Electron microscopy in molecular cell biology I · The wavelength of an electron moving at veolicty v (de Broglie wavelength) is λ = h / m v where h is Planck’s constant, m the

Werner Kühlbrandt

Max Planck Institute of Biophysics

Electron microscopy in molecular cell biology I

Electron optics and image formation

bio

logy

chem

istry

• Galaxy 106 light years 1022 m • Solar system light minutes 1010 m• Planet 20,000 km 107 m• Man 1.7 m 100 m• Fly 3 mm 10-3 m• Cell 50 µm 10-5 m• Bacterium 1 µm 10-6 m• Virus 100 nm 10-7 m• Protein 10 nm 10-8 m• Sugar molecule 1 nm 10-9 m• Atom 0.1 nm = 1Å 10-10 m• Electron 0.1 pm 10-13 m

Objects of interest

• Galaxy 106 light years 1022 m • Solar system light minutes 1010 m• Planet 20,000 km 107 m• Man 1.7 m 100 m• Fly 3 mm 10-3 m• Cell 50 µm 10-5 m• Bacterium 1 µm 10-6 m• Virus 100 nm 10-7 m• Protein 10 nm 10-8 m• Sugar molecule 1 nm 10-9 m• Atom 0.1 nm = 1Å 10-10 m• Electron 0.1 pm 10-13 m

Which objects can we see?R

esol

utio

n lim

it w

ith v

isib

le li

ght

0.1nm=1Å 1nm 10nm 100nm 1µm 10µm 100µm 1mm 10mm

AFM

x-ray crystallography

electron microscopy

light microscopy

organisms

cellsproteins

small molecules

atomsviruses

bacteria

Structural methods in molecular cell biology

NMR

Part 1:

Electron optics

Microscopy with electrons

• Electrons are energy-rich elementary particles

• Electrons have much shorter wavelength than x-rays (~ 3 pm = 3 x 10-12 m for 100kV electrons)

• Resolution not limited by wavelength

• but by comparatively poor quality of magnetic lenses and radiation damage

• High vacuum is essential - no live specimens!

Electron microscopes300 kV 120 kV

The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

viewing screen

projection chamber

specimen

camera

vacuum pump

electron source

specimen

objective lens


focus plane

diffraction plane

image plane

electron source

specimen

objective lens


focus plane

diffraction plane

image plane

Electron sources Thermionic emitters Tungsten filament: emits electrons at ~ 3000 °C (melts at 3380 °C) Large source size, poor coherence of electron beam

Lanthanum hexaboride (LaB6) crystal: single crystal, emits electrons at ~ 2000 °C Electrons are emitted from crystal tip -> smaller source size, better coherence Energy spread ΔE ~ 1 - 2 eV, depending on temperature

Field emitters Oriented tungsten single crystal, tip radius ~ 100 nm Small source size -> high coherence of electron beam, high brilliance Requires very high vacuum (10 -11 Torr) Electrons extracted from crystal by electric field applied to tip (extraction voltage)

Schottky emitter: Zr plated, heated to ~ 1200 °C to assist electron emission and prevent contamination, energy spread ΔE ~0.5 eV

Cold field emitter: not heated, very low energy spread, best temporal coherence, but contaminates easily (frequent ‘flashing’ with oxygen gas)

LaB6 vs FEG

FEG

LaB6

Field emission tip

LaB6 filament field emission tip

incoherent electron beam coherent electron beam

Coherence

temporal coherence: all waves have the same wavelength (they are monochromatic)

spatial coherence: all waves are emitted from the same point source (as in a laser)

According to the dualism of elementary particles in quantum mechanics, electrons can be considered as particles or waves. Both models are valid and used in electron optics.

Electrons can be regarded as particles to describe scattering.

The wave model is more useful to describe diffraction, interference, phase contrast and image formation.

Electrons: particles or waves?

Rest mass m0 = 9.1091x10−31

kgCharge Q = −e = −1.602x10−19

CKinetic energy E = eU,1eV = 1.602x10

−19Nm

Rest energy E0 = m0c 2= 511keV = 0.511MeV

Velocity of li ght c = 2.9979x108ms −1

Planck’s constant h = 6.6256x10−34Nms

Non-relativistic (E << E0) Relativistic (E ~ E0)

Mass m = m 0 m = m0 1− v 2 / c2( )−1 / 2

Energy E = eU =12

m0v2

mc 2= m0c2 + eU = E 0 + E

m = m0 1 + E / E0( )

Velocity v = 2E / m0( )1 / 2

v = c [1−1

1+ E / E0( )2

1 / 2

Momentum p = m 0v = 2m 0E( )1/ 2

p = mv = 2m0E 1+ E /2E0 ( )[ ]1 / 2

=1c

2EE0 + E2

( )1 / 2

deBrogli e wavelength λ =hp = h 2m0E( )

−1 / 2

λ = h 2m0 E 1+ E / 2E0( )[ ]− 1 / 2

=hc 2EE0 + E2

( )−1 / 2

Electron parameters

]

Electrons in an electrostatic (Coulomb) field

The force produced by an electrostatic field E on a particle of (negative) charge e- is

F = -eE

This means that the electron is accelerated towards the anode. The kinetic energy E [eV] of the electron after passing through a voltage U [V] is

E = eU

The wavelength of an electron moving at veolicty v (de Broglie wavelength) is

λ = h/mv where h is Planck’s constant, m the rest mass of the electron. Both the mass and the velocity of the electron are energy dependent. Relativistic treatment is necessary for acceleration voltages > 80 kV.

Electron beams are monochromatic except for a small thermal energy spread arising from the temperature of the electron emitter.

de Broglie wavelength

acceleration velocity wavelength voltage

100 kV 1.64 x 108 m/s 3.7 pm = 0.037 Å

200 kV 2.5 pm = 0.025 Å

300 kV 2.0 pm = 0.02 Å

Highest possible resolution:

1/d = 2/λ

d = λ/2

θ = 90°

Resolution limit in crystallographyReciprocal space!

2 sin θ = (1/d)/(1/nλ)

Resolution limit

Diffraction (Rayleigh) limit: ~ 0.5 λ

• ~150 nm = 1,500 Å for visible light• 0.5 Å for x-rays• 0.01 Å = 1 pm for 300 kV electrons

electron source

specimen

objective lens


focus plane

diffraction plane

image plane

E

E-ΔE

E E -ΔEE E -ΔE

hν

Z = 0

Z = t

E-ΔE

Incident beam of energy E

elastic inelastic

Electron-‐specimen interactions

R. Henderson, Quart. Rev. Biophys. 1995

scattering cross section

radiation damage

Scattering of electrons, X-rays and neutrons

~105

10 x 8000 eV

4 x 20 eV

carbon

~105

10 x 8000 eV

4 x 20 eV

Large elastic cross section • ~106 higher than for x-‐rays high signal per scattering event

… but even larger inelastic cross section • σinel/σel ≈ 18/Z high radiation damage, especially for light atoms, e.g. C: 3 electrons scattered inelastically per elastic event

Good ratio of elastic/inelastic scattering • 1000 times better than for 1.5 Å x-‐rays

carbon

Energy deposited per elastic event

Electrons 80-‐500 keV

X-‐rays 1.5 Å

Neutrons 1.8 Å

Ratio inelastic/elastic 3 10 0.08

Energy deposited per inelastic event

20 eV 8000 eV 2000 eV

Effective energy deposited per elastic event

60 eV 80 000 eV 160 eV

Considerations for biological specimens

• Biological specimens consist mainly of light atoms (H, N, C, O), which suffer most from radiation damage

• This makes it essential to minimize the electron dose to about ~25 e/Å2 for single-‐particle cryoEM

• Low electron dose means low image contrast

electron source

specimen

objective lens


focus plane

diffraction plane

image plane

Electrons in a magnetic field Lorentz force on a charge e- moving with velocity v in a magnetic field B (vectors are bold): F = -e (v x B)

The direction of F is perpendicular on v and B. v, B, and F form a right-handed system (right-hand rule, where v = index finger, B = middle finger and c = thumb)

In a homogenous magnetic field, if

v = 0 then F = 0

v is perpendicular to B: the electron is forced on a circle with a radius depending on the field strength |B|

v is parallel to B: no change of direction

Note that unlike the electric field, the magnetic field does not change the energy of the electron, i. e. the wavelength does not change!

Electromagnetic field

Passing a current through a coil of wire produces a strong magnetic field in the centre of the coil. The field strength depends on the number of windings and the current passing through the coil.

Electromagnetic Lens

Pole pieces of soft ironconcentrate the magnetic field

Electromagnetic Lens

The magnetic field is rotationally symmetric and changes gradually

A light beam is refracted (bent) when the wave enters a medium of a different optical density. The refractive index changes abruptly at the interface.

Refraction in light optics

Path of electron moving with velocity v in magnetic field B of a ring coil, exerting force F

Diagram by Elena Orlova, Birkbeck College London

Magnetic lens

Magnetic lenses are electromagnetic coils with soft iron pole pieces. The strength of the magnetic field depends on the number of windings NI and on the pole piece gap. The field is cylindrically symmetrical about the optical axis. The field strength B(z) along the optical axis is bell-shaped. Upon passing through the lens, the image rotates around the optical axis.

The light microscope is focussed by varying the object distance. The electron microscope is focussed by varying the focal length of the lens f.

The lens diagram looks the same for glass and magnetic lenses

grid square with holey carbon film

20 µm x 20 µm

600x

EM grid

2mm x 2mm

hole

2µm x 2µm

6,000x60x 60,000x

FAS in vitreous buffer

0.2 µm x 0.2 µm

Magnification steps

electron source

specimen

objective lens


focus plane

diffraction plane

image plane

Electron diffraction

diffraction angle

2θ

Bragg’s law: nλ = 2d sinθ

d

Electron diffraction

diffraction angle

2θ


d

Elastic scattering by atoms (Rayleigh scattering)

n is an integer: scattered waves are in phase, resulting in constructive interference


n is not an integer: scattered waves are out of phase, resulting in destructive interference

θd

path length difference nλ x 2 (sin θ = nλ/d)

Interference

constructive interference destructive interference

superposition

1st wave

2nd wave

Highest possible resolution:

1/d = 2/λ

d = λ/2

θ = 90°

Resolution limit in crystallographyReciprocal space!

2 sin θ = (1/d)/(1/nλ)

Large 2D crystal of LHC-II

2D crystals of LHC-II

Kühlbrandt, Nature 1984

Electron diffraction pattern: amplitudes, no phases

3.2 Å

The image is generated by interference of the direct electron beam with the diffracted beams

Part 2:

Phase contrast and image formation

electron source

specimen

objective lens

diffraction plane

image plane


focus plane(UNDERfocus, i.e. weaker objective lens)

E

E-ΔE

E E -ΔEE E -ΔE

hν

Z = 0

Z = t

E-ΔE

Incident beam of energy E

elastic inelastic

Electron-‐specimen interactions

Contrast mechanisms

no interaction

phase object

amplitude object

modified phase

electron wave

modified amplitude, unchanged phase

modified phase, unchanged amplitude

Image

amplitude contrast

phase contrast shorter

wavelength within object

The scattered electron wave experiences three phase shifts

by the imaged object this carries the information about the structure and is the one we need to determine

by the inner potential of the specimen this is 90° for all scattered waves

by the spherical aberration of the objective lens this depends on the scattering angle (= resolution) and gives rise to the oscillation of the contrast transfer function (CTF)

(note that this only holds for the weak phase approximation!)

Origin of the phase shift

€

The potential V(x,y,z) experienced by the scattered electron upon passing through a sample of thickness t causes a phase shift

The vacuum wavelength λ of the electron changes to λ` within the sample:

When passing through a sample of thickness dz, the electron experiences a phase shift of:

€

Vt (x,y) = V (x,y,z)dz0

t

∫

€

λ =h2meE

€

" λ =h

2me(E +V (x,y,z))

€

dϕ = 2π (dz$ λ −dzλ)

€

dϕ =πλE

V (x,y,z)dz =σV (x,y,z)dz

ψincident

ψexit

V(x,y,z)

(how many times does λ fit into the thickness dz?)(by how many degrees out of 2π = 360° does this change the phase φ?)

The potential V of the specimen shift modifies the phase of the incident plane wave ψincident to:

àFor thin biological specimens there is a linear relation between the transmitted exit wave function and the projected potential of the sample:

€

ϕ =σ V (x,y,z)dz∫ =σVt (x,y)As we have seen, the phase shift is proportional to the projected potential of the specimen:

€

Ψexit =Ψincidente− iϕ =Ψincidente

−iσVt

€

Ψexit ≈ Ψincident

≡1(background )! " # (1− iϕ) =1− iϕ =1− iσVt

€

Ψexit =Ψincidente− iϕ

≈ Ψincident (1− iϕ) =Ψincident − iΨincidentϕ =Ψincident − iΨscattered

For Vt<<1 and thus φ<<2π as for thin biological specimens:

€

i = eiπ2 = cos(π2 ) + isin(π2 )

ψincident

ψexit

V(x,y,z)

The weak phase approximation

Taylor series

€

f (n )(a)n!n=0

∞

∑ (x − a)n

€

sin(x) ≈ x − x3

3!+x 5

5!−x 7

7!+O(xn )

€

ex ≈1+ x +x 2

2!+x 3

3!+O(xn )

€

e−iϕ ≈1− iϕ +ϕ2

2!− iϕ

3

3!+O(ϕn )

≈1− iϕ if φ << 1 !

taken from Wikipedia...

Any function can be approximated by a finite number of polynomial terms

à

à

à

à

Phase contrastexit wave = scattered wave + unscattered wave

exit wave

imaginary

real

Ψexit = ψunscattered + iψscattered

Vector notation. Vector length shows amplitude (not drawn to scale)

Cosine wave notation (amplitudes not drawn to scale)

Scattered wave is 90° out of phase with unscattered wave.

Phase contrastexit wave = scattered wave + unscattered wave

à Positive phase contrast A perfect lens in focus gives minimal phase contrast

Since ψscattered = φψunscattered with φ << 1:

ψscattered << ψunscattered

Iimage= |ψexit|2 =| ψunscattered + iψscattered|2 ≈ |ψunscattered|2 = Iincident

exit wave


Lens aberrations cause an additional phase shift that generates phase contrast

Minimal contrast

Minimal contrast

Maximal contrast

Defocus phase contrast

exit wave = scattered wave + unscattered wave

Phase contrast

Lens aberrations

Spherical aberration Chromatic aberration

Lens with spherical aberration

Image plane

Perfect lensImage plane

f1f2

χ = e.g. 3λ/2

Spherical aberration causes a path length difference χ which depends on the scattering angle

In a perfect lens, the number of wavelengths from object to focal point is the same for each ray, independent of scattering angle (Fermat’s principle)

Defocus generates an additional phase shift

Depending on its phase shift, the scattered wave reinforces or weakens the exit wave


exit wave = scattered wave + unscattered wave

scattered and unscattered wave are in phase: exit wave gets stronger

scattered and unscattered wave are out of phase by π = 180° (half a wavelength): exit wave gets weaker

scattered wave is shifted by π/2 = 90° (quarter of a wavelength) relative to unscattered wave: exit wave is nearly same as unscattered wave, therefore phase contrast is minimal

The phase shift χ(θ) introduced by the spherical aberration Cs of the objective lens depends on the scattering angle θ and on the amount of defocus Δf.

χ(θ) = (2π/λ) [Δf θ2/2 - Cs θ4/4]

The spatial frequency R (measured in Å -1) is the reciprocal of the distance between any two points of the specimen. For small scattering angles, R ≈ θ / λ. With this approximation, the phase shift expressed as a function of R is

χ(R) = (π/2) [2Δf R2 λ - Cs R4 λ3] (Scherzer formula)

At Scherzer focus, the terms containing Δf and Cs nearly add up to 1, and a positive phase shift of χ ≈ +π/2 applies to a wide resolution range.

Optimal (Scherzer) focus

Defocus and spherical aberration have similar but opposite effects

exact focus

Scherzer focus

high underfocus

Contrast transfer in

(Erikson and Klug, 1971)

The phase Contrast Transfer Function B(θ) of the electron microscope is defined as

B(θ) = -2 sin χ(θ)

For any particular defocus Δf, the CTF indicates the range of scattering angles θ in which the object is imaged with positive (or negative) phase contrast.

For scattering angles where CTF = 0, there is no phase contrast.

As spherical aberration Cs is constant, Δf can be adjusted to produce maximum phase contrast of ±π/2 for a particular range of scattering angles.

The contrast transfer function (CTF)

Fourier transforms of amorphous carbon film images

Contrast Transfer Function, CTF

underfocus underfocus

FT FT

CTF as a function of defocus

Movie by Henning Stahlberg, UC Davis

Phase contrast at high resolution is reduced (damped) by incoherence

Spatial and temporal incoherence of the electron beam damp the CTF due to lens aberrations:

Edamping(k) = Espatial(Z,Cs,α,k) • Etemporal(Cc,ΔE,k) • Especimen• Edetector• ….

CS is the spherical aberration coefficient, CC is the coefficient for chromatical aberration

spatial frequency [1/nm]

Z=100 nm, CS= 3 mm, CC = 3 mm Z=100 nm, CS= 0 mm, CC = 3 mm Z=100 nm, CS= 0 mm, CC = 0 mm

Chromatic aberration (Cc)

corrector

Max Haider

electron source

specimen

objective lens

diffraction plane

image plane


focus plane

Janet Vonck, Deryck Mills

Film image

CCD vs. direct detector

300 kV electrons

conventional CCD detector

Direct electron detector

Richard Henderson, Greg McMullan (MRC-LMB Cambridge, UK)

Scintillator

Fiber optics

CCD chip

Peltier coolerSeparate vacuum

-35°C

50 µm• CMOS active pixel sensor• 5 - 15 µm pixels• radiation-hard• back-thinned• rapid readout (17 - 400 Hz)

DQE of electron detectors (300 kV)

McMullan, Faruqi and Henderson http://arxiv.org/pdf/1406.1389.pdf

DQE = SNR2o/SNR2i

counting mode,

CCD

Greg McMullan, MRC LMB Cambridge

http://arxiv.org/pdf/1406.1389.pdf

Matteo Allegretti, Janet Vonck, Deryck Mills

Fast readout overcomes drift

Frame displacement in Å

3.36 Å

Science, 28 March 2014Science, 28 March 2014

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Electron microscopy in molecular cell biology I · The wavelength of an electron moving at veolicty v (de Broglie wavelength) is λ = h / m v where h is Planck’s constant, m the