Molten globule-like transition state of protein barnase measured with calorimetric force spectroscopy.

SignificanceUnderstanding the molecular forces driving the unfolded polypeptide chain to self-assemble into a functional native structure remains an open question. However, identifying the states visited during protein folding (e.g., the transition state between the unfolded and native states) is tricky due to their transient nature. Here, we introduce calorimetric force spectroscopy in a temperature jump optical trap to determine the enthalpy, entropy, and heat capacity of the transition state of protein barnase. We find that the transition state has the properties of a dry molten globule, that is, high free energy and low configurational entropy, being structurally similar to the native state. This experimental single-molecule study characterizes the thermodynamic properties of the transition state in funneled energy landscapes.

Protein folding stands as one of the major open questions in biophysics. In the 1950–1960s decade, Anfinsen introduced the thermodynamic hypothesis claiming that proteins spontaneously fold to a free-energy minimum under appropriate conditions (1, 2). In 1969, Levinthal (3) noticed a polypeptide chain could not fold into the native state (hereafter denoted as N) by random search in configurational space. Protein folding is akin to finding a needle in a haystack and must be driven by molecular forces (4). In an effort to solve the paradox, Ptitsyn (5) proposed the molten globule hypothesis (MGH) where folding is similar to solid formation from a gas: A molten globule state must precede protein folding, similarly to the metastable liquid phase preceding solid formation during gas deposition. The dry molten globule is a necessary intermediate (6, 7) to form the N that is structurally similar to it but with most native bonds not yet formed. For years, scientists have searched for folding intermediates, the most natural solution to Levinthal’s paradox. While these have been identified in large proteins, many small globular proteins fold in a two-states manner, raising the question whether such a molten globule intermediate does exist. Methods such as the phi-value analysis have shown that the transition state (hereafter referred to as TS) of two-state globular proteins is structurally similar to the N (8–10). The TS of two-state folders is a disguised molten globule of very short lifetime whose thermodynamic properties reflect those of the molten globule intermediate. In contrast to an intermediate state, defined as a local minimum in the free-energy landscape, the TS corresponds to a local maximum in the free-energy landscape.

A new direction of thought emerged in the late 1980s from Wolynes and coworkers (11, 12), who proposed the energy landscape hypothesis (ELH): Proteins fold in a funnel-like energy landscape by following different and productive folding trajectories. Albeit not excluded, intermediates are not obligatory folding steps. In both scenarios, MGH and ELH, the thermodynamics of the TS has generic and unique properties: On the one hand, a large energy barrier separates TS and N; on the other hand, there is a large configurational entropy loss upon forming the TS from the random coil or unfolded state (hereafter denoted as U). More recently, the alternative foldon hypothesis (FH) has gained considerable attention (13, 14), based on the accumulated evidence gathered from hydrogen exchange, NMR, and mass spectrometry studies. In the FH, proteins fold following a unique pathway by the cooperative and sequential formation of native structure domains (denoted as foldons). Folding amounts to the productive tinkering of amino acids and foldons rather than the diffusion of a polypeptide in a funnel-like energy landscape.

To evaluate the different hypotheses, computer simulations and experiments are employed (15, 16). For the latter, it is crucial to have tools for accurately measuring the thermodynamics and kinetics of folding. Besides bulk techniques (NMR, mass spectrometry, calorimetry, etc.), single-molecule fluorescence and force spectroscopy offer complementary insights on the protein folding problem. With these, individual proteins are manipulated and monitored with enough temporal resolution to detect short-lived intermediates (17–19), and measure transition path times along kinetic barriers (20). Key results are the demonstration that the ribosome promotes the efficient folding of the nascent polypeptide chain (21), and the role of protein mechanical properties on nuclear translocation (22). Single-molecule evidence of protein folding intermediates has been reported for RNAseH (23, 24), the coiled-coil leucine zipper (25, 26), and calmodulin (27). Recently, the molten globule of apomyoglobin has been shown to be highly deformable under force (28), and an off-pathway molten globule has been observed in apoflavodoxin (29). In other cases, proteins fold in a two-states manner without detectable intermediates [e.g., PrP protein (26)], and a molten globule of very short lifetime transiently forms along the folding pathway (30, 31).

Over the last decade, there has been much effort in determining the thermodynamic properties of intermediate states and TSs in globular proteins. How much do the enthalpy and entropy of the TS differ from the N? What is the heat capacity change () between the TS and the N and U conformations? How do the TS properties change by varying the external conditions (e.g., temperature, ionic strength, and pH)? Answering these questions is essential to understand the features of the different hypotheses (e.g., the liquid-like properties of the molten globule in the MGH or the funnel’s shape in the ELH), and the nature of the folding process itself.

Upon heating, proteins melt at a characteristic temperature T at which the heat capacity at constant pressure, C, shows a peak (32, 33). The heat capacity change upon folding, , can be used to determine temperature-dependent enthalpy and entropy differences (34, 35). Moreover, is directly related to the change in the number of degrees of freedom (dof), , across the transition, . Therefore, quantifies the configurational entropy loss, the main contribution to the folding entropic barrier.

Laser optical tweezers (36) are suitable for calorimetric measurements; however, most studies have been carried out at ambient temperature K (37–39), due to the difficulty of controlling temperature (40, 41). For many years, this limitation has challenged direct enthalpy and entropy measurements over a wide range of temperatures, rendering inaccessible to single-molecule assays. We have recently built a temperature-jump optical trap suitable for single-molecule force spectroscopy above and below , thus providing a calorimetric force spectroscopy (CFS) tool for molecular thermodynamics (42, 43). Here, we investigate the folding thermodynamics and kinetics of protein barnase, a paradigmatic model in protein folding studies (Fig. 1). Barnase is a 110-aa bacterial ribonuclease globular protein secreted by Bacillus amyloliquefaciens, which, in physiological conditions, degrades RNA in the absence of its protein inhibitor barstar (45). The high solubility and stability of barnase makes it an excellent model to investigate the folding kinetics of globular proteins, by combining phenomenological approaches (e.g., the phi-value analysis) with protein engineering and site-directed mutagenesis methods (8). Barnase reversibly folds in a two-states manner between the unfolded and native conformations. It has been suggested that barnase folds via a short-lived intermediate (31, 46, 47) and two TSs (48). However, pulling experiments at room temperature could not find evidence of intermediates down to milliseconds (49). Here, we pull barnase in the range 7 ∘C to 37 ∘C and derive thermodynamic quantities by combining fluctuation theorems for free-energy prediction and kinetics. We determine the temperature-dependent folding free energy (), entropy (), and enthalpy (), to derive (∼1,000 cal/molK). Our results are consistent with calorimetry studies under similar ionic strength and pH conditions.

Fig. 1.

CFS of protein barnase. (A) Three-dimensional view of native barnase obtained with X-ray diffraction with 1.50- resolution (44). Four external α-helices (helix 1: Phe7-Tyr17 [red]; helix 2: Lys27-Leu33 [blue]; helix 3: Ala37-Lys39 [cyan]; helix 4: Leu42-Val45 [orange]) contain a total of 25 amino acids surrounding four β-strands (purple) located in the protein core. (B) Schematics of the temperature-jump optical trap setup. The diode lasers (red and blue) form a single optical trap, while the collimated heating laser (green) passes through the microfluidic chamber. CCD, charge-coupled device; PBS, polarizing beam splitter. (C) Illustration of the molecular construct and experimental setup: barnase is flanked by two identical 500-bp dsDNA handles and tethered between two beads. One bead is captured in the optical trap while the other one is kept fixed by air suction on the tip of a glass micropipette.

We also determine the entropy, enthalpy, and of the TS, finding that it is structurally similar to N. Upon folding, most of the enthalpy and entropy change occurs between N and TS, where roughly 80% of the native bonds are formed (from molten to native). In contrast, most of the folding occurs between U and TS, with ∼90% of configurational entropy loss. The collapse from TS to N mostly contributes to the enthalpy and entropy of folding, but residually to . Our results demonstrate that the TS has the properties of a dry molten globule: a large entropy and enthalpy relative to the N and a low configurational entropy. Albeit structurally similar to the N, the molten globule is a high-energy state with most bonds not formed.

Results

Force–Distance Curves.

Barnase was inserted in a molecular construct that was tethered between two beads and mechanically pulled with a temperature-jump optical trap (Fig. 1 and ). Force–distance curves (FDCs) were measured by repeatedly pulling barnase between minimum and maximum force values at different temperatures. Fig. 2 shows FDCs of five selected pulling cycles (unfolding and refolding) for the six investigated temperatures (7 ∘C, 14 ∘C, 18 ∘C, 25 ∘C, 32 ∘C, and 37 ∘C). It is apparent that the lower the temperature, the higher the unfolding force and the FDC hysteresis. In Fig. 2, we show a single pulling cycle at 25 ∘C. During stretching (red curve), barnase unfolding is observed as a sudden force rip ( pN) in the FDC at forces 15 pN to 30 pN. Upon force release (blue curve), a folding transition is detected as a small force jump (∼0.5 pN) at forces 5 pN. The encircled zoomed Fig. 2 , Left Inset shows the two force branches where barnase is folded (N branch, black dashed line) and unfolded (U branch, gray solid line). N and U branches describe the elastic response of the molecular construct where barnase is in N and U, respectively. The relative trap position (λ) in the two branches contains the trap bead displacement plus the handles extension and the molecular extension. The difference between the two branches at a given force, (Fig. 2 , Right Inset), is the difference of molecular extensions between the polypeptide chain and the projection on the force axis of the dipole formed by the N and C termini of barnase.

Fig. 2.

Pulling experiments of protein barnase. (A) (Left) Unfolding (dark) and folding (light) FDCs at different temperatures, 7 ∘C (purple), 14 ∘C (blue), 18 ∘C (green), 25 ∘C (yellow), 32 ∘C (red), and 37 ∘C (brown). FDCs at each temperature have been shifted along the x axis for clarity. Horizontal dashed lines denote the most probable unfolding force at each temperature. (Right) Unfolding force distributions for all the studied temperatures. For sake of clarity, we show Gaussian fits to histograms. For higher temperatures, less force is required to unzip the protein. (B) Unfolding (red) and refolding (blue) trajectories at 25 ∘C. (Left Inset) Zoomed refolding event (arrow) from the U branch (gray solid line) to the N branch (black dashed line). (Right Inset) Zoomed unfolding event (arrow) highlighting . (C) Scheme of the different thermodynamic steps to measure . Stretching of the folded protein (1 → 2), unfolding at a given force (2 → 3), releasing of the unfolded protein (3 → 4), and unfolding at zero force (1 → 4).

Folding Free Energy, Entropy, and Enthalpy.

Here we describe how to extract the temperature-dependent folding free energy of barnase at zero force, , from the measured free-energy difference . equals the (positive) free-energy difference between the native conformation (N) and the random coil state (unfolded, U). The nomenclature for free-energy differences employed throughout the paper is the standard one in single-molecule and calorimetric bulk studies. can be measured in bulk assays at zero force, whereas CFS experiments measure free-energy differences at a given force, . To derive from , it is necessary to subtract contributions coming from the experimental setup, such as the displacement of the bead from the center of the optical trap and the stretching of the handles and the polypeptide chain (50). The procedure is illustrated in Fig. 2 where the different stretching contributions correspond to free-energy differences measured over three distinct steps (). can be derived () from the measured and the coexistence force in equilibrium, (defined by or ),

From Eq. , we derive the folding entropy and enthalpy, and . For the entropy, we find

This is analogous to the Clausius–Clapeyron equation for first-order phase transitions, where f and λ are the equivalent of pressure and volume (42). The second term in Eq. is the entropic contribution to stretch and orient the protein dipole from zero force to f. Eqs. and require measuring over the full integration range, [0, ], and . While is directly obtained from the FDCs (Fig. 2 , Right Inset), the value of is unknown due to the strongly irreversible FDCs. The might be extracted from equilibrium hopping experiments; however, this is not possible in barnase, due to the exceedingly long hooping times (25, 26). Here we derive by measuring using the fluctuation theorem () and using Eq. .

Elastic Response of the Polypeptide Chain.

The temperature-dependent elastic properties of the polypeptide chain were determined using the molecular extension obtained from (Fig. 2) and the dipole contribution from . In Fig. 3, we show f versus measured at three selected temperatures (7 ∘C, 25 ∘C, and 37 ∘C). To extract from Eq. , we modeled the dipole extension with the freely jointed chain elastic model (Fig. 3 , Inset), assuming that the distance between the N and C termini for the folded barnase (the dipole length taken equal to 3 nm) is constant with temperature. By comparing Fig. 3 and Fig. 3 , Inset, we observe that , as expected, since the dipole length is much shorter than the polypeptide extension. Therefore, increases with T at a given f, making the polypeptide chain stiffer with temperature. The is well described by the inextensible worm-like chain (WLC) model and its interpolation equation (51),where is the persistence length, is the number of residues (110 for barnase), and is the distance between consecutive amino acids. The data relative to each investigated temperature were fit to Eq. , as shown in Fig. 3, with L and as free parameters. As reported in Fig. 3, L shows a strong T dependence, which is well approximated by a linear function of slope 0.011 ±0.001 nm/K. Similar results are obtained if we fit the data with a perturbative expansion of the WLC model (52) instead of Eq. . Moreover, presents a weak T-linear dependence of slope 0.0008 ±0.0002 nm/K (Fig. 3), which is one order of magnitude smaller than for . Therefore, can be taken as constant, ∼0.37 nm. Both fitting parameters at room temperature (298 K) agree with previous results (47, 49).

Fig. 3.

Elastic response of the polypeptide chain. (A) Force versus difference in trap position () at three temperatures: 7 ∘C (purple), 25 ∘C (yellow), and 37 ∘C (brown). (Inset) Elastic response of the folded protein modeled as a dipole of 3 nm with the FJC model. (B) Fits of the measured force versus polypeptide chain extension (x) to the WLC model (solid lines). (C) Persistence length (L) and (D) amino acid distance () of the polypeptide chain calculated from the fits in B. Solid lines are linear fits to the experimental points.

Measurement of Folding Free Energy.

To derive , we use the thermodynamic relation illustrated in Fig. 2, that relates with the elastic contributions of the polypeptide chain , the dipole term , and (Eq. in ),

From the results of the previous section, we can readily determine and ; however, how to measure remains unanswered. In optical tweezers experiments, the relative trap position λ is the control parameter, rather than the force which fluctuates depending on the molecular state. A thermodynamic relation similar to Eq. holds by a Legendre transforming to the λ-ensemble (). is determined by measuring the free-energy difference, , between minimum and maximum trap positions where barnase is folded () and unfolded (),where stands for the elastic contributions of the setup (bead, handles, polypeptide chain, and protein dipole) that must be subtracted to ().

We used the work fluctuation theorem (work-FT) (53, 54) to determine from irreversible work (W) measurements by integrating the FDC between the two selected trap positions, (Fig. 4). Let and denote the unfolding and folding work distributions measured over many pulling cycles. The work-FT is given bywhere equals the reversible work. The minus work sign in in the left-hand-side of Eq. is a consequence of the fact that W < 0 in the folding process. A corollary of Eq. is the Jarzynski equality, , where is the average over many (infinite) realizations. In practice, the number of pulls is finite, and the Jarzynski equality is strongly biased (55). From Eq. , work distributions cross at ; that is, . However, the crossing point is not observed, due to the hysteresis of the FDCs, quantified by the area enclosed between the unfolding and folding FDC (Fig. 4). The dissipated work is in the range 50 k to 100k, and much larger than the value of (see below). In the absence of crossing, one can use the matching method (53), that gives reasonable free-energy estimates and is simpler than other mathematical approaches (55, 56). In this method, the value of is determined by matching the functions and . In practice, the leftmost (rightmost) tails of () are fitted to the generic form, , to extract the values of (55). In Fig. 4 , Top, we show at 25 ∘C and the fitted tails. Rightmost (P) and leftmost (P) tails are well fitted with , indicating Gaussian-like tails (δ = 2). Therefore, we simultaneously fitted and to a single Gaussian distribution (black dashed line in Fig. 4 , Bottom) to find the best matching the value of . The fact that the generic and Gaussian distributions are nearly the same and a single Gaussian distribution (dashed line in Fig. 4 , Bottom) simultaneously fits and demonstrates that is an excellent approximation to and tails around ().

Fig. 4.

Folding thermodynamics of barnase. (A) Work measurements from unfolding (red) and folding (blue) FDC. The unfolding and folding work W is the area below the FDC limited by λ and λ (red and blue areas). (B) (Top) Unfolding (red full squares) and folding (blue empty squares) work distributions at 25 ∘C. (Bottom) and times . The black dashed line is a Gaussian fit to determine . (C) Unfolding (solid line) and folding (dashed lines) work distributions at different temperatures (with the elastic contribution being subtracted from the total work). The dotted vertical line indicates for each temperature. (D) Solid squares are the estimated versus T, and black dashed line is a fit to . Empty squares are the estimated coexistence force, and dashed gray line is a linear fit to f. (E) Solid and empty squares are the entropy and enthalpy differences at zero force. Dashed lines are fits to Eqs. and , respectively.

To determine from in Eq. , we subtract the elastic contributions as follows: The bead contribution was calculated by modeling the optical trap with Hooke’s law, that is, a constant optical trap’s stiffness equal to 0.07 pN/nm throughout the explored force range (50, 57); the DNA handles term was calculated by integrating the WLC model with the temperature-dependent elastic parameters from ref. 58; and contributions were calculated using the elastic parameters from . In Fig. 4, we show the and at different temperatures. Distributions are plotted versus instead of W, to directly determine with the matching method (Fig. 4, dotted vertical lines). The values of present a clear temperature dependence (Fig. 4, filled black squares) as expected from the relation .

Derivation of entropy and enthalpy.

From Eq. and the measured values of and (), we derive in the range 7 ∘C to 37 ∘C (gray empty squares in Fig. 4). The decreases linearly with T, thus defining an f – T phase diagram separating the N and U. We note that the linear trend observed in Fig. 4 does not agree with predictions by lattice models (59–61) for the critical force at which the fraction of native contacts equals 0.5 (). The line crosses the T axis at ∘C, in agreement with bulk experiments (see below). Finally, from Eq. and , we derived (black solid squares in Fig. 4). It changes by roughly 22% in the whole temperature range, indicating a finite . Notice that the numerical T derivative of is roughly constant (Fig. 4), which confirms Eq. as the most reliable way to estimate . Folding enthalpies, , are shown in Fig. 4 (empty squares).

Heat capacity change.

Bulk assays have shown that barnase has a finite . The marked temperature dependence in and (Fig. 4) allows us to extract across the melting transition. To do so, we expand and around the melting temperature T,where and are the entropy and enthalpy at T, and is the heat capacity change between N and U. and were fitted to Eqs. and (dashed lines in Fig. 4) with , and T fitting parameters. We obtain cal/molK, cal/molK, and kcal/mol.

Kinetics.

The results derived from the work-FT are confronted with those derived from the temperature-dependent unfolding and folding kinetic rates using the same experimental FDCs. Fig. 5 shows the unfolding and folding force distributions (, ) at three selected temperatures (7 ∘C, 25 ∘C, and 37 ∘C). We extract the unfolding and folding kinetic rates, and , from the corresponding survival probabilities. If f is ramped at constant loading rate , the following relations hold:with and being the survival probabilities of N and U, respectively. Our measurements of and are shown in Fig. 5 in a log-normal scale, for all the studied temperatures. The force dependence of the kinetic rates is described by

Fig. 5.

Folding kinetics of barnase. (A) Unfolding (right) and refolding (left) force distributions at 7 ∘C (purple), 25 ∘C (yellow), and 37 ∘C (dark brown). (B) Force-dependent unfolding (empty symbols) and folding (solid symbols) kinetic rates for all temperatures, 7 ∘C (purple), 14 ∘C (blue), 18 ∘C (green), 25 ∘C (yellow), 32 ∘C (red), and 37 ∘C (brown). The solid (refolding) and dashed (unfolding) lines are fits to the Bell–Evans model. (C) Schematics of the folding free-energy landscape in the Bell–Evans model. (D) Folding free energy and coexistence force (Top) and entropy and enthalpy (Bottom) as a function of temperature. Dashed lines are fits to Eqs. and . (E) Unfolding (Top) and refolding (Bottom) kinetic rates at zero force versus . Dashed lines are simultaneous fits to Eqs. and .

with k as the attempt rate and () as the force-dependent kinetic barrier relative to N (U). An illustrative free-energy landscape is shown in Fig. 5, highlighting , and . Notice the hysteresis between unfolding and folding (a minimum of a ∼10-pN gap is observed between the measured unfolding and folding rates; Fig. 5). This fact precludes us from determining the coexistence force and the folding free energy at zero force using Eq. . To circumvent this problem, we have determined the force dependence of the unfolding and folding kinetic rates beyond the Bell–Evans model, where distances of N and U to the TS are taken as force independent (62, 63). To do so, we have used the detailed balance condition,

that relates the unfolding/folding kinetic rates with the energy difference between N and U at force f, (compare Eq. ). For a given trial value of , the energy is calculated using Eq. with the elastic contributions determined in . Then we used Eq. to reconstruct the unfolding (folding) kinetic rates in the region of forces where folding (unfolding) transition events are observed in the pulling experiments. To estimate the quality of the continuity of the rates between the two force regimes, we have fitted the unfolding kinetic rates (measured and reconstructed) to a single quadratic function in the log-normal plot (dashed lines in Fig. 5). From these fits and Eq. , we inferred the folding kinetic rates (solid lines in Fig. 5). This procedure has been repeated by varying in the range of 2 kcal/mol to 20 kcal/mol in steps of 0.2 kcal/mol. To determine the that best fits the data, we have minimized the between the predicted force-dependent kinetic rates and the experimental and reconstructed rates (). The curvature of the reconstructed vs. force plots at low forces (Fig. 5) has been also observed for the ligand binding rates to filamin protein (64). The derived values for are shown as black circles in Fig. 5 , Top. In addition, using the extrapolated kinetic rates (solid and dashed lines in Fig. 5), we determined the coexistence force f as the force value at which (empty gray circles in Fig. 5 , Top).

From , derived in , and the values of and obtained from the kinetics analysis, we used Eq. to calculate and (Fig. 5 , Bottom, black circles and empty gray circles, respectively). These values agree with those obtained independently from the work-FT analysis (). Fitting and to Eqs. and gives cal/molK, cal/molK, and kcal/mol.

Furthermore, we determined the TS enthalpy and entropy differences, and , relative to states N and U (Fig. 5). To this aim, we rewrite the kinetic rates in Eqs. and at zero force, , , in terms of the TS entropies and enthalpies,

We performed a simultaneous fit of and to Eqs. and to derive the values of , and . Interestingly, we found that is strongly T dependent, while depends weakly on T, hinting at an entropy-driven folding process. The four-parameters fit was done by imposing two constraints: and . For the fits to Eqs. and , the values of and have been taken as the mean values obtained from the FT (Fig. 4) and kinetics (Fig. 5 , Bottom). Moreover, we used the attempt rate previously obtained on the same molecular system in similar experimental conditions (49), . Fits are shown as dashed lines in Fig. 5. TS entropies and enthalpies are shown in Fig. 6 and in Table 1 at all studied temperatures and at the average T (50 ∘C). Fitting them to Eqs. and permits us to extract the heat capacity change between N and TS () and between TS and U (). We obtain 150 cal/molK and 900 cal/molK, which gives the folding 1,050 cal/molK.

Fig. 6.

Barnase free-energy landscape. (A) Entropy (full symbols) and enthalpy (empty symbols) differences between N and TS (, black); and between U and TS (, gray). (B) Illustration of the free-energy (, Top), enthalpy (, Middle), and entropy (, Bottom) landscape relative to N. (C) Golf course folding free-energy landscape of barnase highlighting the molten globule structure formed in the TS.

Table 1.

Thermodynamic properties of barnase at three selected temperatures

T	ΔG_0	ΔH_0	ΔS_0	ΔG^‡	ΔH^‡	ΔS^‡	ΔG^*	ΔH^*	ΔS^*
7	17 ±3	94 ±11	273 ±40	12 ±4	101 ±2	318 ±14	−6±4	6 ±3	45 ±15
14	15 ±2	104 ±12	310 ±40	11 ±4	103 ±2	324 ±14	−4±3	0 ±2	14 ±11
18	13 ±2	108 ±13	327 ±43	9 ±4	104 ±2	326 ±14	−3±3	−3±2	−2±10
25	10 ±2	116 ±12	353 ±38	8 ±5	105 ±2	328 ±15	−3±3	−10±2	−25±11
32	8 ±2	122 ±12	372 ±39	6 ±4	106 ±2	329 ±13	−3±3	−16±2	−42±11
37	6 ±2	126 ±10	387 ±29	4 ±3	108 ±2	332 ±12	−2±3	−19±2	−56±10
50	0	140 ±2	434 ±10	2 ±2	111 ±2	337 ±4	2 ±3	−30±2	−100±8

T in degrees Celsius; and in kilocalories per mole; and in calories per mole kelvin. Thermodynamic potentials differences: 0, N–U; ‡, N–TS; and *, U–TS. Note that or in our notation.

The Folding Funnel.

Fig. 6 and Table 1 summarize our main results: The energy differences between states N and U (, and ); the barrier energies to unfold, N-TS ( and ); and the barrier energies to fold, U-TS (, , and ). The energy parameters of the free-energy landscape of barnase (Fig. 5) are illustrated in Fig. 6. Results show that barrier entropies, enthalpies, and free energies to fold (U TS) are one order of magnitude smaller than the corresponding barriers to unfold (N TS): , and . This difference suggests a folding process in two steps (Fig. 6). In the first step, the unfolded protein reaches a TS with a few H bonds (∼20%) formed relative to U. In the second step, the protein collapses into N by forming the rest of the native bonds (∼80%). These bond percentages are estimated from the different enthalpy values for the TS relative to N and U (109 kcal/mol and -30 kcal/mol at T).

A remarkable difference is found in between TS and N or U (Fig. 6). The main contribution to the total cal/molK comes from TS and U ( cal/molK), which is ∼9 times larger than between N and TS ( cal/molK) (). The value of is directly proportional to the change in the number of dof (), , which gives 1 per cal/Kmol unit in . This gives 900, showing that the main configurational entropy loss occurs upon forming the TS from U. This result depicts the TS as a molten globule of high free energy () and low configurational entropy (), which is structurally similar to the N: The major change in and occurs between U and TS.

Discussion

We have used calorimetric optical tweezers to measure the FDCs of protein barnase in the range 7 ∘C to 37 ∘C and derived the folding thermodynamics at the single-molecule level. An analysis based on the Clausius–Clapeyron equation (Eq. ) was used to extract the temperature-dependent values of and , and . These agree with those obtained from bulk experiments in our conditions of ionic strength (20 mM monovalent salt) and neutral pH (7.0). The mean values for the enthalpy, entropy, and T derived from the work-FT and kinetic analysis ( kcal/mol, cal/molK, and ∘C) agree with those reported in the literature and collected in ref. 65, kcal/mol to 145 kcal/mol and cal/molK (summarized in ). Our estimation of cal/molK also agrees with values obtained from differential scanning and isothermal titration calorimetry assays, cal/molK (), as well as with recent atomistic numerical simulations (66). Measurements of in calorimetric experiments often require determining the temperature dependence of with pH, ionic strength, or the denaturant concentration. In contrast, with CFS, we directly measure thermodynamic potentials and kinetics at a given temperature. In , we also compare the protein stability curve of barnase derived from the fits to Eqs. and with that reported in calorimetric studies (67, 68) and numerical simulations (66).

Remarkably, entropies and enthalpies between the TS and the U (, and ), are ∼10 times lower than the corresponding ones between TS and the N (, and ). In fact, the low value of correlates with the high compliance of the molten globule upon stretching, as has been shown for apomyoglobin (28). A general feature of free energies () in Table 1 is the compensation observed between entropy () and enthalpy () contributions; that is, . In contrast, the major contribution to occurs between U and TS (). The value of is proportional to the reduction () in the number of dof between TS and U, at a rate of ∼1 dof/(cal/Kmol).

Our results agree with the folding funnel scenario of the ELH (Fig. 6), where a large configurational entropy loss (∼90%) occurs upon forming the TS from the U. Such entropy loss is accompanied by a low enthalpy change (∼20% of the total folding enthalpy). The large configurational entropy loss between U and TS demonstrates that folding is an entropically driven process in a golf course energy landscape, where the TS is the native hole. The collapse from TS to N forms most of the native bonds, accounting for most of the entropy and enthalpy of folding (). Overall, our results also validate the main predictions of the ELH.

Are these results applicable to other proteins? Assuming an equal average enthalpy per native bond in N and TS, their structural similarity implies a low fraction of native bonds at TS, . For barnase, 0.2 at T, a value that decreases as T is lowered being 0 at T = 7 ∘C. These results are at odds with the observation often made in computational studies that = 0.5 at the TS. Previous atomic force microscopy studies on the ddFLN4 protein domain in the range 5 ∘C to 37 ∘C show that increases with T, reaching a maximum () at 37 ∘C (69). Apparently, the increase of TS stability with T facilitates the collapse from TS to N.

The TS of barnase features the properties of a dry molten globule: a native-like expanded structure with the backbone formed, but with side chains loosely packed (70). The dry molten globule has a large enthalpy relative to N () but a low , the major contribution to being , in agreement with our results. We hypothesize that folding proceeds in two steps. First (U to TS), hydrophobicity (71) drives the formation of the barnase backbone by the stabilization of the four α-helices and β-strands and the expulsion of water from the protein core. The small difference between the net number of H bonds between U and TS leads to small values of relative to . Next (TS to N), the dry molten globule collapses to N, stabilized by the liquid-like Van der Waals interactions between the loosely packed side chains in TS. The dependence of Van der Waals interactions implies a large even for a short-distance collapse. is compensated by (enthalpy–entropy compensation), a generic feature of weak interactions. Pushing the analogy further, protein folding resembles planet formation, where the mantle forms first and the core solidifies afterward.

Alternatively, in a wet molten globule, the backbone is formed at TS, but water remains inside the protein core. A wet molten globule would lead to a high free-energy barrier to folding, , and a two-state folding scenario. Instead, kcal/mol at T and is negative below T (Table 1). Remarkably, our results support the downhill folding (rather than two states) scenario (72–74), correlating it with the MGH.

Our results do not exclude the FH picture, where proteins fold along a well-defined pathway by the sequential formation of foldons. In the FH, the free-energy landscape resembles the ELH golf course shown in Fig. 6, except for the fact that it incorporates a specific folding pathway (groove) along the ground with one or more intermediates (). Discriminating between the ELH and FH would require monitoring folding paths in configurational space. Bulk measurements have addressed this question by combining hydrogen exchange (75) and NMR (76). In single-molecule experiments, one might measure reaction coordinates other than the molecular extension, for example, in multiple-color fluorescence resonance energy transfer (77), atomistic simulations (78, 79), and changing environmental conditions and mutations (80). Reproducible patterns in the folding trajectories are evidence of a preferential folding pathway in the energy landscape, supporting the FH.

Summing up, CFS permits measuring folding entropies and enthalpies over a broad temperature range, with the accuracy necessary to determine heat capacity changes. In conjunction with a detailed kinetics study, this permits us to determine barrier entropies, enthalpies, and heat capacity changes relative to the N and U. Three thermodynamic inequalities summarize our results: , and . These are key inequalities for molecular folding in line with predictions of the molten globule and energy landscape hypotheses. In particular, accurate measurements of are crucial to quantify to what extent configurational entropy loss drives intermediates formation and folding. Our study might be extended to other proteins, RNAs, and ligand–substrate binding (64, 81, 82). In the latter, the ligand docks into the binding site of the substrate by searching in configurational space, similarly to finding the TS in protein folding. Docking is then followed by the assembly of the ligand–substrate complex, analogously to the TS–N collapse in protein folding.

Materials and Methods

Molecular Construct and Experimental Setup.

For pulling experiments, barnase is expressed between two identical double-stranded DNA (dsDNA) (500 bp) handles, which are attached to the N and C termini via cysteine thiol chemical reduction (details in ref. 49). The 5 end of one handle is labeled with a biotin, while the 3 end of the other handle is labeled with a digoxigenin. The biotin- and digoxigenin-labeled ends specifically bind to spreptavidin (SA) and antidigoxigenin (AD) coated beads. For the pulling experiments (Fig. 1), one end is attached to the SA bead, which is kept fixed at the tip of a glass micropipette by air suction, whereas the other end is attached to the AD bead captured in the optical trap. Force changes by varying the relative distance λ between the center of the optical trap and the bead in the pipette. In pulling experiments, the optical trap is moved up and down at a given speed and force ramped between an initial force (1 pN to 2 pN), where the molecule is folded, and a maximum force (∼30 pN), where barnase is unfolded. The unfolding transition is detected as a force rip in the force versus λ curve (FDC). Moreover, due to the folding reversibility of barnase, it refolds upon reducing λ and the force.

To perform CFS experiments, we used the temperature-jump optical trap described in ref. 42. Briefly, a collimated laser at 1,435-nm wavelength (heating laser) is used to heat uniformly a ∼100-µm-radius area in the center of the fluidics chamber where the experiments are carried out. The wavelength is chosen to maximize the absorption by the water in the buffer solution (10 mM of Na HPO and NaH PO at pH 7.0) to heat the surrounding medium to locally raise the temperature from 25 ∘C (room temperature) to 40 ∘C. Moreover, we can place the miniaturized optical tweezers instrument inside an icebox kept at 5 ∘C and heat from this basal temperature up to 25 ∘C using the heating laser. In this way, the available temperature ranges from 5 ∘C to 40 ∘C.

Folding Free Energy at Zero Force.

To derive the folding free energy at zero force, , from force experiments, we consider four different states of the protein (Fig. 2, circled numbers): state 1, folded barnase at zero force; state 2, folded barnase at a given force f; state 3, unfolded barnase at the same force f; and state 4, unfolded barnase at zero force. Notice that the direct unfolding pathway at zero force 1 → 4, observed in bulk experiments, can be decomposed as the sum of three sequential steps (1 → 4 = 1 → 2 + 2 → 3 + 3 → 4). The three steps are as follows. For step 1 → 2, folded barnase is reversibly pulled from zero force to force f along the native branch of the FDC (Fig. 2 , Left Inset, black dashed line). The free-energy difference equals the sum of the reversible work to orient a dipole of length equal to the distance between the N and C termini of folded barnase [ 3 nm (49)], , and the reversible work of stretching the handles and displacing the bead in the optical trap, . For step 2 → 3, folded barnase is reversibly unfolded (denatured) at a constant force f. In this step, the free-energy difference, , equals the free energy of the stretched polypeptide chain minus the folding free energy of native barnase, at force f. For step 3 → 4, the stretched polypeptide chain is reversibly relaxed from f to zero force along the unfolded branch of the FDC (Fig. 2 , Left Inset, gray solid line). The free-energy difference equals the reversible work of releasing the polypeptide chain from force f to 0 (–) plus the reversible work of relaxing the handles and the bead in the optical trap from f to zero (equal to –, from step 1 → 2). Thermodynamic energy differences are path independent, so . This gives

The same balance equation holds for , and . Notice that does not appear in Eq. , as it cancels out when adding steps 1 → 2 and 3 → 4 (see for details).

The unfolding transition at constant force can be measured in instruments where the intensive variable, that is, the force, is the natural control parameter (e.g., in magnetic tweezers). In contrast, in optical tweezers, force cannot be controlled unless force feedback is applied (43). As a consequence, the unfolding transition does not occur at fixed force f but at fixed λ. Indeed, when pulling with optical tweezers, the unfolding transition is observed as a force rip in the FDC (dark arrows in Fig. 2) which occurs at fixed λ. Therefore, free-energy differences in the force ensemble, , are Legendre transforms of those measured in the λ-ensemble (83).

A major contribution in Eq. is the elastic term for the polypeptide chain, which is often modeled as a semiflexible polymer. The term stands for the elastic energy of aligning a molecular-sized dipole along the force axis. As the dipole extension is much shorter than the contour length of the polypeptide chain, at all forces. The relative magnitude of and depends on the difference between f and the coexistence force f, which is defined as the force at which the folded and unfolded barnases have equal free energies; that is, . Eq. gives, for f = f,

The stretching free energy of the different elastic elements at a given force f can be obtained by using the well-known expression (83)where i p,d, whereas the difference in the trap position between the unfolded and native branches at a given force f (Fig. 2 , Right Inset) equals

Combining the previous equations, we obtain the relationshowing that the knowledge of and the measured permits determination of at a given temperature T. Eq. is the basic thermodynamic formula we will use to determine , , and in barnase folding.