Thermal Denaturation of Proteins and Chemical Equilibrium

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2. Materials and Methods

α-Chymotrypsinogen A was purchased from Sigma and used without further purification. Following the experimental protocol ^[14], 2-5 mg of the protein were added to 1 mL of a buffer solution, 10 mM glycine and 10 mM NaCl, at pH = 3.0. The solution was diluted with buffer in order to have a protein concentration ≈ 0.3 mg/mL. The absorption spectra of the α-Chymotrypsinogen A solution were measured as a function of the temperature on a Cary 5 spectrophotometer in the spectral range 320 – 240 nm. The temperature of the absorption cell was controlled with a closed-circuit Haake L thermostat working between room temperature and 75°C.

3. Results

The spectral data acquired through absorption measurements in the range 20°C - 75°C have been collected and the equilibrium constants between 50°C and 60°C have been calculated. In Figure 2 the temperature dependence of UV spectra of α-Chymotrypsinogen A at selected temperatures is reported, showing the intensity decrease of the band maximum at ≈ 281 nm and the isosbestic point at ≈ 269 nm.

Figure 2. UV absorption spectra of α-Chymotrypsinogen A in aqueous dilute solution at selected temperatures between 20^oC and 75^oC. Optical path, 1 cm. The arrow indicates the spectral sequence with increasing temperature

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As reported ^{[10, 14]}, the equilibrium constants K are determined from the observed absorbance maxima A(T) and the corresponding absorbances A_N(T) and A_D(T) of the native and denatured protein, respectively, as [A_N(T) – A(T)]/[A(T) – A_D(T)]. In the inset of Figure 3, the A_N(T) and A_D(T) values are obtained by linear regression extending the intensity behaviour with temperature of the native and denatured form in the temperature interval, ≈ 10°C, where the thermal unfolding is completed. The equilibrium constants are plotted as a function of the temperature in Figure 3.

Figure 3. Equilibrium constants K calculated from the observed absorbance maxima A(T) and the corresponding absorbances A_N(T) and A_D(T) as explained in the text. In the inset A_N(T) and A_D(T) values are obtained in the 50°C – 60°C temperature range by linear regression

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4. Discussion

The denaturation process of α-Chymotrypsinogen A, assumed to occur in the solution volume V at temperature T and pressure p = 1 bar, may be discussed by means of the following two-state model ^[14]

(1)

where N and D are the native and denatured forms of the protein, respectively, with unit stoichiometric coefficients. At constant temperature and pressure the reaction solution goes spontaneously toward the equilibrium state, where reactant (N) and product (D) concentrations are unaltered in time. As the solution approaches equilibrium, the Gibbs free energy G decreases and at equilibrium is minimum or, in other words, the Gibbs free energy of the reactant, G_N, and of the product, G_D, are equal. The effect on G of the irreversible changes occurring in the solution due to denaturation is described by means of the chemical potentials of N and D. At mechanical and thermal equilibrium the free energy G is given for any composition of the two species by the expression ^{[1, 7, 21]}

(2)

where n_N, n_D are the number of moles and μ_N, μ_D are the chemical potentials. Having initially n₀moles of the native species and being unitary the stoichiometric coefficients, n_N and n_D may be expressed in terms of the degree of process advancement ξ as

(3)

where ξ units are mol. As the composition of the solution varies, the chemical potentials change accordingly, depending on the activity of the native and denatured forms, by

(4)

where μ^o_i is the standard potential at temperature T and pressure p = 1 bar and a_i is the activity of the i-th species. For a sufficiently dilute solution, a_i may be approximated ^[5] to the ratio of molar concentrations c_i/c^o, with c^o = 1M. Substituting eqs. (3) and (4) into eq. (2) and grouping the RT, the result is

(5)

where G₀ = n_0∙{[μ^o_N + RT ln[(n₀/V)/c^o]} is the free energy of n₀ moles of the native species in the volume V, ΔG^o = μ^o_D - μ^o_N is the so-called standard free energy of denaturation, i.e., the rate of change of G with respect to ξ ^{[22, 23, 24]}, and RT{(n₀-ξ) ln[(n₀ - ξ)/n₀] + ξ ln(ξ/n₀)} is the free energy of mixing of the two species. ^[21] Eq. (5) extends the expression valid for ideal gas mixtures ^{[1, 7, 21]} to the case of dissolved species in dilute solution. The G₀ term of this equation is independent of ξ. The linear term, ξΔG^o, would lead, in absence of the third term to the total conversion from the species of highest to that of lowest chemical potential. The equilibrium state, where both N and D are present, is reached since the third term further lowers G with respect to the linear profile. Alternatively, the irreversible course of the reaction toward equilibrium may be followed considering (∂G/∂ξ)_T,p, i.e., by differentiating G with respect to ξ at constant p and T.

Figure 4. Gibbs free-energy of protein denaturation (G-G₀, left) and derivative (∂G/∂ξ)_T,p, (right) as a function of the degree of advancement ξ at the denaturation temperature T_d (middle), below (upper) and above (lower) T_d according to eqs. (5) and (6), respectively. The calculation refers to the α-Chymotrypsinogen A denaturation with T_d = (273+55) K, T_below = (T_d – 2) K, T_upper = (T_d+ 2) K and with the equilibrium constant values determined according to the Results Section. For the sake of simplicity n₀has been taken equal to 1 mol. The equilibrium state is specified by the (G-G₀) minimum on the left and by the zero value of the (∂G/∂ξ)_T,p derivative (the horizontal dashed lines) on the right. The equilibrium composition of the D form is indicated in correspondence on the two abscissa scales

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The well-known van’t Hoff isotherm is easily obtained from eq. (5)

(6)

where ΔG^o=− RT ln[ξ_eq/(n₀ – ξ_eq)] = − RT ln K is determined at equilibrium through the condition (∂G/∂ξ)_T,p = 0.

4.1. Classroom Activity

(G – G₀) and (∂G/∂ξ)_T,p vs. ξ graphs are generated in Figure 4 according to eqs. (5) and (6), respectively, starting with n₀ = 1 mol of native protein in the solution volume V, at three (or more) temperatures; one is the denaturation temperature T_d and the remaining are distributed below and above T_d. The denaturation temperature T_d is defined ^[14] as the temperature at which protein at equilibrium is half native and half denatured, i.e., ξ_eq = n₀ – ξ_eq and c_N,eq = c_D,eq. At this temperature ΔG^o = 0 and K = 1. The volume V is assumed sufficiently large so that activities may be approximated to concentrations for both protein forms. Typically, data are stored on a few columns of a worksheet:

1. in the first column, the X-axis, the ξ values between 0 and 1 mol are given with 0.05 step;

2. in the second and third column the (G – G₀) and (∂G/∂ξ)_T,p values are tabulated according to eqs. (5) and (6), respectively.

Alternatively, the total (G – G₀) result is resolved into two contributions, the linear term in ξ and the nonlinear (logarithmic) term. Then the (G – G₀) values fill the fourth column using the second column for the linear term in ξ and the third the logarithmic term. Splitting (G - G₀) into two contributions clarifies a point, not often mentioned in textbooks, i.e., the purely chemical term ξ∆G^o would lead to the complete conversion from one protein form to the other and only mixing is responsible for the occurrence of the two-component equilibrium solution. ^[3]. The three graphs of Figure 4 result from the equilibrium constants of Figure 3 on α-Chymotrypsinogen A, i.e., T_d = (273+55) K, K = 1; T_below = (T_d – 2) K, K = 1/3; T_upper = (T_d+ 2) K, K = 3. Since denaturation is known to be an endothermic (∆H^o > 0) and disordering (∆S^o > 0) process ^{[3, 5]}, at T < T_d ΔG^o > 0 (K < 1) and at T > T_d ΔG^o < 0 (K > 1). As the temperature increases, the composition at equilibrium shifts from native-rich to denatured-rich, crossing the c_N,eq = c_D,eq equality at the denaturation temperature. This can be seen from Figure 4 looking at the ξ value of the (G – G₀) minimum (on the left) or of the zero value of the (∂G/∂ξ)_T,p derivative (on the right).

4.2. Le Châtelier Principle Exemplified

For the two-state model of eq. (1), it is rewarding also to view the equilibrium constant K = c_D,eq/c_N,eq as the proportionality factor relating the concentrations of the native and denatured species at equilibrium, i.e., c_D,eq = K · c_N,eq. Therefore, in a plot c_D vs. c_N the equilibrium states lie on a straight line graph of slope K, which in Figure 5 is 1 for the particular case presented here at the protein’s denaturation temperature.

Figure 5. Effect of concentration change on the protein denaturation equilibrium at T_d. Lines AA’ and AA’’, additions of native and denatured protein, respectively; lines A’B and A’’B, paths to reach the new equilibrium state. The temperature is kept constant during the two additions

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This diagram offers a vivid application of the Le Châtelier principle. ^[8] All the equilibrium states of the two-component solution are points on the straight line; any other point does not correspond to an equilibrium state. What happens when 2∙10^-5 moles of native protein are added to the dilute equilibrium solution at point A (c_N,A= c_D,A = 1∙10^-5 M, see Figure 5)? Soon after the addition, the system reaches point A’ which, not being on the straight line, corresponds to a non-equilibrium state. The system shifts spontaneously toward equilibrium minimizing, according to the Le Châtelier principle, the effect of the disturbance. Since for each mole of native protein that unfolds at the denaturation temperature T_d, one mole of denatured protein appears at equilibrium, and the concentration of N decreases by the same amount that the concentration of D increases (i.e., -∆c_N = +∆c_D), the path moving toward equilibrium is a straight line of slope (-1) equal to +∆c_D/-∆c_N, intersecting the equilibrium line at point B. It is graphically evident from Figure 5 that the final concentration of native protein, (c_N,A+ c_N,A’)/2, is greater than the initial concentration, c_N,A = 1∙10^-5 M, but less than c_N,A’ = 3∙10^-5 M, the concentration that would have been if no response from the system had occurred. It is equally clear from Figure 5 that, given the same addition of denatured protein (point A’’), the system will move to the same equilibrium state, point B, reached before and that the changes in concentration for native and denatured forms are the same but of opposite signs. In the second case the denatured protein is assumed to undergo renaturation to form more native protein, which may not be necessarily true for a large part of actual proteins. It is left to the students as a geometrical exercise to discuss the more general case, K ≠ 1.

The effect of a temperature change on the denaturation equilibrium can be also visualized on the graphical plot of Figure 6.

Figure 6. Effect of temperature change on the protein denaturation equilibrium at T_d. The equilibrium states at T_d and T’ (with T’ > T_d) are represented on the two straight lines, respectively. For the denaturation process (ΔH⁰ > 0) the temperature increase from T_d to T’ corresponds to the counterclockwise rotation

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