Theoretical minimum

1. Cryo-EM: The Theoretical Minimum
Hans & Dominika Elmlund
Homepage: http://simplecryoem.com

2. First Born approximation,
interpretable quantity is:
Image at t and 2t should
look the same!
Preliminaries—Harmonic Waves
Angular
Freq.
PhaseAmplitude

3. Preliminaries—The New Complex Wave
A complex exponential
behaves better!
(physical meaning: rotation
in the complex plane, as we
will see shortly)

4. It should look like a wave-shape in 3D and
oscillate at every point (time-dependency)

5. Preliminaries—the Fourier Transform

6. Preliminaries—the Fourier Transform

7. Ernst Ruska, 1931 FEI, 2013
The Electron Microscope
 Very bright &
coherent electrons
from FEG:s
 Reduced specimen
movement from very
stable stages
 Movie recording
detectors with better
DQE than film
 The technology
required for atomic
resolution single-
particle cryo-EM is
finally here!

8. Image Formation in the TEM
Specimen = uniform collection
of atoms with nuclei centered at
λ=wavelength
λ’
 When the electron wave is
traversing the specimen its
wavelength is altered in a manner
that depends on the specimen
potential. The incident wave
undergoes a phase change.
 Can view the TEM as an
interferometer
 The out-of-focus image is a
coherent interference pattern
 The image is band-limited
 If the specimen is thin, the
Fourier coefficients represent the
projected electric potential

9. Expression for the Altered Wavelength
λ
λ’

10. Preliminaries—Taylor Expansion
The Taylor series of a real or complex-valued
function ƒ(x) that is infinitely differentiable at a real
or complex number a is the power series:
Specifically, we will expand functions around zero,
so the Taylor series simplifies to the MacLaurin
series:

11. Weak potential approximation: since the specimen potential is much
weaker than U (few V vs. hundreds of kV) we can truncate the Taylor series
Describing the Phase Change
Phase change of wave that traversed specimen:
Small angle approximation: With high electron
energies and thin specimens, it is safe to assume that
the scattered wave is close-to-parallel to the incident
wave (i.e. dz ≈ dz’)

12. The Weak Phase Approximation
The interaction constant σ describes the strength
of the interactions of electrons with matter.
The exit wave is a phase-shifted incident wave
Since the scattering angle is small, the phase change is small (<< 1).
Hence, we can develop the exit wave into a converging Taylor series
The weak phase object approximation refers to truncation of the expansion after
the second term, assuming no higher order scattering. The model enforces a
simple linear relationship between the exit wave function and the specimen

13. The Fourier Transform of the Exit Wave
Rewriting the expression using Euler’s formula
we see that the amplitude of the scattered wave is proportional to the amplitude of
the structure factor, while the phase is shifted by 90 degrees with respect to the
structure factor phase.
structure
factor

14. This means that all the scattered electrons coming from one point in the object plane
will be focused onto one point in the image plane with exactly the same phase.
Consequently: NO IMAGE CONTRAST! Luckily, we have aberrations…
First Born Approximation of the Exit Wave
Gives us NOTHING!
Because the
interaction constant
is very small

15. (exponential of the radially symmetric lens
aberration function (Scherzer 1949))
The Contrast Transfer Function
(CTF)
Electron waves propagating along different directions are focused to
different planes. This phenomenon is called spherical aberration.
Standard optical transfer theory can be used to describe the effects of
lens aberrations by additionally shifting the phase of the exit wave
according to a defocus-dependent function (the CTF)
What does the ctf mean in real life???

16. The CTF–dependent Parameters
Used for Processing Cryo-EM
Images
 Sampling distance or pixel size (smpd/apix) in
units of Å (derived from mag and detector
pixel-size)
 Acceleration voltage (U) in kV (e.g. keV, where
1 eV is 1.6e10-19 J)
 Spherical aberration constant (Cs) in units of
mm
 Defocus (D) in units of micrometers or Å
 Astigmatism parameters (that we ignore 4
now)
Constant machine-
dependent parameters
Parameters that vary
in somewhat
controllable ways
But how do these parameters influence the
image? You wouldn’t run an assay without
knowing how it works, would you?

17. WITH
Aberrations
the First Born
Approximation
of the Exit
Wave gives us
something
sensible!
Let’s go
through the
calculation…

18. We can equivalently express the
real exit wave with the inverse
Fourier transform of the Fourier
transformed exit wave
The complex conjugation of the
exponential term shifts the sign
Remember the formula for F-1?

19. The complex conjugation
shifts the sign
Plugging in the expression for the Fourier
transform of the exit wave with aberrations

20. Taking variables that do not depend
on the integration variable outside of
the integration sign
Integration of a delta
function over all space is 1

21. First we do the variable substitution u to -u in
the right-hand integral. The complex conjugate
of the structure factor then becomes
Then we use Friedel’s symmerty, which is a
property of all Fourier transforms of real-
valued objects
We can leave the
sign of Scherzer’s
aberration function
because it is radially
symmetric
The exponential
becomes sign-shifted
by the variable
substitution

22. Re-express according to Euler’s formula:
The complex terms
cancel in the
multiplication
We remove the square terms
because the interaction
constant is small

23. We recognize this as the inverse Fourier transform
of the structure factor multiplied with the sine of the
lens aberration function

24. The Fourier transform of the square
modulus of the exit wave follows directly
Central spot of the
Fourier transform
Interaction
constant
Lens
Aberration
Function
Structure
factor

25. Assumptions That Went Into This Model
 Real-valued scattering amplitudes. Not true for
heavier elements.
 Consequently: model describes phase-
contrast only. Works well for light elements
(CHNO) (where the fraction of amplitude
contrast is around 5%) but not for heavier
elements.
 All atoms of the same (light) kind.
 Kinematic scattering (can truncate the series
expansion of the exit wave after the linear
terms). Well-known not to be true for heavy
elements or thick samples. Even a single gold
atom breaks this.

26. What About the Lens Aberration
Function (Scherzer 1949)?
?
We can describe it using scalar spatial frequency
because we assume (here) radial symmetry:
Defocus Electron
wavelengh
t
Spherical
Aberration
Constant
Spatial
frequency
 First term: linear dependency
on D and λ. Quadratic
dependency on u
 Second term: Linear
dependency on Cs, cubic
dependency on λ, and fourth
power dependency on u
 But why???

27. Recording an image with the objective lens
at an under-focus value of –D is equivalent
to moving the detector closer to the lens by
the distance D.
Path Difference
Due to
Underfocus

28. (Small angle
approximation)
The path difference ΔS(θ)
between a ray traveling from the
under-focused image plane to the
ideal image plane at an angle θ
and a ray travelling between the
same planes along the optical
axis is given by

29. (Small angle
approximation)
Path Difference
Due to
Underfocus

30. The corresponding phase difference between a ray traveling from the under-focused
image plane to the ideal image plane at an angle θ
Using the condition for constructive interference of waves scattered by a grating we
can express the phase-difference as a function of spatial frequency u in the object
plane.
Phase difference due to under-focus:
Diffraction by a Grating:
Phase-shift Due to Underfocus

31. rays travelling through the lens far
from the optical axis will be
deflected more strongly than rays
travelling close to the optical axis
Path Difference
Due to Spherical
Aberration

32. Path Difference
Due to Spherical
Aberration

33. Path Difference
Due to Spherical
Aberration
Analogous the the path difference due
to defocus, the path difference due to
spherical aberration becomes:

34. Path Difference
Due to Spherical
Aberration
How does S(θ) depend on the
scattering angle θ?
 In the small-angle approximation
we can develop this dependency
into a power series
 The field of the lens is radially
symmetric (not true but we don’t
take astigmatism into account
here), so we have symmetry
properties: If θ=0 then S(θ)=0
and if S (θ)=x then S(-θ)=x.
 Linear function doesn’t work but
a quadratic function would do
just fine!

35. Path Difference
Due to Spherical
Aberration

36. The Total Phase Shift
(Defocus+Spherical Aberration)
 Phase shift due to defocus:
 Phase shift due to spherical aberration:
 Total phase shift:
 Introducing Cs=2C you recognize Scherzer’s aberration function:

37. Shot noise+
detector noise
modelled well
by these two
Gaussian noise
sources
Added an
envelope function
due to temporal
coherence loss
All the other
stuff you know
by now!
The Platonic Cryo-EM Image
This is why you should
high-pass filter
images used for
alignment
Pink
noise
White
noise

38. Why Bother?
 The model dictates what can be done. Perhaps
your problem domain is not well-described by this
model.
 Most software packages are black boxes (and no
good descriptions of exactly what is done are
available—you need to read the code)
 We cannot allow this knowledge to die and it is
already happening. Most books contain painfully
flawed descriptions and most microscopists have
no clue.
 Analogy to the biochemist: you can always
follow Qiagen-kit protocols without understanding
but what happens if you need to troubleshoot?

39. Two GroEL Maps,
Which One Is Better?
Side-view
Top-view

40. F20 200 kV
Camera: DDD
70,000 images
FSC(0.5) = 6.1 Å
Software: EMAN
F20 200 kV
Camera: CCD
50,000 images
FSC(0.143) = 6.8 Å
Software: PRIME
Carragher B &
co-workers 2013
Elmlund H,
unpublished
Software Matters!

41. Elmlund H,
unpublished
Fit of the X-ray
Structure to the
PRIME GroEL
Map

42. Bragg’s Law