It's come undone

Classical Mechanics Level 3

Proteins are molecules responsible for catalyzing chemical reactions, regulating gene expression, sensing changes in the extracellular environment, giving superstructure to the genome, and many other important tasks. They take the form of long chains that fold into compact structures by seeking out shapes that minimize the energy of self-interaction.

Consider a simple protein that has two states, completely folded, and completely unfolded, which have a free energy gap of $\Delta = 3.74$ kJ/mole .

You have an ensemble of several million copies of this protein dissolved in buffer. At temperature $T = 37^{\circ}$ C, what fraction of the proteins are folded?

Note: You may wish to read about partition functions .

The answer is 0.81.

1 solution

A K
Apr 26, 2014

The equilibrium constant $K$ is given by the formula $\Delta G = -RTlnK$ which can be re-arranged to give $K = e^{\frac{\Delta G}{-RT}}$ .

Substituting in $\Delta G = 3740Jmol^{-1}, R = 8.3145JK^{-1}mol^{-1}$ and $T = 37 + 273.15 = 310.15K$ yields an answer of:

$K = 0.2344953605$ .

Therefore: $\frac{X}{Y} = 0.2344953605$ where $X$ = proportion of folded proteins and $Y$ = proportion of unfolded proteins.

We can also say that $X + Y = 1$ since the total fraction must be 1.

Substituting in $X = 1 - Y$ yields $\frac{1 - Y}{Y} = 0.2344953605$ .

Re-arranging and solving gives $Y = 0.1899524032$ .

Therefore, $X = 1 - 0.1899524032 = \boxed{0.8100475968}$ .

It's come undone

The answer is 0.81.

1 solution

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