Characterize protein stability curves as a function of temperature. In the lecture, we derived an expression that describes the free energy unfolding of proteins as a function of temperature. This expression includes the unfolding/melting temperature of the protein (Tm), the enthalpy of unfolding (?Hu(Tm)), and the difference in heat capacities between the folded and unfolded proteins. These parameters can be obtained from standard differential scanning calorimetry experiments. For example, ?G(T) = ?H(Tm) (1-T/Tm) + ?Cp [T-Tm/(Tm)]. The main assumption underlying this expression is that ?Cp can be considered constant, independent of temperature. Here, we assume a protein with a melting/unfolding temperature of 78°C, an enthalpy of unfolding of 400 kJ/mol, and a ?Cp of 9 kJ/(mol·K). Determine the temperature of maximum stability for the protein, i.e., the temperature at which the unfolding free energy reaches its maximum. Provide your answer with the appropriate number of significant figures. The margin of error is 29%. Additionally, estimate the number of unfolded proteins per million at the temperature of maximum stability. Provide your answer with the appropriate number of significant figures. The margin of error is 29%. Lastly, consider cold denaturation. We observed that the protein stability curve described by the derived expression indicates that the native fold of proteins is only stable within a temperature window. This means that denaturation not only occurs at high temperatures but also at low temperatures (although the latter typically lies below the freezing temperature of water). Estimate the cold denaturation temperature of the proteins in this example. Note: Look into the Newton-Raphson method (https://en.wikipedia.org/wiki/Newton%27s_method) to numerically determine the roots of the function f(x) which satisfy f(x)=0. Choose a starting temperature lower than the temperature of stability, and iterate until you obtain a converged numerical result. Provide your answer with the appropriate number of significant figures. The margin of error is 10%.