Question

This problem will relate some of the ideas of microstates, ensemble, statisticalmechanical entropy, and our ideal model of many identical oscillators, to intuitive notions of entropy and disorder in a very complex system – a protein. A protein is basically a long chain of amino acid residues that folds into a regular shape and performs a huge variety of biological functions. Biochemistry and biology students will already know a lot about proteins, you may know a lot after having taken organic chemistry. If you don’t know anything about them, it’s a good idea to learn some. The protein in its folded state is called the “native” state. There are beautiful pictures of folded proteins that are easily accessible. Google [ protein folded unfolded ] then go to “images” and you will find many pictures, e.g. the top row on the left and right are good. Intuitively, one says the properly folded structure is “ordered” and therefore low entropy, the unfolded structure is “disordered” and therefore high entropy. In terms of statistical mechanics, if we think in the microcanonical ensemble (ignore any heat bath), there should be an ensemble W_f for the folded condition and W_u for the unfolded condition. Then S_f = lnW_f and S_u = lnW_u. What does this have to do with order anddisorder?

Think in terms of an oscillator model and its degeneracies. The protein with N atoms in either its folded or unfolded state has 3N vibrational modes i.e. consists of 3N oscillators. The oscillators do not have identical frequencies by any stretch of the imagination. But think of each state as having some “average frequency” ωf or ωu, and then imagine the 3N oscillators to all be identical. Each state, folded or unfolded has the same energy E in the microcanonical ensemble.

Now, try to argue, qualitatively but as persuasively as possible why the “ordered” protein has a much smaller ensemble Wf than the unfolded protein with ensemble Wu, and therefore a lower entropy.

(Hint: the folded protein sort of “snaps together” with non-covalent bonds. Think about

what this does to the average frequency. (Hint: see the frequency equation for an oscillator

in the lecture notes. Think about the bonding in the folded protein in connection with the

oscillator force constant in the frequency expression. Think about what this means for the

number of quanta. Think about what this means when you plug into the combinatorial

formula for the number of microstates in the oscillator model

Answer 1