Measuring the free energy of protein folding

Protein folding is, of course, a very interesting problem for biochemists. An important question is, how much more stable is the folded protein than the unfolded one? But how does one go about getting the answer to such a question? We need to find some conditions under which we can measure an equilibrium between the folded and unfolded forms. A simple model might be that only two forms exist (that is, that there are no significantly populated intermediates). But how can we detect them? All proteins absorb UV light at a wavelength of about 280 nm. This is because of the absorbances of the aromatic amino acids tyrosine, phenylalanine and tryptophan. This absorbance is generally not very different for folded and unfolded forms. However, these amino acids, especially tryptophan, also fluoresce. It is expected that the fluorescence will depend significantly on the chemical environment of the amino acid side chain. Most of these aromatic amino acids tend to be buried in the protein’s hydrophobic core, where fluorescence is quenched, perhaps by alternate pathways for return to the ground state, perhaps by nonradiative energy transfer to another chromophore. Either way, it would not be surprising to find that the two forms showed different fluorescence spectra. We might expect that it would be easy to determine the fluorescence output of the folded form, since that is the form in which, we hope, the protein is isolated and stored. Thus, the difference between the fluorescence of a given sample, and the fluorescence of the folded or native protein, should be related in some way to the proportion of unfolded protein.

Let us assume, then, that the completely unfolded protein has fluorescence F_u and that the native form has fluorescence F_f. Then a given sample containing a mixture of the two would have fluorescence,

F = f_f * F_f + (1 – f_f) * F_u.

Solving for f_f, the fraction in the folded form, we get

f_f = (F – F_u)/(F_f – F_u).

If F_f < F_u it is probably more convenient to solve for the fraction unfolded,

f_u = (F – F_f) / (F_u – F_f)

The equilibrium constant for denaturation, K_D, then, is given by:

K_D = f_u / f_f = f_u / (1 – f_u), and DG_D = -RT ln K_D

It is certainly possible, if not probable, that we will be unable to get complete unfolding under achievable conditions. Therefore, we would like to have some method for estimating how close we are to complete unfolding (and for that matter, whether our “native” protein has any unfolded portion). The best way to do this is to model the unfolding process, fit our flourescence data to the model, and calculate the best fit values for F_f and F_u. The simplest model is the Hill model, as used with hemoglobin and other cooperative proteins. Folding is considered to be a cooperative process (hence, few intermediates). The mathematical form for such a model is

F = F_f + ((F_u - F_f)*x^b)/(c^b + x^b)

where x is the concentration of whatever we are using as a denaturant (usually urea or guanidine HCl), c is the concentration where we have equal amounts of folded and unfolded protein and b is the exponent necessary to fit the data (the Hill coefficient). By fitting the fluorescence data to the above equation, we can determine values for F_f, F_u, c and b. Of course, we can also use the linear form of the Hill equation and fit to that.

All the fluorescence data together, then, give us values for F_f and F_u. We can use those values to obtain DG_D at the different concentrations of denaturant. These values can then be extrapolated to a concentration of 0, or the absence of denaturant, to give us an estimate for the difference in energy between folded and unfolded forms. The actual plot of the DG’s will tell us the form the extrapolation should take.

Urea is soluble to about 10 M and guanidine HCl to about 8 M, so these are the highest concentrations of denaturant we can reach. Clearly it will be best to buffer the solutions to maintain constant pH; you should try at least 3 different pH's to begin. Maintaining constant osmoticum or ionic strength would be more difficult, but should be considered. Clearly, it will not be useful to measure the fluorescence until equilibrium has been reached. This can be determined with just a few samples spanning a range of denaturant concentrations and pH. You can assume practical equilibrium when the fluorescence stops changing.

What concentration of protein should be used? This depends on the fluorescence signal of the protein, but typically a protein concentration of 0.1 mg/mL should be sufficient. This would have an absorbance of about 0.1. Fluorescence is usually more sensitive than absorbance so we should get plenty of signal. The concentration may need to be adjusted up or down for any particular protein.

Which protein should be used? Something that is inexpensive and readily available. We have lysozyme, ribonuclease and chymotrypsin, for instance.

References

F. Ahmad, C.C. Bigelow (1982) Estimation of the free energy of stabilization of ribonuclease A, lysozyme, a-lactalbumin and myoglobin, J. Biol. Chem. 257, 12935-38.