Equilibrium Thermodynamics Analysis Versus Kinetic Analysis

Possibly the most basic issue regarding DSC data analysis is whether the experimental heat capacity values reflect an equilibrium denaturation process or not. By "equilibrium process" we mean that thermodynamic equilibrium between the significantly populated protein states (native, unfolded, partly unfolded, etc.) is established throughout the temperature scan. In other words, the concentrations of the populated states at any given temperature are determined by values of the relevant equilibrium constants at that temperature. Of course, equilibrium constants do change with temperature according to simple thermodynamic relationships (i.e., according to the van't Hoff equation and the process enthalpy change), thus leading to the shift in populations of states with temperature that results in transitions ("peaks") in the DSC thermogram. For instance, for an equilibrium two-state process with a sufficiently high denaturation enthalpy value, the denaturation equilibrium constant changes from a low value that favours the native state to a high value that favours the denatured state in a comparatively narrow range. It must be noted that processes that occur through a series of equilibrium states are usually called "reversible" (or quasistatic) in thermodynamics.

tte analysis of equilibrium DSC thermograms is well established and has been described in detail in several reviews (Freire et al. 1990; Freire 1994, 1995; Sanchez-Ruiz 1995). In principle, it is possible to use textbook thermodynamics to obtain the number and the thermodynamic parameters (values of free energy, enthalpy, entropy and heat capacity) for all the significantly populated states. Furthermore, this detailed energetic description of the denaturation process can be interpreted in structural terms using structure-energetics correlations that have been developed and refined over the years (Robertson and Murphy 1997; Luque and Freire 1998). More recent advances in the equilibrium analysis of DSC thermograms (such as those related to the characterization ofligand effects and barrierless folding) will be described in subsequent sections of this review.

Powerful though it is, the equilibrium thermodynamics analysis of DSC thermograms cannot be applied to DSC thermograms that are distorted by the occurrence of processes of a kinetic character (Sanchez-Ruiz 1992, 1995; Plaza del Pino et al. 2000). tte reason is that the concentrations of states involved in kinetic processes are determined by rate equations (not by a temperature-dependent equilibrium constant). tte two common sources of kinetic distortions in DSC of proteins are "slow equilibrium" and the occurrence of irreversible protein alterations (aggregation, autolysis in the case of proteolytic enzymes, etc.)

Irreversible alterations take the protein to a "final" state which is unable to fold back to the native state ( under the conditions used in the DSC experiment). Irreversible alterations are, therefore, essentially irreversible processes and must be described in terms of rate equations. In experimental DSC, the occurrence of irreversible alterations is easily detected by the absence of a calorimetric transition in a reheating run (i.e., a second DSC scan performed after cooling from the first one, without removing the protein sample from the calorimetric cell; Fig. 2.3). It is, of course, important to stop the first scan immediately after the transition has been completed to avoid detecting the effect of irreversible alterations that may occur at higher temperatures.

A DSC transition that is not reproduced in the reheating run is said to be calori-metrically irreversible (or just "irreversible"). Calorimetric irreversibility suggests that the DSC transition may be kinetically distorted to a significant extent. A more detailed assessment may be derived from DSC experiments carried out at different scan rates. Since kinetic processes are by definition time-dependent processes, the state of a kinetically determined system at a given temperature will depend on the time required to reach that temperature. It follows that kinetically distorted DSC transitions must show a significant scan-rate dependence (Sanchez-Ruiz 1992,1995; Plaza del Pino et al. 2000). In fact, most calorimetrically irreversible DSC transitions

Fig.2.3. Experimental example of a calorimerically irreversible DSC transition. A Original DSC recording for the denaturation of porcine pancreas procarboxipeptidase B at 2 K/min and pH 9.0; protein concentration 1.13 mg/mL; Zn2+ concentration 0.37 mM. B Reheating run showing no sign of a transition. C Buffer-buffer baseline (showing a 25-|iW calibration mark). (Reprinted with permission from Conejero-Lara et al., 1991 Biochemistry 30:2067-2072, Copyright (1991), American Chemical Society)

Fig.2.3. Experimental example of a calorimerically irreversible DSC transition. A Original DSC recording for the denaturation of porcine pancreas procarboxipeptidase B at 2 K/min and pH 9.0; protein concentration 1.13 mg/mL; Zn2+ concentration 0.37 mM. B Reheating run showing no sign of a transition. C Buffer-buffer baseline (showing a 25-|iW calibration mark). (Reprinted with permission from Conejero-Lara et al., 1991 Biochemistry 30:2067-2072, Copyright (1991), American Chemical Society)

Fig. 2.4. Scan-rate effect on the DSC transitions for the thermal denaturation of porcine pancreas procarboxipeptidase B at pH 9.0 and in the presence of several Zn2+ concentrations: A 0.26 mM; B 1.48 mM; C 2.96 mM. Note that the transitions shown are "excess heat capacity" profiles that have been corrected for the instrumental baseline and the so-called chemical baseline (a smooth sigmoidal-like baseline that connects the pretransition and posttransition heat capacity levels). The numbers alongside the transitions stand for the scan rate in kelvins per minute. (Reprinted with permission from Conejero-Lara et al., 1991 Biochemistry 30: 2067-2072, Copyright (1991), American Chemical Society)

studied so far have been found to be strongly scan-rate dependent (Fig. 2.4). Furthermore, in many cases, these irreversible DSC transitions obey closely a two-state irreversible model, which assumes that only the native and the final irreversible-denatured states are significantly populated during the DSC scan and that the conversion from the native to the final state is described by a first-order rate equation (Sanchez-Ruiz 1992). ttis simple model is actually a limiting case of more complex situations and it is reached when the irreversible alteration is very fast and the population of states other than native and final becomes negligible. If a DSC transition follows the two-state irreversible model, equilibrium thermodynamics analysis is not possible and information about the equilibrium denaturation mechanism is not available from the experimental DSC profile.

In a few cases, irreversible DSC transitions have been found to show a negligible scan-rate effect (Vogl et al. 1997; ttorolfsson et al. 2002). tte likely interpretation is that the irreversible alteration occurs with little thermal effect at temperatures somewhat above those of the calorimetric transition or, perhaps, overlapping slightly with the high-temperature side of the transition. In these cases, equilibrium thermodynamics analysis is acceptable with some obvious precautions. For instance, a reliable value for the denaturational change in heat capacity cannot be possibly obtained, since the denatured, high-temperature baseline is almost certainly distorted by the irreversible alteration, even if the transition is not.

Overall, the DSC studies on irreversible protein denaturation have contributed to the realization that protein stability has both thermodynamic and kinetic aspects. In particular, it has made clear that thermodynamic stability (a positive value for the denaturation free energy at physiological temperature) does not guarantee that the protein will remain in the native state during a given time scale, since irreversible alterations (even if they occur through lowly populated unfolded or partially unfolded states) may deplete the native state in a time-dependent manner (Plaza del Pino et al. 2000). It is clear, therefore, that many proteins (in particular, complex protein systems) must be "designed by evolution" (i.e., naturally selected) to have significant kinetic stability when confronted with the destabilizing effect of the numerous irreversible processes that may occur in vivo. Provided that irreversible processes occur mainly from non-native states (unfolded, partially-unfolded, etc.), the required kinetic stability may be achieved through the design of a sufficiently high free-energy barrier for unfolding (the barrier that "separates" the native state from the non-native states). It must be noted that, in recent years, the stability of many complex protein systems of interest has been shown to be of kinetic origin (Schnyrov et al. 1997; Cunningham et al. 1999; Persikov and Brodsky 2000; Jaswal et al. 2002; Mehta et al. 2003; Forrer et al. 2004; Lynch et al. 2004; Manning and Colon 2004; Jaswal et al. 2005; Jaramayan et al. 2005a, b) and that some emerging molecular approaches to the inhibition of amyloidogenesis focus on the increase of the kinetic stability of the protein native state (Foss et al. 2005; Petrassi et al. 2005; Wiseman et al. 2005), as we suggested a few years ago (Plaza del Pino et al. 2000).

When the DSC transition is reproduced with significant amplitude in the reheating run we say that the denaturation process is calorimetrically reversible. Ms situation is usually found with "small" proteins (i.e., the kind of "model" proteins often used in fundamental studies on folding and stability). Strictly speaking, calorimetric reversibility does not necessarily imply thermodynamic reversibility (i.e., that the process occurs through equilibrium states, so that equilibrium is always established during the DSC scan), tte reason is that there is the possibility that the rates of the folding and unfolding processes could be comparatively slow in such a way that the scanning (at the usual scan rates) is too fast for equilibrium to be established. Ms situation is known as "slow equilibrium" (Freire et al. 1990). Slow equilibrium does not appear to be a serious problem for most comparatively small proteins, which are expected to fold and unfold fast at the transition temperature [note that the folding of small proteins is usually fast compared with the time-scale of the DSC experiment (Jackson 1998) and that, at the transition temperature and assuming a two-state process, the rates of folding and unfolding must be equal, so that both processes will be fast]. However, DSC studies carried out in the presence of concentrations of dénaturants (urea, guanidine) close to those of the "bottom" of the kinetic chevron plot may show significant slow-equilibrium distortion, since the rates of folding-unfolding may be slow in those conditions (Plaza del Pino et al. 1992). In the absence of dénaturants, slow-equilibrium distortions are expected to be small (for a possible exception, see Kaushik et al. 2002) and calorimetric reversibility constitutes good evidence for thermodynamic reversibility. Of course, if in a given particular case it is suspected that the rates of folding-unfolding may be unusually slow, it is advisable to carry out DSC experiments at different scan rates (even if the denaturation process is calorimetrically reversible). Obviously, the slow-equilibrium, kinetic distortion will become less important at the lowest scan rates.

tte analyses described in the following sections assume equilibrium thermodynamics is applicable.

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