Wild-type thioredoxin was over-expressed in E. coli JF521 and purified using previously described gel-filtration and ion-exchange chromatographic procedures , and reverse-phase chromatography. The protein was proteolytically cleaved in two complementary fragments: (1–73) and (74–108), called the N fragment and the C fragment , respectively. These fragments were purified by reverse-phase chromatography and were stored in lyophilized form. Purity of the samples was checked by SDS/PAGE , sequencing, amino-acid analysis, and electrospray mass spectroscopy. Protein and peptide concentrations were determined spectrophotometrically using the following molar extinction coefficients: ε280 = 1.41 × 104m−1·cm−1 for the N fragment , ε215 = 3.97 × 104m−1·cm−1 for the C fragment  and ε280 = 1.4 × 104m−1·cm−1 for Trx .
Trx solutions for DSC experiments were prepared by exhaustive dialysis against 10 mm potassium phosphate at the desired pH. The peptide fragments were dissolved in 8 m guanidine, 10 mm potassium phosphate buffer at pH 7 and, subsequently, were extensively dialyzed against phosphate buffer. The NC complex was prepared by mixing stoichiometrically the two fragments at low concentration, followed by dialysis against phosphate buffer, and a concentration step. All samples, isolated fragments, cleaved and uncleaved Trx, were characterized by gel-filtration FPLC to confirm the absence of oligomeric species.
Fitting of theoretical models to the DSC transitions
All fittings were done to the absolute molar heat capacity values calculated using subroutines provided by MicroCal. The specific volumes required for the calculation of absolute heat capacities were determined according to Makhatadze et al.  and found to be 0.736 mL·g−1 (for the N fragment), 0.770 mL·g−1 (for the C fragment) and 0.750 mL·g−1 (for Trx and the NC complex). It should be noted that, while absolute heat-capacity values calculated from individual DSC thermograms suffice for the fitting purposes described in this section, we describe below a more accurate procedure to calculate absolute heat capacities based on several thermograms corresponding to different protein concentrations.
The equations used in the fitting procedures are based on two-state models and are briefly described below (see also Ibarra-Molero et al.).
For any two-state model the excess enthalpy (taking the native state as reference) is given by:
where ΔH is the denaturation enthalpy and α stands for the fraction of protein that is denatured. The protein heat capacity can then be written as:
where ΔCp is the denaturation heat-capacity change.
The two first terms in the right-hand-side of Eqn (2)[ + α · ΔCp] describe a sigmoidal-like curve that smoothly connects the heat capacity levels of the native and the denatured states; this curve is usually called the chemical baseline. The third term on the right-hand side is significantly different from zero only when α changes significantly with temperature; that is, within the comparatively narrow temperature range in which the protein denatures. Therefore, the ΔH · (dα/dT) term describes the transition (or ‘peak’) in the thermogram.
The α value at a given temperature is related with the corresponding equilibrium constant for the denaturation process. If denaturation does not involve changes in association state, this relation is:
which is valid for thioredoxin but not for the NC complex, as, in this latter case, denaturation involves dissociation of the complex into unfolded N and C fragments. Thus, at equilibrium at a given temperature, the concentrations of complex, N fragment and C fragment would be C(1 − α), Cα and Cα, respectively (C being the total protein concentration expressed in moles of complex); hence, the equilibrium constant for NC complex denaturation is:
Given a value of K, the appropriate equation (Eqn 3 or 4) can be easily solved for α. In addition, expressions allowing the calculation of the temperature dependence of α (required for Cp calculation using Eqn 2) can be easily derived by differentiating in Eqns (3 and 4) solving for dα/dT and using the van’t Hoff equation for the temperature dependence of K. The results are:
from Eqn (3) (denaturation not coupled to dissociation), and:
from Eqn (4) (unfolding coupled to dissociation of the NC complex).
Denaturation occurs (i.e. α goes from 0 to 1) in a comparatively narrow temperature range, within which the temperature dependence of the denaturation enthalpy can be neglected. Thus, we will assume that the ΔH value explicitly shown in Eqns (3), (5) and (6) is a temperature-independent value, assign this value to a temperature characteristic of the transition (T1/2, the temperature at which α = 0.5): ΔH(T1/2). Within the same approximation, the temperature dependence of the denaturation equilibrium constant can be written as:
where K1/2 is the value of K at the temperature T1/2. If denaturation is not coupled to changes in association state, K1/2 = 1. For NC complex denaturation (coupled to fragment dissociation) substitution of α= 0.5 into Eqn (4) gives K1/2 = C/2. It must be noted that neglecting the temperature dependence of K for purposes of α and dα/dT evaluation only causes a very small distortion in the shape of the predicted DSC transitions, distortion which is below the uncertainty caused by chemical baseline definition.
In all cases, the experimental temperature range to be fitted was chosen in such a way that the pre- and post-transition baselines were clearly defined, but also that linear temperature dependencies for these baselines could be reasonably assumed. Hence, we used the following equations for and ΔCp in Eqn (2):
where a, b and d are constants and ΔCp(T1/2) is the denaturation heat capacity at the temperature T1/2.
Eqn (2) (together with Eqns 3 or 4, 5 or 6, 7, 8 and 9) was fitted to the experimental heat-capacity data using programs written by us in the mlab environment (Civilized Software Inc.). In each case, the fitting parameters were: T1/2 and ΔH1/2 (which describe the transition) and a, b, d and ΔCp,1/2 (which describe the chemical baseline).
Fitting of the concentration dependence of the denaturation temperature for the NC complex
Denaturation temperature values for the NC complex increase with protein concentration, as is to be expected for a denaturation process coupled to dissociation. Let T0 be the denaturation temperature at unit protein concentration (C = 1) and T1/2 the denaturation temperature at any given concentration, C. Let K0 and K1/2 be the denaturation equilibrium constants at the temperatures T0 and T1/2. Under the assumptions used above, these two equilibrium constants may be related through the following van’t Hoff equation:
In addition, K1/2 and K0 values can be obtained from Eqn (4) by substituting α= 0.5 and the corresponding protein concentration: K1/2 = C/2 and K0 = 1/2. Substituting these values into Eqn (10) and solving for T1/2 yields:
This equation was fitted to the experimental T1/2 versus C data for NC complex denaturation using programs written by us in the MLAB environment. Fitting parameters were T0 and ΔH. Clearly, this type of analysis neglects the temperature dependence of ΔH; this however, is again permissible as the experimental T1/2 values usually span a narrow temperature range (about eight degrees for the NC complex).
The calculation of the absolute heat capacities for native and unfolded states from the experimental thermograms
Accurate values for the absolute heat capacities of native and unfolded states were derived from the concentration dependence of the apparent heat-capacity function (ΔCapp), defined as the heat capacity of the protein (or peptide) solution taking the buffer-buffer baseline as reference. It is a well-known fact that ΔCapp may be a negative number, as the amount of water (for which heat capacity is larger than that of a protein or peptide) is lower in the sample cell. Under suitable assumptions, ΔCapp for a solution of a protein or peptide may be expressed as:
where C is the concentration of peptide (or protein) in mm, V0 is the volume of the calorimetric cell (in mL), Cp, P and Cp, W stand for the heat capacities of the protein and water (buffer), respectively, and Vp and VW are the molar volumes of peptide and water (Vp is easily obtained from the specific volumes calculated using known procedures: see Makhatadze et al.). According to Eqn (12), a plot of ΔCapp versus protein concentration for a given temperature should be linear with zero intercept and with a value for the slope equal to V0 · 10−6 · Cp,P, from which the absolute heat capacity can be easily calculated. Clearly, this procedure requires several thermograms corresponding to different concentrations; however, it shows several important advantages when compared with the usual methods for absolute heat-capacity calculation. These advantages should be evident from the analyses reported under Results but, in any case, they will be summarized at the beginning of the Discussion (see also Kholodenko and Freire ).
Theoretical estimates for the heat capacities of unfolded and native states
The heat capacity expected for fully solvated, unfolded states of thioredoxin and both fragments was calculated from the amino-acid composition using :
where N is the number of amino-acid residues, the index i refers to the 20 amino acids, ni is the number of amino acids of type i, Cp, i is the molar heat capacity of the side chain of amino acid i, and Cp, NH-CO,Cp, COOH and Cp, NH2 stand for the heat capacities of the peptide unit, the terminal carboxyl and the terminal amino groups, respectively. Actually, we used for the temperature-dependent Cp, i, Cp, NH-CO,Cp, COOH and Cp, NH2 values the polynomial representations of the Privalov and Makhatadze data  given by Freire  (see Häckel et al. for a different set of group heat capacity values for use in protein calculations). Theoretical values of the heat capacity of the native proteins were calculated as a function of temperature using the equation given by Freire :
where Mr is the relative molecular mass of the protein.