In this example we're looking at a protein that's being considered as a potential drug
and so some measurements were made of it's denaturation at 25 degrees C, we found that
the protein was 80 percent folded when the urea, the denaturant, had a concentration
of 1.5 molar, and it was only 35 percent folded when the urea concentration was 4 molar. So
we want to use these data to make an estimate of what is the stability of the protein in
the native condition, namely when there's no urea present. By calculating delta G of
unfolding in pure water, and we'll make some assumptions about the relation between delta
G and the urea concentration and these assumptions then involve a linear plot and so we want
to determine a term that's referred to as the m-value of the protein, and this is absolute
value of the slope of delta G unfolding versus this urea concentration. So we're going to
use the notation which will indicate this denaturant, it's concentration. So this is going to be
the concentration of the urea, the denaturant and we have condition where the concentration
is 1.5 molar then it's 80 percent folded and when the concentration is 4 molar then it's
only 35 percent folded and so what we're going to do in this calculation is determine what's
delta G of unfolding at these two conditions and then we'll assume this linear dependence
to estimate delta G when there's no urea present, and to calculate delta G we're going to take
advantage of in thermodynamics that delta G, in this case delta G of unfolding, is related
to gas constant, absolute temperature and the log of the equilibrium constant. Let's
look at essentially the reaction that we're dealing with here. And so what we're doing,
we have the natural state, the folded version of the protein, and it can go to the unfolded
version, and we're this notation to indicate, in this case, relative concentrations, we
don't know the actual concentration of the protein, all we know is the fraction that's
folded. And so in equilibrium constants, since these are in equilibrium, the equilibrium
constant would then be products over reactants, that's the equilibrium constant then in this
delta G relation, so delta G of unfolding is minus RT the log of concentration of unfolded,
concentration of folded. And so we have the fractions, so this means for our condition
D1, so concentration D1, 1.5 molar, this ratio U over N, it's 80 percent folded, 20 percent
unfolded, or 0.2, and for the higher urea concentration of 4 molar, then this ratio,
65 percent is unfolded and 35 percent is folded. So we're going to substitute these numbers
into our equilibrium expression, so I've started the substitution here, so this is K1 and this
is K2, so our relation between Gibbs free energy change for unfolding, gas constant,
absolute temperature, the log of the equilibrium constant, put in the units and this ratio
of 0.2 over 0.8. So delta G1 is 820 calories per mole, if we do the calculation, or 0.82
kilocalories per mole so we now can do the exact same calculation for delta G2, so I've
substituted in the numbers the exact same way, we get delta G2, notice delta G1 is positive
but delta G2 is negative, and so now we're assuming there's a linear relationship of
this delta G of unfolding and the concentration of urea, and so what I've done is made an
excel plot just to make it clear, here's delta G of unfolding, kcals per mole, here's the
concentration of the urea, straight line, and essentially we want to extend this straight
line to the native condition so the native condition where the concentration is zero,
and so we can do a least squares fit to this line, just a linear regression, and if we
do that we get an equation that delta G of unfolding is related to the slope of the line,
so this is the slope, times the concentration of the urea plus the intercept, 1.531, so
this is the Y-axis intercept. So the question the problem statement is asking what's the
value when the urea concentration is zero, the native condition, well that's just the
intercept, that's delta G of unfolding is, and I'll just write two significant figures
since that's all we can justify from the data, so this is the native delta G and we can see
it's a positive value but go back and substitute in the same way we have just done, so we can
calculate this ratio to determine stability and I won't present the numbers here but it's
a straightforward substitution of the exact same calculation and this ratio ends up being
0.079, which means this U is 0.073 and N then is 0.927, or approximately 93 percent folded
at the conditions of zero urea concentration, and then the last question asks for the absolute
value of the slope which is referred to as the m-value, so the m-value of the absolute
value, again 2 significant figures, units, kilocalories per mole and then molar so this
is the m-value for this protein, and then so we've now calculated the information about
stability of the protein and the presence and absence of denaturant and then the value
that's used in characterizing the stability as a function of the urea concentration.