Predicting coupling limits from an experimentally determined energy landscape
- T. C. Jenkins Department of Biophysics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218
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Edited by S. Walter Englander, University of Pennsylvania School of Medicine, Philadelphia, PA, and approved January 11, 2007 (received for review October 5, 2006)
Abstract
Repeat proteins are composed of tandem structural modules in which close contacts do not extend beyond adjacent repeats. Despite the local nature of these close contacts, repeat proteins often unfold as a single, highly coupled unit. Previous studies on the Notch ankyrin domain suggest that this lack of equilibrium unfolding intermediates results both from stabilizing interfaces between each repeat and from a roughly uniform distribution of stability across the folding energy landscape. To investigate this idea, we have generated 15 variants of the Notch ankyrin domain with single and multiple destabilizing substitutions that make the energy landscape uneven. By applying a free energy additivity analysis to these variants, we quantified the destabilization threshold over which repeats 6 and 7 decouple from repeats 1–5. The free energy coupling limit suggested by this additivity analysis (≈4 kcal/mol) is also reflected in m-value analysis and in differences among equilibrium unfolding transitions as monitored by CD versus fluorescence for all 15 variants. All of these observations are quantitatively predicted by analyzing the response of the experimentally determined energy landscape to increasing unevenness. These results highlight the importance of a uniform distribution of local stability in achieving cooperative unfolding.
The origin and limits of cooperativity in protein folding remain an open and important question (1, 2). The structures of repeat proteins, which are elongated and are composed of a repeated structural motif (3), are well suited for investigating cooperativity in folding. One key difference between repeat and globular proteins is that in repeat proteins the close contacts are local in primary sequence, whereas in globular proteins the close contacts are often distant in primary sequence. The abundance of sequence-distant contacts in globular proteins is consistent with the commonly observed two-state unfolding transitions (lacking partially folded intermediates) that couple distant regions of globular protein structures. Thus, the lack of sequence-distant contacts in repeat proteins might be expected to produce noncooperative equilibrium unfolding transitions. As such, it is surprising that many repeat proteins also appear to unfold in a highly cooperative manner (4–7). One example of a repeat protein that unfolds in a two-state manner is the ankyrin domain of the Drosophila Notch receptor (Fig. 1), which contains seven ankyrin repeat sequences, six of which (sequence repeats 2–7) adopt the canonical ankyrin fold (8). The Notch ankyrin domain exhibits a single unfolding transition that appears to be two-state, both by spectroscopic and calorimetric criteria (9, 10).
The Notch ankyrin domain. A ribbon representation of Nank1–7Δ (8) is shown with repeats 1–5 (red) and repeats 6 and 7 (green). Proline (gray) and alanine (yellow) residues at consensus positions in repeats 5, 6, and 7 are shown in CPK representations. This figure was made with PyMOL.
Nearest-neighbor statistical thermodynamic (Ising) models (11, 12) indicate that coupling in repeat proteins depends on the presence of strongly stabilizing interactions between neighboring repeats. In addition, we expect that local stability§ must be uniformly distributed among repeats so that groups of locally coupled repeats do not unfold under different conditions (9). The distribution of folding energy among repeats can be clearly depicted on an energy landscape diagram. We have previously determined an energy landscape for the Notch ankyrin domain experimentally by measuring folding energies of a series of overlapping deletion constructs (12). This dissection allows the folding energy contribution of each repeat to be quantified, which in turn allows the free energies of various partly folded conformations to be determined (including conformations too high in energy to be measured directly). When the energies of these partly folded conformations are arrayed as a function of the number of folded repeats and of the location of structure from the N to C terminus, a surface (or landscape) results that provides a sensible coordinate system to evaluate the energetics and kinetics of folding (12, 13). Although the seven repeats of the Notch ankyrin domain make different contributions to the overall folding energy, these variations largely average out over pairs of repeats, producing an even energy landscape from the N to C terminus, reflecting a roughly uniform distribution of folding stability (Fig. 2 A). It is this uniformity of local stability, combined with strong nearest-neighbor coupling, that likely underlies the observed globally cooperative folding transition.
Examples of increasingly uneven energy landscapes for the Notch ankyrin domain. The free energies of partially folded species with variable numbers of folded, paired repeats are indicated on folding energy landscapes (12). Negative energies, relative to the denatured ensemble, are represented by color saturation. Each horizontal tier (extending from left to right) shows partially folded species with the same number of folded repeats. (A) Although there is local variation, the energy landscape for the wild-type Notch ankyrin domain is roughly uniform from N to C terminus (left to right). The energy minimum for the wild-type construct is the fully folded 1–7 species. (B) Destabilizing the sixth and seventh repeats each by 2.2 kcal/mol results in a less uniform energy distribution with the 1–7 species at approximately the same free energy as the 1–5 species. (C) At increasing destabilization (4 kcal/mol for each of the sixth and seventh repeats), the energy landscape is uneven and the energy minimum is the partly folded 1–5 species.
Here we seek to investigate the relationship between energy landscape evenness and coupling limits for cooperative folding. We perturb the shape of the energy landscape by combining locally destabilizing substitutions at one end of the molecule and monitor the coupling between the destabilized region and the rest of the molecule. We analyze 15 variants to approximate a continuum of destabilization. Destabilization is directed to the C-terminal two repeats (repeats 6 and 7), because earlier studies suggested that C-terminal destabilization could disrupt coupling and induce multistate unfolding (9). The degree of coupling for these 15 variants is assessed by three observable characteristics: thermodynamic additivity, m-value analysis, and agreement between different structural probes during unfolding. Changes in these three observable characteristics are quantitatively reproduced by statistical thermodynamic analysis introducing unevenness to the experimentally determined energy landscape. These findings show that accurately determined energy landscapes have a high degree of predictive value, capturing subtle aspects of the unfolding transition. Our findings also highlight the importance of energy landscape unevenness in addition to nearest-neighbor coupling for generating all-or-none equilibrium folding transitions.
Results
To evaluate the relationship between energy landscape evenness and coupling limits in folding, local structural perturbations must have corresponding local perturbations to the energy landscape. Owing to the lack of long-range contacts within the Notch ankyrin domain, its energy landscape has this property. Destabilizations produced by the C-terminal substitutions studied here perturb one side of the landscape and are predicted to result in a loss of cooperativity and ultimately an uncoupling of repeats 1–5 from repeats 6 and 7. In the absence of C-terminal destabilization, the energy landscape has a single, well defined minimum that corresponds to the fully folded molecule (referred to as the 1–7 species) (Fig. 2 A). In contrast, moderate levels of C-terminal destabilization produce two energy minima, one centered on a species with only the first five repeats folded (called the 1–5 species), and one centered on the 1–7 species (Fig. 2 B). High levels of C-terminal destabilization produce a very uneven energy landscape in which a single minimum is centered on the 1–5 species (Fig. 2 C). If this energy landscape description is accurate, changes in the energy landscape depicted in Fig. 2 should quantitatively predict the experimentally determined threshold to decouple repeats 6 and 7 from repeats 1–5.
To make the energy landscape of the Notch ankyrin domain increasingly uneven, we have destabilized the C-terminal repeats with single and multiple sequence perturbations. Specifically, Ala→Gly and Pro→Ala substitutions were introduced at consensus sequence positions [9/33 and 5/33, respectively (3)] in the sixth and seventh repeats. Contacts made at these substitution sites are localized within the same repeat; thus, substitutions are not likely to affect interfacial coupling. These substitutions were introduced into two constructs: the full-length Notch ankyrin domain (Nank1–7*) and a destabilized construct with the C-terminal nine residues deleted (Nank1–7Δ). The residues at these substitution sites are neither in physical contact with other sites, nor is there an obvious interaction pathway among sites (Fig. 1). This physical separation suggests that substitution at these sites should be additive, so long as the repeats containing these substitutions do not become unstructured (14).
The effects of C-terminal perturbations on the folding energy of the Notch ankyrin domain were determined by urea denaturation (Tables 1 and 2; see, for example, Fig. 3 A). Variants unfold over a range of urea concentrations, providing the range of C-terminal destabilization required to incrementally perturb the energy landscape (Tables 1 and 2).
Thermodynamic parameters of Notch ankyrin domain variants with single destabilizations
Thermodynamic parameters of Notch ankyrin domain variants with multiple destabilizations
Unfolding transitions and additivity of C-terminally destabilized variants suggest multistate equilibrium unfolding. (A) Experimentally determined urea-induced unfolding transitions of Nank1–7*, PA6*, AG6*, and AG67* (triangles, circles, squares, and diamonds, respectively), monitored by CD at 222 nm. Transitions are baseline-normalized. The increasing breadth of these unfolding transitions at progressively higher levels of destabilization (lower midpoints) is consistent with a shift to a multistate mechanism by which partially folded species are populated in the unfolding transition. The solid lines represent the best fit of Eq. 1 to the data. (B and C) The measured destabilization for single substitutions (left stacked histogram bars; total height represents the sum of destabilization from the single substitutions, ΣΔΔG°) is compared with the measured destabilization of the variant containing both substitutions (ΔΔG°CD, right histogram bar). The moderately destabilized PA67* variant (B) maintains additivity, whereas the highly destabilized AG67* variant (C) is nonadditive. Bars represent standard errors on the mean. The measurement conditions included 25 mM Tris·HCl/150 mM NaCl (pH 8.0) at 20°C.
Using Additivity to Probe Coupling Limits in the Notch Ankyrin Domain.
One means to investigate the coupling limit between repeats 1–5 and repeats 6 and 7 is to evaluate the degree to which multiple substitutions have additive effects on folding free energies. Substitutions are additive if the free-energy cost of multiple substitutions within the same molecule is equal to the sum of the costs of the individual substitutions. If repeats 6 and 7 remain substantially coupled to repeats 1–5 despite multiple C-terminal substitutions, those substitutions should be additive. If, instead, repeats 6 and 7 become unstructured from multiple substitutions, those substitutions should be nonadditive. The destabilization free energy required to produce nonadditivity defines the threshold over which repeats 6 and 7 decouple from repeats 1–5.
To evaluate additivity, we compared the sum of the changes in unfolding free energies (as determined from CD in the α-helical region) of multiple single substitutions, ΣΔΔG°, to the measured change in unfolding free energy of the variant containing the corresponding combined substitutions (ΔΔG°CD; the two Δs indicate an unfolding energy difference compared with the parent construct, Nank1–7*). Thus, ΣΔΔG° values represent the expected free energy differences based on additivity, whereas the ΔΔG°CD values are the apparent measured free-energy differences (for both single and multiple substitutions). Some substitution combinations (such as PA6* and PA7* compared with PA67*) (Fig. 3 B) are additive, whereas others (AG6* and AG7* compared with AG67*) (Fig. 3 C) are clearly nonadditive, with ΔΔG°CD values that are less than the corresponding sum from single substitutions. The additivity seen for the PA67* variant implies that repeats 6 and 7 remain structured in species that make a significant contribution to the PA67* unfolding transition. In contrast, the lack of additivity seen for the AG67* variant suggests that repeats 6 and 7 are substantially decoupled from repeats 1–5 in the unfolding transition for this variant. Accordingly, the coupling limit between repeats 6 and 7 and repeats 1–5 must be between the ΣΔΔG° value for PA67* (2.8 kcal/mol) and AG67* (6.3 kcal/mol).
To obtain a more accurate estimate of the destabilization threshold over which coupling is disrupted, we compared the extent of additivity for eight doubly and triply destabilized variants, all located in the sixth and seventh repeats (Fig. 4 A and Tables 1 and 2). There is a clear biphasic relationship between expected free energy change based on additivity (ΣΔΔG°) and the measured free energy change (ΔΔG°CD). Additivity is maintained below destabilizations of ≈4 kcal/mol (purely additive behavior is indicated by the dashed red line in Fig. 4), whereas additivity is lost above ≈4 kcal/mol, and measured ΔΔG°CD values remain at this threshold level regardless of the magnitude of the destabilization produced by the corresponding single sequence perturbations.
Coupling limits indicated by additivity, m-values, and differences between spectroscopic probes are quantitatively reproduced by energy landscapes that are increasingly uneven. (A) Measured free energy changes (ΔΔG°CD) for the PA6Δ (black), PA67* (red), PA7Δ (green), AG6Δ (blue), PA67Δ (yellow), AG7Δ (brown), AG67* (gray), and AG67Δ (violet) variants versus expected free energy changes based on additivity (ΣΔΔG°). Additivity is maintained up to a threshold of ≈4 kcal/mol, in agreement with simulated unfolding transitions (solid black line). The red dashed line indicates additive behavior. Because the AG57Δ variant has destabilizations on both sides of the interface between repeats 5 and 6, it exhibits approximate additivity at 7 kcal/mol destabilization (×), well above the 4 kcal/mol threshold. (B) Measured m-values for single and combined substitution variants (circles; see Tables 1 and 2) decrease and plateau with increasing C-terminal destabilization, in agreement with simulated values (solid line). (C) Measured deviations between ΔΔG°FL and ΔΔG°CD for single and combined substitution variants (circles; see Tables 1 and 2) increase and then decrease with increasing C-terminal destabilization, in agreement with simulated values (solid line). Bars represent standard error on the mean. (D) Simulated 1–7 (black circles), 1–5 (red squares), 2–7 (blue triangles), 2–5 (dashed green), and unfolded (magenta) species populations at the unfolding midpoint as a function of destabilization in the sixth and seventh repeats.
The above observations indicate that additivity is lost above an ≈4 kcal/mol limit if the destabilizations are localized to the sixth and seventh repeats. However, we expect that if the destabilizations are shared between repeats 1–5 and repeats 6 and 7, then additivity should be maintained above the ≈4 kcal/mol limit. To test this prediction, we combined a strongly destabilizing substitution in repeat 5 (AG5Δ) (9) with a strongly destabilizing substitution in repeat 7 (AG7Δ). In agreement with the above prediction, the combined variant, (AG57Δ) (Fig. 4 A, ×) exhibits roughly additive behavior at 7 kcal/mol total destabilization.
Using m-Values to Probe Coupling Limits in the Notch Ankyrin Domain.
Another means to investigate the coupling limits between repeats 1–5 and repeats 6 and 7 is to evaluate the effect of C-terminal destabilization on the breadth of the unfolding transition. As described above, variants with increasing C-terminal destabilization have increasingly uneven energy landscapes and are expected to show increasing levels of multistate unfolding (Fig. 2), broadening the unfolding transition. Consistent with these predictions, the increasingly destabilized PA6*, AG6*, and AG67* variants show progressively broader unfolding transitions compared with Nank1–7* (Fig. 3 A and Tables 1 and 2; see also ref. 9). The magnitude of the m-value, a quantitative measure of the breadth of the unfolding transition, reflects the average size of the cooperative unfolding unit (15). Accordingly, an increase in multistate unfolding that includes partially folded species (such as the 1–5 species) should result in a decreased m-value. The observed decrease in m-value suggests that the average size of the cooperative unfolding unit is reduced with increasing C-terminal destabilization, consistent with an increasing population of the 1–5 species.
The relationship between the observed m-values and the magnitude of C-terminal destabilization (represented by ΔΔG°CD values for single substation variants, and ΣΔΔG° values for multiple substitution variants) is complex. Observed m-values show a sigmoidal decrease and become insensitive to further C-terminal destabilization above 4 kcal/mol (Fig. 4 B). As described above, the decrease in m-value with increasing destabilization below 4 kcal/mol likely reflects the partial decoupling of repeats 6 and 7 from repeats 1–5. The insensitivity of m-value to C-terminal destabilization in excess of 4 kcal/mol (which matches the threshold for additivity identified in Fig. 4 A) likely reflects full decoupling of repeats 6 and 7 from repeats 1–5.
Comparing Different Spectroscopic Signals to Probe Coupling Limits in the Notch Ankyrin Domain.
A third means to evaluate the coupling limits between repeats 1–5 and repeats 6 and 7 is to compare independent spectroscopic probes. Comparison of independent spectroscopic probes is a common means to establish multistate unfolding. To further examine the relationship between coupling limits and energy landscape unevenness, we compared fluorescence-monitored unfolding transitions of C-terminally destabilized Notch ankyrin variants with transitions monitored by CD. The fluorescence signal of the Notch ankyrin domain is dominated by a single tryptophan residue that is buried in the fifth repeat. In multistate unfolding, where both the 1–7 and 1–5 species are populated in the unfolding transition, changes in the relative populations of these species will affect the CD signal (because of their differing number of folded repeats) but not the fluorescence signal (both species have the fifth repeat structured) (9). Thus, the difference between the unfolding free energy monitored by fluorescence and CD (ΔG°FL − ΔG°CD) should reach a maximum for multistate unfolding transitions (in which the 1–7 and 1–5 species are populated), with a lower apparent free energy reported by CD than by fluorescence. In agreement with these predictions, ΔG°FL − ΔG°CD values are small at very low (fully additive, high m-values) and very high (nonadditive, limiting low m-value) levels of C-terminal destabilization, whereas they become large (and positive) at intermediate levels of destabilization (Fig. 4 C). These results are consistent with the unfolding transition being dominated by the 1–7 and 1–5 species at low and high levels of destabilization, with multistate unfolding at intermediate levels of destabilization.
Predicting Coupling Limits from Uneven Energy Landscapes.
The central concept behind this study is that energy landscape unevenness can give rise to multistate folding and can be used to predict coupling limits. The substitutions studied here, which were designed to introduce energy landscape unevenness, clearly demonstrate multistate unfolding and decoupling by three separate criteria (additivity, m-values, and CD versus fluorescence). To rigorously test whether the experimentally determined energy landscape of the Notch ankyrin domain quantitatively reproduces these experimentally measured observations, we used the energy landscape to generate CD- and fluorescence-monitored unfolding transitions associated with increasing C-terminal destabilization (see Materials and Methods). The apparent free energies fitted from these simulated unfolding transitions (Fig. 4 A, solid black line) are in good agreement with the experimentally measured ΔΔG°CD values (Fig. 4 A, circles), including a break from additivity to nonadditivity at the 4 kcal/mol threshold. The m-values from these simulated unfolding transitions (Fig. 4 B, solid line) also show agreement to the experimentally measured values (Fig. 4 B, circles). Finally, the deviations between CD- and fluorescence-monitored apparent free energies from the simulated unfolding transitions (Fig. 4 C, solid line) show agreement to the experimentally measured values (Fig. 4 C, circles). These results indicate that the energy landscape quantitatively predicts coupling limits in the Notch ankyrin domain.
Based on the energy landscape, the breakdown in additivity, the corresponding decrease in m-value, and the discrepancy between ΔG°FL and ΔG°CD result directly from increasing populations of the 1–5 species in the transition region. At moderate levels of C-terminal destabilization, both the 1–5 and 1–7 species are populated in the unfolding transition (Fig. 4 D), decreasing the m-value, and increasing ΔG°FL − ΔG°CD. At high levels of C-terminal destabilization (>4 kcal/mol), the fully folded 1–7 species is not populated; thus, the m-value plateaus at a value of ≈2 kcal·mol−1·M−1 (Fig. 4 B, solid line) and ΔG°FL − ΔG°CD approaches zero, corresponding to a partial unfolding transition involving repeats 1–5 (Fig. 4 D).
In the absence of C-terminal destabilization, the equilibrium energy landscape predicts a small contribution of the 2–7 species to unfolding. Although the predicted concentration of this intermediate is non-zero, suggesting multistate unfolding, the contribution of the 2–7 species is modest, building to ≈3% in the absence of urea (data not shown) to a maximum of ≈9% at the midpoint of the transition (Fig. 4 D, blue triangles). At higher levels of C-terminal destabilization, the contribution of the 2–7 species is diminished, because the transition occurs at lower urea concentrations. Moreover, the calculated increase in the 2–7 species concentration near the transition is likely to be artificially high, because we ascribe the same m-value for the first repeat (0.4 kcal·mol−1·M−1; see Materials and Methods) as we do for repeats 2–7. Because the first repeat is largely disordered in the crystal structure, its m-value is likely to be smaller than for the fully structured C-terminal repeats, suppressing the population of the 2–7 intermediate. Thus, the contribution of the 2–7 species is likely to be too low to detect.
Discussion
The folding of globular proteins often shows remarkably high cooperativity, despite the large number of interactions that are formed upon folding. Recently, this high cooperativity has been shown to extend to repeat proteins. Although the origins of this cooperativity are not fully understood, it is clear that there must be substantial thermodynamic coupling between neighboring units. This coupling can be demonstrated in the folding of repeat proteins, for which the individual units and their interfaces can be separated, both conceptually and experimentally (4, 12).
A less obvious requirement for cooperative folding is a uniform distribution of stability among structural units. Although the distribution of stability is difficult to measure in globular proteins, the modular architecture of repeat proteins allows the distribution to be measured, experimentally perturbed, and used to construct a quantitative energy landscape for folding. Here we demonstrate that, as the distribution of stability across the Notch ankyrin domain becomes uneven, cooperativity is lost. This breakdown of cooperativity is manifested in a loss of additivity of destabilizing substitutions, in m-values for unfolding, and in deviations between spectroscopic probes during unfolding. The behavior of all of these observable characteristics is quantitatively predicted by an experimentally determined energy landscape derived from deletion studies of whole repeats. It is important to point out that this agreement, which is depicted by the solid lines in Fig. 4 A–C, does not involve the fitting of any adjustable parameters, demonstrating that accurately determined energy landscapes can indeed predict subtle aspects of the folding reaction.
How do these results extend to globular proteins? We would expect that the increased number of interactions among the major structural elements of globular proteins should contribute to a uniform energy landscape. However, the variation in native-state hydrogen exchange rates in globular proteins suggests substantial variation in local stability, consistent with an uneven energy landscape (16, 17). For example, hydrogen exchange studies of RNase H indicate two discrete structural regions: a core region of high stability and a peripheral region with lower stability (16). This stability variation is enough to produce a kinetic folding intermediate (as is seen for the kinetics of Notch ankyrin domain folding) involving the stable core region, yet an equilibrium two-state folding mechanism for RNase H is well supported. However, by introducing destabilizing substitutions that include the peripheral region of RNase H, multistate equilibrium unfolding results (18). These results are qualitatively similar to those described here for the Notch ankyrin domain.
There are a number of other globular proteins for which native-state hydrogen exchange data can be used to obtain landmarks on the energy landscape for folding (19). Despite variations in hydrogen exchange rates, most of these proteins fold by a two-state mechanism (19). This observation suggests that the unstable regions of these proteins are either uniformly distributed among stable elements or have local stabilities that differ by a modest amount. One notable exception to this trend is cytochrome-c, for which macroscopic optical measurements of unfolding free energy differ significantly from that determined by hydrogen exchange (20). For this protein, the distribution of stability is uneven (21, 22), with one end of the protein (the N- and C-terminal helices) showing substantially greater stability than the other end (the omega loops).
We note that, although the two-state folding properties that result from even energy distributions among highly coupled structural elements is convenient for determining protein stabilities experimentally, it may not always be biologically advantageous. For example, allosteric proteins have been suggested to use long-range modulation of stability as a means to generate cooperativity among distant binding sites (23). Because repeat proteins are particularly amenable to analysis of coupling and energy distribution, those that exhibit allosteric interactions with multiple partners provide a good system to explore this potential relationship (24).
Materials and Methods
Unfolding free energies were determined for variants of a seven-ankyrin-repeat domain from the Drosophila Notch receptor in which a single cysteine residue at position 2100 (10, 25) was replaced with serine to promote reversible unimolecular folding. In addition, the polypeptide contains an N-terminal His6 tag to facilitate purification. All variants were expressed in the Escherichia coli BL21(DE3) cell line and purified as described previously (10).
Site-directed mutagenesis was performed by using a Stratagene (La Jolla, CA) QuikChange mutagenesis kit. Individual variants are named by their type of substitution (PA and AG indicate Pro→Ala and Ala→Gly substitution, respectively) and the repeat number of the substitution. Variants were constructed in the full-length construct (designated by an asterisk) or in a construct that lacks the nine C-terminal residues (designated by a Δ) (9). Thus, PA6* indicates Pro→Ala substitution at the consensus position of repeat 6 in the full-length construct, and AG67Δ indicates Ala→Gly substitutions at the consensus position in repeats 6 and 7 combined with the C-terminal deletion. Partially folded species are referred to by the location of their folded repeats. Thus, the fully folded species is designated 1–7, and the species with only the first five repeats folded is termed 1–5.
Urea-Induced Unfolding Transitions.
Ultrapure urea was obtained from ICN (Aurora, OH) and was dissolved in water and deionized by using mixed bed resin from Bio-Rad
(Hercules, CA). Urea concentrations were measured refractometrically (26). Urea-induced unfolding transitions were measured on an Aviv 62A DS spectropolarimeter (Aviv Associates, Lakewood, NJ) equipped
with a thermoelectric cell holder. CD was monitored at 222 nm. Fluorescence emission was recorded through a perpendicular
320-nm cutoff filter after excitation at 280 nm. Unfolding transitions were measured in 150 mM NaCl/25 mM Tris·HCl (pH 8.0)
at 20°C, with protein concentrations between 2.0 and 5.0 μM, as described previously (9). Free energies of unfolding were determined from urea-induced unfolding transitions by using the linear extrapolation method
(26–28). Unfolding transitions were fitted by using the nonlinear least-squares algorithm of the Kaleidagraph program (Synergy Software,
Reading, PA) with the following equation:
where Y
obs is the observed signal, and ΔG°H2O is the free energy of unfolding in the absence of urea. We denote ΔG°H2O values that were determined from transitions monitored by CD and fluorescence as ΔG°CD and ΔG°FL, respectively. The sum of free energy changes, ΣΔΔG°, was calculated from ΔG°CD values determined from single substitutions and C-terminal deletion (Δ) in the Nank1–7* construct, except for Ala→Gly substitution
in the fifth repeat, which was determined in the Nank1–7Δ construct.
Numerical Calculations of Unfolding Transitions from Uneven Energy Landscapes.
To compare experimentally determined unfolding free energies to those predicted from energy landscapes of uneven shape, we simulated unfolding transitions for the Notch ankyrin domain with various levels of destabilization shared equally between the sixth and seventh repeats. Estimates for the folding energies of partially folded species were taken from thermodynamic measurements of truncations of the Notch ankyrin domain (12). Truncation free energies were measured in the absence of NaCl, whereas the unfolding transitions for the C-terminal variants presented in this study were measured at 150 mM NaCl. To account for the effects of NaCl on unfolding free energy, we applied a linear transformation to the free energies of partly folded species based on the observed free energy of unfolding of the full-length Notch ankyrin domain in the absence and presence of 150 mM NaCl. Unfolding m-values for partly folded species were taken to be 0.4 kcal·mol−1·M−1 per folded repeat. By using these predicted m-values, the free energies of partially folded species were calculated as a function of urea by a standard linear energy dependence (26, 28).
Unfolding free energies of the full ensemble of partly folded species were used to simulate CD signals of unfolding transitions by Boltzmann-weighting each species to produce populations and then weighting populations by the number of folded repeats in each species. All repeats were assigned an equal contribution to the CD signal except repeat 1, which was treated as having no effect on CD, because this repeat is largely unstructured in the crystal structure (8). Simulated fluorescence signals were determined by assigning a full signal to species that have both the fourth and fifth repeats folded and no signal otherwise. The resulting simulated transitions were fitted by a two-state equation (Eq. 1, with the slopes of the native and denatured baselines fixed at values of zero) to estimate the apparent unfolding free energies and m-values that would be determined from urea titration. Additive estimates for C-terminal destabilization ΣΔΔG° were calculated from the sum of two ΔΔG°CD values associated with equal C-terminal destabilization. These numerical simulations were performed in the R statistical computing package (29).
Acknowledgments
This work was supported by National Institutes of Health Grant GM068462 (to D.B.) and a Burroughs Welcome predoctoral fellowship (to T.O.S.).
Footnotes
- ‡To whom correspondence should be addressed. E-mail: barrick{at}jhu.edu
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Author contributions: T.O.S., C.M.B., and D.B. designed research; T.O.S. and C.M.B. performed research; T.O.S., C.M.B., and D.B. analyzed data; and T.O.S., C.M.B., and D.B. wrote the paper.
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↵ †Present address: U.S. Patent and Trademark Office, 600 Dulaney Street, Alexandria, VA 22314.
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The authors declare no conflict of interest.
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This article is a PNAS direct submission.
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↵ § Here we define local stability to include contributions from both intrinsic and nearest-neighbor interfacial interactions.
- © 2007 by The National Academy of Sciences of the USA
