Joachim Frank

1. The Quest for High(er) Resolution

2. SNR and SSNR Estimates for EM Noise Processes

HHMI, Health Research, Inc., Wadsworth CenterStarting April 1, 2008: Department of Biochemistry and Molecular Biophysics and Department of Biology, Columbia University

Before going into the subject matter, a few introductory slides on single-particle reconstruction.

(“Single particle” here means “single, unattached” – the number of particles or molecules is normally huge)

For details on this technique, see J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies, Oxford University Press 2006

This is the kind of data we are dealing with in “single-particle reconstruction”

Once a reference is established, orientation determination can be done by 3D projection matching

Projection matching normally starts with 83 projections on the half-sphere

Angular Refinement, by Iterative 3D Projection Matching

Subsequently, angular refinement is achieved by iteration, with smaller and smaller angular increments.

(refinement by defocus groups)

Part of the Bill Baxter’s ctfgroup interface: grouping by defocus

Defocus groups are used for expediency, but they need to be narrow enough so as to avoid blurring of high-res information.

Reconstruction by Defocus Groups

At the very end, the reconstructions from all the defocus groups are merged in a CTF correction step (by Wiener Filtering)

Gabashvili et al. (2000) Cell

Cryo-EM 11.5 Å resolution; 73,000 particles

The processing method described is now standard. It was used to obtain this reconstruction of the E. coli ribosome

Math Problem, in a Nutshell, at Current Resolutions (~10 Å)

Computation of 3D density map:

100,000 x (200 x 200) measurements with SNR ~0.1 – 0.4

# of particles # of pixels

(projections)

200 x 200 x 200 + 5 x 100,000 unknowns

# of voxels # of parameters (in 3D density map) (shifts, angles)

What happens if we want to go to 3 Å resolution?

Resolution measures & criteria -- Fourier ring/shell correlation

k = spatial frequency

Δk = ring width or shell thickness

*1 2

[ , ]

2 2 1/ 21 2

[ , ]

| (k) (k) |

( , )[ | (k) | | (k) | ]

k k

k k

Re F F

FSC k kF F

Conservative; SNR =1

R. Henderson, X-ray

Too optimistic

Resolution criteria based on FSC

Extrapolation to 3 Å --Millions of particles?

We’re facing an obstacle!

What can be controlled/improved experimentally?

versus

What can be controlled/improved by image processing with a given dataset?

(or: how do we want to spend our time?)

Processing by defocus groups, or variations in height

These envelope functions show how the final resolution is affected by the width of the defocus range, or by variations in height of particle within the ice

Question of Window Size

Reconstruction of uncorrected data is done under the assumption that all information is present. But part of the information lies outside the particle, because of the finite width

of the point spread function!

Radius of point spread function has to be accommodated in the window.

Window size = particle diameter + diameter of the point-spread function = point-spread function

associated with the CTF

Exploration of reconstruction strategy“High-resolution project”

Use small dataset (50,000) to optimize processing, with the idea to switch to larger dataset (130,000)

Parameters of image processing:• Sampling (strategy to switch from coarse to fine, to save CPU)• Window size (to avoid CTF effects)• Angular spacing (how to reduce)• Amplitude correction in each step of refinement vs. at the very end

Final parameters emerging from this exploration: angular step 0.5 degrees, angular search range 2 degrees7 iterations of refinement: 920 hours on a 48-node clusterRegular window size OKSampling (decimation) can be switched mid-way from coarse to fine

Resolution measurement issues

• Apply soft mask to reconstruction to get true resolution!• Evidence for dependence of resolution R vs. log(N)• Is lin-log dependence general?• Is it allowed to extrapolate from half to full dataset?• Verification of molecular detail: phosphorus atoms show

up as bumps

“Clutter”

outside

To be replaced – first two curves are incorrect! Other curves OK.

Final reconstructiion, at 7.5 A:

Progression of resolution, from the beginning to the end of the project.

E. coli 70S•aa-tRNA•EF-Tu•GDP•kir at 7.5 Å

EF-Tu

130,000 particles 7.5 Å (FSC=0.5)

APE

EF-Tu

Protein S2

X-ray

missing helix

Except for a missing helix at the periphery, protein S2 is exactly as it appears in the X-ray structure.

Extrapolation of FSC resolution to full set

65,000130,000

Resolution is a linear function of Log(Number of particles) -- Ribosome.

GroEL (Stagg et al.)

Resolution is a linear function of Log(Number of particles) -- GroEL.

6.7 Å

6.7 Å (LeBarron et al., in prep.) 10 Å (Valle et al., NSB 2003)

Cutoff Cryo-EM X-ray Cryo-EM X-ray

6.7 Å

5.0 Å

6.7 Å

5.0 ÅPhosphorus atomsrecognized as bumps!

Definition of EF-Tu domains (stereo)

• SNR and SSNR estimation

• Why?

• We need better noise models for phantoms, to fine-tune classification

• William Baxter

Given: two images representing independent realizations of a signal embedded in noise. Align the two images, then compute the cross-correlation coefficient. Estimate the SNR α from the cross-correlation coefficient by

α = ρ/(1-ρ) ρ: α:

0 0 1 ∞ 0.5 1

(Frank and Al-Ali, Nature 1975)

• Compare the images of two molecules that were identified, by 3D projection matching, as having the same orientation.

• These images differ by (i) structural noise, (ii) shot noise, and (iii) digitization noise.

• How to disentangle the three different noise portions?

• Structural noise:

the irreproducible portion of the object.

(i) Conformational changes

(ii) Superimposed structure of substrate (ice, thin carbon)

• Experimental design:

• Three experiments:(i) take images of two molecules deemed in the same

orientation (as determined by projection matching)

(ii) take two images of the same field

(iii) digitize the same image of a molecule twice

These are the kinds of images we are using in the SNR estimation

Projection matching: two particles falling into the same orientation bin represent the same molecule structure plus “structural noise”

Using projection matching to a template, we can find molecules that nominally have the same orientation, and can bye compared in the SNR estimation.

SNR estimation by double experiments

Molecules insame view

Same fieldimaged 2x

Same micrographdigitized 2x

Formula for the case of two noise subprocesses: obtaining SNR of subprocess #I from measured SNRs of complete process (αcomp) and subprocess #2 (αsub2):

1αsub1 = --------------------------------

(1+1/ αcomp )/(1+1/αsub2)-1

Results of Estimation:

• SNR for digitization noise: 27 [for ZI microdensitometer -- will be different for different scanners]

• SNR for shot noise: 0.09• [Is in the range measured before, e.g. 0.1 (Fu et al. JSB 2007)]

• Note: for CCD recording, digitization and shot noise cannot be separated.

• SNR for structural noise: ~2 [Reasonable, since the “noise” contains all irreproducible

components: conformational differences, thin carbon]

SSNR Estimates, using the same formula in Fourier space along rings

double scan

double exposure

double structure

The curves show how the SNR is distributed as a function of spatial frequency

SSNR for “true” film-recorded Shot Noise (averaged over 4 defoci)

SSNR for “true” Structural Noise (averaged over 3 defoci)

Outlook

• Atomic resolution soon in reach. How soon is soon? Asymmetric structures much more tough than those with symmetry.

• Efficient classification is key!• Extrapolation of resolution might be possible in

general – does lin-log rule always hold?• Develop strategy with subsets is a good idea!

People in my group who have contributed to this research:

• William Baxter (SNR)• Robert A. Grassucci (HR)• Jamie LeBarron (HR)• Haixiao Gao (HR)• Tapu Shaikh (HR)• Jayati Sengupta (HR)

• Funding: HHMI, NIH, NCRR

HR = high-resolution project SNR = SNR and SSNR estimation