# Protein Stability (BIO)

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.