HOW TO CALCULATE DGO FROM
PROTEIN DENATURATION CURVES:
Multiple methods can be used to investigate the denaturation of a protein. These include UV, fluorescence,
CD, and viscosity measurement. In all these methods the dependent variable (y) is measured as a function of the independent
variable, which is often temperature (for thermal denaturation curves) or denaturant (such as urea, guanidine hydrochloride)
concentration. From these curves we would like to calculate the standard free energy of folding or unfolding (DGO) for the protein (for the reaction N <=> D).
The denaturation curves usually show a sigmoidal, cooperative transition from the native to the denatured state. The dependent
variable can also be normalized to show fractional denaturation (fD). Example of these curves are shown in the
CURVES FOR PROTEINS
For each curve, the value of y (either A280, Fluorescence intensity, viscosity, etc) can be
thought of as the sum contributed by the native state and from the denatured state, which are present in different fractional
concentrations from 0 - 1. Hence the following equation should be reasonably intuitive.
1. y = fN yN + fD yD where fN is the fraction
native and yN is the contribution to the dependent variable y from the native state, and fD is the fraction
denatured and yD is the contribution to the dependent variable y from the denatured state. Conservation give equation
2. 1 = fN + fD or fN = 1 - fD. Substituting 2 into
- y = ( 1 - fD) yN + fD yD .
- y = yN - fDyN + fD yD .
- y - yN = fD (yD - yN). Rearranging this equation gives
3. fD = (y - yN) / (yD - yN)
Notice the right hand side of the equations contains variables that are easily measured.
By substituting 2 and 3 into the expression for the equilibrium constant for the reaction N<=> D
Keq = [D]eq/[N]eq = fD/fN or
4. Keq = fD/(1 - fD) and
5. DGo = -RTlnKeq = -RTln[ fD/(1
Remember that DGO (and hence Keq)
depends only on the intrinsic stability of the native vs denatured state for a given set of conditions. They vary as
a function of temperature and solvent conditions. At low temperature and urea/guanidine HCl concentration, the native
state is favored, and for the N <=> D transition, DGO < 0. At high temperature and urea/guanidine HCl concentration, the denatured
state is favored, and DGO >
0. tAt some value of temperature or urea/guanidine concentration, both the native
and denatured state would be equally favored. At this point, Keq = 1 and DGO = 0. cIf temperature is the denaturing agent, the temperature at
this point is called the melting point (Tm) of the protein, which is analogous to the Tm (in the heat capacity vs temperature
graphs) for the gel to liquid crystalline phase transition of phospholipid vesicles.
Figure: gel to liquid crystalline phase transition of phospholipid vesicles
Ordinarily, at a temperature much below the Tm for the protein, so little of the protein would be in the
D state that it would be extremely difficult to determine the concentration of the protein in the D state.
Hence it would be difficult to determine the Keq or DGo
for the reaction N <=> D. However, in the range where the protein denatures (either with urea or increasing temperature),
it is possible to measure fD/fN.and hence D Go
at each urea or temperature. A plot of DGo vs [urea] is linear,
and by extrapolating the line to [urea] = 0, the D Gow,
the free energy for denaturation of the protein in water, can be estimated. (See above figure.) The same method is not usually
used when increasing temperature is the denaturant since D G0
for protein folding usually changes with T (remember the D Cp for protein