Lysozyme (MW = 14.313 kDa) undergoes a phase transition from a native folded (N) to unfolded state (U) that can be considered a reversible reaction, N_(aq) ⇆ U_(aq), where the subscript (aq) indicates that the protein is in an aqueous solution. At high temperature the protein is in state U and at low temperature, it is in state N. The melting temperature (T_m) is where the concentrations are equal. The melting temperature can be changed by pH, ionic strength (added salt), denaturant (solvents), or pressure. The results of problem 17.24 apply. The protein unfolding has been studied by Cooper, et al. The heat to unfold 1 mg/mL of protein in 2 mL of pH = 4 solution at T_m = 78°C is 0.0755 J, ΔCP = 6.3 kJ/mol-K. The ΔV = –40 mL/mol.

(a) What is the Gibbs energy change for the unfolding at the melting temperature?

(b) Show that at the melting temperature at pH = 4, ΔH = 540 kJ/mol and ΔS = 1.54 kJ/ mol.

(c) Evaluate the contributions to ΔG at 23°C and pH = 4. Is the process driven by enthalpy or entropy under these conditions?

(d) Show that the relation for [N]/[U] is

For a solution of overall concentration 1 mg/mL, plot [U] as a function of temperature for 20 ≤ T ≤ 110 °C, and provide a tabular summary of [U] at 60°C and 90°C.

(e) When pH is varied, the net charges on the protein change, resulting in different T_m and ΔH. In the range 20 ≤ T_m ≤ 78 °C, the relation is ΔH(kJ/mol) = 48.6 + 6.3 T_m(°C). Derive the relation between ΔS and T_m.

(f) At pH = 1, T_m = 43°C. Provide a plot of plot [U] as a function of temperature for 20 ≤ T ≤ 110 °C, and provide a tabular summary of [U] at 25°C and 55°C.

(g) (advanced) Lysozyme has four disulfide bonds that are important in the folding behavior. The authors created a mutant protein with only three disulfide bonds. The relation in (e) still holds. At pH = 2.5, T_m = 23°C for the mutant. Evaluate the contributions of enthalpy and entropy to the folding.

(h) (advanced) Provide an interpretation for the sign of ΔV. At pH = 4, how much pressure is needed to lower the melting temperature 5°C to 73°C?

Data from problem 17.24

The enthalpy of reaction for many biological reactions and surfactants is strongly temperature- dependent, but instead of using full heat capacities, the temperature dependence can be characterized by differences in heat capacities ΔC_P = C_P,prod – C_P,react, typically assumed to be independent of temperature.

The heat of reaction is represented by ΔH = ΔH_R + ΔC_P(T – T_R). Show that

Provide an equation consistent with part (a) for the temperature dependence of the entropy of reaction.

Transcribed Image Text:

Խ[U] [M] -ACP Тр R T [(¹-4) - ()]-[¹-4] 1- Inf T TRASR AS₂ TR Tr R