 of σ⁷⁰ in the rate-determining isomerization step

The dependences of the rate constant k₂ for the rate-determining conformational change in forward direction on solute concentrations and temperature provide thermodynamic clues about the process of converting I₁ to the subsequent (I₁I₂)^‡ transition state. In principle, these data can tell us whether this step is rate-determining because DNA opening is initiated, or because conformational changes in polymerase precede or nucleate opening. Unfortunately, but not surprisingly, the solute and temperature dependences suggest that multiple conformational changes occur in this step. The two unambiguous signatures of these conformational changes are the very large, facilitating effect of temperature on the rate constant (corresponding to a temperature-independent activation enthalpy of 34±2 kcal mol^-1 (2)) and the large retarding effect of GB concentration found in this study (∂lnk₂/∂m_GB = -0.6±0.1 m^-1).

The sign of the GB effect on k₂ indicates that anionic biopolymer surface is exposed in this step. Since •OH footprinting studies show that DNA phosphate surface is buried in interactions with polymerase during open complex formation, it is plausible to propose that GB detects the exposure of the highly negatively charged region of sigma region 1.1 in the (I₁-I₂)^‡ transition state. Release of 1.1 from its interactions with β and/or β′ subunits in the jaws (17) is required to unmask the active site and allow interactions of start site DNA with these regions of polymerase. Movement of 1.1 may be the key step for initiating untwisting and opening of promoter DNA. Clearly the step that exposes the active site is a logical target for regulation; in this proposal unmasking would occur after promoter recognition is established and DNA downstream is placed in the jaws.

Intriguingly, this step exhibits no heat capacity change, no urea effect, and only a small destabilizing effect of salt concentration (WSK, in preparation). No significant heat capacity change is expected if the overall change in surface is predominantly (∼70%) polar or charged (2, 75). The absence of a urea effect means that no net exposure or burial of polar amide ASA occurs. The temperature, [salt], urea and GB data taken together suggest to us that the exposure of 1.1 in the (I₁-I₂)^‡ transition state is coupled to the folding of another region on polymerase. A logical candidate is SI1 of β. Positioned near the top of the downstream lobe of β (Figure 9), SI1 may be partially disordered in E. coli core: only ∼50% of the expected volume is visible in the EM structure (20), and PONDR predicts that 47 of the ∼ 125 residues in the N-terminus of SI1 are disordered. If binding of 1.1 in the jaws props the downstream jaw open (15) to allow DNA downstream of −5 to enter, release of 1.1 could lead to closure of the downstream lobe on DNA and thus to ordering of SI1 in (I₁-I₂)^‡ (see schematic in Figure 9). Ordering of SI1 in the conversion of I₁ to (I₁-I₂)^‡ would be consistent with the observation that deleting SI1 does not affect the urea dependence of later steps (k_d). The exchange of interactions with region 1.1 for contacts with downstream DNA in the jaws in this step is also consistent with the small [salt]-dependence of k₂. This mechanistic hypothesis predicts that urea will facilitate the isomerization step (k₂) with the mutant ΔSI1 polymerase at λP_R. Similarly, deletion of σ⁷⁰ region 1.1 should eliminate the GB dependence of k₂, and introduce a urea dependence of k₂. These predictions are currently being investigated.

Supplementary Material

si20051123_071. Supporting Information Available.

Association kinetic data describing CR complex formation between Eσ⁷⁰ RNAP and λP_R promoter DNA as a function of urea (0, 0.21, 0.42, 0.63 m) at 17.1 °C in ${\text{BB}}_{\text{urea}}^{\text{dissoc}}$ available free of charge via the Internet at http://pubs.acs.org.

Abbreviations

ASA

water-accessible surface area

bp

base pair

DBD

DNA binding domain

Eσ⁷⁰

E. coli RNA polymerase holoenzyme

EM

electron microscopy

GB

glycine betaine

I₁

first kinetically significant intermediate

I₂

second kinetically significant intermediate

•OH

hydroxyl radical

RNAP

RNA polymerase

RP_o

open complex

wt

wild type

Appendix: Conversion of (∂lnK_obs/∂m₃)_m4,m5 to (∂lnK_obs/∂m₃)_a4,a5

Here we obtain eq 10 in the text, which relates the experimentally-determined derivative (∂lnK_obs/∂m₃)_m4,m5 at constant temperature, pressure and molalities of KCl and MgCl₂ (components 4 and 5, respectively) to the derivative (∂lnK_obs/∂m₃)_a4,a5 at constant salt activities and thereby eliminate the effect on lnK_obs of changes in salt activity brought about by changing urea or GB (component 3) concentration at constant salt molality. The basic mathematical relationship between (∂lnK_obs/∂m₃)_m4,m5 and (∂lnK_obs/∂m₃)_a4,a5 in an excess of solute components 3, 4, and 5 (relative to biopolymer participants) is:

$${\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_3}\right)}_{{\text{a}}_4,{\text{a}}_5}={\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_3}\right)}_{{\text{m}}_4,{\text{m}}_5}+{\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_4}\right)}_{{\text{m}}_3,{\text{m}}_5}{\left(\frac{\partial {\text{m}}_4}{\partial {\text{m}}_3}\right)}_{{\text{a}}_4,{\text{a}}_5}+{\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_5}\right)}_{{\text{m}}_3,{\text{m}}_4}{\left(\frac{\partial {\text{m}}_5}{\partial {\text{m}}_3}\right)}_{{\text{a}}_4,{\text{a}}_5}$$ (A1)

The additive correction terms in A1 are evaluated using experimental data. For component 4 (KCl):

$${\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_4}\right)}_{{\text{m}}_3=0}\cong {\text{SK}}_{\text{obs}4}^{\text{o}(3)}/{\text{m}}_4$$ (A2)

$$\text{where}{\text{SK}}_{\text{obs}4}^{\text{o}(3)}\equiv {\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial \text{ln}[\text{4}]}\right)}_{{\text{m}}_3=0,{\text{m}}_5}\cong {\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{lnm}}_4}\right)}_{{\text{m}}_3=0,{\text{m}}_5}$$

Based on Euler reciprocity:

$${\left(\frac{\partial {\text{m}}_4}{\partial {\text{m}}_3}\right)}_{{\text{a}}_4,{\text{a}}_5}=-\frac{\partial {\mu }_4/\partial {\text{m}}_3}{\partial {\mu }_4/\partial {\text{m}}_4}=-\frac{\partial {\mu }_3/\partial {\text{m}}_4}{\partial {\mu }_4/\partial {\text{m}}_4}$$ (A3)

$$\text{where}\partial {\mu }_3/\partial {\text{m}}_4={\text{RT}\Delta \text{Osm}}_{34}{/\text{m}}_3{\text{m}}_4$$ (A4)

$$\text{and}\partial {\mu }_4/\partial {\text{m}}_4=2\text{RT}(1+{\epsilon }_{\pm }{)}_{{\text{m}}_3,{\text{m}}_5=0}/{\text{m}}_4$$ (A5)

where ε_± = ∂lnγ_±/∂m₄ (where γ_± is the mean ionic activity coefficient of the salt)

An analogous derivation (parallel to eqs A2-5) holds for component 5 (MgCl₂) involving ${\text{SK}}_{\text{obs}5}^{\text{o}(3)}\equiv {\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial \text{ln}[\text{5}]}\right)}_{{\text{m}}_3=0,{\text{m}}_4}$ except that ∂μ₅/∂m₅ = 3RT(1+ε_±)_m3,m4=0/m₅

Combining equations A2-5 with equation 1 yields:

$${\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_3}\right)}_{{\text{a}}_4,{\text{a}}_5}\cong {\left(\frac{\partial {\text{lnK}}_{\text{obs}}}{\partial {\text{m}}_3}\right)}_{{\text{m}}_4,{\text{m}}_5}+\left(\frac{{\text{SK}}_{\text{obs}4}^{\text{o}(3)}\Delta {\text{Osm}}_{34}}{2{\text{m}}_3{\text{m}}_4{(1+{\epsilon }_{\pm })}_{{\text{m}}_3{\text{m}}_5=0}}\right)+\left(\frac{{\text{SK}}_{\text{obs}5}^{\text{o}(3)}\Delta {\text{Osm}}_{35}}{3{\text{m}}_3{\text{m}}_5{(1+{\epsilon }_{\pm })}_{{\text{m}}_3{\text{m}}_4=0}}\right)$$ (A6)

The correction is shown here for an equilibrium constant K_obs, but is applicable to either equilibrium or rate constants. Eq A6 is only applicable to the relevant experimental situation where urea, GB, KCl, and MgCl₂ are in excess of the biopolymers and to the analysis of the initial dependence of lnK_obs on solute concentration (m₃).

Analysis of data from a previous study on the [KCl] dependence of the kinetics (in 10 mM MgCl₂ at 37 °C) (1) yields Sk_{a 4} (= SK₁ + Sk₂) ≅ -4.0 ± 0.7 and Sk_{d 4} ≅ 3.2 ± 0.7 over the ranges 100 to 180 mM KCl and 120 to 200 mM KCl, respectively. A later study showed that the log-log dependence of k_a on [MgCl₂] in the absence of a 1-1 salt is approximately one-half that observed on [KCl] in the absence of MgCl₂ (Sk_{a 5} ≅ (1/2)Sk_{a 4}) and that k_d is much less dependent on [MgCl₂] than it is on [KCl] (76); thus, we assume Sk_{a 5} = -2.0 ± 0.4 and Sk_{d 5} = 0.4 ± 0.1 here. Recent experimental data shows that the dependence of k₂ on salt concentration is much smaller than the dependence of K₁ (WSK, in preparation); here, no significant errors are introduced by assuming that Sk₂ ≅ 0 and Sk_a ≅ SK₁.

In eq A4 and A6, ΔOsm₃₄/m₃m₄ = Osm(m₃,m₄) − Osm(m₃) − Osm(m₄) where Osm(m₃) and Osm(m₄) are osmolalities of two component solutions at the same molality as that of these solutes in the three component solution with osmolality Osm(m₃,m₄). (The analogous definition applies to ΔOsm₃₅.) Values of ΔOsm₃₄/m₃m₄ (0.09 ± 0.02 for urea-KCl and 0.12 ± 0.01 for GB-KCl (37)) and ΔOsm₃₅/m₃m₅ (0.32 ± 0.09 for urea-MgCl₂ and 0.6 ± 0.1 for GB-MgCl₂ were obtained by vapor pressure osmometry (M. Capp, unpublished)). For KCl solutions in the range 0.2-0.43 m, (1+ ε_±)_m3,m5=0 = 0.893±0.001 (77); for MgCl₂ in the range 0.053-0.546 m, (1+ ε_±)_m3,m4=0 = 0.99± 0.02 (calculated from published data (78)).