Temperature and pressure dependence of protein stability: The engineered fluorescein-binding lipocalin FluA shows an elliptic phase diagram

Abstract

We have measured the equilibrium constant for the denaturation transition of the engineered fluorescein-binding lipocalin FluA as a function of pressure and temperature, taking advantage of the fact that the ligand's fluorescence is almost fully quenched when complexed with the folded protein, but reversibly reappears on denaturation. From the equilibrium constant as a function of pressure and temperature all of the involved thermodynamic parameters of protein folding, in particular the changes in entropy and volume, compressibility, thermal expansion, and specific heat, were deduced in a global fitting procedure. Assuming that these parameters are independent of temperature and pressure, we can demonstrate from the ratio of Δβ, Δα², ΔC_p that the phase diagram of protein folding assumes an elliptic shape. Furthermore, we can show that the thermodynamic condition for such an elliptic phase diagram is related to the degree of correlation between the fluctuations of the changes in volume and enthalpy at the phase boundary. For the protein investigated this correlation is low, as generally expected for highly degenerate systems. Our study suggests that the elliptic phase diagram is a consequence of the inherent conformational disorder of proteins and that it may be viewed as the thermodynamic manifestation of the high degeneracy of conformational energies that is characteristic for this class of macromolecules.

Since the pioneering work on the thermodynamic stability of proteins by Brandts (1, 2) and Hawley (3), efforts to understand the processes that drive a folded protein under nonphysiological thermodynamic conditions, in particular, by changes in temperature and pressure, into the denatured state have gained ongoing interest (4–7). One persistent problem, for instance, concerns the nature of the denatured state. Although it seems generally accepted (8) that the heat-denatured polypeptide essentially assumes a random coil conformation, this is not necessarily true for the pressure-denatured state (9). Heat denaturation is well understood on the basis of the so-called oil droplet model. This model was developed by Baldwin (10, 11) on the basis of caloric experiments on a large series of proteins (12–14). In essence, this model views the thermal denaturation process as the solvation of the hydrophobic side chains, which are normally packed in the interior of a folded protein.

However, this concept cannot account for pressure-induced denaturation processes. In this case the observed volume change on unfolding is negative, at least above a certain threshold pressure, whereas the dissolution of hydrophobic molecules or chemical groups in water at ambient pressure is generally accompanied by a positive volume change. This discrepancy was first pointed out by Kauzmann (15) and a possible escape from this dilemma was proposed by Hummer et al. (9, 16). Guided by results from molecular dynamics simulations, these authors suggested that denaturation under pressure occurs by pressing water into the interior of the protein, which retains a high degree of compactness. This was also verified experimentally for the pressure-denatured staphylococcal nuclease protein (17). As a consequence, the folded polypeptide chain swells, internal hydrogen bonds and other noncovalent interactions become disrupted, but it retains a largely globular shape. Interestingly, a recent theoretical study indicated that α-helical structures seem to be stable against pressure (18), whereas β-hairpin structures behave similarly to globular proteins (19).

Already in the first attempts to measure pressure/temperature stability diagrams of proteins (3) it was observed that the boundary that separates the native and denatured states in the p/T plane may be described by an ellipse. However, a major problem with the unambiguous determination of the shape of the p/T-phase diagram lies in the fact that for many proteins a sufficiently large parameter range of the respective p/T values may not easily be accessible experimentally, either because the required pressure is negative or equilibrium cannot be reached within reasonable time scales. Despite this, the few experiments published so far suggest that the phase diagrams may indeed be elliptic (3, 20) (for reviews see refs. 4, 21, and 22). At least, they can be viewed as distorted ellipses (23), meaning that one or more of the relevant thermodynamic material parameters, namely the changes of the specific heat ΔC_p, compressibility Δβ, and thermal expansion Δα, depend on pressure and/or temperature (24).

In fact, an elliptic shape implies a strict relation between ΔC_p, Δβ, and Δα (3, 25). A challenging question in this context is: Why were only elliptic shapes observed so far, and what is the physical explanation of this finding? Are parabolic and hyperbolic shapes forbidden and, if so, why? In the present study, we will show that the pairing of organization and randomness is fundamentally bound to produce elliptically shaped phase diagrams. To this end, we investigate the thermodynamic stability of an engineered globular protein, the fluorescein-binding “anticalin” (FluA) (26).

FluA was derived from the 174-aa bilin-binding protein (BBP) of Pieris brassicae, a member of the lipocalin protein family, which is characterized by a β-barrel made of eight antiparallel strands that wind around a central axis and form a circularly closed hydrogen bond network (27). At one end, the β-barrel is open to the solvent and four loops connect the strands in a pairwise fashion to form a binding pocket for the ligand. By using targeted random mutagenesis in the loop region and molecular selection by phage display, a variant of BBP, dubbed FluA, was identified, which specifically complexes fluorescein as a ligand with high affinity (K_D = 35.2 ± 3.2 nM) (26). X-ray crystallographic analysis revealed that fluorescein is deeply buried in the β-barrel of the engineered lipocalin, thus replacing the original ligand biliverdin IXγ (28). Interestingly, the characteristic stationary fluorescence of fluorescein becomes almost completely quenched on complex formation with FluA (26). This phenomenon has been attributed to an extremely fast and efficient electron transfer between the excited fluorescein dianion and a tryptophan side chain that is tightly packed against its xanthenolone moiety in the ligand pocket, followed by a radiationless backtransfer (29).

Thus, FluA provides attractive properties for investigating protein stability by fluorescence spectroscopy. In the native state emission of the complexed fluorescein is completely quenched, whereas in the denatured state fluorescein regains its high quantum yield (29). Consequently, measurement of the relative fluorescence intensity as a function of pressure at a defined temperature yields the equilibrium constant for denaturation as a function of pressure or vice versa and, hence, the boundary of the phase diagram.

Results and Discussion

Thermodynamics of Proteins: Fundamental Aspects.

Although proteins are isolated molecular systems, their number of conformational states is extremely high so that a thermodynamic treatment at the level of a single protein seems to be justified. The usual thermodynamic treatment of the folding–denaturation transition is based on a first-order phase transition, which is described by the Clausius–Clapeyron equation with volume and entropy changes ΔV and ΔS, respectively (3):

In the following, we assume that the specific heat C_p, the compressibility β = −∂V/∂p, and the thermal expansion α = −∂V/∂T are characteristic material parameters for the denatured and the native state, respectively, and do not significantly change with pressure or temperature in the ranges investigated. Then, straightforward integration of the Clausius–Clapeyron equation leads to a general curve of second order for the shape of the phase boundary,

which can, in principle, be of elliptic, hyperbolic, or parabolic shape. T₀ and p₀ are arbitrarily chosen reference values for pressure and temperature, for which we chose standard conditions (Table 1).

Table 1.

Thermodynamic parameters characterizing the folding stability of FluA complexed with fluorescein

Parameter*	FluA·fluorescein	apo-FluA†
ΔG0, kJ/mol	10.8 ± 1.1	11.9 ± 0.3
ΔS0, kJ/K·mol	0.28 ± 0.09	0.43
ΔV0, ml/mol	−8 ± 3
ΔH0, kJ/mol	83 ± 26	139.1 ± 3.8
Δα, ml/K·mol	−1.0 ± 0.3
Δβ, ml/MPa·mol	0.65 ± 0.04
ΔCp, kJ/K·mol	5.7 ± 1.8
ρ*2	0.07
Th, °C	47	52.9 ± 0.2
Tc,°C	−27
pd,MPa	171

*Subscript 0 refers to ambient conditions (p₀ = 0.1 MPa and T₀ = 298 K). T_h, T_c, and p_d are the heat denaturation temperature, the cold denaturation temperature, and the denaturing pressure at ambient temperature, respectively.

^†apo-FluA data are from circular dichroism measurements of the thermal denaturation (at ambient pressure) of the same protein, but in the absence of the fluorescein ligand (32).

The mathematical condition for an elliptic shape of the general second-order curve, Eq. 2, is determined by the changes ΔC_p, Δβ, and Δα at the transition (3, 25):

If this ratio were >1, the shape would be hyperbolic, if it were equal to 1, the shape would be parabolic (3, 30) (for reviews, see refs. 4, 21, and 22).

One experimental approach to obtain information on the behavior of the thermodynamic parameters in Eq. 3 is to measure the phase diagram itself. However, as stressed earlier, in most cases the phase diagram is experimentally accessible only in a limited range, whereas a larger part of it may lie at negative pressure or temperature ranges that are hard to achieve. Alternatively, in the present case we measured the equilibrium constants, K_p(p, T_i) and K_T(T, p_i), as functions of temperature and pressure, respectively (see below). From

the Gibbs free energy is obtained by combining the results from different experiments either at constant pressure or at constant temperature and, thus, global fit parameters for ΔG₀, ΔS₀, ΔV₀, ΔC_p, Δα, and Δβ are obtained. From the latter three parameters the left-hand side of Eq. 3 can be evaluated and, if the mathematical condition holds, the phase boundary is an ellipse and its geometric parameters are completely determined (3, 30).

Fluorescence Monitoring of Temperature- and Pressure-Induced Denaturation Transitions of FluA.

From the absorption and emission spectra of free fluorescein and of the FluA·fluorescein complex at ambient temperature and pressure (Fig. 1) it is apparent that the residual fluorescence of the native protein complex is negligible. However, the fluorescence of a solution of the FluA·fluorescein complex evolves with increasing pressure at ambient temperature (Fig. 1b) and with increasing temperature at a constant pressure (Fig. 1c), indicating denaturation of the protein.

Fig. 1.

Spectral properties of free fluorescein and of the FluA·fluorescein complex. (a) Absorption and emission spectra of free fluorescein (solid lines) in comparison with the absorption spectrum of FluA·fluorescein (dashed line) in buffer at ambient conditions. Note that the fluorescence of the native FluA·fluorescein complex is fully quenched. (b and c) Fluorescence spectra of a solution of FluA·fluorescein as a function of either increasing pressure at ambient temperature (b) or of increasing temperature at a pressure of 50 MPa (c).

The fluorescence spectrum obtained from the FluA complex at elevated, that is, denaturing temperature and/or pressure conditions coincides with the spectrum of free fluorescein in solution (Fig. 2). Furthermore, the small pressure-induced shift of the fluorescence maximum obtained from a solution of the FluA·fluorescein complex is indistinguishable from the one observed for free fluorescein (Fig. 2b). These findings indicate that the molecular environment of the ligand in the pressure- and/or temperature-denatured complex is dominated by the pure solvent.

Fig. 2.

Comparison of the fluorescence spectrum of the FluA·fluorescein complex (dashed line) with the fluorescence spectrum of free fluorescein in solution (solid line) at the same pressure (200 MPa). (Inset) Investigation of the pressure-induced shift of the band maximum for both samples (crosses, fluorescein; filled circles, FluA·fluorescein).

To obtain the phase diagram of FluA the total fluorescence intensity integrated over the whole band was determined for each p/T point on a relative scale by fitting a log-normal distribution to the shape of the fluorescence band (Fig. 3). This fit reproduces the integrated fluorescence intensity to within 5% error (Fig. 3a Inset). The quality of the fit did not change under different pressure and/or temperature conditions, that is, the fluorescence spectrum did not depend on pressure or temperature except of small variations in the position and amplitude of its maximum.

Fig. 3.

Measurement of the pressure- and temperature-induced denaturation transitions. (a) Typical denaturation of the FluA·fluorescein complex induced by pressure. Plotted is the band-integrated fluorescence as a function of pressure. (Inset) Exemplary fluorescence spectrum after baseline subtraction (solid line) and fit of a log normal distribution (dashed line). The small residuals (dotted line) demonstrate the accuracy of the resulting intensity measurements. (b) Typical denaturation of the FluA·fluorescein complex induced by temperature. Data refer to the band-integrated fluorescence as a function of temperature. Note that both curve fits were obtained by an iterative global fit to all data points from 18 measured transitions altogether (see Materials and Methods), which explains the deviation in this example.

A typical denaturation transition of the FluA·fluorescein complex at increasing pressure had the following appearance (Fig. 3a): At ambient pressure, the complex is intact and no ligand fluorescence is observed; at >100 MPa, there is a steep increase of fluorescence intensity with pressure, indicating a highly cooperative denaturation transition, whereby the maximum is reached at ≈250 MPa; beyond this pressure, a slight linear decay of fluorescence intensity is observed. The denaturation pressure p_d is the pressure at which 50% of the FluA·fluorescein complex has dissociated, which can be determined from the observed fluorescence intensity of the free fluorescein after correction for its small, linear negative shift with increasing pressure. A similar behavior was observed during the temperature-induced denaturation (Fig. 3b). From a global fit of these curves measured at various fixed temperatures and pressures, respectively, according to the Law of Mass Action (see Materials and Methods) a series of equilibrium constants K_T(p_i) and K_p(T_i) could be deduced and all of the necessary parameters that determine the phase diagram were obtained, namely ΔG₀, ΔS₀, ΔV₀, Δα, Δβ, and ΔC_p (Table 1).

The Elliptic p/T Phase Diagram of FluA.

The stability phase diagram of FluA·fluorescein resulting from our measurements is depicted in Fig. 4. Most data points were obtained by pressure-induced denaturation at different temperatures; a few data points (Fig. 4a, squares) were obtained from temperature variation at fixed pressure. The pressure values p_d and (melting) temperatures T_d, respectively, at 50% denaturation were taken as the phase boundary.

Fig. 4.

Properties of the p/T phase diagram for protein denaturation. (a) The stability phase diagram of FluA·fluorescein (circles, data obtained at constant temperature; squares, data obtained at constant pressure). Note that the scatter of the data, which represent denaturation midpoints, is largely caused by the uncertainty of experimental pressure offset for the different measurements, which are indicated by error bars. The errors in temperature are smaller than the symbols. The parameters of the ellipse, however, are determined alone by the global fit of Eq. 2 to the 18 individually measured transition curves. (b) Sketch of the free energies G_D, G_N of the denatured state |D> and of the native state |N > as well as their difference ΔG (dashed line) as a function of pressure. p_d,h and p_d,l denote the pressure levels for high and low pressure denaturation.

Denaturation of the FluA·fluorescein complex is well described by the simple two-state model. The quenching of the fluorescence of the protein-bound ligand is suddenly abolished with increasing temperature or pressure concomitantly with protein denaturation. Apparently, protein denaturation is directly coupled to the dissociation of the ligand, because the recovered fluorescence is indistinguishable from the one of free fluorescein in solution. As stressed above, this conclusion is also supported by the lineshift experiments as a function of pressure (Fig. 2b), which yield exactly the same results for both situations, namely 6.5 nm/GPa. Notice that at moderate pressure levels, the pressure shift of the fluorescence emission is supposed to be proportional to the solvent shift (31). This implies that, after denaturation, there is no significant influence of the protein on the chromophore.

As a consequence, it seems to be justified to assign the structural change of the protein associated with the release of the fluorophore to the transition from the native to a functionally denatured state. This interpretation is in agreement with previous temperature-dependent circular dichroism (CD) experiments (32): The melting temperature as deduced from the change in the CD signal at the β-sheet-indicative wavelength of 212 nm (326.0 ± 0.2 K) coincides within the error margin with the transition temperature obtained from our fluorescence measurements (324 ± 4 K), even though the CD data were obtained in the absence of the ligand.

Regarding the pressure-dependent measurements, this interpretation is further supported by the observation that appearance of the ligand fluorescence was not reversible on a time scale of minutes. However, from kinetic studies of the association between FluA and fluorescein (under native conditions) we know a k_on value of (5.28 ± 0.05) × 10⁶ M⁻¹ s⁻¹, which means that the association equilibrium is quickly attained for the native protein. Thus, it must be concluded that we have actually observed the denaturation transition of the protein complex, for whose renaturation a much slower kinetics is expected. In addition, there is no apparent discrepancy between the temperature- and pressure-induced transitions (Fig. 4a), and during the entire course of pressure denaturation (e.g., Fig. 3a) there was no indication of intermediate states. Thus, it was justified to indirectly monitor the denaturing transition of FluA via dissociation of its fluorophore.

By using Eqs. 2 and 4, the measured fluorescence intensity data from the series of denaturation experiments, I(p, T_i) and I(p_i, T), were globally fitted by one set of values for ΔG₀, ΔS₀, ΔV₀, Δα, Δβ, and ΔC_p (see Materials and Methods). The resulting parameters are summarized in Table 1. Notably, these fit parameters account for the shape of the entire transition diagram (Fig. 4a), not just for the individual transitions defined by the 50% denaturation criterion for each measurement as explained above.

Although this global fit procedure per se is independent of the observed transition midpoints, they fall nicely onto the resulting elliptic phase boundary, which validates our approach. The detectable deviation of some data points from the shape of the transition diagram is attributable to the fact that the experimental determination of the absolute pressure (i.e., the frequency of the ruby emission that corresponds to a pressure of 1 bar; see Materials and Methods) has a larger error margin than the measurement of the pressure shift. From Fig. 4a, it appears that almost any second-order curve could have been fitted to the set of measured transition midpoints; however, the ellipse is unambiguously defined by the independent evaluation of the left-hand side of Eq. 3.

It is interesting to compare the p/T-phase diagram of FluA with the one of cytochrome c (20). For cytochrome c, the slope of the high-temperature transition at the 0.1-MPa line is positive, meaning that ΔV is positive. Accordingly, there is a narrow positive temperature range where the volume in the denatured state is larger than in the native state and, hence, pressure stabilizes the native state. A possible explanation could arise from the hydrophobic effect: if the denatured state is a random coil (a reasonable assumption for this p/T-parameter range), the hydrophobic amino acids come into contact with the solvent molecules and repel them, thereby increasing the volume. For FluA the tilt of the phase diagram lies in the opposite direction, meaning that the respective slope at the 0.1-MPa line is negative, which implies a smaller volume in the denatured state. As a consequence, higher pressure stabilizes the denatured state, possibly because of rather large voids in the native state that are squeezed on denaturation. Large voids in the native state could be a consequence of the barrel-type fold of FluA (28).

Implications of the Elliptic Phase Diagram.

As pointed out earlier the phase boundary of the native state of FluA has to be a general second-order curve in p and T. This holds strictly if Δβ, ΔC_p, and Δα are system constants. Under this assumption all of the necessary parameters can be deduced from the series of experiments described in this study. Our results show that the parameter ratio from Eq. 3 amounts to 0.07 (Table 1). Even though the estimated error is of the same order, this value clearly is significantly <1, confirming an elliptic shape of the diagram. The phase boundary of the denaturation transition is given by the equilibrium points of the observed transitions, which were individually determined from the pressure- or temperature-induced measurements and fall nicely in place (see Fig. 4a), so that its elliptic shape bears high confidence despite the comparatively small parameter range investigated.

Considering the physical origins behind the elliptic shape of the observed phase diagram, we have previously shown that, for proteins, the left-hand side of the relation (Eq. 3) can be interpreted as the square of a degree of correlation, ρ*², between the fluctuations of the changes of volume and enthalpy, δΔV and δΔH, at the folding/denaturation transition (33):

The notion that ΔV and ΔH are fluctuating rather than fixed quantities originates from the high dimensionality of the structural phase space; there may be many pathways into and out of the folded state, each characterized by its individual parameters. ρ*² can never be >1 and, consequently, a hyperbolic shape of the phase boundary should be forbidden. It is also rather unlikely that ρ*² = 1, which would imply a parabolic shape. In this case volume and enthalpy changes would have to be strictly correlated.

From the present experiments and the literature (3, 33) we know that the value for ρ*² is at the other end of the correlation scale. Typical numbers determined so far for ρ*² are of the order of 0.1 or even smaller, as in the present case with

Such a low degree of correlation means that a given ΔH may correspond to a broad range of ΔV values, and vice versa. This is the thermodynamic consequence of a high degeneracy of the conformational states of a protein, a characteristic phenomenon of materials with intrinsic disorder.

A situation like this is also inherent in energy landscape diagrams (34, 35). Typically, in these diagrams “energy” is plotted over some global conformational coordinate. Energy in this context means the total free energy of the protein, including its hydration shell, yet without the contribution from the conformational entropy of the polypeptide chain, which is represented by the degrees of freedom along the conformational coordinate. In this representation energy can be viewed as enthalpy with respect to the individual conformations of the solvated protein, averaged over the degrees of freedom of the hydration shell. A certain energy corresponds to many conformational coordinates, reflecting its high degeneracy. Although each conformation may not be distinguished by a characteristic volume, many of them certainly are. Hence, an immediate conclusion from the energy landscape representation is that the correlation between enthalpy and volume is low.

However, if a certain volume is chosen, it may correspond to a large set of different conformations and, accordingly, a large spectrum of energies. Along these lines of reasoning the low correlation between enthalpy and volume is an immediate consequence of the intrinsic disorder of proteins, which is also reflected by the random features of their energy landscapes. As a matter of fact, the left-hand side of Eq. 3 is the inverse of the so-called Prigogine–Defay ratio, well known in the physics of the glass transition. It was Jäckle (36) who associated this ratio with the correlation of enthalpy and volume fluctuations that become frozen at the glass transition. This ratio serves as a characteristic feature to thermodynamically distinguish the glass transition from a purely second-order phase transition.

Proteins, however, have their own specific features: the folding-denaturation transition is a first-order transition characterized by ΔH and ΔV, whereas both are zero for a second-order transition and for the glass transition. Accordingly, apart from the fluctuations of V and H, we have to consider fluctuations of the respective changes ΔV and ΔH, too. As pointed out previously, a low correlation between ΔH and ΔV means that a definite value of one of these parameters is related to a wide range of the other. The consequence is that the sharp changes characteristic for a first-order transition are washed out and, thus, the transition itself shows features of a second-order transition and even like a glass transition (37).

Although we interpret the elliptic phase boundary as a special feature of a general property of proteins (33), namely their intrinsic disorder and the resulting low correlation between enthalpy and volume fluctuations, the implications are severe: for the ellipse condition to generally hold true (Eq. 6), the changes in compressibility Δβ and in the specific heat ΔC_p are bound to have the same sign. As far as ΔC_p is concerned, there is experimental evidence on numerous proteins (12–14) indicating that the specific heat in the denatured state is, as a rule, larger than in the native state. This is easy to understand for high-temperature denaturation, but it seems to be true for cold-denaturation processes, as well. In the latter case it is the melting of the so-called iceberg (38), that is, the ordered structure of water around the solvent-exposed hydrophobic amino acids, which causes the higher specific heat of the denatured state.

Concerning β, however, it is much less straightforward to find general arguments that support a larger compressibility in the denatured state compared with the native state. Nevertheless, in our study of FluA, Δβ clearly shows a positive sign (Table 1). Fig. 4b illustrates the situation for a pressure-induced denaturation. Both curves for the free energies G(p) of the native and the denatured states, respectively, must have a negative curvature, that is, a decreasing slope with increasing pressure, because this slope reflects the volume, and the volume must decrease as pressure increases. The respective curvature has to be larger for G_D than for G_N to display the crossing points both at high and low pressures, where denaturation occurs. The transition is a consequence of Le Chatelier's principle: in the pressure-denatured state the voids that are present in the protein–solvent system are squeezed such that its volume falls below the one of the native state. Because the state with the lower volume is thermodynamically preferred at high pressures, denaturation occurs.

Low-pressure denaturation, whenever it can be experimentally observed, is likely to occur in a range of the phase diagram where the denatured state adopts a kind of random coil. In this case the hydrophobic amino acids are highly solvated and induce local structuring of the water molecules. As is well known from corresponding experiments on solutions of liquid hydrocarbons, this structuring process is associated with an increase in volume (15). In addition, the hydration shell associated with the hydrophilic surface of the folded protein, which has a significantly higher density than bulk water, melts away and, as a consequence, the overall volume increases further. Under these circumstances the denatured state has a larger volume than the native state and, thus, lowering of the pressure—in the present case, even to negative values (39)—favors denaturation. It should be noted that closed-phase diagrams are also predicted by simple microscopic models like the heteropolymer-collapse model (HPC) (40, 41).

After qualitatively understanding the occurrence of the transitions both at high and low pressures, a general argument becomes evident as to the sign of Δβ: Because G_D and G_N are approximately parabolic in p, ΔG = G_D − G_N has to be a parabola that opens toward negative values. The second-order derivative of such a parabola is negative for all values of p. As

Δβ is bound to be positive.

Materials and Methods

Protein Sample.

The mutant R95K of the anticalin FluA—a version that shows improved yields on heterologous expression with otherwise essentially unchanged properties (32)—was produced in Escherichia coli and isolated from the bacterial periplasm as a correctly folded protein by affinity chromatography as described (26). The sample contained 2.4 mg of protein (21,001 Da) per ml (corresponding to 0.114 mM) and an equimolar concentration of fluorescein in a 50 mM Tris·HCl buffer at pH 7.5. The buffered control sample of fluorescein had the same concentration.

Experimental Setup.

The sample was placed in a temperature-stabilized diamond anvil cell (DAC) with a diameter of ≈0.4 mm and a thickness of ≈0.1 mm. Fluorescence was exited at 420 nm by a pulsed dye laser pumped by an excimer laser (Lambda Physics LPX 100, LPD 3000). Spectra were recorded with a Jobin Yvon Triax 180 spectrograph equipped with a CCD for signal acquisition. Because of the design of the DAC, a linear optical setup was chosen, whereby the transmitted laser light was blocked by a beam stop and/or an appropriate filter.

Pressure was determined in situ by measuring the shift of the R1 band of ruby (42). The position of the respective emission line depends on both temperature and pressure (43). However, most of our experiments were performed at constant temperature so that the temperature shift could be neglected. The error margin of the pressure variation was ≈10 MPa, but there was an additional uncertainty in pressure offset of ≈40 MPa; the accuracy of the temperature was ≈1 K. To ensure thermodynamic equilibrium, pressure and temperature were changed in small steps and after each change the sample was equilibrated for at least 30 min. Under these conditions no temporal change of the fluorescence spectra was observed.

Evaluation of the Fluorescence-Based Denaturation Measurements.

Denaturation of the FluA·fluorescein complex can be described by a dynamic chemical equilibrium according to the reaction scheme

with N being the FluA·fluorescein complex in its native state, D the FluA protein in its denatured state, and F the fluorophore. In our equimolar setup the concentration [D] is identical to the concentration [F]. According to the Law of Mass Action the bimolecular equilibrium constant of the (pressure- and temperature-dependent) combined denaturation/dissociation is defined as

where γ denotes the degree of denaturation/dissociation and [P] is the total protein concentration, that is, 0.114 mM in our experiments. Consequently, at the condition of 50% denaturation/dissociation K(p,T) assumes a value of 0.057 mM. Eq. 9 can be solved for

which can be directly determined from the fluorescence intensity ratio I/I₀ as a function of temperature at constant pressure or vice versa. However, the fluorescence of the free ligand, and thus I₀, shows a small linear pressure dependence that can be expressed as

where I′₀ corresponds to the fluorescence signal of the free ligand at the total protein = ligand concentration at standard pressure and temperature. In contrast, the temperature dependence of I appeared to be negligible so that no linear correction was applied.

Thus, the measured fluorescence intensity as a function of the temperature- and pressure-dependent equilibrium constant was obtained as

whereby b_n was included as a baseline constant. By employing Eqs. 4 and 2, K(p, T) was expressed as a function of the global parameters ΔG₀, ΔS₀, ΔV₀, Δα, Δβ, and ΔC_p and iteratively fitted against the data points of 15 pressure-induced and 3 temperature-induced denaturing transitions altogether (see Fig. 3) by using gnuplot 4.0 software until convergence was achieved.