Abstract: In this talk we present a reconstruction algorithm for the 3D atomic structure of randomly oriented proteins using modified graph laplacians. The 2003 Nobel Prize in Chemistry was co-awarded to R. MacKinnon who was the first to structure a protein channel (the potassium channel) in 1998 by crystallizing the protein and using the classical X-ray computed tomography (CT). However, most membrane proteins cannot be crystallized and the classical CT cannot be used.

In our experimental setup, the data consists of thousands of real noisy 2D projections (electron microscope images) of the protein given in random unknown directions, because the proteins are randomly oriented rather than being aligned as in MacKinnon's setup. Still, we show that the reconstruction is made possible by a certain modification of the images' graph laplacian coupled with the 3D Fourier slice theorem. The reconstruction is a particular case of a more general spectral non-linear independent component analysis (ICA) algorithm that combines local principal component analysis (PCA) with the graph laplacian.

No prior knowledge of graph laplacians, tomography and proteins is needed for this talk.

This is a joint work with Ronald Coifman, Yoel Shkolnisky and Fred Sigworth (Yale University).