Bonnet Transformation: Explaining the Alpha-Helix to Beta Barrel Transition

[akoska]

Hello,

Can anyone explain to me what this is? I can't seem to find any good references on this.

I'm looking into protein transformations from a helix structure to a catenoid structure through the Bonnet transformation (ie, alpha-helix to beta barrel transition)

 

[Chris Hillman]

Well, you probably don't mean the first thing I thought of, the Gauss-Bonnet map.

You might be looking for a local diffeomorphism between two surfaces (as Riemannian two manifolds) in $E^3$, namely the tangent developable surface of a helix and the surface of rotation generated by a catenoid curve. If so, see Lectures on classical differential geometry, by Dirk Struik, available as a Dover reprint.

Are you thinking of proteins as something like two-dimensional ribbons by any chance? If so, you might be looking for a diffeotopy of ribbons, which might be related to the diffeomorphism between the two surfaces I mentioned.
Unfortunately, I don't seem to be familiar with this "barrel", although I've heard of Pauling's alpha helix structure.

(Diffeotopy versus diffeomorphism: see http://planetmath.org/encyclopedia/Diffeotopy.html and create your very "own" [hah!] WP article)

Additional: searching the arXiv, these abstracts suggest that your Bonnet transformation does indeed deform one minimal surface into another, so my guess about the two surfaces (which are both minimal surfaces) is probably about right.

http://arxiv.org/abs/cond-mat/0605617

http://arxiv.org/abs/cond-mat/0102466

http://arxiv.org/abs/cond-mat/0109512