Non-unique Games over Compact Groups and Orientation Estimation in - PowerPoint PPT Presentation

Non-unique Games over Compact Groups and Orientation Estimation in Cryo-EM Amit Singer Princeton University Department of Mathematics and Program in Applied and Computational Mathematics July 28, 2016 Amit Singer (Princeton University) July 2016 1 / 23

Single Particle Reconstruction using cryo-EM Schematic drawing of the imaging process: The cryo-EM problem: Amit Singer (Princeton University) July 2016 2 / 23

New detector technology: Exciting times for cryo-EM www.sciencemag.org SCIENCE VOL 343 28 MARCH 2014 1443 BIOCHEMISTRY Advances in detector technology and image The Resolution Revolution processing are yielding high-resolution electron cryo-microscopy structures of biomolecules. Werner Kühlbrandt P recise knowledge of the structure of A B C macromolecules in the cell is essen- tial for understanding how they func- tion. Structures of large macromolecules can now be obtained at near-atomic resolution by averaging thousands of electron microscope images recorded before radiation damage accumulates. This is what Amunts et al . have done in their research article on page 1485 of this issue ( 1 ), reporting the structure of the large subunit of the mitochondrial ribosome at 3.2 Å resolution by electron cryo-micros- copy (cryo-EM). Together with other recent high-resolution cryo-EM structures ( 2 – 4 ) (see the fi gure), this achievement heralds the beginning of a new era in molecular biology, where structures at near-atomic resolution are no longer the prerogative of x-ray crys- tallography or nuclear magnetic resonance Near-atomic resolution with cryo-EM. ( A ) The large subunit of the yeast mitochondrial ribosome at 3.2 Å (NMR) spectroscopy. reported by Amunts et al . In the detailed view below, the base pairs of an RNA double helix and a magnesium Ribosomes are ancient, massive protein- ion (blue) are clearly resolved. ( B ) TRPV1 ion channel at 3.4 Å ( 2 ), with a detailed view of residues lining the RNA complexes that translate the linear ion pore on the four-fold axis of the tetrameric channel. ( C ) F 420 -reducing [NiFe] hydrogenase at 3.36 Å ( 3 ). genetic code into three-dimensional proteins. The detail shows an α helix in the FrhA subunit with resolved side chains. The maps are not drawn to scale. Mitochondria—semi-autonomous organelles Amit Singer (Princeton University) July 2016 3 / 23

Cryo-EM in the news... March 31, 2016 • REPORTS Cite as: Sirohi et al ., Science 10.1126/science.aaf5316 (2016). The 3.8 Å resolution cryo-EM structure of Zika virus Devika Sirohi, 1 * Zhenguo Chen, 1 * Lei Sun, 1 Thomas Klose, 1 Theodore C. Pierson, 2 Michael G. Rossmann, 1 † Richard J. Kuhn 1 † 1 Markey Center for Structural Biology and Purdue Institute for Inflammation, Immunology and Infectious Disease, Purdue University, West Lafayette, IN 47907, USA. 2 Viral Pathogenesis Section, Laboratory of Viral Diseases, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD 20892, USA. Amit Singer (Princeton University) July 2016 4 / 23

Method of the Year 2015 January 2016 Volume 13 No 1 Single-particle cryo-electron microscopy (cryo-EM) is our choice for Method of the Year 2015 for its newfound ability to solve protein structures at near-atomic resolution. Featured is the 2.2-Å cryo-EM structure of β-galactosidase as recently reported by Bartesaghi et al . ( Science 348, 1147–1151, 2015 ). Cover design by Erin Dewalt. Amit Singer (Princeton University) July 2016 5 / 23

Big “Movie” Data, Publicly Available http://www.ebi.ac.uk/pdbe/emdb/empiar/ Amit Singer (Princeton University) July 2016 6 / 23

E. coli 50S ribosomal subunit 27,000 particle images provided by Dr. Fred Sigworth, Yale Medical School 3D reconstruction by S, Lanhui Wang, and Jane Zhao Amit Singer (Princeton University) July 2016 7 / 23

Orientation Estimation: Fourier projection-slice theorem R i c ij c ij = ( x ij , y ij , 0) T ( x ij , y ij ) Projection I i ˆ Projection I i I i 3D Fourier space ( x ji , y ji ) Molecule φ R i c ij = R j c ji R i ∈ SO(3) ˆ Projection I j I j 3D Fourier space Electron source Cryo-EM inverse problem: Find φ (and R 1 , . . . , R n ) given I 1 , . . . , I n . n = 3: Vainshtein and Goncharov 1986, van Heel 1987 n > 3: S, Shkolnisky (SIAM Imaging 2011) � � R i c ij − R j c ji � 2 min R 1 , R 2 ,..., R n ∈ SO (3) i � = j Amit Singer (Princeton University) July 2016 8 / 23

Maximum Likelihood Estimation The images contain more information than that expressed by optimal pairwise matching of common lines. Algorithms based on pairwise matching can succeed only at “high” SNR. We would like to try all possible rotations R 1 , . . . , R n and choose the combination for which the agreement on the common lines (implied by the rotations) as observed in the images is maximal. Computationally intractable: exponentially large search space, complicated cost function. n � f ij ( g i g − 1 min ) j g 1 ,..., g n ∈ G i , j =1 Amit Singer (Princeton University) July 2016 9 / 23

3D Puzzle G = SO (3) n � f ij ( g i g − 1 min ) j g 1 , g 2 ,..., g n ∈ G i , j =1 Amit Singer (Princeton University) July 2016 10 / 23

Non-Unique Games over Compact Groups Optimization problem: n � f ij ( g i g − 1 min ) j g 1 , g 2 ,..., g n ∈ G i , j =1 G is a compact group, f ij : G → R smooth, bandlimited functions. Parameter space G × G × · · · × G is exponentially large. For G = Z 2 = {− 1 , +1 } this encodes Max-Cut, which is NP-hard. 1 w 14 w 12 4 w 24 2 w 45 w 25 w 34 5 3 w 35 Amit Singer (Princeton University) July 2016 11 / 23

Why non-unique games? Max-2-Lin( Z L ) formulation of Unique Games (Khot et al 2005): Find x 1 , . . . , x n ∈ Z L that satisfy as many difference eqs as possible x i − x j = b ij mod L , ( i , j ) ∈ E This corresponds to G = Z L and � − 1 x = b ij f ij ( x ) = 0 x � = b ij in n � min f ij ( x i − x j ) x 1 , x 2 ,..., x n ∈ Z L i , j =1 Our games are non-unique in general, and the group is not necessarily finite. Amit Singer (Princeton University) July 2016 12 / 23

Fourier transform over G Recall for G = SO (2) ∞ � ˆ f ( k ) e ı k α f ( α ) = k = −∞ � 2 π 1 f ( α ) e − ı k α d α ˆ f ( k ) = 2 π 0 In general, for a compact group G ∞ � � � ˆ f ( g ) = d k Tr f ( k ) ρ k ( g ) k =0 � f ( g ) ρ k ( g ) ∗ dg ˆ f ( k ) = G Here ρ k are the unitary irreducible representations of G d k is the dimension of the representation ρ k (e.g., d k = 1 for SO (2), d k = 2 k + 1 for SO (3)) dg is the Haar measure on G Amit Singer (Princeton University) July 2016 13 / 23

Linearization of the cost function Introduce matrix variables (“matrix lifting”) X ( k ) = ρ k ( g i g − 1 ) ij j Fourier expansion of f ij ∞ � � � ˆ f ij ( g ) = d k Tr f ij ( k ) ρ k ( g ) k =0 Linear cost function n n ∞ � f ij ( k ) X ( k ) � � � � ˆ f ij ( g i g − 1 f ( g 1 , . . . , g n ) = ) = d k Tr j ij i , j =1 i , j =1 k =0 Amit Singer (Princeton University) July 2016 14 / 23

Constraints on the variables X ( k ) = ρ k ( g i g − 1 ) ij j 1 X ( k ) � 0 2 X ( k ) = I d k , for i = 1 , . . . , n ii 3 rank( X ( k ) ) = d k X ( k ) = ρ k ( g i g − 1 ) = ρ k ( g i ) ρ k ( g − 1 ) = ρ k ( g i ) ρ k ( g j ) ∗ ij j j   ρ k ( g 1 ) ρ k ( g 2 ) X ( k ) =   � ρ k ( g 1 ) ∗ ρ k ( g 2 ) ∗ ρ k ( g n ) ∗ � · · ·  .  .   .   ρ k ( g n ) We drop the non-convex rank constraint. The relaxation is too loose, as we can have X ( k ) = 0 (for i � = j ). ij and X ( k ′ ) Even with the rank constraint, nothing ensures that X ( k ) ij ij correspond to the same group element g i g − 1 . j Amit Singer (Princeton University) July 2016 15 / 23

Additional constraints on X ( k ) = ρ k ( g i g − 1 ) ij j The delta function for G = SO (2) ∞ � e ı k α δ ( α ) = k = −∞ Shifting the delta function to α i − α j ∞ ∞ ∗ e ı k α e − ı k ( α i − α j ) = e ı k α X ( k ) � � δ ( α − ( α i − α j )) = ij k = −∞ k = −∞ The delta function is non-negative and integrates to 1: ∞ ∗ ≥ 0 , e ı k α X ( k ) � ∀ α ∈ [0 , 2 π ) ij k = −∞ � 2 π ∞ ∗ d α = X (0) ∗ = 1 1 e ı k α X ( k ) � ij ij 2 π 0 k = −∞ Amit Singer (Princeton University) July 2016 16 / 23

Finite truncation via Fej´ er kernel In practice, we cannot use infinite number of representations to compose the delta function. Simple truncation leads to the Dirichlet kernel which changes sign m � e ı k α D m ( α ) = k = − m This is also the source for the Gibbs phenomenon and the non-uniform convergence of the Fourier series. The Fej´ er kernel is non-negative m − 1 m F m ( α ) = 1 � 1 − | k | � � � e ı k α D k ( α ) = m m k =0 k = − m The Fej´ er kernel is the first order Ces` aro mean of the Dirichlet kernel. Amit Singer (Princeton University) July 2016 17 / 23