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1 Department of Chemistry and
2 Swedish NMR Centre, Göteborg University, SE 405 30 Göteborg, Sweden
3 Department of Chemistry and Bioscience, Chalmers University of Technology, SE 405 30 Göteborg, Sweden
Reprint requests to: B. Göran Karlsson, Swedish NMR Centre, Göteborg University, Box 465, SE 405 30 Göteborg, Sweden; e-mail: goran{at}nmr.se; fax: +46-31-773-3880.
(RECEIVED May 7, 2004; FINAL REVISION July 1, 2004; ACCEPTED July 2, 2004)
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Abstract |
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 TOP Abstract IntroductionResultsDiscussionMaterials and methodsReferences |
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GU–NH2O) of the C112S and C112A mutants were measured to 36.8 ± 1 kJ mole–1 and 26.1 ± 1 kJ mole–1, respectively, and the m-value of the transition to 23.5 ± 0.7 kJ mole–1 M–1. The difference in folding free energy (GU–NH2O) is largely attributed to the intramolecular hydrogen bonding properties of the serine O
in the C112S mutant, which is lacking in the C112A structure. Furthermore, only the unfolding rates differ between the two mutants, thus pointing to the energy of the native state as the source of the observed GU–NH2O. This also indicates that the formation of the hydrogen bonds present in C112S but absent in C112A is a late event in the folding of the apo-protein, thus suggesting that formation of the metal-binding site occurs after the rate-limiting formation of the transition state. In both mutants we also noted a burst-phase intermediate. Because this intermediate was capable of binding 1-anilinonaphtalene-8-sulfonate (ANS), as were an acid-induced species at pH 2.6, we ascribe it molten globule-like status. However, despite the presence of an intermediate, the folding of apo-azurin C112S is well approximated by a two-state kinetic mechanism.
Keywords: protein folding; intermediate; 
-sandwich; Greek key; azurin; cupredoxin
Abbreviations: ANS, 1-anilinonaphtalene-8-sulfonate • TS, Tanford -value for the transition state • I, Tanford -value for the intermediate state • CD, circular dichroism • [D], denaturant concentration • [D]50%, mid-point of denaturation • GdmCl, guanidinium chloride • I, intermediate state • N, native state • TS, transition state • U, unfolded state.
Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.04848204.
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Introduction |
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TOPAbstractIntroductionResultsDiscussionMaterials and methodsReferences |
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Recently, there has been considerable focus on proteins dominated by long-range interactions, such as structures with primarily -secondary structural elements. The all- group of proteins is dominated by -barrels and -sandwich structures which usually have Greek-key topology (Chothia et al. 1997). Its prevalence is an indication that the folding of the polypeptide into a coherent all- structural entity is limited by geometrical and/or kinetic-mechanistic factors. Clarke and coworkers (Clarke et al. 1999; Fowler and Clarke 2001) undertook the first extensive experimental folding analysis of the Greek-key immunoglobulin-like domains, where they propose a common nucleation-condensation mechanism for this group of proteins. They concluded that the folding nucleus is centered on a common structural core made up of nonlocal residues that are also important in defining the structure of these proteins. Intriguingly, nearly all (94%) -sandwich-like proteins do contain an invariant supersecondary substructure defining the core which is even more conserved than the Greek-key topology (Kister et al. 2002). Hence, analysis of the folding behavior of -sandwich proteins other than the Ig-like domains may reveal whether there also is a "super pathway" in the folding of these proteins. Herein lies our main interest in the folding of the cupredoxins, which is a structurally homogeneous but sequentially inhomogeneous family of proteins.
Our model protein for analysis of the folding of Greek-key proteins is the cupredoxin azurin from Pseudomonas aeruginosa. This is a fairly small protein of 14 kDa with 128 residues wrapped into an eight-stranded antiparallel -sandwich structure (Fig. 1
). The native protein binds a copper ion (but can also accommodate other metals) with a 5[N2S(OS)] trigonal bipyramidal coordination at the top of the tightly wound structure (Gray et al. 2000). The high hydrophobicity of the azurin core is indicated by the fact that the only tryptophan (W48) has a structured emission in the native protein resembling that of indole in cyclohexane. Its emission maximum at 308 nm is probably the least red-shifted tryptophan fluorescence found in proteins (Eftink 1991). Hence, the integrity of the native tertiary structure can uniquely be monitored by the tryptophan fluorescence at this wavelength with great sensitivity. In addition, there is a disulfide bond that stabilizes one of the two sheets, but there are no cis prolines in this protein.
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-sandwich proteins, for example, the fibronectin type III module and other Ig-like domains (Clarke et al. 1999), also display at least three-state kinetics. In contrast, there are several reports stating that apo-azurin folds via a two-state mechanism (see Pozdnyakova and Wittung-Stafshede 2003 and references therein). We find it rather unlikely for a protein of such complex topology dominated by long-range interactions to fold by a two-state mechanism, and thus decided to investigate this further. In the present study we analyzed the folding properties of two apo forms of this protein. In order to eliminate distortion of the data by metal contamination and/or cysteine oxidation of the apo form, we constructed mutants in which the metal ligating C112 residue was replaced with serine and alanine. These azurin mutants were then subjected to equilibrium and kinetic folding studies using guanidinium chloride (GdmCl) as denaturant.
As expected for a protein of such size and complex topology, we found the existence of multiphasic kinetics and molten globule-like burst phases in these two apo-mutants. Thus, we obtained evidence suggesting that apo-azurin does not fold by a two-state mechanism with observed curvature being attributed to movement of the transition state (TS) alone (Pozdnyakova and Wittung-Stafshede 2003), although a two-state approximation is a fairly reasonable approximation under the physicochemical conditions studies here. We also conclude, based on 
-value analysis, that the formation of the metal site is a late event in the folding pathway.
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Results |
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TOPAbstractIntroductionResultsDiscussionMaterials and methodsReferences |
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, and the parameters obtained by least-squares fitting of a two-state model to the data are presented in Table 1. Table 1 also contains parameters from an apo-wild-type (WT) control experiment (data not shown). The transitions are fully reversible and conform to two-state behavior irrespective of the probe monitored, indicating an apparent global conformational change in GdmCl. The mU–N values are also similar for both mutants, as well as for the apo-WT protein, with an average of 23.5 ± 0.7 kJ mole–1 M–1 for the seven independent measurements. The C112S mutant is the more stable of the two mutants, with a midpoint of denaturation ([D]50%) centered at 1.55 M and an average free energy of folding of 36.8 ± 1 kJ mole–1 for the three different measurements. This indicates a slight destabilization of C112S with respect to the apo-WT protein by 3.0 ± 2 kJ mole–1. Replacing the serine with an alanine, that is, the C112A mutant, apparently destabilizes the C112S protein by 9.9 ± 1 kJ mole–1 by shifting the midpoint of denaturation to 1.13 M.
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shows a typical trace of C112S refolded in 1.40 M GdmCl as monitored by fluorescence. At [D]<<[D]50% the amplitude of the minor phase (k2) is typically less than 10% of the total amplitude (Fig. 4). The minor phase reaches a maximum in relative total amplitude of 20% at ~1.40 M GdmCl for both fluorescence and CD data. This minor phase is also evident in the unfolding limbs within the transition region up to ~2.0 M GdmCl. At [D]<<[D]50%, however, the unfolding traces are all monophasic.
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and 5A,B), and this [D]50% does not equal the [D]50% of the major phase. Similarly, in the C112A mutant, a minor phase with nonlinearity in its dependence of k2 on [D] is also present (Fig. 5C). The relative amplitudes of the minor and major phases are similar to that for C112S, although the maximum of the minor phase (also 20% of the total phase) is shifted to lower GdmCl concentrations (1.0 M; data not shown). Furthermore, there were no changes in the relative amplitudes of the two phases when refolding C112S from different start concentrations of denaturant (the initial conditions test; see Wallace and Matthews 2002), so the slow phase is probably not a sequential process. The latter experiment also rules out the possibility that significant residual structure is a cause of the observed biphasic kinetics.
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, A for the fluorescence data and B for the CD data. The dependence of the natural logarithm of the rate constant associated with the major phase (k1) on GdmCl concentration is clearly nonlinear for both mutants. This curvature was not caused by transient aggregation (Silow and Oliveberg 1997b) as determined by additional refolding experiments at 1, 2, 5, and 10 µM protein concentrations (data not shown). The electrolyte nature of the denaturant as causing the curvature (Monera et al. 1994; Makhatadze 1999) was ruled out by preliminary data obtained by urea denaturation (A. Sandberg, unpubl.). Nonlinearity in the activation free energies of folding has been observed in a number of proteins and is usually interpreted in terms of movement of the TS across a broad energy barrier as the physicochemical properties of the solvent change upon addition of denaturant (Matouschek and Fersht 1993; Silow and Oliveberg 1997a). Alternatively, downward curvature in the Chevron plots may also be due to a denaturant-dependent switch between two or a few distinct TSs on a sequential pathway (Sanchez and Kiefhaber 2003).
Inspection of the extrapolated start values of the fluorescence refolding traces as shown in Figure 5D reveals the presence of a burst-phase intermediate becoming accumulated at denaturant concentrations below 0.70 M GdmCl. This burst phase is not evident in the CD data (Fig. 5E), possibly because this data is generally noisier. The existence of refolding intermediates is also implicated from the additional curvature in the refolding limb (Cf) of the major phase compared to the unfolding limb (Cu) in the Chevron plots. For the fluorescence data, the refolding limb had a squared parameter of –(1.45 ± 0.2)[D]2, whereas for the unfolding limb the same value was – (0.99 ± 0.2)[D]2. This additional curvature of –0.46 ± 0.2 is not significant enough to allow for equation 4 to converge properly, however. For the CD data the additional curvature in the refolding limb was well within experimental error.
Using the rate of unfolding kinetics and the equilibrium data, it is possible to calculate the expected refolding kinetics by using the requirements of a two-state transition (Jackson and Fersht 1991a). The expected folding rate in water is determined by:
 | (1) |
and the full expression for the expected two-state refolding rates is (Jackson and Fersht 1991a; Silow and Oliveberg 1997a):
 | (2) |
Hence, using this expression to calculate a theoretical Chevron profile (as kobs = kf, calc + ku) one expects the kinetic refolding m-value and curvature to be equal to those obtained from the kinetic unfolding data (here mu and Cu, respectively) because unfolding usually approximates two-state. This is shown in Figure 5, A and B, for the major phases of the C112S mutant as dashed lines. In our opinion, a two-state mechanism with curvature-ascribed movement of the TS is a good approximation for the folding of this mutant, although it appears that intermediates may be present.
Kinetic analysis of C112A folding
In order to probe the degree of involvement of the copper site in the folding of apo-azurin, fluorescence folding kinetics were collected for the C112A mutant. The results are nearly identical to those obtained with C112S, although the C112A mutant is somewhat more unstable, as reflected in the increased unfolding rate (see Table 2). There is also a more pronounced curvature in the dependence of the natural logarithm of the refolding rate constant k1 on denaturant concentration when fitting equation 3 to the data: The re-folding limb had a parameter of – (2.35 ± 0.2)[D]2 defining the curvature, which is significantly more than the – (0.88 ± 0.1)[D]2 for the unfolding limb. This additional curvature of –1.46 ± 0.3 is also evident from the two-state approximation of refolding data (dashed line in Fig. 5C). A function describing the folding behavior with intermediates (equation 4), which ascribes all of the additional curvature to the accumulation of an intermediate, did converge properly in this case without affecting the quality of the fit (data not shown; this fit is virtually indistinguishable from the solid line in Fig. 5C), presumably because of the more pronounced curvature (parameters in Table 2). A burst phase is also evident from the extrapolated start values presented in Figure 5F. This intermediate starts to accumulate at approximately the same GdmCl concentration as does the C112S intermediate (~0.70 M).
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), and its amplitude increases exponentially (or sigmoidally) at denaturant concentrations under approximately 0.70 M GdmCl (data not shown). Furthermore, these refolding traces could only be described satisfactorily by a triple-exponential decay (Fig. 6). Notably, both mutants have very similar ANS-binding kinetics. In addition, there was no detectable ANS fluorescence at [D] above 
0.7 M GdmCl or in any of the unfolding traces measured.
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, with a strong increase in fluorescence quantum yield of the ANS probe at this pH (Fig. 7A,B insets). The fluorescence spectrum of the C112S mutant at pH 2.6 resembles a mixture of native and denatured states in having a significantly red-shifted yet structured emission. A red shift is expected upon contact of the indole side chain with hydrogen-bonding partners as the hydrophobic core relaxes around the fluorophore, but the blue-shifted emission suggests a persistent hydrophobic environment. Molten globules may indeed have significant hydrophobic interactions resistant to solvent penetration (Denisov et al. 1999). The C112A mutant, on the other hand, has an unstructured fluorescence spectrum at pH 2.6 that is almost identical to solvent-exposed tryptophan with a broad peak at 355 nm, and is thus indistinguishable from denatured protein. That these mutants differ in this respect may be due to the different hydrogen-bonding properties of the serine and alanine residues of the copper site, which is located directly above the tryptophan (Fig. 1). The original spectra of both mutants are regained upon increasing the pH to 7.0 again (data not shown).
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Discussion |
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Compared to wild-type apo protein, the effect of the cysteine to serine substitution is very small, resulting in a GU–N of only 3.0 ± 2 kJ mole–1. A larger effect was seen when removing the hydrogen-bonding properties of the 112 residue: 9.9 ± 1 kJ mole–1 was lost upon the serine to alanine substitution. We note a significant discrepancy of 12 ± 2 kJ mole– between our data for apo-WT and data previously published for the same species (Pozdnyakova et al. 2001). We believe our data to be a more correct estimate, as only 10 data points were used in defining the transition curve in the earlier publication, of which only three fell within the actual transition region. Our data are also in accord with that obtained in this study for the C112S mutant with several different optical probes, and this species should realistically be nearly isoenergetic with apo-WT.
Folding pathway
The importance of the hydrogen bonds to the Ser112 O in the folding process can be elucidated with -value analysis (Fersht 1999). However, it is important to ascertain that the removal of these hydrogen bonds does not affect the TS. An estimate of the relative exposure of the hydrophobic surfaces of the two TSs relative to the folded state can be obtained from the Tanford -value (TS). For both mutants the TS obtained for the fluorescence data is close to 0.4, indicating that significant compactness is lacking, and suggesting a very similar nature of the TS in both mutants.
A comparison of the folding limbs in Figure 5 clearly demonstrates that nearly all of the difference in folding stability between the mutants lies on the kinetic unfolding limb. Indeed, the f-value for the fluorescence data at 0.7 M, where the contribution of folding intermediate is negligible, is virtually zero (f = 0.04). This means that the hydrogen bonding to the O of Ser112 is a late event which occurs after the formation of the TS in the folding pathway. Based on this, it follows that the metal site is not a stabilizing factor in early folding events, but rather forms very late in the folding process.
Folding by the minor phase
There is a significant heterogeneity in the refolding reaction of both mutants. That both fluorescence and CD report similarly on the slow-folding population indicates that it is not an artifact of fluorescence, nor is it a local folding event. Evidence of this refolding heterogeneity comes mainly from three observations. First, there is biexponential kinetics at all [D]. Second, this results in two vertically and horizontally displaced Chevron curves. Third, there is a nonlinear dependence of the relative amplitudes of the two phases. The initial conditions test (Wallace and Matthews 2002) indicates no dependence on start conditions of folding, which implies that there is probably not a significant equilibrium between different macroscopic compact states in the denaturing solutions used for the refolding studies, despite 2–3 M GdmCl being a bad solvent for polypeptide chains (Dill and Shortle 1991). Although most of these observations may be attributed to the existence of a parallel folding pathway (Wallace and Matthews 2002), the horizontally displaced Chevron profile of the slow phase relative to the fast phase cannot. Instead, it suggests the presence of an alternative species folding in parallel.
The occurrence of a slow kinetic phase (k2) with a maximum relative amplitude of ~20% somewhat displaced from the midpoint of the U 
N folding transition of the faster kinetic phase (k1) is consistent with isomerization of cis proline residues being rate-limiting for the folding of a fraction of the population (Kiefhaber et al. 1992). However, that the rate of the isomerization reaction is dependent on [D], and the observation that its amplitude decreases in magnitude under increasingly stronger native conditions, indicate that the rate of conversion of cis into trans configuration is increased by the conformational strain exerted by the native or intermediate structure. Such folding-assisted proline isomerization has been observed previously, for example in the refolding kinetics of chymotrypsin inhibitor 2 (Jackson and Fersht 1991b). As the appearance of the burst-phase intermediate occurs only after the relative amplitude of the slow phase has lost most of its magnitude, the molten globule-like intermediate appears not to be a result of incorrect isomers impeding the formation of the correct tertiary interactions.
Three-state folding of the major phase
Interestingly, the burst-phase intermediate in the fluorescence implies a compact core with high hydrophobicity, because fluorescence at 308 nm is unusual in proteins as it indicates a very small Stokes shift (Eftink 1991). Even though a burst phase is not resolvable in the noisier CD data, such significant core packing also implies a satisfied hydrogen-bonding pattern in this region and, hence, significant secondary structure. This is corroborated by its ability to bind ANS, which selectively monitors the I N transition. Burst-phase intermediates are, however, controversial in that they may be a result of experimental artifacts (Krantz et al. 2002), but the concomitant formation of a species capable of binding ANS suggests that it is a proper intermediate. It is also possible that the acid-induced molten globules identified in this study (Fig. 7) resemble this kinetic intermediate, as has been demonstrated for several other proteins (Arai and Kuwajima 2000), but the lack of fluorescence of this state for the C112A mutant suggests that it is more compact in the high-salt conditions of guanidinium refolding experiments. Because of the identical burst appearance of a fluorescent intermediate and the identical disappearance of the molten-globule intermediate in both of these mutants, there is no reason to suspect that the intermediate differs between the two mutants. This reasoning is corroborated by the -value analysis, which points to the copper site as being detached from the folding of the rest of the -sandwich structure. From our data it is not discernible whether the burst-phase species and the molten globule-like species are the same, but judging from their co-appearance they probably are.
The kinetic parameters from fitting indicate that there is curvature of comparable magnitude in the unfolding limbs of both mutants (Table 2). Whether this movement of the TS is due to continuous Hammond behavior across broad energy barriers or to a discrete change in the rate-limiting step of the reaction remains to be investigated. Nevertheless, for the fluorescence data of both mutants there is additional curvature in the refolding limbs that, in light of the presence of intermediates, may not be attributed to the movement of the TS along the reaction coordinate a priori. This warrants analysis of these Chevron profiles with rate equations that take intermediates into account (Parker et al. 1995). However, the "kink" resulting from the shift in the rate-limiting step as the intermediate accumulates is not well resolved in the C112S mutant, and a three-state analysis of the kinetic parameters is hampered by this fact. The C112A mutant is, on the other hand, less stable and kinetic data can be collected at even stronger native-like conditions (0.2 M GdmCl as opposed to 0.3 M GdmCl for C112S), which may be one of the factors contributing to the kink being better resolved in this mutant. It would thus appear from the data presented above that only the C112A mutant reports qualitatively on the three-state folding mechanism of azurin. As the Tanford -value can be regarded as a reaction coordinate by reflecting the progression of compaction during folding, an energy diagram for the U I N scheme can be constructed for the folding of this mutant (Fig. 8).
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Materials and methods |
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Expression and purification
The two azurin mutants in this study (C112S and C112A) have been described (Sandberg et al. 2002). They were expressed and purified by a method analogous to wild-type azurin, which is described by Karlsson et al. (1989), with the exception that the cultivation medium was not supplemented with 100 µM CuSO4 and all additions of CuSO4 during the purification procedure were omitted. Expression yields of the apo-mutants were comparable to wild-type protein using the cited protocol, that is, approaching 100 mg per L culture.
Equilibrium data collection and analysis
Fluorescence equilibrium denaturation was monitored by excitation at 295 nm and emission at 308 nm on a SPEX Fluorolog 
2 spectrofluorometer at 20°C. The concentration of protein was 5 µM in 0.5 mM potassium phosphate buffer (pH 7.0). Different concentrations of GdmCl were obtained by successive addition of azurin denatured in 7 M GdmCl to a 0 M GdmCl solution of native azurin. Each point was incubated at 20°C for several minutes prior to collection of the emission.
An apo-WT equilibrium curve (data not shown) was measured by first incubating 30 µM Zn2+-coordinated WT azurin (the A280:A628 absorption ratio for this sample (which reflects the amount of bound copper) was zero, thus indicating 100% Zn2+) in 5.0 M GdmCl and 1.0 mM EDTA for ~20 min. The formed apo-azurin was then refolded by a 1:10 dilution to native conditions (0.5 M GdmCl) prior to titration by successive addition of GdmCl, with 1 mM EDTA present throughout the experiment. All holo-species were converted into the apo form by this treatment as judged from the obtained equilibrium curve. Dilution of the fluorescence intensity by the GdmCl additions was corrected for prior to curve fitting. The buffer was 0.5 mM potassium phosphate (pH 7.0), and the temperature was 20°C. Excitation was set to 295 nm, and the emission was collected at 308 nm.
Equilibrium CD data were obtained on a Jasco J-810 spectropolarimeter thermostated at 20°C. All points were prepared separately and incubated at room temperature overnight before data collection. Far-UV CD was collected at 220 nm using a 3-mm path-length cuvette, and near-UV CD was monitored at 281 nm using a 1-cm cuvette. The protein concentration was 2.5 µM for the far-UV measurements and 0.10 mM for the near-UV measurements. The buffer was 0.5 mM K phosphate (pH 7.0).
Parameters describing the equilibrium unfolding transitions were obtained as described (Santoro and Bolen 1988; Clarke and Fersht 1993) using the Igor Pro program (Wavemetrics).
The acid-induced molten globule-like states and their ability to bind ANS were analyzed in a SPEX Fluorolog 2 spectrofluorometer thermostated at 20°C. The protein concentration was 5 µM, the ANS concentration was 0.5 mM, and the buffer was 0.5 mM potassium phosphate. Spectra of azurin were recorded by excitation at 295 nm and emission collected from 300 to 460 nm, and spectra of ANS were recorded by excitation at 388 nm and emission collected from 400 to 550 nm. The pH was set to 7.0 for the native azurin references, and to 2.6 for the acid-induced intermediate states.
Kinetic data collection and analysis
Kinetic unfolding and refolding traces were obtained using a BioLogic µSFM-20/MPS-52 stopped-flow apparatus. The instrumental dead time was 10.7 msec in all experiments. For the fluorescence, the observation head was connected to a Bio-Logic MOS-250 spectrometer, and the emission was recorded at 308 ± 2 nm with excitation set to 295 ± 2 nm. However, for the initial conditions test and the concentration-dependency test, the fluorescence was collected through an interference filter at 310 ± 2 nm (Melles Griot) using a Bio-Logic MOS-450 spectrometer coupled to the same stopped-flow instrument. CD data were collected at 220 nm also using the Bio-Logic MOS-450 spectrometer.
Refolding and unfolding of azurin monitored by fluorescence were analyzed by a 1:10 dilution of a 20 µM azurin solution to different end concentrations of denaturant. For the concentration-dependency test, the start concentration of protein was instead made 15, 20, 50, 80, 110, 140, 170, 200, 250, and 300 µM. Fluorescence refolding was initiated from a 3.0 M and a 2.0 M GdmCl solution for the C112S and C112A mutants, respectively, except for the initial conditions test of C112S, which was refolded in 0.5 M from different start conditions (4.0, 3.7, 3.4, 3.1, 2.9, 2.6, 2.3, 2.0, and 1.7 M GdmCl). The CD measurements were measured similarly, albeit with a starting concentration of 200 µM azurin. Prior to refolding, the denatured protein was incubated at room temperature for at least 2 h.
For the refolding experiments in the presence of ANS, both buffers to be mixed were made 0.5 mM ANS. The excitation was set to 388 nm, and the emission at 480 ± 5 nm was collected through an interference filter (Melles Griot) using the Bio-Logic MOS-450 configuration of the instrument. The end concentration of protein was 5 µM in these experiments, and the buffer was 0.5 mM potassium phosphate (pH 7.0).
The unfolding and refolding traces were averaged (typically five traces for the fluorescence and 15 traces for the CD data) and fitted to an exponential decay with one to three exponents using the Igor Pro program (Wavemetrics). The start values were obtained by extrapolation to time zero. Chevron plots (ln kobs versus [GdmCl]) were constructed, and a complete description of the observed rate constants for the major phases was obtained by least squares fitting of the following equation to the experimental data:
 | (3) |
The curvature parameter Cf,u reflects the nonlinear dependence of the activation free energy of folding and unfolding on denaturant concentration. The Chevron profiles were also analyzed in terms of a U I N three-state model assuming a rapid U I pre-equilibrium (Parker et al. 1995):
 | (4) |
Here, kni denotes unfolding from the native state to the intermediate state, and kin denotes folding from the intermediate state to the native state. The m values associated with these transitions are similarly denoted. KU–I is equivalent to the rapid pre-equilibrium constant of the I U transition. Because of this rapid pre-equilibrium, an on-pathway U I N model is kinetically indistinguishable from an off-pathway I U N model (Fersht 1999). However, a distinction between on- and off-pathway models can sometimes be made on the basis of how reasonable the ln kobs H2O is for an off-pathway model for which kobsH20 = KU–I x kinH2O (Capaldi et al. 1999). The curvature in the dependence of ln kobs on [D] is reflected in the C[D]2 parameter. This curvature was set equal in both unfolding and refolding limbs during the curve fitting, thereby ascribing additional curvature to refolding intermediates becoming accumulated.
The free energies of folding from the kinetic data were taken as –RTxln(ku/kf) or –RTxln(kni/kin) and the Tanford -values, reflecting buried surface area of the TS relative N, were calculated as:
 | (5) |
For the three-state model, the same parameter was instead calculated as:
 | (6) |
The corresponding parameter relating the intermediate (I) to N was obtained by:
 | (7) |
All m parameters were taken at the equilibrium [D]50% as their respective first derivatives in folding free energies on denaturant concentration (
G/ [D]) on account of the curvature in the Chevron plot (Fersht 1999).
The f value is a measure of the relative energetic sensitivity of the transition state and the unfolded state as resulting from the mutation (Fersht 1999). It is calculated as:
 | (8) |
Here, GU–TS = RT ln kf/kf'(the prime denotes mutant parameter), where kf is the folding rate. The height of the TS barrier in Figure 8 was estimated using the folding rate and the following equation (Fersht 1999):
 | (9) |
where kB is the Boltzmann constant and h is the Planck constant.
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Acknowledgments |
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References |
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