Likelihood-based optimization of cryo-em data. Sjors Scheres National Center for Biotechnology CSIC Madrid, Spain

Transcription

1 Likelihood-based optimization of cryo-em data Sjors Scheres National Center for Biotechnology CSIC Madrid, Spain

2 Life based on molecular machines DNA replication Protein synthesis Dynein motion

3 Molecular machines m 15 m F1-ATPase: Abrahams et al., 1994 Dutch windmill

4 Studying these machines The different states tell much about the way these machines work! Different conformations of (chemically identical) molecules are very hard to purify Biophysical techniques that study the bulk, average-out information about these conformations

5 The promise of 3D-EM In 3D Electron Microscopy individual molecules are visualized Trapped in ice, these molecules are free to adapt many conformations

6 Classification alignment & classification are strongly intertwined! noise forms a serious problem!

7 Structural heterogenity Our approach: Combine classification & alignment in a single optimization process multi-reference refinements Use maximum-likelihood principles

8 Why maximum likelihood?

9 Conventional data models Maximum cross-correlation (~least squares) X = P V i ϕ k =?

10 Conventional data models Maximum cross-correlation (~least squares) X = CTF P V But i what about i ϕ experimental noise?! =? k

11 Statistical data models Maximum likelihood X + i = P V N ϕ k i white =? Gaussian noise

12 Statistical model k = 1 k = 2 k = 3 Each image is a projection of one of K underlying 3D objects k with addition of white Gaussian noise Unknowns: the 3D objects k, orientations

13 Statistical model for each pixel j: data: X σ P(X j A j ) exp ((X j A j ) 2 ) -2σ 2 A j X j model: A White noise = independence between pixels! P(data image model image) ~ Π P(X j A j ) j

14 Log-likelihood function Adjust model to maximize the log-likelihood of observing the entire dataset: L ( model) = ln P( image ) i model = N N i= 1 K ln i= 1 k = 1 orient P ( image k,orient., model) P( k,orient. model) i The model comprises: estimates for the underlying objects estimate for the amount of noise (σ) statistical distributions of k & orient. Optimization algorithm: Expectation Maximization

15 Two cases Alignment & classification in 2D: align images and calculate 2D averages for the distinct classes Alignment & classification in 3D align images and calculate 3D reconstructions for the distinct classes

16 The 2D algorithm estimates for K 2D objects k=1 k=2 sampled rotations 360 for each image, calculate all P ( image k, rot) i calculate new 2D average as probability weighted averages

17 Parallel to ML-phasing in XRD M P α Probability-weighted average over all (phase) assignments! (Figures by P. Gros)

18 ML2D classification Scheres et al. (2005) J. Mol. Biol., 348, Scheres et al. (2005) Bioinformatics 21 (Suppl. 2), ii243-ii244

19 The 3D algorithm estimates for K 3D objects k=1 k=2 project into all (discretely sampled) orientations for each image, calculate all P ( image k,orient., model) i calculate new 3D estimates as probability weighted 3D reconstructions

20 Prelim. ribosome reconstruction 91,114 particles; 9.9 Å resolution fragmented (depicted at a lower threshold) blurred

21 Seed generation 80 Å filter 4 random subsets; 1 iter ML

22 ML3D-classification 4 references 91,114 particles 64x64 pix (6.2Å/pix) 25 iterations 10 angular sampling 64 CPUs in parallel: 3 days 6 CPU-months BSC

23 ML-derived classes no ratcheting; no EF-G; 3 trnas differences: overall rotations ratcheting, EF-G, 1 trna

24 ML3D classification Scheres et al. (2007) Nat Methods, 4, 27-29

25 Back to the algorithm design table

26 Statistical model for each pixel j: data: X σ P(X j A j ) exp ((X j A j ) 2 ) -2σ 2 A j X j NOT TRUE! model: A White noise = independence between pixels! P(data image model image) ~ Π P(X j A j ) j

27 An improved data model Maximum likelihood X P V + i = CTFi ϕ k N i =? spatially stationary Gaussian noise coloured, pink,

28 Coloured noise model for each Fourier pixel h 2D-Gaussian in complex plane imaginary σ h C h i [ B V ] h ϕ k h X i real Assuming independence of noise between all Fourier terms: P ( X k, ϕ, Θ) i = [ ] h 2 h h 1 CTFi Pϕ Vk X i exp ( ) ( ) = h 2 h 2 1 2π σ 2 σ H h resolution-dependent noise model!

29 Simulated data (3,000 images)

30 70S E.coli ribosome (20,000 images) (kindly provided by Haixao Gao & Joachim Frank)

31 Coloured noise!! (for different defocus groups) σ resolution (Å -1 )

32 SV40 large T-antigen (7,718 sub-images) Scheres et al. (2007) Structure, 15,

33 Xmipp protocols Standardized workflow Standardized environment Automated logging Graphical user interface Now in Xmipp Scheres et al., Nature Protocols (in press)

34 Acknowledgements Ribosome data Haixiao Gao Joachim Frank LTA data Mikel Valle Rafael Núñez-Ramírez MCM data Yacob Gómez-Llorente Carmen San Martín Mathematics Gabor T. Herman Paul P.B. Eggermont Computing Barcelona Supercomputer University of Almeria CNB José María Carazo Other data Jorge Cuellar JoseMari Valpuesta Jaime Martín-Benito Andres Leschziner

35 Future challenges Improve robustness Explore multivariate t-distributions Decrease computational burdens FFT-dominated: use GPUs? Overcome model bias Ideas???