Big Numbers

Representation Theory for cryo-EM April 14, 2010

I’m going to a seminar on representation theory taught by Shamgar Gurevich, in particular as it applies to the cryo-electron microscopy problem.

Cryo-EM is actually what I’ve been spending this whole year on, for my senior thesis. The short story is this. One method of determining protein structure is to encase the proteins in a thin film of ice, stick them in an electron microscope, and take pictures. (Really they aren’t photographs but measurements of electrical potential; but this gives a picture of the density of a cross section of the molecule.) The trouble is, these are two-dimensional images, and we want a three-dimensional model.

It would be straightforward to do this if we knew the direction the images were taken at; we could reconstruct the 3-d structure using the Fourier Projection Slice Theorem, the same way tomography works. But we don’t know the angles. Another hurdle is that experimentally, the images are extremely noisy, with an SNR of about 1/60. So, we need mathematics to come to the rescue. It turns out to be very pretty math, involving graph theory, the eigenvalues of large sparse matrices, and some random matrix theory.
For more thorough explanations, check out some of my advisor’s recent papers.

Anyhow, all that is by way of introduction. Cryo-EM is also intimately related to representation theory, since it deals with the symmetries of groups. For example, if the molecule happens to have internal symmetries, and many such molecules do, then the simplest version of the reconstruction algorithm falls apart. We need representation theory to explain generalizations of the problem.

So I’m learning. At the moment we’re only working with finite groups; today we proved Schur’s Lemma, which gives an important relationship — a little like orthogonality, to my mind — between irreducible finite-dimensional representations of a group. If the two representations are not isomorphic then there is no linear map between them that commutes with the action of the group; if they are isomorphic, then the linear map is a scalar operator.

For us, the motivation is that this allows us to diagonalize matrices. If an operator T from V to V is diagonalizable, and V has a group representation, and T commutes with the representation, then T preserves all the irreducible subrepresentations. This, and Schur’s lemma, allows us to conclude that T restricted to a subrepresentation is some (complex) eigenvalue multiple of the identity on that subrepresentation.

We like diagonalizing big matrices for cryo-EM because a key method is to make enormous matrices based on the correlations between images, and determine the viewing directions of the images from the eigenvectors of those matrices. These matrices are sparse, which makes computation easier, but they have to be very large to achieve accuracy. So methods of diagonalization — and methods of identifying irreducible subrepresentations, a related problem — are invaluable.