Titration Properties and Thermodynamics of the Transition State for Folding: Comparison of Two-state and Multi-state Folding Pathways

J. Mol. Biol. (1996) 264, 377–389

Titration Properties and Thermodynamics of the Transition State for Folding: Comparison of Two-state and Multi-state Folding Pathways Yee-Joo Tan, Mikael Oliveberg and Alan R. Fersht* Cambridge Centre for Protein Engineering, Hills Road Cambridge CB2 2QH, UK

CI2 folds and unfolds as a single cooperative unit by simple two-state kinetics, which enables the properties of the transition state to be measured from both the forward and backward rate constants. We have examined how the free energy of the transition state for the folding of chymotrypsin inhibitor 2 (CI2) changes with pH and temperature. In addition to the standard thermodynamic quantities, we have measured the overall acid-titration properties of the transition state and its heat capacity relative to both the denatured and native states. We were able to determine the latter by a method analogous to a well-established procedure for measuring the change in heat capacity for equilibrium unfolding: the enthalpy of activation of unfolding at different values of acid pH were plotted against the average temperature of each determination. Our results show that the transition state of CI2 has lost most of the electrostatic and van der Waals’ interactions that are found in the native state, but it remains compact and this prevents water molecules from entering some parts of the hydrophobic core. The properties of the transition state of CI2 are then compared with the major folding transition state of the larger protein barnase, which folds by a multi-state mechanism, with the accumulation of a partly structured intermediate (Dphys or I). CI2 folds from a largely unstructured denatured state under physiological conditions via a transition state which is compact but relatively uniformly unstructured, with tertiary and secondary structure being formed in parallel. We term this an expanded pathway. Conversely, barnase folds from a largely structured denatured state in which elements of structure are well formed through a transition state that has islands of folded elements of structure. We term this a compact pathway. These two pathways may correspond to the two extreme ends of a continuous spectrum of protein folding mechanisms. Although the properties of the two transition states are very different, the activation barrier for folding (Dphys : ‡) is very similar for both proteins. 7 1996 Academic Press Limited

*Corresponding author

Keywords: chymotrypsin inhibitor 2; barnase; heat capacity; activation energy; solvation

Introduction A complete understanding of the folding pathway of a protein requires the determination of the structures of all conformational states on the Present address: M. Oliveberg, Department of Biochemistry, Chemical Center, Box 124, 221 00 Lund, Sweden. Abbreviations used: CI2, chymotrypsin inhibitor 2; m, ionic strength; GdmCl, guanidinium chloride. 0022–2836/96/470377–13 $25.00/0

pathway, which are the denatured state, possible intermediates, the native state, as well as the transition states linking them (Fersht, 1995a). Stable conformations can be studied at equilibrium or in the pre-equilibrium of the folding reactions. The properties of the high energy and metastable transition state, on the other hand, can be inferred only from kinetic studies. A successful way for studying the transition state is the protein engineering method, which allows the interactions in the transition state to be mapped out at the level 7 1996 Academic Press Limited

378

Figure 1. The pH-dependence of the refolding and unfolding kinetics of CI2 at T = 298 K and different ionic strengths (m). The refolding kinetics were obtained from mixing denatured protein in 32 mM HCl with high pH buffers to induce refolding, and the unfolding kinetics were obtained from mixing folded protein in water with low pH buffers to induce unfolding. The rate constants have units of s−1. A, The rate constants of the fastest refolding phase, kf1obs (W), and the fastest unfolding phase, kulobs (w), at m = 50 mM. The rate constants of the fastest refolding phase, kf1obs (R), and the fastest unfolding phase, kulobs (r), at m = 200 mM. B, The changes in the amplitudes of these phases with pH at m = 50 mM (Af1 (W) and Aul (w)) and at m = 200 mM (Af1 (R) and Aul (r)).

of individual residues (Matouschek et al., 1989; Fersht et al., 1992; Fersht, 1995a). By this method, the transition states for the folding of two proteins, CI2 and barnase, have been characterised extensively in this laboratory (Serrano et al., 1992; Itzhaki et al., 1995). The folding pathways of these two proteins appear to be distinctively different: CI2 folds by a two-state process (Jackson & Fersht, 1991a), while barnase folds by a multi-state process from its denatured state with the accumulation of one or more intermediates (Matouschek et al., 1990; Oliveberg & Fersht, 1996b). Tertiary and secondary structure are formed simultaneously in the folding of CI2 (Otzen et al., 1994) whereas certain secondary structural elements in barnase form before tertiary (reviewed by Fersht, 1993). To complement what is already known about the transition states of these two proteins at the level of

Expanded and Compact Folding Pathways

individual residues from the protein engineering method, we investigate here how the gross properties of the transition state respond to changes in external parameters such as pH, temperature and chemical denaturants. The pH-dependence of the activation energy for folding/unfolding gives the titration properties of the transition state and is a reflection of the extent of electrostatic interactions in the transition state. The temperature-dependence of the activation energy yields the entropy, enthalpy and heat capacity of the transition state. These thermodynamic properties reflect the flexibility of the polypeptide backbone and side-chains, as well as the solvation of protein moieties in the transition state. The slope of the activation energy versus denaturant concentration is related to the difference in the exposure of hydrophobic surfaces in the transition state and the native/denatured state. Finally, we compare the titration and thermodynamics properties of the transition state and Dphys, the denatured state under physiological conditions, for the folding of CI2 and barnase. The folding pathway of CI2 can be described as an ‘‘expanded’’ pathway, since it involves a Dphys with very little native or compact structure, which is energetically similar to the denatured state in a high concentration of denaturant (D), and a transition state with few fully formed interactions. Barnase, on the other hand, folds by a ‘‘compact pathway’’, where there is a significant amount of native interactions in both Dphys and the transition state. The folding pathways of CI2 and barnase may be taken to represent two extreme ends of a continuous spectrum of folding mechanisms, where there is a variation in the degree of consolidation of structures in the Dphys and transition state for the folding of different proteins. Despite these differences in folding characteristics of the two proteins, the activation barrier for folding under physiological conditions (Dphys : ‡), such as T = 298 K, is very similar for both proteins.

Results Refolding kinetics at 298 K The refolding of CI2 consists of a very fast phase and a few slower phases. The fastest phase corresponds to the folding of denatured species with all the proline residues in the same configurations as in the native state and constitute 080% of the total change in fluorescence. The rate constant of the fastest refolding phase (kf1obs ), from about pH 2, increases with increasing pH until around pH 5, where the pH-dependence levels off and kf1obs reaches a maximum of 067 s−1 (Figure 1A). The amplitude of this phase, Af1 , decreases with decreasing pH as the occupancy of the native state decreases (Figure 1B): at 298 K, the midpoint for the unfolding transition is pH 1.5 for

379

Expanded and Compact Folding Pathways

m = 50 mM and pH 1.0 for m = 200 mM (Tan et al., 1995). Unfolding kinetics at 298 K The unfolding of CI2 is also dominated by a major phase with rate constant of 02 s−1 at pH 1.3, m = 50 mM. The rate constant of this fastest phase (kulobs ) has only a small dependence on pH (Figure 1A). The amplitude of this phase, Aul , decreases when the pH is increased because the occupancy of the denatured state decreases with increasing pH (Figure 1B). Calculating kul and kf1 from k1obs (=kul + kf1 ) for a two-state process It has been demonstrated previously that the folding of CI2 can be described by a simple two-state mechanism with cis-trans isomerization in the denatured state (Jackson & Fersht, 1991a): Scheme (1)

Figure 2. The pH-dependence of kf1 (continuous line) and kul (dotted line) calculated from the pH-dependence of the observed rate constant at T = 298 K, k1obs (w), using equations (3) and (4). At a particular pH, the observed rate constant (k1obs ) for the reaction Ntrans F Dtrans is the sum of the forward and reverse rate constants, i.e. k1obs = kul + kf1 , but k1obs can be deconvoluted into kul and kf1 using equations (3) and (4) when the equilibrium constant, K(pH) = [Dtrans ]/[Ntrans ], is known.

Ntrans F Dtrans F Dcis where Dtrans is the denatured state with all its proline residues in a trans conformation, Dcis is the denatured state with at least a proline in a cis conformation, and Ntrans is the native state. The observed rate constant (k1obs ) for the reaction Ntrans F Dtrans is always the sum of the forward and reverse rate constants, which in this case are kul and kf1 : obs 1

k

K=

= kul + kf1

(1)

[Dtrans ] kul = [Ntrans ] kf1

(2)

At pH-values well below the transition region, where mainly the denatured state is present under equilibrium conditions, the observed rate constant (k1obs ) is dominated by the unfolding rate constant (k1obs1kul ). Conversely, k1obs1kf1 at higher pH-values where mainly the native state is populated. In the transition region, where both the native and the denatured protein are present under equilibrium, then both kul and kf1 will contribute to k1obs . As shown in Figure 1A, the observed relaxation from the unfolding (kulobs ) and refolding (kf1obs ) experiments are the same in the overlapping pH region. Using equations (1) and (2), k1obs can be separated into kul and kf1 by:

kul =

[Dtrans ] + [Dcis ] [Dcis ] and Kiso = = 0.3 [Ntrans ] [Dtrans ]

(Jackson & Fersht, 1991a). Using equations (3) and (4), values of kul (pH) and kf1 (pH) at T = 298 K were calculated from k1obs (pH) and DGD−N (pH) determined previously by thermal denaturation experiments (Tan et al., 1995). The results for m = 50 mM are shown in Figure 2. pH-dependence of the major refolding/unfolding reaction and its relationship to the titration of the transition state The pH dependence of the free energy difference between two protein conformations, A and B, is related to the difference in the number of bound protons between these two conformations (Tanford, 1968, 1970): 1DGA−B (pH) = 2.3RT[QA (pH) − QB (pH)] 1pH = 2.3RTDQA−B (pH)

k1obs 1+K

(3)

K × k1obs 1+K

(4)

kf1 =

KD−N , 1 + Kiso

where: KD−N =

and the ratio of kul and kf1 is determined by the equilibrium constant, K, between Ntrans and Dtrans : K=

Owing to the cis-trans isomerization, K in equations (3) and (4) is given by:

(5)

where DGA−B is the difference in the free energy between states A and B. QA (pH) and QB (pH) are the number of moles of protons bound to per mole of states A and B, respectively, and DQA−B (pH) is the

380

Expanded and Compact Folding Pathways

change in number of protons taken up on the transition from B to A. The rate constants for the major refolding/unfolding reaction are related to the free energy difference between the denatured/native state and the transition state by transition state theory: k=

0

1

kB T − DG ‡ exp h RT

(6)

where kB is Boltzmann constant, h is Planck’s constant, T is the absolute temperature, R is the gas constant and DG ‡ is the activation energy. Therefore, the pH-dependence of log kf1 (pH) (or log kul (pH)) is directly proportional to the pHdependence of DG‡−D (pH) (or DG‡−N (pH)), which, in turn, is related to the degree of protonation in the different states (Oliveberg & Fersht, 1996a). For example, from the refolding rate constants: 1 log kf1 (pH) 1 1DG‡−D (pH) = − 1pH 2.3RT 1pH = − DQ‡−D (pH)

(7)

where DQ‡−D is the number of moles of protons taken up going from the denatured state to the transition state. Using equation (7), DQ‡−D (pH) and DQ‡−N (pH) can be calculated from the values of kf1 (pH) and kul (pH) obtained above. The protonation of a state is related to the pKA-values of residues in the state, for example, DQD−N can be calculated directly from pKANi and

pKADi , the pKA-values of acidic residues in the native and denatured states, respectively (Oliveberg et al., 1995a): Ni

n

Di

n 10pH−pKA 10pH−pKA −s pH−pKNi pH−pKDi A A i = 1 1 + 10 i = 1 1 + 10

DQD−N (pH) = s

For CI2, fDQ‡−N d(pH), the integral of DQ‡−N with respect to pH, i.e. area under the DQ‡−N (pH) curve, is positive just like fDQD−N d(pH) (Table 1). The fDQ‡−N d(pH) is the additional free energy required to protonate native state compared with the transition state, therefore, the average pKA-values of acidic residues in the transition state are greater than those in the native state. On the other hand, fDQ‡−D d(pH) is negative because the average pKA-values of acidic residues in the transition state are lower than those in the denatured state (Table 1). Refolding rate constants as a function of temperature The rate of refolding at pH 6.3 (m = 50 mM) was followed over a wide range of temperatures below the midpoint for the thermal unfolding transition (Tm ), which is 358 K at this pH. The rate constant of the major refolding reaction, kf1obs , increases with temperature until it reaches a maximum value of 0160 s−1 at T = 323 K, then it decreases as the temperature is raised further (Figure 3). The temperature-dependence of kf1obs reflects the tem-

Table 1. Thermodynamic parameters (pH 6.3, 298 K) and protonation energies (298 K) of CI2 from refolding, unfolding and equilibrium experiments Refolding (‡ − D) DCp (cal mol−1 K−1 ) DH(298 K) (kcal mol−1 ) DS(298 K) (cal mol−1 K−1 ) DG(298 K) (kcal mol−1 ) 5,0 f1.3 DQd(pH)(50 mM) (kcal mol−1 ) 5,0 f1.0 DQd(pH)(200 mM) (kcal mol−1 ) a

−490210

c

Unfolding (‡−N) d

Unfolding -Refoldinga

Equilibrium (D − N) e

‡/N(%)b

970230

970240

51

24.020.7

2723e

—

−7.020.7g

480220 (4502130)f 36.820.7d (3122)f 47.322.4g

54.322.5

5926e

13

14.920.1g

22.720.1g

7.820.1

—

−1.9i

4.9i

6.8

7.620.1h (9.221.5)e 6.8i

−1.6i

4.9i

6.5

6.5i

25

12.820.2c

28

Column 2 minus column 1. Column 1 divided by (column 3). c From ln(kf1obs /T ) against 1/T (equation (10), Figure 3). d From the linear fit of the plot of DH‡−N against the mean temperature of measurement (Figure 5). e DCp,D−N was obtained from the slope of DHD−N (Tm ) against Tm at different pH values; DHD−N (Tm ) was extrapolated to 298 K in the absence of denaturant from measurements in 3 to 5 M urea and Tm between 342 K and 348 K (see Tan et al., 1995). Error in DGD-N was calculated as described by Matouschek et al. (1994). f From ln(kulobs /T ) against 1/T (equation (10), Figure 4). The errors are large because the curvature is only marginal in the experimentally accessible temperature range. g DS ‡ and DG ‡ were calculated from the refolding and unfolding rate constants (pH 6.3, 298 K) using equation (6). h From GdmCl denaturation curves (Itzhaki et al., 1995). i Areas under the DQ‡−D , DQ‡−N and DQD−N against pH curves. b

(8)

381

Expanded and Compact Folding Pathways

Figure 3. The plot of ln(kf1obs /T ) against 1/T, obtained from pH-jump at pH 6.3, shows a strong curvature and fits well to equation (10) to give DCp,‡−D = −490(210) cal mol−1 K−1.

perature-dependence of the free energy of activation (DG‡−D ) (Oliveberg et al., 1995b): DG‡−D (T ) = DH‡−D (T ) − TDS‡−D (T ) = DH‡−D (To ) + DCp,‡−D (T − To ) − T[DS‡−D (To ) + DCp,‡−D ln(T/To )]

(9)

where T is the absolute temperature, To is a standard temperature (e.g. 298 K), DS‡−D (To ) is the activation entropy at To , DH‡−D (To ) is the activation enthalpy at To and DCp,‡−D is the heat capacity difference between the transition state (‡) and the denatured state (D). When the heat capacities of the transition state and the denatured state are different, i.e. DCp,‡−D$0, then the plot of DG‡−D against T will be non-linear (Oliveberg et al., 1995b). At the temperature where DG‡−D /RT is minimal, the refolding rate constant (kf1obs ) reaches a maximal value, and below and above this temperature, the rate constant decreases. Substitution of equation (9) into equation (6) gives (cf Hagerman & Baldwin, 1976; Pohl, 1976; Oliveberg et al., 1995b): ln

01 $ 01

kf k DS (T ) DH‡−D (To ) = ln B + ‡−D o − T h R RT −

0 1%

DCp,‡−D (T − To ) DCp,‡−D T + ln RT R To

(10)

The fit of equation (10) to the plot of ln(kf1obs /T ) against 1/T (Figure 3) gives: DCp,‡−D = −490(210) cal−1 mol−1 K−1, DH‡−D (298 K) = 12.8(20.2) kcal mol−1, DS‡−D (298 K) = −7.4(20.6) cal−1 mol−1 K−1. These values differ slightly from those previously reported (Jackson & Fersht, 1991b; Oliveberg et al., 1995b) because the data here are obtained over a much more extended temperature range and, hence, the analysis of the ln(kf1obs /T ) against 1/T plot is more accurate.

Figure 4. A, The plot of ln(kulobs /T ), obtained from GdmCl-jump at pH 6.3, against 1/T fits equally well to a linear function (continuous line) and equation (10) (broken line). The fit of the plot to equation (10) gives: DCp,‡−N of 450(2130) cal−1 mol−1 K−1. B, The residuals for fitting to linear function (R) and equation (10) (r).

Unfolding kinetics as a function of temperature obtained by GdmCl-jump In order to obtain the temperature-dependence of the fastest unfolding phase (kulobs ), the unfolding reaction at pH 6.3 was carried out by GdmCl-jump stopped-flow experiments at different temperatures (293 K to 324 K). At a given pH, the plot of ln(kulobs /T ) against 1/T is linear if there is no difference between the heat capacities of the native and transition states, i.e. DCp,‡−N = 0. On the other hand, if DCp,‡−N$0, there will be a curvature in the temperature-dependence of kulobs (cf curvature in kf1obs and equation (10)). The plot of ln(kulobs /T ) against 1/T fits equally well to equation (10) and a linear function (Figure 4A); the residuals for both fit are similar (Figure 4B). The fit of the data to equation (10) gives: DCp,‡−N of 450(2130) cal−1 mol−1 K−1, DH‡−N (298 K) = 31(22) kcal mol−1, DS‡−N (298 K) = 27(25) cal−1 mol−1 K−1. DCp,‡−N obtained from activation enthalpies at different pH To obtain the activation enthalpies of unfolding at a fixed pH, the unfolding rate constants were

382

Expanded and Compact Folding Pathways

Figure 5. A, Eyring plots of the rate constant of the fastest unfolding phase (kulobs ) at different pH (m = 50 mM); pH 1.6 (W), pH 1.8 (w), pH 2.0 (R), pH 2.2 (r), pH 2.4 (Q), pH 2.6 (q) and pH 2.8 (R). At each pH, the data were fitted to a straight line whose slope is equal to (−DH‡−N /R) at that pH-value. B, The plot of DH‡−N against the mid-temperature of each Eyring plot fits well to a linear function (continuous line) to give DCp,‡−N = 480 (220) cal mol−1 K−1 and DH‡−N (298 K) = 36.8 (20.7) kcal mol−1. The data can also be fitted to a quadratic function (broken line), which suggests that DCp,‡−N may be temperature-dependent.

obtained as a function of temperature. For unfolding, the activation energy in equation (6) is the difference in free energy between the native state and the transition state, DG‡−N (T ), therefore: ln

01 01 01

ku k DG‡−N (T ) = ln B − T h RT

temperatures must always be substantially above the thermal unfolding mid-point, Tm , at that pH to ensure that the protein is at least 99% unfolded in the final solution. Over the short temperature range for each pH (3 to 5 K), the Eyring plots are linear and the activation enthalpies are given by the slopes of the linear fits to the data (equation (11)). It has been observed in equilibrium unfolding studies that the slope of a plot of DHD−N (Tm ) against Tm , obtained from thermal denaturation at different acid pH values, coincides with the value of DCp,D−N derived from calorimetric measurements at a fixed pH (Privalov & Khechinashvili, 1974). This is presumed to be a consequence of DHD−N not changing with pH in the acid region because the only groups ionizing here are the carboxylates that have enthalpies of ionization close to zero. Thus, changing pH is a simple way of lowering Tm and extending the range of temperature over which measurements of DHD−N can be made. Analogously, the plot of DH‡−N against the mid-temperature of the measurements at each pH fitted well to a straight line to give DCp,‡−N of 480 (220) cal mol−1 K−1, although there appears to be a slight downwards curvature (Figure 5B). Privalov & Makhatadze (1990) have suggested that DCp,D−N may have some degree of temperature-dependence and is lower at high temperature due to the diminishing influence of the hydration of non-polar groups on DCp,D−N . A similar effect might cause a downwards curvature in the plot of DH‡−N against mid-temperature of measurement. A fit to a quadratic function gives DCp,‡−N = 6030 − 16.8 × T cal mol−1 K−1. This implies that DCp,‡−N becomes zero at 360 K, which is not inconsistent with the equilibrium values. The value of DCp,‡−N obtained by this method is in good agreement with that obtained from fitting the temperature-dependence of the unfolding rate constants (kulobs ) at pH 6.3 to equation (10) (Figure 4). Therefore, it has been shown by two independent methods that the DCp,‡−N for CI2 is 0480 cal mol−1 K−1. Thus, the conclusion from equilibrium unfolding studies that DHD−N (Tm ) is a function of Tm and not of pH when the enthalpies for ionization of buffer are small in the acidic region and are likely to compensate each other if suitable buffers are used (see Privalov & Khechinashvili, 1974) seems to apply to DH‡−N as well, i.e. the pH-dependence of DH‡−N is a consequence of the temperature at which the Eyring plot is obtained rather than of pH per se.

(11)

Thermodynamics of the transition state for the folding of CI2

where DH‡−N is the activation enthalpy and DS‡−N is the activation entropy. The plots of ln(kulobs /T ) against 1/T (Eyring plots) for the unfolding of CI2 at different values of pH are shown in Figure 5A. At each pH, the final

The enthalpy changes for both unfolding and refolding are positive (Table 1), i.e. the enthalpy of the transition state is higher (weaker interactions) than that of both the native and denatured states. When a protein unfolds, there are three contributions to the enthalpy of the system (Kauzmann,

= ln

kB DS (T ) DH‡−N (T ) + ‡−N − h R RT

383

Expanded and Compact Folding Pathways

1959): (1) a gain in enthalpy from the weakening/ loss of interactions, primarily the van der Waals’ interactions and hydrogen bonds between residues within the protein, (2) a loss in enthalpy from the additional interactions between the water molecules and the surfaces that become exposed to the solvent only upon unfolding, and (3) a gain in enthalpy from the disruption of the interactions between water molecules when they solvate the freshly exposed protein surfaces. It is found that when a non-polar molecule is present in water, the water molecules in the immediate vicinity arrange into ‘‘icebergs’’ in which there is less randomness and better hydrogen bonding than in ordinary liquid water at the same temperature; therefore, when hydrophobic surfaces become exposed to the solvent upon the unfolding of a native structure, the overall hydration effect gives rise to a loss in enthalpy (gain in hydrogen bonding; Kauzmann, 1959). With CI2, there is a large gain in enthalpy upon the unfolding of the native state to the transition state suggesting that the gain in enthalpy resulting from the weakening or breaking of internal interactions in the native state is not completely compensated by the hydration effect. From the transition state to the denatured state, there is a loss in enthalpy as the denatured state is more solvated than the transition state and the hydration effect is dominant in this case. The entropy of the transition state of CI2 is in between that of the native and denatured state (022% relative to the native state). The main contributions to the entropy change for unfolding include changes in the entropy of the conformational freedom of the polypeptide backbone and side-chains, and the entropy change when water molecules become ordered around non-polar groups. Therefore, the difference in entropy between the native state and the denatured state results from a delicate balance between at least two counteracting effects, which may be temperaturedependent as well (Sturtevant, 1977). Similar considerations apply to the difference in entropy between the transition state and the native state or between the transition state and the denatured state. In CI2, the activation heat capacity changes for the unfolding and refolding processes are similar, implying that the heat capacity of the transition state is approximately half-way between the native and denatured states (Table 1). Although there are other contributions to the heat capacity change for the unfolding of a native structure, like the increase in the configuration freedom of the polypeptide chain upon the disruption of the rigid native structure, it has been shown that the hydration of hydrophobic groups that become exposed upon the unfolding of the native structure, amounts to more than 80% of the total heat capacity change of unfolding (Privalov, 1979; Privalov & Makhatadze, 1990). It appears, therefore, that the transition state is more hydrated than the native state but is not as hydrated as the denatured state.

It may be noticed that (DCp,‡−N − DCp,‡−D ) agrees well with DCp,D−N from equilibrium studies (Table 1). This suggests that, under physiological conditions, the protein is folding from a denatured state (Dphys ) that is as hydrated as the denatured state under strongly denaturing conditions (D).

Discussion Proteins can be broadly classified into two classes according to their folding behaviour; proteins that fold by a two-state process and proteins that fold by a multi-state process with the accumulation of one or more intermediates. CI2 is one of the small proteins that belongs to the former class (Jackson & Fersht, 1991a) whereas barnase, which is nearly twice the size of CI2, folds by a non-two-state pathway (Matouschek et al., 1990). Here, we have measured the overall acid-titration properties of the transition state for the folding and unfolding of CI2, measured its heat capacity relative to the denatured state and, by a novel method, relative to the native state, as well as extending the earlier measurements of Jackson & Fersht (1991b) of the overall thermodynamic properties. A comparison of the overall properties of the transition state for the folding of CI2 and barnase leads us to introduce the terms, expanded pathway and compact pathway. CI2 folds by an expanded pathway, which involves a denatured state that does not collapse significantly under physiological conditions, and a transition state with few fully formed interactions. Barnase, on the other hand, folds by a compact pathway, where there is a significant number of native interactions, including a buried salt-bridge, in both the Dphys, denatured state under physiological conditions, and the transition state. This more compact pathway may be the norm for larger proteins that generally fold by multi-state processes. It is possible that the formation of residual structures in Dphys on the compact pathway helps to reduce the entropy of the longer polypeptide of larger proteins in the early events of folding. Surface interactions in the transition state are poorly formed The pKA of a particular residue in a protein reflects the extent of its electrostatic interactions (Russell & Warshel, 1985; Bashford & Karplus, 1990). Generally, the acidic residues in the native state have lower pKA-values than the model compounds because of interactions with the charges and dipoles of the protein. On the other hand, in an ‘‘extended’’ polypeptide chain, all the residues are well-separated from each other and are fully solvated and so the pKA-value of each residue is similar to the model compound values. The pKA-values of the ten acidic residues in the native state of CI2 have been obtained by NMR and most of them are two to three units lower than the model compound values, whereas in the much less

384

Figure 6. Titration behaviour of the native state (N) and transition state (‡), relative to the denatured state (D), of CI2 and barnase. A, CI2: DQD−N at m = 50 mM (thick continuous line) and m = 200 mM (thin continuous line), DQD−‡ at m = 50 mM (thick broken line) and m = 200 mM (thin dotted line). B, Barnase: DQD−N at m = 50 mM (thick continuous line) and m = 600 mM (thin continuous line), DQD−‡ at m = 50 mM (thick broken line) and m = 600 mM (thin dotted line).

compact denatured state, the pKA-values are, on average, 0.3 unit lower than the model compound values (Tan et al., 1995). The difference in the pKA-values in the native and denatured state is reflected by DQD−N (pH), the number of protons that are taken up when the protein unfolds (Figure 6A): DQD−N is nearly zero at pH 5 and increases sharply below pH 5 because the denatured state starts to become protonated. DQD−N reaches a maximum value at around pH 3 and drops rapidly below this pH because the native state starts to become protonated while the denatured state is already fully protonated (Tan et al., 1995). On the other hand, the DQD−‡ (pH), which reflects the uptake of protons on going from the transition state to the denatured state, has little pH-dependence and has a maximal magnitude of less than 1 mol H+ per mol of protein (Figure 6A). This shows that the transition state retains only a small degree of electrostatic interactions: the equivalent of
Expanded and Compact Folding Pathways

(Figure 6A). Both DQD−N and DQD−‡ show only small dependence on ionic strength (Figure 6A). The major transition state for the folding of barnase, on the other hand, maintains a higher degree of the native pKA-shifts: at m = 50 mM, the equivalent of 0two residues in the transition state is protected at pH 3.0 compared with 0four residues in the native state (Figure 6B, Oliveberg & Fersht, 1996a). The DQ profiles of barnase also show more pronounced dependence on ionic strength (Figure 6B). The urea dependence of the free energy difference between the native and denatured state, mD−N = 1DGD−N /1[urea], is generally believed to be a measure of the relative solvent accessibility of hydrophobic residues in the two states (Tanford, 1968, 1970). Similarly, the urea dependence of the free energy for refolding, mD−‡ , would be related to the difference in the exposure of hydrophobic residues in the transition state and the denatured state, and the ratio of these two m-values, bU = mD−‡ /mD−N , is a measure of the ‘‘degree of consolidation’’ of the transition state (Tanford, 1968, 1970). Analogously, the ratio of the areas under the DQD−‡ and DQD−N curves, bpH = (fDQD−‡ d(pH))/(fDQD−N d(pH)), is a measure of the amount of electrostatic interactions in the transition state (relative to the native state) (Oliveberg & Fersht, 1996a). From the DQ profiles at m = 50 mM (Figure 7A), bpH for CI2 is 0.28 and bpH for barnase is 0.47 (Oliveberg & Fersht, 1996a). The higher bpH for barnase is indicative of the greater amount of native electrostatic interactions in the transition state of barnase. This difference, however, may be linked to a buried salt-bridge, D93-R69, in the native state of barnase, which is almost fully formed in the transition state (Oliveberg & Fersht, 1996c). The pKA of D93 in the transition state is 01.3, which is similar to that in the native state. Therefore, D93 contributes one unit to both DQD−N and DQD−‡ at low pH because it is almost fully protonated in the denatured state but not protonated in either the native state or the transition state. If this is subtracted from the DQD−N and DQD−‡ of barnase, the DQ profiles for barnase without D93 and CI2 become similar (Figure 7B): at pH 3 (m = 50 mM), DQD−N 03.5 and DQD−‡01. The bpH for barnase after the subtraction of one unit from DQD−N and DQD−‡ decreases to 0.24, which is similar to the value for CI2 (bpH = 0.28). In conclusion, the more structured transition state of barnase involves a strong semi-buried salt-bridge. When this salt-bridge is removed, the transition state has little electrostatic interactions compared with the native state, just like the transition state for CI2. This is consistent with the transition state being compact and hydrophobic in the interior but rather loose on the surface. The surface interactions, which are mainly electrostatic in nature (Chothia, 1976), are thus poorly formed. It might, therefore, appear that surface interactions play little direct role in the early folding events.

Expanded and Compact Folding Pathways

Figure 7. A comparison of the titration behaviour of the transition states for the folding of CI2 and barnase (m = 50 mM). A, DQD−N (thick continuous line) and DQD−‡ (thick broken line) for barnase, DQD−N (thin continuous line) and DQD−‡ (thin dotted line) for CI2. B, DQD−N (thick continuous line) and DQD−‡ (thick broken line) for barnase after subtraction of one unit, DQD−N (thin continuous line) and DQD−‡ (thin dotted line) for CI2 as in A.

However, the ‘‘hydrophobic collapse’’ of the interior has to be accompanied by the expulsion of unpaired charge residues to the surface of the protein, since these will be energetically very unfavourable in a hydrophobic interior. Such a rearrangement seems to be formed to some extent already in the physiological denatured state as indicated by the perturbed pKA-values in denatured states (Oliveberg et al., 1995a; Swint-Kruse & Robertson, 1995; Tan et al., 1995). The transition state of CI2 is compact but both the surface electrostatic interactions and the hydrophobic interactions in the interior of the protein are poorly formed The results for CI2 show that the heat capacity of the transition state is half-way in between that of the native and denatured states (Table 1). This suggests that the hydrophobic core in the native state becomes loosened up in the transition state and allows some water molecules to penetrate the protein, resulting in the solvation of hydrophobic

385 moieties (cf. Segawa & Sugihara, 1984; Shakhnovich & Finkelstein, 1989). In comparison with the low degree of native-like electrostatic interactions in the transition state (bpH = 0.28), however, the compactness of the transition state is much more native-like, as reflected by the higher ratio of DCp,D−‡ / DCp,D−N00.5 (Table 1). This is consistent with the transition state of CI2 being an expanded form of the native state, where it would be expected that there is a greater increase in the distance between residues on the surface of the molecule than in the interior of the molecule. This does not mean that the hydrophobic interactions in the interior are formed to a greater extent than the surface interactions because the attractive potential energy for van der Waals’ interactions decreases much more rapidly with increasing distance (01/r 6 ) than, say, electrostatic interactions between two oppositely charged ions (01/r: Fersht, 1985). Therefore, a small increase in the volume of the hydrophobic core is sufficient to disrupt the van der Waals’ interactions in the interior of the protein. The lack of strong hydrophobic interactions in the interior of the transition state is reflected by the low Ff-value, an average of 0.3, found for transition state of CI2 (Itzhaki et al., 1995). To solvate the hydrophobic moieties in the interior of the protein, however, the volume expansion must be big enough to allow water molecules to enter, hence the higher DCp,D−‡ /DCp,D−N ratio. Consistent with the idea of a compact interior in the transition state of CI2, bU (=(1DGD−‡ / 1[GdmCl])/(1DGD−N /1[GdmCl]))00.6 (Jackson & Fersht, 1991a), which is again higher than bpH and is slightly higher than DCp,D−‡ /DCp,D−N . This suggests that fewer GdmCl molecules, which are in fact larger than water molecules and also highly charged, can penetrate into the interior of the transition state. In summary, the van der Waals’ interactions and electrostatic interactions in native state of CI2 are largely weakened in the transition state. The transition state, however, remains compact with only a small increase in volume of the hydrophobic core(s) and this restricts the amount of solvent/ denaturant entering to solvate/bind the newly exposed hydrophobic moieties in the interior of the protein. The results here are consistent with molecular dynamics simulations of the transition state of CI2, where it is found that the radius of gyration for the transition state increased by 10% from the native state (the crystal structure) compared with a 44% increase for the unfolded state (Li & Daggett, 1994). Also, the simulated transition state has a gap between the hydrophobic residues in the helix and b-sheets; this gap is dynamic and is not filled by water molecules and, therefore, represents a state in which previously favourable packing interactions are disrupted and uncompensated (Li & Daggett, 1994). Theoretical studies by Shakhnovich & Finkelstein (1989) also suggest that the transition state is an expanded form of the native state in which a significant part

386

Figure 8. A, CI2: the degree of formation (the reaction coordinate) of the transition state when measured by bU, heat capacity, enthalpy, entropy and bpH. B, Barnase: the degree of formation (the reaction coordinate) of the transition state and the Dphys (0I) when measured by bU, heat capacity, enthalpy, entropy and bpH.

of the van der Waals’ energy is lost, but the expansion is insufficient to permit rotational isomers of the side-chains or penetration of water. CI2 folds by an expanded pathway whereas barnase folds by a compact pathway Figure 8 shows the difference in the properties of the transition states of CI2 and barnase. The transition state of barnase is more structured than that for CI2 as reflected by the higher bU, DCp,D−‡ /DCp,D−N , and bpH. The enthalpy and entropy of the transition state, relative to the native state, are also different for the two proteins. In addition, the F-value analysis has shown that the transition state of CI2 has few fully formed interactions (Itzhaki et al., 1995), while for barnase, the majority of the secondary structure elements and one of the hydrophobic cores are nearly fully formed (Serrano et al., 1992). In addition, the denatured state under physiological conditions, Dphys, for barnase is also more compact than that for CI2. The folding of CI2

Expanded and Compact Folding Pathways

proceeds by the two-state mechanism both kinetically and at equilibrium (Jackson & Fersht, 1991a) and so there are no partially structured intermediates on the pathway. Good agreement between the equilibrium and kinetic DCp values suggests that Dphys is as hydrated as D, the denatured state under strongly denaturing condition. Hence, Dphys is likely to be largely unstructured and expanded. For barnase, on the other hand, the folding proceeds by a three-state mechanism, with the accumulation of a compact Dphys, which is structurally and energetically different from D (Oliveberg & Fersht, 1996b). The histogram on the right-hand side of Figure 8B shows the properties of Dphys relative to the native state of barnase. The results suggest a poorly hydrated Dphys (high DCp,D−Dphys /DCp,D−N and bU ). Also, the thermodynamic properties of Dphys are consistent with Dphys being structured (Oliveberg & Fersht, 1996b) and this is in agreement with the results from the F-value analysis (Matouschek et al., 1992). In summary, the folding of CI2 proceeds from an expanded Dphys, through a fairly compact but weakly structured transition state, which has very few native interactions that are fully formed. In contrast, barnase folds from a compact and native-like Dphys, through an even more compact transition state, where the structures in Dphys are consolidated further. The folding pathways of CI2 and barnase may be taken to represent two extreme ends of a continuous spectrum of folding mechanisms, where the folding pathways of different proteins are distinguishable by the compactness of the Dphys states and transition states. For example, proteins that fold by a two-state mechanism have unstructured Dphys, while proteins that fold by multi-state mechanisms have compact Dphys and intermediates, with varying degrees of structure formation.

Activation barriers for the folding of CI2 and barnase are similar The rate of refolding is controlled by the energy barrier between the most stable denatured state (Dphys ) and the transition state. With CI2, the activation barrier is the change upon going from an unstructured Dphys to weakly structured transition state, whereas with barnase, the activation barrier is the conversion of a partially structured Dphys into an even more structured transition state. Despite these differences, however, it is interesting to note that the activation barriers for the folding of the two proteins are actually quite similar (Figure 9). For barnase, the activation barrier for folding is given by the difference between Dphys and transition state (Figure 8B), normalised to the difference between Dphys and N. The similarities in the activation barrier for the two proteins are found in the thermodynamic properties (DH, DS and DCp ), as well as the bU (Figure 9). It has been suggested that the two-state folding of CI2 fulfils the criteria for fast folding

387

Expanded and Compact Folding Pathways

Figure 9. A comparison of the degree of formation for the Dphys to ‡ process, which is the activation barrier for folding under physiological conditions, in CI2 and barnase.

(Fersht, 1995b). It might, therefore, be the case that the thermodynamic properties of the energy barrier for the folding of CI2 have been evolved for efficient folding. And barnase, by folding from a partially structured Dphys, has also achieved a folding energy barrier whose properties are similar to that for CI2 and, hence, can fold rapidly in vitro. It appears, in particular, that the exclusion of water from the hydrophobic core(s) has occurred to a large extent in the transition state of both CI2 and barnase, and this gives rise to a compact transition state for folding. Is the barnase folding intermediate on or off pathway? For the past 20 years, it has been thought that folding intermediates are essential elements along the pathway (e.g. Khorasanizadeh et al., 1996). Since the demonstration that CI2 folds without a kinetically detectable intermediate (Jackson & Fersht, 1991a,b), several other proteins have been found to fold without intermediates and it has been argued that folding intermediates are off pathway (e.g. Sosnick et al., 1994). There is thus considerable current debate as to whether the sequence of events for the folding pathways of proteins in which compact states accumulate is: k1

k2

k−1

k−2

U gh I gh N

(12)

(where U is the highly unfolded state which exists under denaturing conditions, I the compact state and N the native) or is: k1

k2

k−1

k−2

I gh U gh N

(13)

for the examples where intermediates have been observed. It is very difficult to distinguish between the two schemes by kinetic experiments since, as has been known since the beginning of the century (Lowry & John, 1910), the solution of the kinetic equations

for consecutive reversible reactions is symmetrical with respect to k1 and k2 and also for k−1 and k−2 so that the two mechanisms (12) and (13) cannot be distinguished by simple rate laws (see also Fersht, 1985). Second, it is difficult to distinguish between the two schemes from structure-activity relationships. For example, suppose that I accumulates and that I and U are in rapid equilibrium compared with the formation of N (i.e. k2k1 in both (12) and (13)), a situation that occurs for the folding of barnase and many other proteins. The effective ground state for the reaction is I for both schemes, so procedures such as F-value analysis give the properties of the transition state for the unfolding of N relative to those of I in both cases. The two mechanisms can be distinguished when I and U equilibrate slowly by using a double-jump experiment in which the denatured protein is subjected to a short period of renaturation and the amount renatured measured by the amplitude of a subsequent denaturation reaction (Kiefhaber, 1995). Application of this to barnase failed to detect a direct route of U to N. The best evidence for the sequence of events in the folding of barnase comes from the F-value analysis: the structure of I is very similar to that of the transition state for the unfolding of N but virtually all the interactions are somewhat weaker. The transition state is like a partly expanded form of N, and I is like a partly expanded form of the transition state (Fersht, 1993). There is a build up of native-interactions that is consistent with the sequence U : I : N, rather than I being a seriously misfolded state that has to unfold completely for reaction to occur.

Materials and Methods Materials Protein and buffers were prepared as described by Tan et al. (1995). GdmCl stock solutions were prepared volumetrically. pH-jump stopped-flow experiments The kinetics of unfolding and refolding of CI2 was monitored after rapid mixing of a protein solution with unfolding/refolding buffer in a volumetric ratio of 1:1, using an Applied Photophysics DX-17 MV stopped-flow spectrofluorimeter. The final protein concentration was 7 mM, unless otherwise stated, and the final ionic strength was either 50 mM or 200 mM. Excitation was at 280 nm (slit-width of 9 nm) and the emission collected at wavelengths greater than 335 nm, using a cut-off filter. Temperature was maintained to 20.1 K. The pH-jump stopped-flow experiments can be divided into three classes: (1) studies of the pH-dependence of the unfolding kinetics (kuobs (pH)), where the native protein, dissolved in pure water, was mixed with an acidic denaturing buffer, typically in the range of pH 1 to 3; (2) studies of the pH-dependence of the refolding kinetics (kfobs (pH)), where the denatured protein, dissolved in 32 mM HCl (pH 1.5), was mixed with a high pH buffer to induce refolding, the final pH is typically between pH 2 and 6.3; and (3) refolding/unfolding

388

Expanded and Compact Folding Pathways

kinetics at a particular pH over a range of temperatures. Salt was included in the syringe containing the refolding/unfolding buffer to avoid any association or aggregation processes. GdmCl-jump stopped-flow experiments The unfolding kinetics was followed with a Perkin-Elmer MPF-44B fluorescence spectrophotometer equipped with a rapid mixing head as described by Jackson & Fersht (1991a). Temperature was maintained to 20.1 K. Rate constants for unfolding at a particular temperature were determined in the presence of five different concentrations of GdmCl and then extrapolated linearly to give their value in water as described by Jackson & Fersht (1991a): ln ku[GdmCl] = ln kuH2 O + m‡−N [GdmCl]

(14)

A slight downwards curvature has been observed in the plots of ln(ku ) against [urea] for barnase (Matouschek et al., 1994). The plots of ln(ku ) against [GdmCl] (in the range of 5 to 8 M) for CI2 are linear. If there is any curvature at lower denaturant concentrations, the absolute values of kuH2 O obtained from a linear fit would be offset from the true values, but the temperature-dependence of kuH2 O should not change significantly. The final temperatures and GdmCl concentrations were always high enough to unfold the protein to at least 99%. Data analysis The rate constants and amplitudes of the folding/unfolding events were obtained by fitting a sum of exponential functions to the acquired data, using the data-analysis program Kaleidagraph (Adelbeck Software). When mathematically processing the results, i.e. a series of closely spaced rate constants against pH, a ‘‘smooth function’’ (Kaleidagraph) was fitted to the data, and this smooth function was then used to represent mathematically the experimental values.

Acknowledgements Y.-J. T. was supported by a Glaxo Research Scholarship.

References Bashford, D. & Karplus, M. (1990). pKA ’s of ionizable groups in proteins: atomic detail from a continuum electrostatic model. Biochemistry, 29, 10219–10225. Chothia, C. H. (1976). The nature of the accessible and buried surfaces in proteins. J. Mol. Biol. 105, 1–14. Fersht, A. R. (1985). Enzyme Structure and Mechanism, 2nd edit., W. H. Freeman and Company, New York. Fersht, A. R. (1993). Protein folding and stability: the pathway of folding of barnase. FEBS Letters, 325, 5–16. Fersht, A. R. (1995a). Characterising transition states in protein folding: an essential step in the puzzle. Curr. Opin. Struct. Biol. 5, 79–84. Fersht, A. R. (1995b). Optimization of rates of protein folding: the nucleation-condensation mechanism and its implications. Proc. Natl Acad. Sci. USA, 92, 10869–10873.

Fersht, A. R., Matouschek, A. & Serrano, L. (1992). The folding of an enzyme. I. Theory of protein engineering analysis of stability and pathway of protein folding. J. Mol. Biol. 224, 771–782. Hagerman, P. J. & Baldwin, R. L. (1976). A quantitative treatment of the kinetics of the folding transition of ribonuclease A. Biochemistry, 15, 1462–1473. Itzhaki, L. S., Otzen, D. & Fersht, A. R. (1995). The structure of the transition state for folding of chymotrypsin inhibitor 2 analysed by protein engineering methods: Evidence for a nucleation-condensation mechanism for protein folding. J. Mol. Biol. 254, 260–288. Jackson, S. E. & Fersht, A. R. (1991a). Folding of chymotrypsin inhibitor 2. 1: Evidence for a two-state transition. Biochemistry, 30, 10428–10435. Jackson, S. E. & Fersht, A. R. (1991b). Folding of chymotrypsin inhibitor 2. 2: Influence of proline isomerization on the folding kinetics and thermodynamic characterization of the transition state of folding. Biochemistry, 30, 10436–10443. Kauzmann, W. (1959). Some factors in the interpretation of protein denaturation. Advan. Protein Chem. 14, 1–63. Khorasanizadeh, S., Peters, I. D. & Roder, H. (1996). Evidence for a three-state model of protein folding from kinetic analysis of ubiquitin variants with altered core residues. Nature Struct. Biol. 3, 193–205. Kiefhaber, T. (1995). Kinetic traps in lysozyme folding. Proc. Natl Acad. Sci. USA, 92, 9029–9033. Li, A. & Daggett, V. (1994). Characterization of the transition state of protein unfolding by use of molecular dynamics: chymotrypsin inhibitor 2. Proc. Natl Acad. Sci. USA, 91, 10430–10434. Lowry, T. M. & John, W. T. (1910). Studies of dynamic isomerism. X11. The equations for two consecutive unimolecular changes. J. Chem. Soc. 97, 2634–2645. Matouschek, A., Kellis, J. T., Jr, Serrano, L. & Fersht, A. R. (1989). Mapping the transition state and pathway of protein folding by protein engineering. Nature, 342, 122–126. Matouschek, A., Kellis, J. T., Jr, Serrano, L., Bycroft, M. & Fersht, A. R. (1990). Transient folding intermediates characterized by protein engineering. Nature, 346, 440–445. Matouschek, A., Serrano, L. & Fersht, A. R. (1992). The folding of an enzyme. IV. Structure of an intermediate in the refolding of barnase analysed by a protein engineering procedure. J. Mol. Biol. 224, 819–835. Matouschek, A., Matthews, J. M., Johnson, C. M. & Fersht, A. R. (1994). Extrapolation to water of kinetic and equilibrium data for the unfolding of barnase in urea solutions. Protein Eng. 7, 1089–1095. Oliveberg, M. & Fersht, A. R. (1996a). Formation of electrostatic interactions on the protein-folding pathway. Biochemistry, 35, 2726–2737. Oliveberg, M. & Fersht, A. R. (1996b). Thermodynamics of transient conformations in the folding pathway of barnase: reorganization of the folding intermediate at low pH. Biochemistry, 35, 2738–2749. Oliveberg, M. & Fersht, A. R. (1996c). A new approach to the study of transient protein conformations: The formation of a semi-buried salt link in the folding pathway of barnase. Biochemistry. 35, 6795–6805. Oliveberg, M., Arcus, V. & Fersht, A. R. (1995a). pKA values of carboxyl groups in the native and denatured states of barnase: the pKA values of denatured state are on average 0.4 units lower than

Expanded and Compact Folding Pathways

those of model compounds. Biochemistry, 34, 9424–9433. Oliveberg, M., Tan, Y.-J. & Fersht, A. R. (1995b). Negative activation enthalpies in the kinetics of protein folding. Proc. Natl Acad. Sci. USA, 92, 8926–8929. Otzen, D. E., Itzhaki, L. S., elMasry, N. F., Jackson, S. E. & Fersht, A. R. (1994). Structure of the transition state for the folding/unfolding of the barley chymotrypsin inhibitor 2 and its implications for mechanisms of protein folding. Proc. Natl Acad. Sci. USA, 91, 10422–10425. Pohl, F. M. (1976). Temperature-dependence of the kinetics of folding of chymotrypsinogen A. FEBS Letters, 65, 293–296. Privalov, P. L. (1979). Stability of proteins. Advan. Protein Chem. 33, 167–236. Privalov, P. L. & Khechinashvili N. N. (1974). A thermodynamic approach to the problem of stabilization of globular protein structure: A calorimetric study. J. Mol. Biol. 86, 665–684. Privalov, P. L. & Makhatadze, G. I. (1990). Heat capacity of proteins. II Partial molar heat capacity of the unfolded polypeptide chain of proteins: Protein unfolding effects. J. Mol. Biol. 213, 385–391. Russell, S. T. & Warshel, A. (1985). Calculations of electrostatic energies in proteins. The energetic of ionized groups in bovine pancreatic trypsin inhibitor. J. Mol. Biol. 185, 389–404.

389 Segawa, S.-I. & Sugihara, M. (1984). Characterization of the transition state of lysozyme unfolding. I. Effect of protein-solvent interactions on the transition state. Biopolymers, 23, 2473–2488. Serrano, L., Matouschek, A. & Fersht, A. R. (1992). The folding of an enzyme. III. Structure of the transition state for unfolding of barnase analysed by a protein engineering procedure. J. Mol. Biol. 224, 805–818. Shakhnovich, E. I. & Finkelstein, A. V. (1989). Theory of cooperative transitions in protein molecules. I. Why denaturation of globular proteins is a first-order phase transition. Biopolymers, 28, 1667–1680. Sosnick, T. R., Mayne, L., Hiller, R. & Englander, S. W. (1994). The barriers in protein folding. Nature Struct. Biol. 1, 149–156. Sturtevant, J. M. (1977). Heat capacity and entropy changes in process involving proteins. Proc. Natl Acad. Sci. USA, 74, 2236–2240. Swint-Kruse, L. & Robertson, A. D. (1995). Hydrogen bonds and the pH-dependence of ovomucoid third domain stability. Biochemistry, 34, 4724–4732. Tan, Y.-J., Oliveberg, M., Davis, B. & Fersht, A. R. (1995). Perturbed pKA-values in the denatured states of proteins. J. Mol. Biol. 254, 980–992. Tanford, C. (1968). Protein denaturation. Part A and B. Advan. Protein Chem. 23, 121–282. Tanford, C. (1970). Protein denaturation. Part C. Advan. Protein Chem. 24, 1–95.

Edited by J. Karn (Received 17 July 1996; received in revised form 12 September 1996; accepted 18 September 1996)