APPLIED MATH SEMINAR

Speaker: Amit Singer, Princeton
Title: Mathematical Theory of Cryo-Electron Microscopy
When/where: Tuesday, February 2nd, 4:15PM, LOM 205

Abstract: (see attached pdf too)
The importance of determining three dimensional macromolecular structures for
large biological molecules was recognized by the Nobel Prize in Chemistry
awarded this year to V. Ramakrishnan, T. Steitz and A. Yonath for studies of
the structure and function of the ribosome. The standard procedure for
structure determination of large molecules is X-ray crystallography, where the
challenge is often more in the crystallization itself than in the
interpretation of the X-ray results, since many large proteins have so far
withstood all attempts to crystallize them.

In cryo-EM, an alternative to X-ray crystallography, the sample of
macromolecules is rapidly frozen in an ice layer so thin that their tomographic
projections taken by the electron microscope are typically disjoint. The cryo-EM
imaging process produces a large collection of tomographic projections of the
same molecule, corresponding to different and unknown projection orientations.
The goal is to reconstruct the 3D structure of the molecule from such unlabeled
2D projection images, where data sets typically range from 10^4 to 10^5
projection images whose size is roughly $100 \times 100$ pixels.

I will present a new algorithm for finding the unknown imaging directions of all
projections. Compared with existing algorithms, the advantages of the algorithm
are five-fold: first, it has a small estimation error even for images of very
low signal-to-noise ratio (SNR); second, the algorithm is extremely fast, as it
involves only the computation of a few top eigenvectors of a specially designed
symmetric matrix; third, it is non-sequential and uses the information in all
images at once; fourth, it is amenable to rigorous mathematical analysis using
representation theory of the rotation group SO(3) and random matrix theory;
finally, the algorithm is optimal in the sense that it reaches the information
theoretic Shannon bound up to a constant.

Time permitting, I will discuss generalizations of the algorithm and its
mathematical analysis to other applications in computer vision, structural
biology and dimensionality reduction.