The Sceptical Chymist | Chemiotics: Why should a protein have one shape?

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Well of course they don’t, but the proteins we know the most about (because they can be crystallized and their structure determined by X-ray diffraction) do have a shape. Sperm whale myoglobin, the first protein to have its 3-dimensional structure determined, showed that this couldn’t be the whole story. Sperm whales (air breathing mammals after all) use their myoglobin to carry oxygen during their hour-long dives down to 1000 meters. Kendrew and Perutz’s crystal structure showed no way for oxygen to find its way in to the embedded porphyrin ring. Amazingly, the 153 amino acids of myoglobin must themselves breathe to let the oxygen in.

All it takes to denature (seriously change its tertiary structure so it is no longer functional) a protein of 100 amino acids is 10 kcal/mole (Voet & Voet – Biochemistry 3rd Edition p. 258). That’s two hydrogen bonds – not much.

Sight your eye at the alpha carbon of one of the amino acids of this protein, looking toward the carbonyl carbon. There are three conformational energy minima the carbonyl can adopt. That’s potentially 3^99 = 10^48 conformations (clearly an overestimate because of self intersection, but still, a huge number). Yet to be crystallizable, this protein must choose just one of them, and it must be lower in energy by 2 hydrogen bonds than all the rest.

In addition, to get to this single structure, the protein can’t possibly sample all the conformations available to it. The rotation barrier of ethane is 12 kJ/mole and a barrier of 73 kJ/mole allows a rotation rate of 1 per second, and every 6 kJ changes the barrier by a factor of 10 at 25 deg C (Clayden et al. Organic Chemistry pp. 450-1). So the maximum rate of rotation of ethane is 10^11 per second (at a body temperature of around 37 deg C) rather than 10^10 at 25 deg C. This is clearly an upper bound on the rotation rate as the mass attached to the alpha carbons of a protein will make the rotation far slower, but let it pass (that’s why I chose ethane in the first place). That’s 10^37 seconds to sample the conformations available, far longer than the age of the universe. This is the Levinthal paradox.

So for the crystallizable proteins (all of biological interest so far) one conformation out of all those available must be more stable (but only by two hydrogen bonds) than all the rest, and the particular conformation must be findable quickly (or we’d all be dead).

How likely is this for a ‘random’ sequence of amino acids. We’ll probably never know (but we might if we’re lucky). This is the subject of the next post…

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