Unfolded Protein Equilibrium (BIO)

February 29, 2020

In this example we are given the total protein
concentration in solution, it’s 10 micromolar, and the question we ask is what fraction
of this protein is unfolded at 25 degree C? where we are given the delta G of unfolding
and this value then of course is at 25 degrees C. So what we are looking at is, there is
a folded version of the protein, and it is in equilibrium with an unfolded version, and
the question really is what fraction or what is the concentration that is unfolded. So
we will use the notation were N represents the concentration of folded, and U represents
the concentration of unfolded , and the problem says that the sum of the two types is the
concentration of 10 micromolar. So ten to the minus six molar concentration. So this
is an equilibrium process. We are given delta G, which means we can calculate the equilibrium
constant, and then we’re going to say the equilibrium constant is equal to the concentration of the unfolded. The product over the folded,
and we can use concentrations here, These are low concentrations so we are treating this as
ideal. The equilibrium constant is equal to the Gibbs free energy change divided by the
gas constant and the temperature. So we can calculate the equilibrium constant because
we are given delta G, minus sign. This is 41000 Joules per mole. the gas constant 8.314 Joules per mole per Kelvin. So this temperature must be absolute temperature 273 plus 25 so 298 Kelvin. This is equal to then e to the minus 16.55. So if we do that exponential we get a very small equilibrium constant not
surprising we have a positive delta G that is significant in size, and so our equilibrium
calculation is just that U over N is 6.5 times ten to the minus eight Essentially we can stop here. Essentially
what this says is that essentially there is no unfolded protein. Such a small fraction
of the total if we wanted to calculate the concentration, and then since N is pretty close to
ten to the minus six molar then U is equal to ten to the minus six times 6.5 times ten to the minus eight. So our concentration is 6.5 times ten to the minus fourteen molar. This is the concentration of the unfolded protein, and so this really is insignificant, and we conclude for these conditions, this temperature, all the protein is folded.

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