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The mechanism by which proteins fold to their unique native conformations from an initially disorganized form is one of the fundamental problems in molecular biology. In the study of proteins fold or refold standard Gibbs free energy,is the single most important parameter for quantitating-protein stability and comparing stabilities of closely related proteins. Nearly all theoretical and experimental aspects of protein folding relate in some way to unfolding free energy changes, and much of the current work involving comparisons of mutant proteins is highly dependent on accurate evaluation of unfolding free energy measurements .
The history of evaluation of the quantity known as standard Gibbs free energy spans more than four decades, and at least three procedures involving strong solvent denaturation have been used in evaluation of this quantity. The three procedures are known as the transfer model of Tanford, the denaturant binding model, and the linear extrapolation method. Of the three methods employed in evaluation of the so-called linear extrapolation method appears to enjoy the greatest acceptance. The virtue of the linear extrapolation method over the other two methods resides in the perception that it is more reliable since it gives values that appear to be independent of whether urea or guanidinium hydrochloride is used as denaturant .
The linear extrapolation method is the most frequently used method of determining protein unfolding free energy changes induced by urea or guanidine hydrochloride (GdnHCl). It is based upon the premise that unfolding free energy changes are linearly dependent on denaturant concentration and that extrapolation to zero denaturant concentration gives, the unfolding free energy change in the absence of denaturant. When only the native and unfolded forms of protein are present in significant concentrations at equilibrium, , the unfolded process corresponds to one transition, i.e. two-state model. The can be obtained from, where m reflects the sensitivity of the transition to denaturant concentration [D] by using the linear extrapolation method .
When the native, partially unfolded (stable intermediate) and unfolded forms of protein are present in significant concentrations at equilibrium, , the unfolded process corresponds to two transitions, i.e. three-state model. Moreover, the is considered to be the sum of and [4-8]. By comparing the two-state and a three-state unfolding processes of proteins, the order of the unfolding free energy was always inconsistent with the order of stability observed. This raises the question about the accuracy of the evaluated unfolding free energy values of three-state models. In this chapter, the linear extrapolation method was considered as a standard method in the two-state unfolding model and particular attention is paid to the terms of the three-state unfolding model.
1. The protein unfolding process is determined by the structural element, Ei. Ei can change from a native state, Ei(F) to a denatured, unfolded state Ei(U), in two-state model., , are the measureable signal of the folded and unfolded states for Ei, respectively, which were assumed to not vary with denaturant concentration to minimize the number of parameters.
The equilibrium between Ei(F) and Ei(U) is described in eq (1).
Figure 1A shows that the observed signals of the various experimental techniques (optical rotation, viscosity, UV-Vis absorption, fluorescence, circular dichroism, dynamic light scattering, nuclear magnetic resonance, etc) for structural element, Ei vary with the denaturant concentrations. At low denaturant concentrations (from [D] = 2.0 to [D] = 3.0) the measured signals are constant and can be considered as, the signal of folded state. At high denaturant concentrations (from [D] = 6.5 to [D] = 7.5) the measured signals are constant, too, and can be considered as, the signal of unfolded state. The unfolding curve of Ei, which is the plot of apparent fraction of unfolded structural element,against the denaturant concentrations, can be prepared as Figure 1B by using eq (2).
where is the observed signal at the given denaturant concentration
According to the linear extrapolation method the unfolding curve of Ei can be fitted based on eq (3).
where is the difference in free energy between the folded and the unfolded states of Ei at the given concentration of denaturant, R is the gas constant, and T is the absolute temperature. The equilibrium constant of unfolding transition is described in eq (4).
The free energy difference, can be evaluated following eq (5)
is the single most important parameter for quantitating structural element, Ei stability and comparing stabilities of closely related structural element, Ei. mi is a measure of the dependence of on denaturant concentration.
From Figure 1B the plot ofagainst the concentration of denaturant is shown in Figure 2. and mi can be obtained from Figure 2. They are 56.70 kJ mole-1 and -12.41 kJ mole-1 M-1, respectively.
When is 0.5 or the native and unfolded forms of structural element, Ei are 50%, respectively, the equilibrium constant of unfolding transition, Ki is 1. From eq (5)eq (6) can be obtained.
where [D]1/2 is the midpoint concentration of denaturant required for unfolding of structural element. It means that we can use ether mi and [D]1/2 as two parameters or as single parameter to quantitate the structural element stability and compare the stabilities of closely related structural element. Only one [D]1/2 or mi is not enough to account for the stability of structural element, Ei.
At [D]1/2 = 4.57M the variations of and with the concentration of denaturant are shown in Figure 3. At constant mi the unfolding curves corresponding to different [D]1/2 are only shifts of a unfolding curve.
is a function of mi and [D]1/2, and can be written as (mi, [D]1/2). Therefore, can also be written as (mi, [D]1/2).
2. Protein is consisted of structural elements connected by chemical bonds that there is no contribution to measured signal. When protein contains two kinds of structural elements, Ei and Ej, the three-state unfolding behavior may be observed. The observed three-state unfolding curve is composed of curves () and () mixed at a certain molar fraction.
If the number of structural elements is n, the protein can be express as scheme 1. Typical 1 of protein is consisted of same structural elements and typical 2 and 3 of proteins are consisted of different structural elements.
It is known that the unfolding of structural element, Ei obeys two-state model. The stability can be expressed as (mi, [D]1/2) and the unfolding process can be described by (mi, [D]1/2). If m1 is -24.72 kJ M-1 mole-1 and is 2.57 M we can obtain (-24.72 kJ M-1 mole-1, 2.57M) from eq(3) and eq(5). Similarly, we can obtain (-24.72 kJ M-1 mole-1, 6.57M). There is a protein, in which the structural element, E1 appears for n times. The measured unfolding data (O) of the protein are presented in Figure 4 A.
Denaturant-induced unfolding of the protein is found to be a two-state process, just as that of the structural element, E1.
There is a protein, in which the structural element, E1 appears for n1 times and E2 does for n2 times. The sum of n1 and n2 is n. The measured unfolding data (O) of the protein are presented in Figure 4B. Denaturant-induced unfolding of the protein is found to be a two-step process with accumulation of an intermediate state at around 3.5~5.5 M denaturant concentration. Therefore, denaturant-induced denaturation may be considered as a three-state transition and the mechanism for unfolding of the protein may be represented as:
where F, I and U are the folding, intermediate and unfolding states of the protein, respectively. The first transition which corresponded to the transformation of F state to the I state started at around at 1.5 M denaturant concentration and completed at 3.5 M denaturant with a midpoint occurring at 2.57 M denaturant. In fact the transition corresponds to the unfolding of E1 in the protein.
The second transition, which corresponded to the unfolding of the I state, started at around 5.5 M denaturant concentration and finally sloped off to the U state at 7.7 M denaturant with a midpoint occurring at 5.57 M denaturant. The transition corresponds to the unfolding of E2 in the protein. The intermediate state of the protein is a partial unfolding state, in which E1 exists in completely unfolding form and E2 does in completely folding form. The three-state unfolding process can be considered to be composed of two sequential two-state unfolding processes. Then the measured unfolding data (O) of the protein in Figure 4B can be fitted by using eq(8).
where f1 and f2 are molar fractions of structural elements E1 and E2 in the protein.
From eq(8) it is known that f1 or f2, and are the important factors to determine the three-state unfolding curve of protein. When f2 is zero, the protein is consisted of structural elements, E1, which appear n time in the protein. The three-state unfolding curve changes into two-state unfolding curve,. When f1 is zero, the protein is consisted of structural elements, E2, which appear n time in the protein. The three-state unfolding curve changes into two-state unfolding curve,.
has the scale as f1 and f2. It can be seen that the measured stable intermediate state of the protein appears at = 0.5 from Figure 4B. So we could determine the f1 or f2 value to be 0.5.
Protein is consisted of structural elements, E1, E2, E3, , En-1, which appears n1, n2, n3, , nn-1 times in the protein. The unfolding curves corresponding to E1, E2, E3, , En-1 are, , , , and, respectively, and mi mj, (i and j change from 1 to n-1). The unfolding curve of the protein would be a n-state unfolding curve and can be fitted using eq(10).
So the four-state, five-state unfolding curves can be fitted by using eq(10).
3. The unfolding free energy represented by the curve () and the () areand, respectively, for structural elements E1 and E2. is the unfolding curve of protein which is consisted of structural elements, E1, E2. Then the three-state unfolding free energy, represented by the curve () can be calculated from and using the same molar fraction and eq(12).
If is greater than, /n will change from to. From eq(11) f1 and f2 can change from 0 to 1. The variation of with f2 is shown in Figure 5.
Protein is consisted of structural elements, E1, E2, E3, , En-1, which appears n1, n2, n3, , nn-1 times in the protein. The unfolding free energies corresponding to E1, E2, E3, , En-1 are, , , , and, respectively, and , (i and j change from 1 to n-1). The unfolding free energy, of the protein can be calculated using eq(13).
When n is 4 the four-state unfolding free energy, /n can be calculated from, , and. The plot of against f1 and f2 is shown in Figure 6.
It can be seen that /n is a point located at the plane consisted of, , and in the space of /n, f1, and f2.
Although the unfolding curve of a protein, which is composed of same structural elements, i.e. Ei appears n times in the protein, is same with that of Ei, the of the protein is different with the of Ei.
According to thermodynamics the of the protein should be n times of the of Ei (see eq(14)).
3. Typical multistate unfolding curves
Three-state unfolding curves
From eq(8) three-state unfolding curve is dependence on (m1,)and the (m2,). Some typical three-state unfolding curves are given in Figure 7. When m1 = m2, and , the three-state unfolding curve is shown in Figure 7A. When m1 m2, and , the three-state unfolding curve is shown in Figure 7B. And when m1 m2, and , the three-state unfolding curve is shown in Figure 7C.
Of course, if the difference of andis greatly, stable intermediate state could be easily observed. When m1 m2, and =, the three-state unfolding curve is shown in Figure 7D. The unfolding curves in Figure 7D are often considered as two-state unfolding.
Four-state unfolding curves
When n is 4 the unfolding curve appears three transitions from eq(10). The four-state unfolding curve is a function of, , , and fi. Some typical four-state unfolding curves are shown in Figure 8.
4. Structural element
Structural element is a smallest structural unit in native protein, of which the unfolding obeys two-state model.
Native proteins are only marginally entities under physiological conditions. There are various no covalent bonds to which protein is subject—electrostatic interactions (both attractive and repulsive), hydrogen bonding (both intramolecular and to water), and hydrophobic force, over an entire protein molecule. The low conformational stabilities of native proteins make them easily susceptible to denaturation by altering the balance of the weak nonbonding forces that maintain the native conformation. Native protein structure unfolds in a highly cooperative manner: Any partial unfolding of the structure destabilizes the remaining structure, which must simultaneously collapse to the random coil.
It has been shown that the unfolding or folding of small globular proteins occurs via a two-state process, whereas the unfolding or folding of larger proteins is complex and often involves the formation of intermediate . Thioredoxin is characterized by an active site containing two cysteine residues separated by two other residues. The reversible oxidation of these cysteine residues to the disulfide form serves as a redox couple for a number of biological reactions. It was first characterized in yeast for its role in the reduction of methionine sulfoxide and inorganic sulphate[10, 11]. The backbone of oxidized form and ribbon model of reduced form for Escherichia coli thioredoxin are shown in Figure 9. The protein consists of single polypeptide chain of 108 amino acid residues. Either the oxidized or the reduced forms the protein is a compact molecule with 90% of its residues in -helices, -strands or reverse turns, which a core of twisted -strand is flanked on either side by -helices. The conversion between the oxidized and the reduced forms is accompanied by a change in conformation that is reflected in the fluorescence properties of Trp28.
The equilibrium unfolding of the oxidized form of Escherichia coli thioredoxin at pH 7 was studied by Santoro and Bolen  using guanidine hydrochloride (GdnHCl) and urea as denaturant at 25 C by monitoring the changes in ellipticity at 222nm.
The GdnHCl- or urea-induced unfolding of thioredoxin is a two-state process. The unfolding free energy change is 7.8±0.2 kcal/mole (32.64±0.84 kJ/mole) in the absence of (GdnHCl) or 8.6±0.9 kcal/mole (35.98±3.77 kJ/mole) in the absence of urea. Thermal unfolding measurements give G0(H2O) being 8.1±0.1 kcal/mole (33.89±0.42 kJ/mole) by using differential scanning calorimetry. Within error, G0(H2O) values obtained from GdnHCl- and urea-induced, and thermal unfolding are in agreement, meaning that GdnHCl, urea, and thermal unfolding involve the same native form to unfolding form equilibrium in oxidized form of Escherichia coli thioredoxin.
The free energy required to denature a protein is ~ 0.4 kJ/mole of amino acid residues so that 100-residue protein is typically stable by only around 40 kJ/mole . Since the measured G0(H2O) of thioredoxin is close to 40 kJ/mole, the compact protein molecule can be considered as a structural element.
Goat -lactoglobulin consists of single polypeptide chain of 101 amino acid residues. The protein contains - helices (red) and - sheets (yellow) as shown in Figure 10. The measured G0(H2O) is 11.7±0.8 kcal/mole (48.95±3.35 kJ/mole) at 25 C by measuring optical rotation . -lactoglobulin can be considered as a structural element. Although -chymotrypsin consists of single polypeptide chain of 178 amino acid residues the G0(H2O) is only 8.3±0.4 kcal/mole (34.73±1.67 kJ/mole) at 25 C by measuring optical rotation . If the G0(H2O) of a 100-residue protein were 40 kJ/mole, the G0(H2O) of -chymotrypsin would have been around 70 kJ/mole at 25 C. It means that -chymotrypsin contains same structural elements at least two.
Human serum albumin (HSA) has been used as a model protein for protein folding and ligand-binding studies over many decades [5, 14-19]. The protein is a single chain protein with 585 amino acids, with a molecular weight of ~67,000 Da. Serum albumin homologs with very similar properties are found in other mammals. The protein contains only - helices. The structure of this protein has been determined by X-ray crystallography of high resolution (Figure 11); it includes three homologous domains (I-III) that assemble a heart-shaped molecule. Each domain is formed by two subdomains that possess common structural motifs. Only one cysteine residue located at position 34 (in domain I) with a free sulfhydryl group. HSA plays a special role in transporting metabolites and drugs throughout the vascular system and also in maintaining the pH and osmotic pressure of plasma. Interestingly, the structure and dynamics of HAS are known to be influenced by a number of factors, like pH, temperature, and binding of different ligands.
Denaturation of HSA has been studied by several workers using different methods like absorption difference at 287nm, fluorescence at 340nm upon excitation at 282nm, tryptophan fluorescence upon excitation at 295nm, ellipticity at 222nm, 1-anilinonaphthalene-8-sulfonate fluorescence, differential calorimetry, etc. Denaturation of HSA has been suggested to follow a single-step transition as well as two- or multiple-step transitions depending upon the reaction conditions and probes used. For example, urea denaturation of HSA has been shown to follow a two-step, three-state transition involving one intermediate when studied by fluorescence and ellipticity measurements but became single-step, two-state transition when studied by UV difference spectroscopy. Similarly, guanidine hydrochloride (GdnHCl) denaturation of serum albumin has been reported to follow both single-step and two-step transitions . Meanwhile the measured G0(H2O) has big difference. The measured G0(H2O) is 5.98 kcal/mole (25.02 kJ/mole) and 6.00 kcal/mole (25.10 kJ/mole) at 25 C, 1M KCl pH 7.0 by measuring circular dichroism and fluorescence, respectively . In the presence of 2,2,2-trifluoroethanol the measured G0(H2O) is 8.04 kcal/mole (33.64 kJ/mole) and 8.06 kcal/mole (33.72 kJ/mole) . 2,2,2-trifluoroethanol makes the G0(H2O) increase around 2 kcal/mole. Halim et al  used bromophenol blue as a probe the G0(H2O) of bovine serum albumin is 4.04 and 4.60 kcal/mole, respectively, from urea and GdnHCl two-state denaturation curves at pH 8.0, 25 C. However, the G0(H2O) of HSA is 146.7 kJ/mole from sodium dodecyl sulphate two-state denaturation curves at pH 7.0, 35 C . It means that HAS is a multi-element protein. When the structural elements are same at the experimental conditions the unfolding of HAS is a two-state process, and when the structural elements are different at another experimental conditions the unfolding behaves as a three-state process.
Urea-induced unfolding of bovine serum albumin and one of its fragments containing domain II+III has been studied by difference spectral and fluorescence emission measurements. The unfolding-refolding curves of both the proteins showed the presence of at least one stable intermediate when the transition was monitored at 288 nm. The presence of the intermediate was not detectable at 293 nm where only tryptophan contributed towards the protein absorption. However, both the proteins did show the presence of intermediate when the denaturation was monitored fluorometrically. Since domain III of the albumin is devoid of tryptophan, it is concluded that the formation of intermediate in the unfolding-refolding transition of serum albumin involves (i) unfolding of domain III, (ii) minor structural transformations in domain II, and/or (iii) the separation of the sub-domains of domain III from each other [15-17].
5. The factors affected structural element
Acquisition of native globular conformation (3D structure) of a protein is governed by its structural elements and biological environment surrounding it. The stability of protein’s 3D structure is endorsed by various intramolecular forces, such as hydrogen bonds, van der Waals, electrostatic and hydrophobic interactions involving various amino acid side chains as well as their milieu in the native structure. So protein structure is affected by factors such as experiment conditions (extreme temperature, pH and pressure) and presence of destabilizing agents (salt, alkali, denaturant and surfactant), and of stabilizing agents (metal ions, anions and small organic molecules) [15-19]. In fact they are the factors which structural element is affected. A few examples are given to show the effect of stabilizing agents on structural element.
Anticoagulation factor I (ACF I) isolated from the venom of Agkistrodon acutus is an activated coagulation factor X-binding protein in a Ca2+-dependent fashion with marked anticoagulant activity. The protein is a single chain protein with 129 amino acids, in which there are 7 Trp. The crystal structure measurement shows that the protein has both - helices and - sheets and tryptophan residues locate at -helices, -sheets and loops. Metal ions-induced stabilization and unfolding of ACF I was studied in guanidine hydrochloride solutions by Xu et al  following the fluorescence and circular dichrosim. Metal ions can increase the structural stability of ACF I against guanidine hydrochloride denaturation and change its unfolding behavior. Reproduced unfolding curves of apo-ACF I and holo-ACF I from ref  are shown in Figure 12.
Thermodynamic parameters for unfolding and refolding of ACF I by GdnHCl at 25 C, monitored by measurement of fluorescence at 340nm
The guanidine hydrochloride induced unfolding of apo-ACF I is a two-state process with no detectable intermediate state, whereas the unfolding of holo-ACF I follows a three-step transition, with intermediate state Ia and intermediate state Ib. The thermodynamic parameters for unfolding and refolding of ACF I are shown in Table 1 . The unfolding data can be fitted by using eq (3) for two-state process of apo-ACF I and eq (10) for four-state process of holo-ACF I, respectively.
Using the new method it can be thought that ACF I is consisted of multi-structural elements, and the structural elements are same in apo-ACF I in guanidine hydrochloride. The stability of structural element can be described by parameters [D]1/2 (1.22M) and m (-4.06 kcal/M/mole), i.e. G0(H2O) is 4.96 kcal/mole. After binding Ca2+ the structural elements are changed. At least three-kind structural elements appear in holo-ACF I. Their stability is 4.22 kcal/mole, 7.07 kcal/mole, and 5.95 kcal/mole, for structural element 1, 2, and 3, respectively.
Parameters of [D]1/2 and m values are 0.85M and -4.96 kcal/M/mole, 1.56M and -4.53 kcal/M/mole, 2.56M and -2.32 kcal/M/mole, for structural element 1, 2, and 3, respectively. The unfolding of holo-ACF I can be fitted from eq (10).
where, , and are unfolding transition of structural element 1, 2, and 3, respectively.
The /n of holo-ACF I is 6.00 kcal/mole from eq (13).
If using the stability of structural element in apo-ACF I as standard the binding of Ca2+ makes structural element 1 be destabilized and 2 and 3 be stabilized. Of the various forces, hydrogen bonding and hydrophobic effect have been shown to make significant contribution to the protein stability. Being secondary contributor to protein’s stability through salt bridges, charged residues have also been found important in manipulating protein stability . It is that Ca2+ binding breaks the balance of charged residues makes the charged residues in apo-ACF I re-distribute and form new structural elements in holo-ACF I.
Second example is succinylation-induced conformational destabilization of lysozyme as studied by guanidine hydrochloride denaturation . Figure 13 shows the crystal structure of lysozyme which is made up of a single polypeptide chain of 129 amino acid residues arranged in the form of two domains and stabilized by four disulphide bonds. All six lysine residues are distributed in the molecular surface and all six tryptophan residues are located at -helices and loops. The unfolding of the protein obeys two-state process, meaning that the protein is consisted of same structural element(s). Using fluorescence data upon excitation at 280nm value of /n is 8.56 kcal/mole and ‘m’ is -2,03kcal/M/mole for native lysozyme. When excitation at 295nm /n is 7.96 kcal/mole and ‘m’ is -1.93 kcal/M/mole. Using 100-fold molar excess of succinic anhydride, about 99% of lysine residues of lysozyme were modified. Though the succinylated lysozyme obeys two-state process yet a pronounced decrease in /n value is noticed with succinylated lysozyme preparation compared to native lysozyme. /n value is 4.40 kcal/mole and 4.67 kcal/mole excitation at 280nm and 295nm, respectively. The ‘m’ values are also changed from -2.03 kcal/M/mole and -1.93 kcal/M/mole (for native lysozyme) to -1.54 kcal/M/mole and -1.59 kcal/M/mole for succinylated lysozyme, respectively. The conformational destabilization of the modified protein can be attributed to the effect of succinylate on structural element(s), i.e. it is that the positive charges on lysine residues are neutralized by negative charges on succinylate, makes the structural element(s) be destabilized. Experimental measurements show that the net charge of both native and succinylated lysozyme is +8 and -9, respectively at pH 7.0.
Third example is HAS, whose crystal is shown in Figure 11. In the presence of different concentrations of KCl urea-induced denaturation of HAS displays different pathways. When the concentration of KCl is 1.0M HAS undergoes a single-step transition with no intermediate. At pH 7.0, 25 C the variation of G as a function of urea concentration is presented in Figure 14A (curve 2). According to eq (5) /n and ‘m’ values can be obtained to be 5.98±0.26 kcal/mole and –(1.03±0.04) kcal/M/mole, respectively. In the absence of KCl the unfolding of HAS is a two-step transition process and the urea-induced denaturation curve is shown in Figure 14B, using the data of Figure 1B or 1A in ref . Muzammil et al  considered that first transition corresponded to the formation of intermediate state whereas second transition corresponded to the unfolding of the intermediate. Assuming both the transitions follow two-state mechanism the variations of G as a function of urea concentration are presented in Figure 14A (curve 1 and 3) at pH 7.0, 25 C. According to eq (5) and are 3.66±0.35 kcal/mole and 5.38±0.24 kcal/mole, respectively. The ‘m’ values are –(0.95±0.10) kcal/M/mole and –(0.80±0.04) kcal/M/mole, respectively. From eq (6) the [D]1/2, of which corresponds second transition from intermediate state to unfolding state, should be 6.73M.
However, the authors  reported to be 1.4 kcal/mole. Using ‘m’ value as -0.80 kcal/M/mole the calculated [D]1/2 is only 1.75M from eq (6). The difference of [D]1/2 is 4.98 M, at which the second transition from intermediate state to unfolding state starts. The authors  got from extrapolation of G values upto the starting of second transition. Since /n being a thermodynamic property does not depend on the path free energy change in the absence of salt should be obtained by summing the and. The /n (4.89 kcal/mole) in the absence of KCL is smaller than that (5.98 kcal/mole) in the presence of 1M KCl. The authors attributed the binding of Cl- to the domain III to stabilize the domain .
In my views the calculation of is unreasonable. G0(H2O) is a parameter to describe the stability of a protein. In two-state transition ‘m’ and [D]1/2 values are also two parameters to describe the stability of the protein. When ‘m’ is same the G0(H2O) is proportional to [D]1/2. The protein whose unfolding curve appears at higher denaturant concentrations is more stable than that whose unfolding curve appears at lower denaturant concentrations For the unfolding curve of HAS (Figure 14B) in the absence of KCl, two transitions are observed. The first transition starts at around 2M urea and completes at 4.5M urea, and the second transition starts at around 5.2M urea and finally slopes off at 8.4M urea. The free energy change corresponded to second transition is greater than that corresponded to the first transition if ‘m’ is nearly same. The reported (=1.40 kcal/mole) (=3.40 kcal/mole) . Using the new method HAS is consisted of multi-structural elements. In the absence of KCl HAS contains at least two-kind structural elements, the unfolding shows three-state process. A least squares analysis of the data in Figure 14B according to eq (8) yielded a fitting curve,.
where and are unfolding transition of structural element 1 and 2, respectively.
The /n (4.69 kcal/mole) in the absence of KCL is smaller than that (5.98 kcal/mole) in the presence of 1M KCl. The binding of Cl- to the domain III to stabilize the domain
6. The unfolding of CopC
The protein CopC of Pseudomonas syringae pathovar tomato (CopC) is one of four proteins, CopA, B, C, and D, coded on the copper resistance operon (cop). CopC belonging to periplasmic protein, consists of 102 amino acids. In solution it adopts a fold essentially constituted by nine β sheets forming a barrel motif. The copper (Ⅰ) site located in C-terminal is constituted by His-48 and two or three of the four Met residues (40, 43, 46, 51) [CuI(His)(Met)x (x = 2 or 3)]. On the other hand, the protein is known to bind copper (Ⅱ) in an N-terminal position that is found consistent with a coordination arrangement including His-1, Glu-27, Asp-89, and His-91 [CuII(His)2(Asp)(Glu)(OH2)]. They represent novel coordination environments for copper in proteins and the two copper binding sites are about 30 Å apart. All of these have been confirmed by the method of nuclear magnetic resonance (NMR) and extended X-ray absorption fine structure spectroscopy (EXAFS) [21, 22]. CopC has high affinity for both copper ions although the Cu(II) site has the highest affinity (~1013 M-1) [23, 24]. The binding sites of CopC can also occupied by other transition metal ions. However the affinity is lower than copper ions. For example, Hg(II) ions can bind to the both sites  and Ag(I) only binds to the Cu(I) site . Figure 15 shows the NMR structure of apoCopC (A) and crystal structure of Cu2+-CopC-Cu+ (B) drawn with the Protein Data Bank files 1M42 and 2C9Q, respectively. A -sandwich structure comprised of -strands arranged in a Greek Key topology exists in apoCopC and Cu2+-CopC-Cu+. Single tryptophan residue 83, sandwiched between the two β sheets, locates at a rather hydrophobic microenvironment and has numerous contacts with residues in strand β2 and strand β7 .
The fluorescence properties of tryptophan residues in protein can be considered as three discrete spectral classes: One is the buried in nonpolar regions of the protein, the fluorescence maximum position appears at 310~330 nm and spectral band width is around 48~49 nm: two is located at the surface of protein and completely exposed to water, maximum peak is in 350~353 nm and half bandwidth is in the range of 59~61 nm; the latter is in limited contact with water which is probably immobilized by bonding at the macromolecular surface, the fluorescence maximum peak is located at 340~342 nm and the half bandwidth is about 53~55 nm.
Figure 16 is the spectra of apoCopC in different experimental conditions. In the presence of 9.4 M urea the fluorescence peak of apoCopC locates at 353 nm and the half bandwidth reaches to 60 nm (Figure 16 spectrum 3), meaning that apoCopC exists in unfolding state. In the absence of urea the fluorescence peak of native apoCopC appears at 320 nm and the half bandwidth is 48 nm (Figure 16 spectrum 1), meaning that the tryptophan residue 83 is buried in nonpolar regions of apoCopC. In the presence of 5.0 M urea the fluorescence peak appears at 331 nm and the half bandwidth reaches to 66 nm (Figure 16 spectrum 2).
Obviously spectrum 2 is the summing of spectra produced by two-class tryptophan residue in protein(s). With the increasing of urea concentration an iso-fluorescence point keeps at 332nm from spectrum 1 to spectrum 3, meaning that there are two species of apoCopC in the presence of 5.0 M urea. Curve 4 (red line) in Figure 16 is fitted from spectrum 1 and 2, in which the contribution of spectrum 1 is 0.73 and that of spectrum 2 is 0.27. It can be seen that the fitted curve is nearly coincided with the measured spectrum. In the presence of 5.0 M urea apoCopC exists in native and completed unfolding forms. The fluorescence property of tryptophan residue 83 in apoCopC is an index to show the protein in either native or unfolding states. It can be deduced that the unfolding of apoCopC would obeys two-state mechanism.
When the tryptophan residue 83 is mutated as leucine residue the protein is nearly immeasurable by fluorescence. Meanwhile the mutagenesis make the ability of binding to Cu(II), the thermal stability of apoCopC be decreased . The important role of the tryptophan residue 83 is not only in using as a fluorescence probe but also in keeping the -sandwich hydrophobic structure. The thermodynamic stability apoCopC can be investigated at pH 7.4 by monitoring the fluorescence intensity at around 405 nm [25, 29, 30]. As known Cu2+-CopC and Hg2+-CopC-Hg2+ are prepared by adding Cu(II) or Hg(II). The unfolding of apoCopC, Cu2+-CopC and Hg2+-CopC-Hg2+ are reversible. According to two-state model, the free energy change can be measured, respectively, and the order of stability is Cu2+-CopC > Hg2+-CopC-Hg2+ ~ apoCopC. The binding of Cu(II) makes the stability of CopC be increased greatly. Although Hg(II) can also binding the Cu(II) site the contribution to the stability can be neglected. The reason is the weak binding of Hg(II) to the sites of apoCopC.
Using guanidine hydrochloride as chemical denaturant the unfolding profiles are presented in Figure 17 by monitoring the changes of fluorescence ratio at 400nm and 320nm. ApoCopC unfolds in two-state equilibrium reaction. It means that apoCopC is consisted of a kind of structural element(s). The /n, [D]1/2 and ‘m’ values are listed in Table 2. When used urea as denaturant the obtained parameters are also in Table 2. Compared the data in Table 2 the /n values agree between guanidine hydrochloride and urea as chemical denaturant. The unfolding curve of CopC-Ag+ roughly is a repeat of that of apoCopC. Like Hg(II), the binding of Ag(I) to the Cu(I) site of apoCopC does not also make the stability of the protein be increased. In another word, the structural element(s) in apoCopC don’t affected by the binding of Hg(II) or Ag(I).
The corresponding GdnHCl-induced unfolding with Cu2+-CopC reveals some difference. If the unfolding were considered as two-state process the GdnHCl-concentration at transition midpoint would have been 2.06 M which is higher than 1.55 M, and the dependence of ∆G on GdnHCl concentrations would have been as inset in Figure 17. Indeed the binding of Cu(II) makes the transition midpoint of apoCopC from 1.55 M shift to 2.06 M and stabilize the protein. However, there are evidently two slopes in dependence of ∆G on GdnHCl concentrations (Inset of Figure 17). This indicates that the equilibrium-unfolding reaction is three-state process, involving an intermediate species populated at 1.4 to 1.7 M guanidine hydrochloride .
Thermodynamic parameters for various forms of CopC from chemical denaturant-induced unfolding at pH 7.4, 25 C
According to the new model the structural element(s) of apoCopC is changed by the binding of Cu(II), from one-kind structural element(s) changing to two-kind structural elements, which one is more stable than another. Naturally the equilibrium-unfolding reaction of the protein displays three-state behavior. The data (●) measured guanidine hydrochloride experiment with Cu2+-CopC are fitted well in eq (19).
where and are the unfolding transitions, and are the parameters described the stabilities of structural elements 1 and 2, respectively. A least-squares curve fitting analysis is used to calculate the, , m1, and m2. Theand correspond to and, and m1 and m2 are mFI and mIU respectively, in Table 2. From eqs (8) and (20) free energy change of Cu2+-CopC can be calculated to be 36.82 kJ/mole.
Cu2+-CopC is consisted of two-kind structural elements. Thus the unfolding profile appears to involve two sequential transitions. The first transition corresponds to the unfolding of structural element one, whose free energy change is 12.83 kJ/mole and the guanidine hydrochloride concentration at the transition midpoint is 1.53 M, close to 1.55 M of apoCopC or apoCopC-Ag+. The second transition corresponds to the unfolding of another, whose free energy change is 42.52 kJ/mole and the guanidine hydrochloride concentration at the transition midpoint is 2.12 M. The probability of one appeared in the protein is 0.19 and another is 0.81. This indicates that the binding of Cu(II) makes the secondary structure of 81% integrative apo-CopC be stabilized. The stabilized energy ∆∆G0Cu(II) is 11.84 kJ/mole. Since Cu(II) binds at N-terminal of apoCopC the first transition corresponds to the partial unfolding of apoCopC and it is the part the C-terminal of apoCopC.
Using urea as chemical denaturant the unfolding of Cu2+-CopC shows three-state process. The free energy change is calculated to be 29.16 kJ/mole by using the new model. The stabilized energy ∆∆G0Cu(II) is only 4.18 kJ/mole. At pH 7.4 apoCopC is a protein charged positively since the pI of the protein is 8.27. After binding of Cu(II) the positive charges of the protein is increased and anion ions would make Cu2+-CopC be stable. As an ionic denaturant, guanidine hydrochloride makes Cu2+-CopC be stable.
In a similar way the unfolding of Cu2+-CopC-Ag+ is studied by measuring the fluorescence ratio at 320 nm and 400 nm. Two transitions appear in the unfolding profile (Figure 17). Corresponding to the unfolding of C-terminal the first transition midpoint is shifted from 1.53 M to lower guanidine hydrochloride concentration (1.42 M), while the second transition midpoint is shifted from 2.12 M to higher guanidine hydrochloride concentration (2.38 M). Comparing with Cu2+-CopC the binding of Ag(I) makes structural element one be unstable and structural element two be stable. This means that the structural element one and two are affected by the binding of Ag(I). Although the structural elements are marked as structural element one and two in Cu2+-CopC-Ag+, they are different with that in Cu2+-CopC.
The data (■) of Cu2+-CopC-Ag+ are fitted well in eq (21).
It is noticed that and in eq (21) are different with that in eq (19). From eqs (8) free energy change of Cu2+-CopC-Ag+ can be calculated to be 36.71 kJ/mole. As Ag(I) binding to apoCopC the binding of Ag(I) to Cu2+-CopC doesn’t change free energy change of the protein and affected only the structural elements and the probabilities of structural elements appeared in the protein. Comparing eq (21) with eq (19), the contribution of structural element one is increased and two is decreased.
Hydrogen bonding and hydrophobic forces have major influence to the structure of native protein. Some molecules are often used in researching stability of native proteins since their perturbation on the hydrogen bonding and hydrophobic forces [31, 32]. Apo-Y79W-W83F is a mutant of apoCopC, in which tyrosine residue 79 is mutated as tryptophan and tryptophan residue is did as phenylalanine. As shown in Figure 18A the fluorescence spectrum of apoY79W-W83F is dominated by tryptophan residue at position 79 and centered at 328 nm. Comparing the fluorescence spectrum of apoCopC the maximum peak is shifted toward to red around 8 nm and the tryptophan residue is yet a typical for buried tryptophyl side chains in apolar environment. The protein binds one equivalent of Cu(II) with high affinity and the fluorescence at 328 nm is quenched.
Since the hydrogen bonding and hydrophobic forces in apoCopC are changed, the GdnHCl-induced unfolding profile of apo-Y79W-W83F shows an obvious two transitions. According to the new model apo-Y79W-W83F is consisted of two-kind structural elements, the unfolding curve can be fitted from and from eq (8), and free energy change can be calculated using eq (12). Although free energy change corresponding to apo-Y79W-W83F unfolding is nearly same with that of apoCopC (unpublished data) the structural elements in apo-Y79W-W83F are different with that of apoCopC.
Figure 18B shows GdnHCl-induced unfolding of Cu2+-Y79W-W83F (O). Analyzing the unfolding data according to two-state unfolding model, the free energy changes plotted as a function of denaturant concentration are an obviously inflection point, suggesting that GdnHCl-induced unfolding of Cu2+-Y79W-W83F does not proceed via a simple two-state process but involves one intermediate state. Using eq (8) the unfolding data of Cu2+-Y79W-W83F in Figure 18B can be fitted well. and are the unfolding curves of structural elements 1 and 2, respectively. The obtained thermodynamic parameters are listed Table 2.
Tryptophan residue 83 locates at N-terminal and tyrosine residue 79 locates at C-terminal of apoCopC. After binding Cu(II) the positions of the residues have no large change (Figure 15). Two transitions are observed in GdnHCl-induced unfolding profiles of Cu2+-CopC-Ag+, Cu2+-CopC, and Cu2+-Y79W-W83F. The first transition corresponds to the unfolding of C-terminal and second does to that of N-terminal. It can be seen that the mutation from tryptophan to phenylalanine residue at position of 83 makes the N-terminal stability be increased, while the mutation from tyrosine to tryptophan residue makes the C-terminal stability be decreased a little. This indicates that the aromatic ring stacking at position 83 plays an important role in keeping the secondary native structure of apoCopC.
A new model to calculate the free energy change of proteins unfolding is presented. In the new proposed model proteins are consisted of different structural elements; the unfolding of structural element obeys two-state mechanism and the free energy change of the element can be obtained by using linear extrapolation method; if the protein is consisted of same structural elements unfolding of the protein displays a two-state process, while the unfolding of the protein shows a multi-state behavior; A least-square fitting is used to analyze the contribution of structural element to the protein and free energy change of the protein is the probable summing of all structural element. The model can be used to analyze and compare the stability of proteins, which are similar in size and structure.
Yang BinSheng (September 15th 2011). The Stability of a Three-State Unfolding Protein, Thermodynamics - Physical Chemistry of Aqueous Systems, Juan Carlos Moreno-Piraján, IntechOpen, DOI: 10.5772/21572. Available from:
Thermodynamics of Ligand-Protein Interactions: Implications for Molecular Design
By Agnieszka K. Bronowska
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