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Cp-values in protein folding
1 Department of Life Sciences, Aalborg University, DK-9000 Aalborg, Denmark
2 Department of Biochemistry, Umeå University, S-90187 Umeå, Sweden
Reprint requests to: Daniel E. Otzen, Department of Life Sciences, Aalborg University, Sohngaardsholmsvej 49, DK-9000 Aalborg, Denmark; e-mail: dao{at}bio.aau.dk; fax: +45-98-14-18-08.
(RECEIVED July 14, 2004; FINAL REVISION August 24, 2004; ACCEPTED August 26, 2004)
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Abstract |
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at different denaturant concentrations measured by m-values and by heat capacity values are very similar. These observations are consistent with significant transition state movements within the framework of two-state folding. The basis for TSE movement appears to be enthalpic rather than entropic, suggesting that the binding energy of denaturant–protein interactions is a major determinant of the response of energy landscape contours to changing environments. Keywords: protein folding; kinetics; thermodynamics; transition state ensemble; m-values; heat capacity
Abbreviations: 
CpD–N, heat capacity difference between the native and denatured state • Cpf, activation heat capacity of folding • Cpu, activation heat capacity of unfolding • TSE, transition state ensemble
Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.04991004.
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is compared to D and N. This ratio, here termed , is a simple measure of the position of the TSE on the reaction coordinate between D (D = 0) and N (N = 1).
Most two-state proteins have a constant value (Jackson and Fersht 1991a; Alexander et al. 1992; Milla and Sauer 1994; Kragelund et al. 1995; Schindler and Schmid 1996; Jackson 1998; Perl et al. 1998; Reid et al. 1998), indicating that the structure of the TSE is not sensitive to denaturant concentration. However, we have previously shown that varies with GdmCl concentration for the protein U1A, which otherwise appears to fold by two-state kinetics (Silow and Oliveberg 1997; Otzen et al. 1999). This proposal was based on pronounced curvature in the refolding and unfolding limbs of U1A’s chevron plot (the log of the observed rate constant kobs vs. [GdmCl]). Curvature may also be induced by suitable mutations. The protein S6 from the small ribosomal subunit of Thermus thermophilus manifests a straight V-shaped chevron plot, giving a constant . However, the mutant LA30 shows very pronounced unfolding curvature, leading to a gradual increase in the value of with GdmCl concentration (Otzen et al. 1999). We have interpreted these phenomena in terms of structural changes, or movement, in the TSE. In other words, the TSE is an ensemble of states with structure (in this context, compactness) that varies with solvent conditions; it may also be explained by saying that as the solvent conditions change, so different ensembles of states become the ensemble with the highest free energy on the reaction coordinate. Thus, rising values of , which accompany the increase in [GdmCl], translate to an increasing compactness of the TSE, and structural characteristics of the energy landscape are reflected in the details of the unfolding limb. For example, smooth curvature in the unfolding limb can arise when activation barriers to folding form a shallow top in a rather flat landscape; the landscape surface tilts in response to changes in solvent conditions and thus alters the position of the hilltop (Otzen et al. 1999). Kinks may arise if the landscape is more rugged, so that one peak replaces another as the point of highest energy over a relatively narrow denaturant concentration range.
To analyze the thermodynamic basis for these postulated TSE movements, we have examined the folding kinetics of U1A, wild-type S6, and the S6 mutant Leu 
Ala30 (LA30) in the temperature interval 10°C to 70°C. By introducing temperature as a physical variable in addition to denaturant concentration, we can extract values for the activation enthalpy, entropy, and heat capacity of folding and unfolding, which may provide important complementary information. Generally, thermodynamic parameters for the folding and unfolding of polymers in aqueous environments should be approached with caution, because they result from a combination of very large favorable and unfavorable energetic contributions which almost cancel out (Creighton 1993). The reduction in conformational entropy upon folding is balanced by an increase in the entropy of dehydration as bound water molecules are released (the hydrophobic effect). Conversely, the change in enthalpy upon unfolding is composed of contributions from (1) the gain in enthalpy from loss of van der Waals interactions and hydrogen bonds, (2) the loss in enthalpy from interactions between water molecules and surfaces exposed upon unfolding and the removal of steric repulsion in the native state, and (3) the gain in enthalpy due to the disruption of water interactions upon solvating freshly exposed protein surfaces (Johnson and Fersht 1995). However, the link between thermodynamic changes and conformational transitions becomes less unequivocal in the case of the heat capacity change Cp. This parameter predominantly reflects a negative contribution from desolvation of nonpolar groups (Makhatadze and Privalov 1990; Livingstone et al. 1991; Spolar et al. 1992), with the nuances that heat capacity changes for aliphatic and aromatic groups differ to a certain extent (Spolar et al. 1992) and that the desolvation of polar groups makes a small but positive contribution to Cp (Spolar et al. 1992). However, the effect of these groups on the m-value is less clear. The configurational freedom gained upon protein unfolding accounts for <20% of the total heat capacity increase (Privalov and Makhatadze 1990).
Because heat capacity changes and m-values both reflect the burial of nonpolar groups, they should go hand in hand; indeed, for the cold-shock protein CspB, 90% of the total heat capacity change and 96% of the m-value change occur between the unfolded state and the TSE (Schindler and Schmid 1996). CspB has a fixed ; consistent with this, its activation heat capacities for folding and unfolding (Cpf and Cpu, respectively) are essentially insensitive to denaturant concentration. However, we expect that the activation heat capacity for folding and unfolding of U1A should be sensitive to denaturant concentration and, furthermore, that the Cpu of LA30 should be more sensitive than that of S6 wild type. Both predictions are borne out by the experimental data.
It should be pointed out that kinetic data by their very nature generally lend themselves to more than one mechanistic interpretation. Protein folding is a complex multivariable reaction and is no exception to this rule, particularly when it comes to the interpretation of thermodynamic parameters. In addition to moving TSEs, we certainly do not exclude the possibility that other scenarios may be at work that also lead to curvature. These scenarios include the existence of discrete unfolding intermediates or several TSEs. However, for reasons detailed in Materials and Methods, we regard them as less probable. Instead, we will take the opportunity to examine in what way the temperature data may shed more light on the thermodynamic properties of the moving TSE.
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for the three proteins over the temperature interval 10°C to 70°C, and the data are summarized in Table 1. At all temperatures, the chevron plots for each protein retain the characteristic features previously reported at 25°C (Otzen et al. 1999), namely, symmetrical curvature for U1A, a V-shape for S6, and curvature in the unfolding region for LA30.
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. The rate constants calculated from these parameters fit well to the actual rate constants at all concentrations (Fig. 1). Table 1 only lists those parameters necessary to improve the quality of the fit.
Equivalence of equilibrium and kinetic Cp-values
Having obtained values of Cp from kinetic experiments, we now compare them with values obtained from equilibrium denaturation data. In a two-state system, the sum of Cpf and Cpu equals the heat capacity difference between the native and denatured state CpD–N.
U1A
The CpD–N of U1A has been determined by Lu and Hall (1997) to be 1200 ± 100 cal/mol/K using differential scanning calorimetry. Our kinetic data extrapolated to 0 molar denaturant should be directly comparable to this value. Cpf and Cpu are 350 ± 20 and 760 ± 50 cal/mol/K, respectively. The sum is 1110 ± 54 cal/mol/K, which agrees well with the DSC-value. Both Cpf and Cpu show a significant dependence on the denaturant concentration, but in opposite directions, namely, 55 and –64 cal/mol/K/M, respectively. As a consequence, the sum Cpf + Cpu remains essentially invariant over the entire GdmCl-range (Table 1). An analogous behavior is seen for the kinetic m-values mf and mu of U1A. At 25°C, they depend on [GdmCl] as follows (Silow and Oliveberg 1997):
 | (1a) |
 | (1b) |
Thus, the equilibrium m-value mD–N = mf – mu is essentially constant at –1.67 M–1–0.019 M–2*[GdmCl].
S6
DSC studies at low pH in the absence of denaturant did not yield useful data for S6 due to poor data quality (data not shown). Instead, we have determined CpD–N by two independent approaches. First, we have performed thermal scanning experiments for S6 wild type and LA30 by using circular dichroism (CD) (Fig. 2A). CpD–N can be obtained as the slope of the plot of the enthapy of unfolding HTm versus the denaturation temperature Tm. Different values of HTm and Tm were obtained by carrying out the thermal scans at different GdmCl concentrations. If CpD–N depends on the GdmCl concentration, there will be a systematic deviation between the value obtained by this approach as opposed to measuring it in the absence of denaturant, for example, at low pH values. This is elaborated in the Discussion. For both proteins, there is a reasonable linear correlation between the two parameters (Fig. 2B), which leads to a CpD–N value of 1008 ± 79 and 976 ± 56 cal/mol/K for wild type and LA30, respectively.
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). At each temperature, the free energy of unfolding GD–N is extrapolated back to 0 molar denaturant by using equation 10. When GD–N is plotted versus temperature and the data fitted to equation 11 (Fig. 3B), a CpD–N value of 930 ± 45 cal/mol/K is obtained. The Tm value in 0M GdmCl required for this fit (383.13 ± 1.07 K) is estimated as an extrapolation from the observed linear relationship between Tm and the GdmCl concentration (Fig. 2C). Note that although the error on CpD–N is small, the CpD–N value is very sensitive to the value of Tm, and altering Tm by a few degrees leads to significant changes in CpD–N. This makes the heat capacity determined by this approach less reliable. However, the agreement with the CpD–N value obtained from CD scans is reassuring.
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Cpf and Cpu are summed to give CpD–N in 0 molar denaturant (815 ± 72 and 1153 ± 84 ca/mol/K for wild type and LA30, respectively), the dependence completely cancels out for LA30 and is reduced for wild type (Table 1). The correspondence with the values obtained from thermal scans is reasonable. If we extrapolate Cpf + Cpu to the midpoint of the denaturation of S6 wild types (3.3M GdmCl), the agreement becomes even better, namely, 960 ± 70 cal/mol/K. Overall, the accuracy of our data is comparable to that of calorimetric measurements (Pace and Laurents 1989; Lu and Hall 1997), although the errors are certainly not <10%.
Temperature optima of the three proteins and cold denaturation
Based on the free energy of unfolding in a two-state system (GD–N = RT*ln10*log(kf/ku)) and the temperature dependence of the rate constants (equation 7), we can calculate the free energy of unfolding as a function of temperature (Fig. 4) and estimate the temperature at which each protein is most stable (Table 1). It is not surprising that U1A has a somewhat lower TGD–N max than does S6 (Table 1) because U1A is from a mesophilic organism and S6 from a thermophile. By contrast, it may seem odd that LA30 has a higher TGD–N max than does S6 wild type. However, the changes in the thermodynamic parameters compared with S6 wild type simply shift the position of the minimum. Although LA30 unfolds at significantly lower denaturant concentrations than does S6 wild type, the difference in stability between S6 wild type and LA30 is close to 0 at room temperature. This is because of the curvature in the unfolding limb of LA30, which is absent in S6 wild type. The curvature means that the unfolding rates of the two proteins are very similar when extrapolated to 0 molar denaturant (Otzen et al. 1999).
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 | (2) |
Because we already know the GdmCl-dependence of Tm (Fig. 2C), HTm (Fig. 2D) and CpD–N (Table 1), we can calculate Tc as a function of denaturant concentration. For S6 wild type, Tc only rises to 2.5°C around the midpoint of denaturation, whereas the corresponding value for LA30 is 7.7°C. In practice, this means that cold denaturation is not experimentally accessible for these two proteins.
The nature and temperature-dependence of the activation barrier
Although the enthalpic and entropic parameters obtained in this study do not lend themselves so easily to a structural interpretation as the heat capacity, they merit some comments. At 25°C, all three proteins manifest a massive enthalpic barrier to both folding and unfolding. The barrier is largest for unfolding, because this process primarily represents the breaking of non-covalent bonds stabilizing the native state. This lopsidedness is also seen for other two-state proteins such as CI2 (Jackson and Fersht 1991b; Oliveberg et al. 1995) and CspB (Schindler and Schmid 1996). At all temperatures, there is very clear "entropy-enthalpy compensation" with regard to the effect of the mutation LA30, resulting in a very small net change in free energy (Table 2). However, at 25°C, LA30 shows a significant reduction in the equilibrium enthalpy of unfolding (calculated as the difference between the activation entalpy of unfolding and of refolding; 1.4 ± 1.2 kcal/mol) compared to S6 wild type (12.4 ± 1.3 kcal/mol). This loss of 11 kcal/mol most probably reflects the loss of stabilizing interactions due to the mutation. At 25°C, the average change in activation enthalpy caused by the removal of a core methylene group in the protein CD2.d1 is 3.7 ± 1.8 kcal/mol (Lorch et al. 2000). For a Leu Ala mutation, this translates into 11.1 kcal/mol, very close to the above data. Most of the enthalpy reduction translates into a decrease in the activation enthalpy of unfolding, showing that the enthalpically favorable side-chain interactions involving Leu30 in the native state are largely broken in the TSE. An extensive 
-value analysis (Otzen and Oliveberg 2002) of S6 shows that the TSE has a very diffuse folding nucleus, centered around four residues, including Leu 30 (Otzen and Oliveberg 2002). Because of the diffuse nature of the nucleus, the -values of these four residues are rather low. -Values are a reflection of the environment of the side chain in the TSE, with high -values indicating a native-like structure (Fersht et al. 1992). This is consistent with our thermodynamic data, which suggest that most stabilizing side-chain interactions are only formed very late in the folding process after the rate-limiting step.
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Increasing the temperature does not alter the roles allocated to enthalpy and entropy in the unfolding activation barrier. The barrier to unfolding becomes increasingly enthalpic, whereas the entropic component favors the unfolding process. The temperature-associated increase in the magnitude of the enthalpic and entropic values is largest for LA30, because of its large Cpu. However, for refolding, the entropy–enthalpy compensation changes between 10°C and 70°C (Table 2), as seen for CspB (Schindler and Schmid 1996). At high temperatures, the activation barrier for refolding becomes entropic, probably because the entropy of desolvation becomes less favorable: the enthalpy and entropy of desolvation of hydrophobic residues are very large at low temperatures but decrease strongly with increasing temperature (Murphy et al. 1990).
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Cp
Variations in Cpf and Cpu of U1A show a remarkable correlation with the variation in the m-values (Fig. 5A). For S6 wild type, Cpu is essentially invariant with denaturant concentration, whereas Cpf changes to a small extent. Encouragingly, Cpu of LA30 has become sensitive to unfolding, and the sensitivity of Cpf has increased (Table 1). This generally correlates well with the mobilization of the TSE deduced from the chevron plot at 25°C (Otzen et al. 1999).
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CpD–N values for all three proteins are essentially denaturant independent. Generally, HD–N decreases with urea, whereas Cp increases (Johnson and Fersht 1995). This is because denaturant–protein interactions are exothermic, in contrast to the intrinsically endothermic enthalpy of unfolding. Thus, unfolding in urea is a composite of two opposite enthalpic effects, and at high urea concentrations, the enthalpy of unfolding at a given Tm is significantly lower in urea than in the absence of urea (Johnson and Fersht 1995). This leads to a larger variation of HD–N with Tm when urea rather than pH is used as a perturbant, and as a result, the apparent Cp can be significantly overestimated. For barnase, the intrinsic Cp is 1.7 kcal/mol/K, whereas it is 2.7 kcal/mol/K when measured by using urea (Johnson and Fersht 1995). Because the effects of denaturant and temperature on HD–N are linear and thus additive, Cp increases linearly with denaturant concentration, as demonstrated directly for several proteins (Pfeil and Privalov 1976; Griko and Privalov 1992; Makhatadze and Privalov 1992; Johnson and Fersht 1995). Some proteins, however, appear to show low denaturant sensitivity with regards to the heat capacity. For example, the CpD–N of the two-state protein CspB actually decreases slightly with [urea] (Schindler and Schmid 1996), and the CpD–N value obtained by thermal scanning of thioredoxin in different urea concentrations was in good agreement with isothermal titration data (Santoro and Bolen 1992). The good correspondence between the Cp values for S6 wild type and LA30 obtained by different methods, including thermal scans at different denaturant concentrations, suggests that the denaturant insensitivity is a genuine phenomenon for some proteins. Although this invariance may be difficult to explain, it conveniently simplifies the situation for a structural interpretation of the kinetic Cp values.
Movement of the TSE: An enthalpic rather than entropic phenomenon
The variation in mf and mu with temperature for all three proteins is shown in Figure 5B. With these values, we can calculate the position of the TSE on the reaction coordinate by using the relationship = –mf/(mu – mf) (Fig. 5C). The increase in with temperature means that as the TSE becomes less stable at higher temperatures, it moves closer to the native state. This is entirely in accord with the Hammond postulate (Hammond 1955). A similar shift in with temperature has been seen for CI2, barnase, and lysozyme (Matouschek and Fersht 1993; Matouschek et al. 1995). can also be calculated from the heat capacities, viz. = Cpf/(Cpf + Cpu). Importantly, the variation in deduced from heat capacities and m-values is strikingly similar for U1A and also good for S6 wild type and LA30 (Fig. 5D). This provides additional support for the moving TSE model.
The curvature underpinning TSE movement appears to be linked entirely to the activation enthalpy (Table 1), which for U1A and LA30 shows significant nonzero values of m*h (equation 8) for both refolding and unfolding. There is no contribution from the activation entropy, because inclusion of an m*s parameter does not lead to an increase in the quality of the fits and is associated with large errors. We have previously discussed the nature of the moving TSE barrier and the way it shifts in response to changes in the denaturant concentration, but we have not been able to pin it down to any specific thermodynamic phenomenon. In the light of the data presented in this article, its enthalpic origin leads to a straightforward and reasonable conclusion: The plasticity of the energy landscape (that is, its response to denaturant concentration) is primarily governed by the strength of the enthalpic, namely, direct binding, interactions between denaturant molecules and the protein, rather than by any entropic parameters. Thus, the entropic effect of the changes in compaction of the TSE at different denaturant concentrations, and the liberation or immobilization of denaturant molecules with regard to the protein, is overruled by the enthalpic effect.
In summary, we have presented thermodynamic evidence to support our model for moving TSEs. This approach illustrates how complex thermodynamics can be derived relatively simply from kinetics by using linear relationships between energies and denaturant concentrations.
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 | (3) |
 | (4) |
 | (5) |
where x is refolding or unfolding, D is denaturant, amx and bmx are constants (for which bmfS6 wild type = bm uS6 wild type = bmf LA30 = 0). Equation 3 describes the relationship between log k and denaturant concentration (which is curved for bmx = 0), whereas equation 4 derives from the two-state relationship (equation 6).
Equilibrium denaturation experiments
CD thermal scans were monitored for wild-type S6 and LA30 at 225 nm in a 1-mm cell on a Jasco 720 CD spectrometer equipped with a Peltier element over the temperature range 20°C to 100°C at varying [GdmCl] (1 to 3 M) in 50 mM MES (pH 6.3). Protein concentrations were 30 µM. Equilibrium denaturation data between 0 and 6 M GdmCl at a given temperature were performed as described (Otzen et al. 1999). We were unable to obtain reliable data from CD thermal scans of U1A due to extensive aggregation at elevated temperatures (data not shown).
Thermodynamic analysis in a two-state system
For each protein, a global analysis fit was performed with GraFit (Erithacus Software), in which the following two equations were combined:
 | (6) |
where ku and kf are the rates of unfolding and refolding, respectively.
(2) Temperature dependence of the rate constants:
 | (7) |
where To = 298K. The term for vibrational frequency (
kBT/hÅ, ~1013 s–1), which is conventionally used in the analysis of simple chemical reactions, has been replaced with the factor 3356T (which is 106 s–1 at 298K). This figure constant represents the fastest step in protein folding, namely, closing of a loop (Hagen et al. 1996, 1997). The absolute value of this factor only affects the entropy of activation. The thermodynamic parameters had the following denaturant dependencies:
 | (8) |
HTo and STo refer to the activation enthalpy and entropy of either folding or unfolding and Cp to the activation heat capacity of folding or unfolding.
The m-values are constants. In practice, we find that all m-values linked to D2 and higher concentrations can be omitted because they do not improve the quality of the fit, except for mh* (as judged by the value of the 
2 factor and statistical errors of deviation). Although there is some evidence that activation heat capacity is temperature dependent (Oliveberg et al. 1995), we will for simplicity assume that Cp is temperature independent in the range 10°C to 70°C. CD thermal scans were fitted to the following equation (Oliveberg et al. 1994):
 | (9) |
where 
is the raw CD signal in millidegrees, 
N and D are the -values for the folded and denatured states at 298K, N and D are the linear slopes of N and D versus T, HTm is the enthalpy of unfolding at the midpoint of denaturation Tm, and Cp is the heat capacity difference between the native and denatured state. However, the Cp value derived from a single thermal scan is highly inaccurate. A much more reliable value is obtained as the slope of the plot of HTm versus Tm, in which Tm is varied by altering the denaturant concentration (see Results).
Cp was also calculated from equilibrium denaturation experiments in GdmCl at 10°C to 60°C as follows. At each temperature, the free energy of unfolding in water (GD–Nwater) was calculated according to the following equation (Pace 1986):
 | (10) |
where [GdmCl]50% is the midpoint of denaturation and 
mD–N
, the average dependence of the logarithm of the equilibrium constant for unfolding (KD–N) on [GdmCl] based on a large number of mutants, has a value of 1.75 ± 0.044 M–1 (Otzen et al. 1999). We used an average m-value, because individual m-values are typically measured with errors of at least 0.20 kcal/mol/M, and there was no systematic variation of mD–N with temperature (data not shown). The data were then fitted to the equation:
 | (11) |
where Tm is the midpoint denaturation temperature of S6 in 0 molar denaturant (383.13 ± 1.07 K). This value is extrapolated from a plot of Tm versus [GdmCl], based on CD-monitored thermal scans (see Results).
Analysis using other kinetic models
Different kinetic models can often account for the same data. The curvature in the unfolding limb of LA30 could in principle also be explained by the transient accumulation of an unfolding intermediate at high denaturant concentrations, according to the following scheme:
 | (scheme 1) |
We have previously argued against this scheme. First, there is no spectroscopic evidence (e.g., burst-phase effects) for such an intermediate. Second, the high mu-value of LA30 (significantly larger than that of wild-type S6) is difficult to explain from scheme 1 (Otzen et al. 1999), which is highly unlikely, because side-chain interactions are most structured, and thus stabilizing, in N, unless they were all engaged in nonnative interactions in I. Third, kinetic data from a series of S6 mutants fitted to scheme 1 led to the paradoxical conclusion that the mutated side chains generally stabilized I more than N (Otzen et al. 1999). The thermodynamic data do not support Scheme 1 either. Although the individual chevron plots fitted well to scheme 1, there was no consistent pattern in the temperature dependence of the parameters derived from the scheme (data not shown).
Unfolding limb curvature could, however, arise from a discrete jump between two peaks in the TSE (Otzen et al. 1999).
 | (scheme 2) |
Although I* never accumulates, it will give rise to two TSEs during unfolding, one between N and I* (TSE1), the other between I* and D (TSE2). This model has recently been extended to explicitly include parameters defining the stability of the intermediate that by definition resides between the two peaks (Bachmann and Kiefhaber 2001). At low denaturant concentrations, TSE1 will be ratelimiting, whereas TSE2 becomes ratelimiting at high denaturant concentrations (Bachmann and Kiefhaber 2001). This has been proposed to account for the folding behavior of tendamistat (Bachmann and Kiefhaber 2001) and S6 (Sánchez and Kiefhaber 2003). When we analyzed our LA30 data according to this scheme, however, we were unable to determine the parameters relating to the conversion of I*, unlike the case for tendamistat. We were able to vary mIU between around –0.5 and 2.0 M–1 without any effect on the fit’s appearance at different temperatures, provided the difference mIU – mIN remained constant. At all temperatures, the only restriction on kIU and kIN was that kIU/kIN had to be ~5.0 and kIU at least 105 s–1. The model in scheme 2 implies that different degrees of curvature in different mutants can be rationalized by the effects of the mutation on the relative stability of the two TSEs, whereas the m-values remain constant. However, a global fit of scheme 2 to a large number of S6 mutants failed to converge to a common set of m-values (M. Lindberg, D.E. Otzen, and M. Oliveberg, unpubl.). Therefore, we decided not to pursue scheme 2 any further.
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Acknowledgments |
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). The lines represent the best linear fits. The slopes, which represent the equilibrium heat capacity values, are 1050 ± 50 cal/mol/K for S6 wild type and 976 ± 56 cal/mol/K for LA30. (C) The dependence of Tm for S6 wild type (•) and LA30 (
), 40°C (
), 50°C (+), and 60°C (x). Fluorescence emission intensities are normalized to a maximum value of 100. (B) Stability of S6 wild type between 10°C and 60°C measured by isothermal chemical denaturation. The stability at each temperature is calculated from equation 10. The data are then fitted to equation 11 to yield a 
,
). (C) Change in the position of the TSE (