2. Resolution
Power spectrum (I=amp2)
d =
0.61 λ
n sin α
d: The smallest still visible distance between two spots.
Numerical
aperture (NA)
Wavelength
Slide credit: Stefan Raunser
3. Atomic resolution
• Resolve interatomic distances
– Peptide bond length 1.32 Å
– Resolution required to fit an atomic model? Completely
trace a protein chain and resolve all side chains?
– In Cryo-EM you will frequently see the term “near-
atomic” resolution used
4. Measuring resolution
• Why do we measure resolution?
– To allow a quantifiable(?), objective(?), comparison of
maps/models/structures(?)
• IMPORTANT POINT: In Cryo-EM, we never measure
resolution, we estimate it!
– How?
8. Fourier transforms
∗ ∗
• The Fourier transform is a mathematical operation
that deconvolves a complex signal into its
components wave functions
• Any signal can be described by the sum of a series
of wave functions (Fourier synthesis)
• By textbook definition, the mathematical operation
that relates pattern of wave scattering by an object
to the object is the Fourier transform
10. What is Fourier Shell Correlation?
A measure of the self consistency in the data and the
reconstruction process (assuming no artificial bias has
been introduced)
11. What is Fourier Shell Correlation?
Object
A
Object
B
FFT[A] FFT[B]
Compute correlation between
the objects in FFT shells
12. What is Fourier Shell Correlation?
FSC 0.143
FSC 0.5
All FSC plots taken from
Penczek et al. Meth Enzymol 2010
14. Odd-looking FSC curves
All FSC plots taken from
Penczek et al. Meth Enzymol 2010
Problem
FSC never drops to zero
Not necessarily a problem depending
on what you are looking at (e.g.
convergence plot in EMAN2)
May indicate
• Correlation of noise
• Data undersampled (pixel size)
• For any two objects where noise is present and the resolving power of
the imaging system becomes limiting, the FSC should be 1 at some
infinitely low resolution and 0 at some infinitely high resolution
15. Odd-looking FSC curves
All FSC plots taken from
Penczek et al. Meth Enzymol 2010
Problem
FSC has “dips”
The information in Fourier space is
incomplete within certain resolution
shells
What might cause this?
Most commonly, not collecting data
over a range of defocus values
Possibly also orientation bias?
• Check directional FSC
(Lyumkis et al.
doi:10.1038/protex.2017.055)
16. Defocus and the Contrast transfer function
• CTF is introduced by any imaging system that uses a
lens
• Signal is modulated in a way that is dependent on
spatial frequency, defocus, wavelength, Cs
2
s
3 4
s
17. Contrast transfer function
• CTF is observed in the power spectrum of images as
a pattern of ripples
• Zero crossings in contrast vs spatial frequency
(phase inversion) – problematic at high defocus
Power spectrum (I=amp2)
18. Image restoration
• CTF correction initially by phase flipping
• Later, “sharpening” (negative B factor) to restore
amplitudes (correct for the Gaussian decay)
Recovered signal CTF Envelope function
(Gaussian decay)
Ideal signal
19. Collecting data at multiple defocus points
Ranson and Stockley, Emerging Topics in Physical Virology, 2010
20. Masking
• Clearly the mask influences the FSC curve/resolution estimate
• Not necessarily a bad thing
FSC curve from CryoSparc discussion board
22. Features expected at different resolutions
1 Å 2 Å 3.5 Å 5 Å
3.5 Å 5 Å 8 Å 12 Å
Mackay et al. TIBS 2017
23. EM vs X-ray resolution
• Despite efforts to “equalise” resolution estimates,
differences remain in what you can see at
apparently comparable resolutions
• In cryo-EM, we don’t have to guess the phases
• Cryo-EM maps represent the electrostatic potential
(not the electron density)
x,y,z ϕ,θ,ψ Amp Phs
XRD x
SPEM X x
24. The importance of phases
Kevin Cowtan’s structure factor tutorial
http://www.ysbl.york.ac.uk/~cowtan/sfapplet/sfintro.html
25. The interaction of electrons
• Negatively charged side chains tend to disappear
more quickly
BUT…
Bartesaghi et al. Science 2015
45 e/Å2
94,000 ptcls
(half thrown
out)
Only 12 e/Å2 in
the final map!
26. The interaction of electrons
• Negatively charged side chains tend to disappear
more quickly
Bartesaghi et al. PNAS 2014
27. There’s another problem…
• Conventional resolution estimates provide a single
number, which at best implies the “average
resolution” of the map
• Cryo-EM maps vary considerably in resolution
Resmap
Blocres
Monores
Local FSC estimates
calculated in
EMAN2, Sphire
28. Sato et al.
(& Fujiyoshi)
J Mol Biol, 2004
~100,000 particles
Cryo (He-cooled)
FSC0.5 = 20Å
JEM3000 @ 300kV
Film (LeafScan)
IMAGIC
da Fonseca et al.
(& Morris)
PNAS, 2003
~3,500 particles
2% UA stain
FSC3σ = 30Å
CM200 @ 200kV
Film (LeafScan)
IMAGIC
Serysheva et al.
(& Wah Chiu)
J Biol Chem, 2003
~9,500 particles
Cryo (N2-cooled)
FSC0.5 = 30Å
JEOL1200 @ 100kV
Film (Zeiss SCAI)
EMAN
Jiang et al.
(& Sigworth)
Embo J, 2002
~4,500 particles
Cryo (N2-cooled)
FSC0.5 = 24Å
T12 @ 120kV
Film (Zeiss SCAI)
EMAN
Hamada et al.
(& Mikoshiba)
J Biol Chem, 2003
~6,000 particles
1% UA stain
FSC0.5 = 34Å
JEM1200 @ 80kV
Film (???)
EMAN
Structure validation
30. Validation
• Class averages = Projections = Single Particles
• Class averages = Projections = Ref. Free 2D Averages
• Radermacher RCT test (confirms shape, v.low res features)
• Rosenthal & Henderson tilt pair test
• With different packages 20Å LPF volumes look similar
• At high resolution, can discern expected features
IP3R structure
36. Validation by tilt pair test
Bartesaghi (2013) NSMB 20:1352-7
Bartesaghi et al.
6Å resolution
Manual selection
of 114,000 particles
Mao et al.
6Å resolution
Automatic selection
of 600,000 particles
37. • Map validation – is my EM map correct?
• With the resolution revolution, this is becoming less
common
• Model validation
• How well does my model fit my map?
• Is my model of (relatively) good quality?
Validating maps vs validating models
38. • Probably true; why?
– In crystallography, you have an incentive to improve your
model (improved phase estimates > better map > model)
– In cryo-EM, model building commences after the map
refinements is finalised
39. http://www.lander-lab.com/convergence/
• For every structure deposited in EMDB (>5Å)
• Autobuild into density with Rosetta.
• Refine and calculate RMSD for 10 best structures.
• Date represented in the context of a global
validation plot
• Posits an argument for an ensemble of structures
rather than a single atomic model solution
Cryo-EM Convergence server
44. FSCs for validating models
“Gold standard” FSC Model/Map FSC
Rwork – assess resolution by comparing a model to a map built
from data that contributed to the model building process
Rfree – assess resolution by comparing a model to a map built
from data excluded from the original model building process
During gold standard refinement, two independent half maps
are built
Gao et al. Nature 2016
Model/final map FSC
Model/half map 1 FSC
Model/half map 2 FSC
45. Concluding remark
• When we determine structures by any method, we
are often compelled to ask two questions
• What is the structure?
• What is the resolution?
• Cryo-EM maps don’t (extremely rarely) have a
single, consistent resolution throughout the map
• Most biological structures are dynamic, not
constrained by crystal packing
• What is the biological insight provided by your
structure?
• An intangible quantity, probably not adequately
summarised by a single number
