Multireference Alignment, Cryo-EM, and XFEL

Multireference Alignment, Cryo-EM, and XFEL Amit Singer Princeton University Department of Mathematics and Program in Applied and Computational Mathematics August 16, 2017 Amit Singer (Princeton University) August 2017 1 / 32

Joint work with... Afonso Bandeira NYU Tejal Bhamre Princeton -> Apple Tamir Bendory Princeton Nicolas Boumal Princeton Edgar Dobriban Stanford -> UPenn Roy Lederman Princeton Will Leeb Princeton Lydia Liu Princeton -> Berkeley Joao Pereira Princeton Nir Sharon Princeton Teng Zhang Central Florida Zhizhen (Jane) Zhao UIUC Amit Singer (Princeton University) August 2017 2 / 32

Multi-reference alignment of 1D periodic signals =0 =0.1 =1.2 High SNR: pairwise alignment succeeds. Low SNR: pairwise alignment fails. How to use information in many (n > 2) signals? Can we reconstruct the signal accurately while estimating most shifts poorly? How many observations are needed for an accurate reconstruction? Amit Singer (Princeton University) August 2017 3 / 32

Shift invariant features y i = R i x + ε i, x, y i, ε i R L, ε i N (0, σ 2 I L L ), i = 1, 2,..., n 1 Zero frequency / average pixel value: 1 n n ŷ i (0) ˆx(0) as n. Need n σ 2 i=1 2 Power spectrum / autocorrelation: 1 n n ŷ i (k) 2 ˆx(k) 2 + σ 2 as n. Need n σ 4 i=1 3 Bispectrum / triple correlation (Tukey, 1953): 1 n n ŷ i (k 1 )ŷ i (k 2 )ŷ i ( k 1 k 2 ) ˆx(k 1 )ˆx(k 2 )ˆx( k 1 k 2 ) as n. Need n σ 6 i=1 Amit Singer (Princeton University) August 2017 4 / 32

Estimation using shift invariant features The bispectrum Bx(k 1, k 2 ) = ˆx(k 1 )ˆx(k 2 )ˆx( k 1 k 2 ) contains phase information and is invertible (up to global shift) Sadler, Giannakis (JOSA A 1992) Kakarala (1992; arxiv 2009) It is possible to accurately reconstruct the signal from sufficiently many noisy shifted copies for arbitrarily low SNR without estimating the shifts and even when estimation of shifts is poor Notice that if shifts are known, then n 1/SNR is sufficient. Unknown shifts make a big difference: n 1/SNR 3. No method can succeed with fewer measurements! Perry, Weed, Bandeira, Rigollet, S (arxiv 2017) Abbe, Pereira, S (ISIT 2017) Amit Singer (Princeton University) August 2017 5 / 32

Maximum Likelihood vs. Invariant Features Computation time [s] 1.2 1 signal invariant features MRA expectation maximization 2 1.5 1 10 4 10 3 10 2 invariants feature approach expectation maximization 0.8 0.5 10 1 0.6 0 0.4-0.5 10 0 0.2-1 10-1 0-0.2 5 10 15 20 25 30-1.5-2 5 10 15 20 25 30 10-2 10 1 10 2 10 3 10 4 10 5 10 6 10 7 #observations n (n = 10 4, σ = 1) Invariant features: only one pass over data, data can come as a stream and not be stored, parallel computation Expectation-Maximization, Stochastic Gradient Descent: multiple passes over the data, all data needs to be stored, parallel computation. Bendory, Boumal, Ma, Zhao, S (arxiv 2017) Amit Singer (Princeton University) August 2017 6 / 32

Non-uniform but unknown distribution of shifts 1.2 1 0.8 true signal gradient-based algorithm expectation maximization 0.6 0.4 0.2 0-0.2-0.4 5 10 15 20 25 30 (n = 10 5, σ = 2, Pr{R i = t} exp ( 1 2 t2 /5 2) ). Non-uniform is easier than uniform (only second-order moments): n σ 4 Moments-based method is faster and more accurate than EM Amit Singer (Princeton University) August 2017 7 / 32

Heterogeneity from invariant features 1.5 1 estimated signal true signal 4 3 0.5 0-0.5 0 5 10 15 20 25 30 3 2 estimated signal true signal 2 1 0-1 1 0-1 5 10 15 20 25 30-2 -3 0 5 10 15 20 25 30 (n = 10 6, σ = 1.5) Two step procedure: 1 Compute invariant features averages over entire data 2 Find two (or more) signals and their population proportions that agree with the computed averages. No clustering! Why does it work? Count degrees of freedom Amit Singer (Princeton University) August 2017 8 / 32

Main motivation: Cryo-EM THE REVOLUTION WILL NOT BE CRYSTALLIZED B Y E W E N C A L L A WAY I n a basement room, deep in the bowels of a steel-clad building in Cambridge, a major insurgency is under way. A hulking metal box, some three metres tall, is quietly beaming terabytes worth of data through thick orange cables that disappear off through the ceiling. It is one of the world s most advanced cryoelectron microscopes: a device that uses electron beams to photograph frozen biological molecules and lay bare their molecular shapes. The microscope is so sensitive that a shout can ruin an experiment, says Sjors Scheres, a structural biologist at the UK Medical Research Council Laboratory of Molecular Biology (LMB), as he stands dwarfed beside the 5-million (US$7.7-million) piece of equipment. The UK needs many more of these, because there s going to be a boom, he predicts. In labs around the world, cryo-electron microscopes such as this one are sending tremors through the field of structural biology. In the past three years, they have revealed exquisite details of protein-making ribosomes, quivering membrane proteins and other key cell molecules, ILLUSTRATION BY VIKTOR KOEN MOVE OVER X-RAY CRYSTALLOGRAPHY. CRYO-ELECTRON MICROSCOPY IS KICKING UP A STORM IN STRUCTURAL BIOLOGY BY REVEALING THE HIDDEN MACHINERY OF THE CELL. 1 7 2 NAT U R E VO L 5 2 5 1 0 S E P T E M B E R 2 0 1 5 2015 Macmillan Publishers Limited. All rights reserved Amit Singer (Princeton University) August 2017 9 / 32

BIOCHEMISTRY The Resolution Revolution Werner Kühlbrandt P recise knowledge of the structure of macromolecules in the cell is essential for understanding how they function. 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-microscopy (cryo-em). Together with other recent high-resolution cryo-em structures ( 2 4) (see the figure), 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 crystallography or nuclear magnetic resonance (NMR) spectroscopy. Ribosomes are ancient, massive protein- RNA complexes that translate the linear genetic code into three-dimensional proteins. Mitochondria semi-autonomous organelles www.sciencemag.org SCIENCE VOL 343 28 MARCH 2014 1443 A B C Advances in detector technology and image processing are yielding high-resolution electron cryo-microscopy structures of biomolecules. Near-atomic resolution with cryo-em. (A) The large subunit of the yeast mitochondrial ribosome at 3.2 Å reported by Amunts et al. In the detailed view below, the base pairs of an RNA double helix and a magnesium ion (blue) are clearly resolved. (B) TRPV1 ion channel at 3.4 Å ( 2), with a detailed view of residues lining the ion pore on the four-fold axis of the tetrameric channel. (C) F 420-reducing [NiFe] hydrogenase at 3.36 Å ( 3). The detail shows an α helix in the FrhA subunit with resolved side chains. The maps are not drawn to scale. Amit Singer (Princeton University) August 2017 10 / 32

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) August 2017 11 / 32

Main motivation: Cryo-EM Why cryo-electron microscopy? Mapping the structure of molecules without crystallizing them Imaging of heterogeneous samples, with mixtures of molecules or multiple conformations Why now? Advancements in detector technology have led to near-atomic resolution mapping Amit Singer (Princeton University) August 2017 12 / 32

How does it work? Schematic drawing of the imaging process: The basic cryo-em problem: Amit Singer (Princeton University) August 2017 13 / 32

Image formation model and inverse problems Projection images I i (x, y) = T i φ(xr1 i + yr 2 i + zr 3 i ) dz + noise". n images (i = 1,..., n), images of size N N pixels φ : R 3 R is the scattering density of the molecule. Cryo-EM basic problem: Estimate φ given I 1,..., I n. The heterogeneity problem: Estimate φ 1,..., φ n given I 1,..., I n. Amit Singer (Princeton University) August 2017 14 / 32

Kam s method Amit Singer (Princeton University) August 2017 15 / 32

Why was Kam s method mostly forgotten? Idea that was ahead of its time: There was not enough data to accurately calculate second and third order statistics. Requires uniform distribution of viewing directions. Maximum likelihood framework prevailed. Amit Singer (Princeton University) August 2017 16 / 32

Fourier projection-slice theorem Amit Singer (Princeton University) August 2017 17 / 32

Kam s method revisited Spherical harmonics expansion L l ˆφ(k, θ, ϕ) = A lm (k)yl m l=0 m= l (θ, ϕ) PCA / 2D covariance gives for each l the A lm s up to an orthgonal matrix of size (2l + 1) (2l + 1) through C l (k 1, k 2 ) = l m= l A lm (k 1 )A lm (k 2), or C l = A l A l C l is the analogue of power spectrum. Autocorrelation of 3D structure is obtained through Fourier slice theorem. Amit Singer (Princeton University) August 2017 18 / 32

From phase retrieval to orthogonal matrix retrieval C l = A l A T l = A lo l O T l AT l Cholesky decomposition of C l determines A l up to an (2l + 1) (2l + 1) orthogonal matrix O l. Orthogonal matrix retrieval: Bispectrum / homology modelling. Homology: Compute C l = F l Fl T, want A l = F l O l such that A l B l O l = argmin O O(2l+1) F l O B l 2 F Closed form solution using singular value decomposition: O l = V l U T l where B T l F l = U l Σ l V T l Bhamre, Zhang, S (ISBI 2015) Amit Singer (Princeton University) August 2017 19 / 32

Orthogonal matrix retrieval via homology modeling Bhamre, Zhang, S (arxiv, 2017) E EMDB 8118 (a) Homologous structure (a) Ground Truth (b) EMDB 8117 (b) Least Squares (c) Twicing (d) Anisotropic Twicing (e) Synthetic Dataset: TRPV1 with DxTx and RTX. SNR= 1/40, 26000 images, 10 defocus groups. Amit Singer (Princeton University) August 2017 20 / 32

Improved covariance estimation Steerable PCA: Covariance matrix commutes with in-plane rotations, hence block-diagonalized in Fourier-Bessel basis (or any other angular Fourier basis) Zhao, Shkolnisky, S (IEEE Computational Imaging, 2016) CTF correction Eigenvalue shrinkage Bhamre, Zhang, S (Journal Structural Biology, 2016) Amit Singer (Princeton University) August 2017 21 / 32

Application to denoising Bhamre, Zhang, S (Journal Structural Biology, 2016) Raw Closest projection TWF CWF CWF = Covariance Wiener Filter, TWF = Traditional Wiener Filter TRPV1, K2 direct electron detector 35645 motion corrected, picked particle images of 256 256 pixels belonging to 935 defocus groups (Liao et al., Nature 2013) Amit Singer (Princeton University) August 2017 22 / 32

Third order moment tensor for 3D ab-initio reconstruction Work in progress: extension to non-uniform distributions, uniqueness? constraints? (positivity at low frequency?) Why bother? Extremely fast: just one or two passes over the data; single pass is much faster than a typical refinement iteration Streaming? Validation tool: No starting model to refine, no need to worry about rotations estimated correctly Amit Singer (Princeton University) August 2017 23 / 32

Kam s method for XFEL X-ray free electron laser (XFEL) molecular imaging (Gaffney and Chapman, Science 2007) Amit Singer (Princeton University) August 2017 24 / 32

Kam s method for XFEL vs. Cryo-EM Ewald spheres Uniform distribution No CTF, no shifts Low photon count: Poisson noise Non-negativity constraint Amit Singer (Princeton University) August 2017 25 / 32

epca: Exponential family PCA Demo of epca on XFEL images Demo of epca on XFEL images Liu, Dobriban, S (arxiv 2017) (a) Clean intensity maps (b) Noisy photon counts (c) Denoised (PCA) (d) Denoised (epca) Figure: XFEL diffraction images (n = 70, 000, p = 65, 536) XFEL diffraction images (n = 70, 000, p = 65, 536) Amit Singer (Princeton University) August 2017 26 / 32 6 / 35

PCA for Exponential Family Distributions One-parameter exponential family with density f θ (y) = exp[θy A(θ)] No commonly agreed upon version of PCA for non-gaussian data (Jolliffe 2002) Likelihood/generalized linear latent variable models (Collins et al. 2001; Knott and Bartholomew 1999; Udell et al. 2016) lack of global convergence guarantees slow Gaussianizing transforms: wavelet, Anscombe (Anscombe 1948; Starck et al. 2010) unsuitable for low-intensity Amit Singer (Princeton University) August 2017 27 / 32

Problem formulation Sampling model for Poisson Each p-dim latent vector is drawn i.i.d. from distribution D (X(1),, X(p)) = X D(µ, Σ) e.g., the noiseless image. µ and Σ are the mean and covariance of D. Observations Y i Y R p e.g., the noisy image Model for Y : draw latent X R p, then Y = (Y (1),, Y (p)) where Y (j) Poisson(X(j)) Goal: Recover information about the original distribution D, i.e. Σ. Recover X. Amit Singer (Princeton University) August 2017 28 / 32

Summary of epca epca can be seen as a sequence of improved covariance estimators Table: Covariance estimators Name Formula Motivation Sample covariance S = 1 n n i=1 (Y i Ȳ )(Y i Ȳ ) - Diagonal debiasing S d = S diag[v (Ȳ )] Hierarchy Homogenization S h = D 1 2 n S d D 1 2 n Heteroskedasticity Shrinkage S h,η = ρ(s h ) High dimensionality Heterogenization S he = D 1 2 n S h,η D 1 2 n Heteroskedasticity Scaling S s = ˆα i ˆv i ˆv i (S he = ˆv i ˆv i ) Heteroskedasticity Amit Singer (Princeton University) August 2017 29 / 32

Conclusions Method of moments paves the way to signal(s) recovery through one or two passes over the data, no alignment and no clustering. Reconstruction is possible at any SNR, given sufficiently many observations. Qualitative determination of the number of observations needed as a function of the SNR: 1/SNR, 1/SNR 2, 1/SNR 3 Improved high dimensional covariance estimation (shrinkage, steerable, Poisson) Amit Singer (Princeton University) August 2017 30 / 32

ASPIRE: Algorithms for Single Particle Reconstruction Open source toolbox, publicly available: http://spr.math.princeton.edu/ Amit Singer (Princeton University) August 2017 31 / 32

References A. S. Bandeira, M. Charikar, A. Singer, A. Zhu, Multireference Alignment using Semidefinite Programming", in Proceedings of the 5th conference on Innovations in Theoretical Computer Science (ITCS 14), pp. 459 470 (2014). Z. Zhao, A. Singer, Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM", Journal of Structural Biology, 186 (1), pp. 153 166 (2014). T. Bhamre, T. Zhang, A. Singer, Orthogonal Matrix Retrieval in Cryo-Electron Microscopy", in IEEE 12th International Symposium on Biomedical Imaging (ISBI 2015), pp. 1048 1052, 16-19 April 2015. T. Bhamre, T. Zhang, A. Singer, Denoising and Covariance Estimation of Single Particle Cryo-EM Images", Journal of Structural Biology, 195 (1), pp. 72 81 (2016). T. Bendory, N. Boumal, C. Ma, Z. Zhao, A. Singer, Bispectrum Inversion with Application to Multireference Alignment", https://arxiv.org/abs/1705.00641 (2017). E. Abbe, J. Pereira, A. Singer, Sample Complexity of the Boolean Multireference Alignment Problem", IEEE International Symposium on Information Theory (ISIT) (2017). A. Perry, J. Weed, A. S. Bandeira, P. Rigollet, A. Singer, The sample complexity of multi-reference alignment", https://arxiv.org/abs/1707.00943 (2017). L. Liu, E. Dobriban, A. Singer, epca: High Dimensional Exponential Family PCA", preprint. Available at https://arxiv.org/abs/1611.05550 (2017). Amit Singer (Princeton University) August 2017 32 / 32