Characterization of ß-trypsin at acid pH by differential scanning calorimetry

Abstract

Trypsin is a serino-protease with a polypeptide chain of 223 amino acid residues and contains six disulfide bridges. It is a globular protein with a predominance of antiparallel ß-sheet and helix in its secondary structure and has two domains with similar structures. We assessed the stability of ß-trypsin in the acid pH range using microcalorimetric (differential scanning calorimetry) techniques. Protein concentrations varied in the range of 0.05 to 2.30 mg/ml. Buffer solutions of 50.0 mM ß-alanine and 20.0 mM CaCl2 at different pH values (from 2.0 to 4.2) and concentrations of sorbitol (1.0 and 2.0 M), urea (0.5 M) or guanidinium hydrochloride (0.5 and 1.0 M) were used. The data suggest that we are studying the same conformational transition of the protein in all experimental situations using pH, sorbitol, urea and guanidinium hydrochloride as perturbing agents. The observed van't Hoff ratios (deltaHcal/deltaHvH) of 1.0 to 0.5 in the pH range of 3.2 to 4.2 suggest protein aggregation. In contrast, deltaHcal/deltaHvH ratios equal to one in the pH range of 2.0 to 3.2 suggest that the protein unfolds as a monomer. At pH 3.00, ß-trypsin unfolded with Tm = 54ºC and deltaH = 101.8 kcal/mol, and the change in heat capacity between the native and unfolded forms of the protein (deltaCp) was estimated to be 2.50 ± 0.07 kcal mol-1 K-1. The stability of ß-trypsin calculated at 298 K was deltaG D = 5.7 kcal/mol at pH 3.00 and deltaG D = 15.2 kcal/mol at pH 7.00, values in the range expected for a small globular protein.

ß-Trypsin; Thermal denaturation; Differential scanning calorimetry

Braz J Med Biol Res, December 2003, Volume 36(12) 1621-1627

Characterization of ß-trypsin at acid pH by differential scanning calorimetry

E.R. Bittar^1,2, F.R. Caldeira², A.M.C. Santos², A.R. Günther⁴, E. Rogana³ and M.M. Santoro²

¹Instituto de Estudos Avançados em Medicina Veterinária, Universidade de Uberaba, Uberaba, MG, Brasil

Departamentos de ²Bioquímica e Imunologia, and ³Fisiologia e Biofísica, Instituto de Ciências Biológicas, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brasil

⁴Departamento de Ciências Fisiológicas, Centro de Ciências Biológicas, Universidade Federal de Santa Catarina, Florianópolis, SC, Brasil

References

^{Correspondence and Footnotes}

Abstract

Trypsin is a serino-protease with a polypeptide chain of 223 amino acid residues and contains six disulfide bridges. It is a globular protein with a predominance of antiparallel ß-sheet and helix in its secondary structure and has two domains with similar structures. We assessed the stability of ß-trypsin in the acid pH range using microcalorimetric (differential scanning calorimetry) techniques. Protein concentrations varied in the range of 0.05 to 2.30 mg/ml. Buffer solutions of 50.0 mM ß-alanine and 20.0 mM CaCl₂ at different pH values (from 2.0 to 4.2) and concentrations of sorbitol (1.0 and 2.0 M), urea (0.5 M) or guanidinium hydrochloride (0.5 and 1.0 M) were used. The data suggest that we are studying the same conformational transition of the protein in all experimental situations using pH, sorbitol, urea and guanidinium hydrochloride as perturbing agents. The observed van't Hoff ratios (DH_cal/DH_vH) of 1.0 to 0.5 in the pH range of 3.2 to 4.2 suggest protein aggregation. In contrast, DH_cal/DH_vH ratios equal to one in the pH range of 2.0 to 3.2 suggest that the protein unfolds as a monomer. At pH 3.00, ß-trypsin unfolded with Tm = 54^ºC and DH = 101.8 kcal/mol, and the change in heat capacity between the native and unfolded forms of the protein (DCp) was estimated to be 2.50 ± 0.07 kcal mol^-1 K^-1. The stability of ß-trypsin calculated at 298 K was DG_D = 5.7 kcal/mol at pH 3.00 and DG_D = 15.2 kcal/mol at pH 7.00, values in the range expected for a small globular protein.

Key words: ß-Trypsin, Thermal denaturation, Differential scanning calorimetry

Introduction

The biological activity of proteins depends on the structure and three-dimensional conformation of their polypeptide chains. These conformations are the result of the thermodynamically driven folding of the linear amino acid sequence, allowing proteins to change from a nonfunctional and unordered structure to a well-defined, compact and active form.

Although there is a great deal of interest in protein folding, we still have much to learn about the mechanisms involved in the process. The reversible process of protein folding and unfolding can be studied by various sensitive methods capable of detecting changes in protein structure. According to Jackson and Brandts (1), the two-state denaturation process that considers only the conformational extremes, i.e., the native and denatured forms of the protein, can be used for the analysis of small and simple three-dimensional structure proteins. Thermodynamic analysis is simplified if one can describe the denaturation process as a simple transition between two macroscopic states (native and denatured). In contrast, for large proteins, an increase in temperature induces unfolding of isolated domains, an event that cannot be regarded as a simple two-state process (2).

The success of thermodynamic analysis of protein folding and unfolding is directly dependent on the method of determination of the thermodynamic parameters and the characteristics of the protein analyzed. Differential scanning calorimetry (DSC) is an experimental tool capable of determining the calorimetric parameters DG_D, DH_D, DS_D and DCp_D associated with a given conformational change of the protein induced by temperature variation. A special type of microcalorimeter with adequate technical characteristics for data acquisition from aqueous protein solutions of relatively low concentration is necessary for DSC. The DSC apparatus integrates hardware and software which allow data acquisition, analysis, and deconvolution (3).

In this context, ß-trypsin is a very interesting model for the microcalorimetric study of a serine protease since it exhibits 223 amino acid residues that are presented in two similar domains (4), its three-dimensional structure is known, it possesses high reversibility for the type of denaturation utilized (thermal), and shows great stability in its native form. Trypsin stability has been investigated using solvent denaturation (5,6) and calorimetry (7), but these authors did not adequately explore different experimental conditions. Thus, the objective of the present investigation was to study the thermodynamic characteristics of the reversible thermal denaturation of ß-trypsin under several solvent conditions and in the presence of different protein concentrations in the acid range (pH 2.0 to 4.5) using DSC. This is part of an ongoing research project that ultimately aims to compare the stability of ß-trypsin with its zymogen under similar experimental conditions and to describe the activation process of a typical serine protease from a thermodynamic point of view.

Material and Methods

Purification of ß-trypsin. Trypsin was purified by the chromatographic procedure described by Schroeder and Shaw (8) and Higaki and Light (9). For purification, 250 mg of commercial bovine trypsin (Sigma, St. Louis, MO, USA) was dissolved on an ice bath in 5.0 ml Tris-HCl buffer (100.0 mM Tris-HCl, 20.0 mM calcium chloride, 100.0 mM sodium chloride and 1.0 mM benzamidine), pH 7.10. The solution was applied to a SE-Sephadex C-50 column (2.0 x 110.0 cm; 345.0 cm³ volume) and was eluted with the same buffer (isocratic) at a constant flow of 7.0 ml/h. The fractions were collected using an automatic collector (Pharmacia-LKB Frac-100, New York, NY, USA), read at 280.0 nm in a spectrophotometer (Shimadzu UV 160-A, Kyoto, Japan) and submitted to electrophoresis to determine their purity (peaks). The amidase activity of each fraction pool was determined as described below.

Electrophoresis. The purified fractions were submitted to electrophoresis by the method of Laemmli (10). The following standard markers (Sigma) were added to all electrophoretic runs: 116-kDa ß-galactosidase, 97.4-kDa rabbit muscle ß-phosphorylase, 66-kDa bovine albumin, 45-kDa swine heart fumarase, and 29-kDa carbonic anhydrase. The gels were stained with silver by the method of Tunon and Johansson (11) and with Coomassie blue by the method of Hames and Rickwood (12).

Amidase activity was determined by the method of Antonini and Ascenzi (13) using DL-BAPNA as substrate. After SE-Sephadex C-50 chromatography and determination of amidase activity, selected fractions were submitted to dialysis for 8 h at 4ºC using membranes of 6000-8000 Da exclusion limit, against 1.0 mM HCl, pH 3.00. The dialyzed samples were lyophilized and stored at -20ºC.

Active site titration. The active site of the enzyme samples was titrated with small modifications (14). In this method, ß-trypsin catalyzes the hydrolysis of 4-nitrophenyl-4-guanidinebenzoate. The first catalytic step is several-fold faster than the second step, permitting the spectrophotometric detection of the acyl-enzyme at 410 nm. A reference cuvette with Veronal buffer and a sample cuvette with the same buffer plus trypsin were used. The values of 40,000 and 16,595 M^-1cm^-1 were adopted as the molar extinction coefficient (e) for ß-trypsin (read at 280.0 nm) and 4-nitrophenol at pH 8.30 (read at 410.0 nm), respectively. The equation used for calculation was

(Eq. 1)

where f = enzyme dilution factor.

Thermal denaturation of ß-trypsin in acid media. The microcalorimetric study of ß-trypsin denaturation at pH 2.00 to 4.20 was performed by the methods of Privalov and Potekhin (2) and Freire et al. (15) using the MicroCal Ultrasensitive VP-DSC apparatus and standard software for data acquisition and analysis. For analysis in acid media the ß-trypsin samples, resuspended in ß-alanine buffers at pH 2.00, 2.25, 2.50, 2.75, 3.00, 3.25 and 3.50 (50.0 mM ß-alanine and 20.0 mM CaCl₂), 4.00 and 4.20 (50.0 mM sodium acetate and 20.0 mM CaCl₂) and determined spectrophotometrically at 280.0 nm to be approximately 1.0 mg/ml, were degassed for 30 min and then loaded onto the DSC MicroCal apparatus. Runs were performed with a scan rate of 60ºC/h. The changes in enthalpy (DH) did not depend on the scan rate used in the range of 30º to 90ºC per hour. The data obtained were subtracted from a baseline of water against water or buffer against buffer (ß-alanine at the corresponding pH values) before normalization of the protein concentration.

Microcalorimetry of ß-trypsin at pH 3.00 in the presence of a protein denaturant or stabilizer. The microcalorimetric runs of ß-trypsin with a destabilizing or stabilizing agent were carried out by the method described above. Guanidinium hydrochloride was used as a denaturing agent at 0.25, 0.50 and 0.75 M in ß-alanine buffer, pH 3.00 (50.0 mM ß-alanine, 20.0 mM CaCl₂) (16), and sorbitol was used as a stabilizing agent at 0.5, 1.0, 1.5 and 2.0 M in ß-alanine buffer, pH 3.00 (50.0 mM ß-alanine, 20.0 mM CaCl₂) (16).

Results and Discussion

It is not possible to directly assess the stability of ß-trypsin at neutral pH due to the autolysis of this proteolytic enzyme. The alternative is to assess stability within the acid pH range (pH 2.00 to 4.50) and to calculate the stability of the enzyme in the neutral range according to the thermodynamic cycle described below (Figure 1).

The analysis of experimental DSC data (Cp x temperature) allows the evaluation of two types of enthalpy variation (2): i) calorimetric enthalpy (DH_cal), which represents the area under the denaturation curve, minus the contribution of the variation of heat capacity (DCp) between the native and denatured forms of the protein. This is a model-independent parameter and the units are cal/mol of protein (monomer), but can also be expressed as cal/mol dimer, if necessary. ii) van't Hoff enthalpy (DH_vH), which reflects the shape of the denaturation curve and describes the dependence of the equilibrium constant (K_D) as a function of temperature. It is a model-dependent parameter expressed as cal/mol of cooperative units, responsible for the observed conformational transitions.

An important experimental parameter is the DH_cal/DH_vH ratio, which represents the ratio

(Eq. 2)

For a monomer, a two-state transition is described by a DH_cal/DH_vH value = 1.0, i.e., the monomer is the cooperative unit responsible for the observed conformational transition. In the case of a dimer, the ratio is DH_cal/DH_vH = 0.5, i.e., we have 0.5 mol of cooperative units per mol of monomer. For a monomer with two domains of similar size, when uncoupled, a DH_cal/DH_vH ratio = 2.0 should be expected, i.e., 2 mol of cooperative units per monomer would be detected, assuming that the domains make similar contributions to the overall DH_cal value (2).

At pH 3.00 the reversibility of the thermal denaturation of ß-trypsin was 90% or higher. For this protein the denaturation process was evaluated as a two-state transition, as can be seen in Figure 2 which shows a quite symmetrical transition curve. At this pH, the denaturation process was assumed to be a two-state transition because the DH_cal/DH_vH ratio was very close to one.

ß-Trypsin at pH values ranging from 2.00 to 3.00 exhibited a DH_cal/DH_vH ratio always close to one (Figure 3), indicating that the enzyme is a monomer and that the domains are coupled under these conditions. These results agree with those reported by Brumano et al. (5), who showed that the thermal denaturation of ß-trypsin at pH 2.80 was a two-state transition with a DH_cal/DH_vH value close to one.

On the other hand, the DSC runs performed at pH values ranging from 3.20 to 4.20 revealed that the DH_cal/DH_vH ratio ranged from 1.0 to 0.5 (Figure 3), suggesting protein dimerization. Indeed, literature data suggest ß-trypsin dimerization at pH values around 4.00 (17), although Tishchenko and Gorodkov (7), in experiments with ß-trypsin in the 2.5 to 4.0 pH range, observed that the DH_cal/DH_vH ratio was always approximately one.

In an attempt to clarify this point, the DH_cal/DH_vH ratio was determined at constant pH 3.70 (50.0 mM ß-alanine and 20.0 mM CaCl₂). At this pH the DH_cal/DH_vH ratio was about 0.75, i.e., between the extreme values 1.0 (100% of proteins are monomers) and 0.5 (100% of proteins are dimers). Figure 4 shows the results obtained at the protein concentration range of 0.05-2.30 mg/ml. While the protein concentration varied, the protein aggregated under these conditions, since the ratio tended to reach lower values with high protein concentrations, while it tended to reach 1.0 when the protein concentration tended to be zero.

Assuming that under these conditions the protein shows an equilibrium of 2 monomers Û dimer, the dimerization constant can be calculated using the following equation

(Eq. 3)

where [P] is the total protein concentration and f is the fraction of the total protein as a monomer. Solving this equation for f leads to a quadratic equation that can be solved by retaining only the positive f value, whereas the other possible solution involves a negative f value that has no physical meaning. For the observed DH_cal/DH_vH ratio, we can assume that

(DH_cal/DH_vH)_obs = f * (DH_cal/DH_vH)_monomer + (1 - f)*(DH_cal/DH_vH)_dimer

(Eq. 4)

Recall that (DH_cal/DH_vH)_monomer = 1.0 and (DH_cal/DH_vH) _dimer =0.5.

By combining this equation with the f value obtained previously, we obtain the following equation for (DH_cal/DH_vH)_obs as a function of total protein concentration.

(DH_cal/DH_vH)_obs =0.5 - (1/4*K_D*[P]) + (1+4*K_D*[P]^1/2/(4*K_D*[P]))

(Eq. 5)

Simulations showed that the limiting value of this equation was 0.5 when the protein concentration tended to reach high values, and 1.0 when the protein concentration tended to be zero.

When this equation was fitted to the (DH_cal/DH_vH)_obs values (Figure 4) it did fit reasonably well protein concentrations lower than 1.0 mg/ml, but did not fit the experimental data observed at higher protein concentrations. We believe that this departure suggests the presence of higher aggregate states (trimers, tetramers, etc.) that cannot be accounted for by this equation.

For a two-state transition, at the transition temperature (Tm), DG_(Tm) = 0 cal/mol (K_D = 1.0). Thus,

(Eq. 6)

The dependence of DH_(Tm) on Tm allows the evaluation of DCp, which is the variation of the heat capacity between the native and denatured forms of the protein. Hence:

(Eq. 7)

Integrating between the limits (assuming constant DCp in the temperature range studied):

(Eq. 8)

\DH_(T) = DH_(Tm) + DCp (T - Tm)

(Eq. 9)

Analogously, it can be demonstrated that

DS_(T) = DS_(Tm) + DCp 1n (T/Tm)

(Eq. 10)

Once DH_(Tm), Tm and DCp are known, the protein stability DG_(T) in the temperature range of interest can be calculated according to the following equation (18):

DG_(T) = DH_(Tm) (1 - T/Tm) - DCp[(Tm- T) + T * ln (T/Tm)]

(Eq. 11)

At pH 3.00, ß-trypsin denaturation presented a Tm of 54ºC and a DH of 101.8 kcal/mol. The variation in heat capacity between the native and denatured forms of the protein (DCp) was estimated to be 2.50 ± 0.07 kcal mol^-1 K^-1 for the ß-trypsin monomer. The values used for this evaluation presented DH_cal/DH_vH ratios close to one. In some runs, sorbitol was used as the protein stabilizer and guanidinium hydrochloride or urea as a destabilizer (Figure 5). These compounds are known to affect the stability of proteins (16) by changing the relative stabilities of the native and unfolded forms of the protein. The observation of a straight line fitting well all the data suggests that we are studying the same conformational transition of the protein in all experimental situations using pH, sorbitol, urea and guanidinium hydrochloride as perturbing agents. The stability curve calculated for ß-trypsin at pH 3.00 is shown in Figure 6.

The DG value of ß-trypsin denaturation calculated from the DSC at pH 3.00 and extrapolated to 298 K was 5.7 kcal/mol, as can be observed in Figure 6. The result for ß-trypsin was similar to that obtained by Günther (19) by solvent (urea) denaturation at the same pH and temperature (6.10 ± 1.03 kcal/mol).

It is known that protein folding is a thermodynamically directed process depending on the primary protein structure and can produce stable intermediates with structures sufficiently similar to the native form of the protein (molten globule) (20,21). In the case of small proteins the denaturation process is quite simple, with a transition between two states, i.e., native and denatured. For large proteins and with some domains the denaturation process can present some intermediate states between the native and the denatured forms (21). Based on the DH_cal/DH_vH values observed in the 2.00 to 3.00 pH range, the process of thermal denaturation of ß-trypsin occurs as a transition between two states. Although ß-trypsin has two similar domains, the enzyme denatures as a cooperative unit in the denaturation process. This behavior was already shown in studies using DSC for observation of the denaturation of ß-trypsin at pH 2.80. This was attributed to the close similarity and strong interaction between the two domains (5). As the pH increased from 3.20 to 4.20, ß-trypsin tended to aggregate, as suggested by a DH_cal/DH_vH ratio tending to 0.5. This dimerization was the consequence of deprotonation of some carboxyl groups causing an interaction on the surface of molecules. Dimerization can be reversed by dilution of the protein solution (17).

Martins et al. (6) determined the thermodynamic values of the denaturation of ß-trypsin at pH 3.00, with better adjustment for a three-state transition, using solvent denaturation with urea. Ruan et al. (22) showed the presence of the intermediate molten globule in the denaturation of trypsin under pressure. Thus, the results of the cited reports are consistent with the existence of one or more intermediates in the denaturation of ß-trypsin. However, Ruan et al. used pressure denaturation and did not purify the sample of bovine trypsin, and other trypsin isoforms were present, while Martins et al. (6) used solvent denaturation with urea. Bulaj and Otlewski (23) worked with commercial trypsin, trypsinogen and ß-trypsin and concluded that the process of denaturation of these proteins fitted a two-state transition, although they described a kinetically complex denaturation process.

Finally, it is worthwhile to calculate the stability of ß-trypsin at neutral pH. From the thermodynamic cycle (Figure 1) we obtain

¼¼DG_N^Tit + DG _D^{pH 7.00} = DG_D^Tit + DG_D ^{pH 3.00}

(Eq. 12)

\DG_D^{pH 7.00} = (DG_D^Tit - DG_N^Tit) + DG_D^{pH 3.00} = (DDG^Tit) + DG_D^{pH 3.00}

(Eq. 13)

The titration of the native form of this protein from pH 3.00 to 7.00 is affected by any change in the state of the protein, including the aggregation that occurs around pH 4.5, so that the thermodynamical cycle shown in Figure 1 is still valid, and it is known that this enzyme is monomeric at neutral pH. In the present study, the DG_D^{pH 3.00} = 5.71 ± 0.06 kcal/mol was assessed at pH 3.00 and at 25ºC. The DDG^Tit term was found to be 9.51 ± 0.06 kcal/mol (24) when the native and unfolded forms of the protein were independently titrated from pH 3.00 to 7.00 at 25ºC. Based on these values, the calculated stability of ß-trypsin at pH 7.00 was 15.22 ± 0.07 kcal/mol, a value that is typical for small globular proteins at neutral pH. It is important to realize that we cannot obtain this value directly at pH 7.00 due to autolysis of the protein.

We have come a long way from the work of Anson and Mirsky (25), the first description of the effect of temperature on the equilibrium between the native and denatured forms of a preparation of trypsin. Their data were obtained only at pH 2.0 using a quite different technique. Our work is an extensive thermodynamic study of ß-trypsin denaturation at acid pH that significantly extends our knowledge of the thermodynamical behavior of this important protein. It would be instructive to compare the stability of ß-trypsin with its zymogen under similar experimental conditions (at pH 7.00 and 25ºC) and describe the activation process of a typical serine protease from a thermodynamic point of view. This is the subject of an ongoing research project in our laboratory.

Figure 1.
Thermodynamic cycle for ß-trypsin, where: DG_D^{pH 3.00} = measured stability of the protein at pH 3.00; DG_D^{pH 7.00} = calculated stability of the protein at pH 7.00; DG_D^Tit = free energy change of titration of the denatured species from pH 3.00 to 7.00; DG_N^Tit = free energy change of titration of the native species from pH 3.00 to 7.00.

Figure 2.
Acid denaturation curve of ß-trypsin. The proteins were 1.0 mg/ml in 50.0 mM ß-alanine and 20.0 mM CaCl₂ buffer, pH 3.00. Data were obtained by differential scanning calorimetry. The solid line shows data processed after subtraction of the baseline and normalization of protein concentration. The dashed line indicates the fitting of the data to the mathematical model of a two-state transition.

Figure 3.
DH_cal/DH_vH values for ß-trypsin in the acid pH range. Buffers: 50.0 mM ß-alanine with 20.0 mM CaCl_2, except at pH 4.00 and 4.20 where 50.0 mM acetate with 20.0 mM CaCl₂ buffers were used. ß-Trypsin concentration was 1.0 mg/ml.

Figure 4.
DH_cal/DH_vH values for ß-trypsin at pH 3.70 as a function of protein concentration. The buffer was 50.0 mM ß-alanine and 20.0 mM CaCl₂, pH 3.70. Dotted line = fourth degree polynomial fitted to the data to guide the eyes. Solid line = equation 5 fitted to the data to show the predicted values due to a putative dimerization process. Protein concentration was 0.05 to 2.3 mg/ml.

Figure 5.
DH_cal versus Tm (ºC) at different pH values, concentrations of sorbitol at pH 3.00, of guanidinium hydrochloride at pH 3.00, of urea at pH 3.00, and different buffers. Protein concentration was 1.0 mg/ml except when the influence of protein concentration was tested. NaCl (0.2 M) was used in one experiment at pH 3.00. In all experiments the buffer was 0.1 M ß-alanine except when glycine or acetate was used. The slope was DCp = 2.50 ± 0.07 kcal mol^-1 K^-1. Run conditions: 1) 1.00 M guanidinium hydrochloride, pH 3.00; 2) 0.70 M guanidinium hydrochloride, pH 3.00; 3) 0.50 M guanidinium hydrochloride, pH 3.00; 4) 0.50 M urea, pH 3.00; 5) pH 2.70; 6) pH 2.80; 7) pH 2.85; 8) pH 2.90; 9) 0.20 M NaCl, pH 3.00; 10) ß-alanine, pH 3.00; 11) pH 3.20; 12) glycine, pH 3.00; 13) 0.1 mg/ml protein, pH 3.70; 14) acetate buffer, pH 3.00; 15) 1.0 M sorbitol, pH 3.20; 16) 2.0 M sorbitol, pH 3.14. In all experiments, DH_cal was equal toDH_vH.

Figure 6.
Stability curve of ß-trypsin: DG_D as a function of temperature (ºC), at pH 3.00, where Tm is the heat denaturation temperature, Tm' is the theoretical cold denaturation temperature and Ts is the temperature of maximum stability of the protein under these experimental conditions. The value at 25ºC, 6.10 ± 1.03 kcal/mol, was obtained by solvent denaturation with urea as denaturant (21); the data were analyzed as a two-state transition.

Address for correspondence: M.M. Santoro, Departamento de Bioquímica e Imunologia, ICB, UFMG, Caixa Postal 486, 30161-970 Belo Horizonte, MG, Brasil. E-mail: [e-mail]

Received July 19, 2002. Accepted September 2, 2003.

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Correspondence and Footnotes

Publication Dates

• Publication in this collection
  17 Nov 2003
• Date of issue
  Dec 2003

History

• Accepted
  02 Sept 2003
• Received
  19 July 2002