Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I...

97
Werner Kühlbrandt Max Planck Institute of Biophysics Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14

Transcript of Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I...

Page 1: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Werner Kühlbrandt

Max Planck Institute of Biophysics

Electron microscopy in molecular cell biology I

Electron optics and image formation

Donnerstag, 3. Juli 14

Page 2: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

• 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

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Page 3: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

bio

logy

• 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

Donnerstag, 3. Juli 14

Page 4: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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

Donnerstag, 3. Juli 14

Page 5: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

• 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?

Donnerstag, 3. Juli 14

Page 6: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

• 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

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Page 7: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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

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Part 1:

Electron optics

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Page 9: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Microscopy with electrons

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Page 10: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Microscopy with electrons

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Page 11: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Microscopy with electrons

• Electrons are energy-rich elementary particles

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Page 12: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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)

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Page 13: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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!

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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

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Page 15: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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!

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Electron microscopes300 kV 120 kV

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The transmission electron microscope

camera

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Page 18: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

camera

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Page 19: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

condenser

camera

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The transmission electron microscope

electron gun

objective lens

condenser

camera

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The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

camera

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Page 22: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

projection chamber

camera

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Page 23: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

projection chamber

camera vacuum pump

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Page 24: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

viewing screen

projection chamber

camera vacuum pump

Donnerstag, 3. Juli 14

Page 25: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

viewing screen

projection chamber

camera

camera

vacuum pump

Donnerstag, 3. Juli 14

Page 26: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron gun

projector lenses

objective lens

condenser

viewing screen

projection chamber

specimen camera

camera

vacuum pump

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Page 27: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron source

specimen

objective lens

diffraction plane

image planefocus plane

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Page 28: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

electron source

specimen

objective lens

The transmission electron microscope

focus plane

diffraction plane

image plane

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Page 29: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Electron sourcesThermionic emittersTungsten filament: emits electrons at ~ 3000 °C (melts at 3380 °C)Large source size, poor coherence of electron beam

LaB6 crystal: single crystal, emits electrons at ~ 2000 °CElectrons are emitted from crystal tip -> smaller source size, better coherenceEnergy spread ΔE ~ 1 - 2 eV, depending on temperature

Field emitters Oriented tungsten single crystal, tip radius ~ 100 nmSmall source size -> high coherence of electron beam, high brillianceRequires 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)

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Page 30: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

tungsten 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)

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Page 31: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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?

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Page 32: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Rest mass m0 = 9.1091x10

−31kg

Charge Q = −e = −1.602x10−19C

Kinetic energy E = eU,1eV = 1.602x10−19

NmRest energy E0 = m0c 2 = 511keV = 0.511MeVVelocity of li ght c = 2.9979x108

ms −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

de Brogli e wavelength λ =hp = h 2m0E( )

−1 / 2λ = h 2m0 E 1+ E / 2E0( )[ ]

− 1 / 2

=hc 2EE0 + E2

( )−1 / 2

Electron parameters

]

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Page 33: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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.

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de Broglie wavelength

acceleration velocity wavelengthvoltage

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

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Page 35: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Highest possible resolution:1/d = 2/λ

d = λ/2

θ = 90°

Resolution limit in crystallography

Reciprocal space!

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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

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Page 37: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

specimen

The transmission electron microscope

electron source

objective lens

focus plane

diffraction plane

image plane

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Page 38: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

E

E-ΔE

E E -ΔEE E -ΔE

Z = 0

Z = t

E-ΔE

Incident beam of energy E

elastic inelastic

Electron-­‐specimen  interactions

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Page 39: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Elastic and inelastic scattering cross sections for electrons, x-rays and neutrons

R. Henderson, Quart. Rev. Biophys. 1995

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Page 40: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Electron  scattering  factors

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

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Considerations  for  biological  specimens

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

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Page 42: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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

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Page 43: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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

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Page 44: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron source

specimen

objective lens

focus plane

diffraction plane

image plane

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Page 45: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Electrons in a magnetic fieldLorentz 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!

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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.

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Electromagnetic Lens

Pole pieces of soft ironconcentrate the magnetic field

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Page 48: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Electromagnetic Lens

Pole pieces of soft ironconcentrate the magnetic field

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Electromagnetic Lens

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Page 51: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

EM grid

2mm x 2mm

60x

Magnification steps

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Page 52: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

grid square with holey carbon film

20 µm x 20 µm

600x

EM grid

2mm x 2mm

60x

Magnification steps

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Page 53: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

grid square with holey carbon film

20 µm x 20 µm

600x

EM grid

2mm x 2mm

hole

2µm x 2µm

6,000x60x

Magnification steps

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Page 54: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

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

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Page 55: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

The transmission electron microscope

electron source

specimen

objective lens

focus plane

diffraction plane

image plane

Donnerstag, 3. Juli 14

Page 56: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Electron diffraction

diffraction angle

Bragg’s law: nλ = 2d sinθ

d

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Page 57: Electron microscopy in molecular cell biology I · Electron microscopy in molecular cell biology I Electron optics and image formation Donnerstag, 3. Juli 14 ... light microscopy

Electron diffraction

diffraction angle

Bragg’s law: nλ = 2d sinθ

d

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Interference

constructive  interference destructive  interference

superposition

1st  wave

2nd  wave

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Elastic scattering by atoms (Rayleigh scattering)

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

Bragg’s law: nλ = 2d sinθ

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

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2D crystals of LHC-II

Kühlbrandt, Nature 1984

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Electron diffraction pattern: amplitudes, no phases

3.2 Å

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

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Part 2:

Image formation

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electron source

specimen

objective lens

diffraction plane

image plane

The transmission electron microscope

focus plane

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E

E-ΔE

E E -ΔEE E -ΔE

Z = 0

Z = t

E-ΔE

Incident beam of energy E

elastic inelastic

Electron-­‐specimen  interactions

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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

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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?)

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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

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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

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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

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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.  

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Phase  contrastexit  wave                        =              scattered  wave        +            unscattered  wave        

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

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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

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exit  wave

Ψexit  =  ψunscattered  +  iψscattered

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

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Lens  aberrations

Spherical  aberration Chromatic  aberration

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Perfect lensImage plane

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

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Lens with spherical aberration

Image plane

Perfect lensImage plane

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

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Lens with spherical aberration

Image plane

Perfect lensImage plane

f1

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

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Lens with spherical aberration

Image plane

Perfect lensImage plane

f1f2

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

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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

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

Ψexit  =  ψunscattered  +  iψscattered

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  =  90°:    exit  wave  gets  weaker

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

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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

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Defocus  and  spherical  aberration  have  similar  but  opposite  effects

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exact focus

Scherzer focus

high underfocus

Contrast transfer in

(Erikson and Klug, 1971)Donnerstag, 3. Juli 14

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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)

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Fourier transforms of amorphous carbon film imagesunderfocus underfocus

FT FT

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Fourier transforms of amorphous carbon film images

Contrast Transfer Function, CTF

underfocus underfocus

FT FT

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CTF as a function of defocus

Movie by Henning Stahlberg, UC Davis

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CTF as a function of defocus

Movie by Henning Stahlberg, UC Davis

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electron source

specimen

objective lens

diffraction plane

image plane

The transmission electron microscope

focus plane

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Janet Vonck, Deryck Mills

Film image

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DQE of direct electron detectors

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

DQE = SNR2o/SNR2i

counting mode, super-resolution

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CCD vs. direct detector

300 kV electrons

conventional CCD detector

Falcon-II direct electron detector

50 µm

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

Scintillator

Fiber optics

CCD chip

Peltier coolerSeparate vacuum

-35°C

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CCD vs. direct detector

300 kV electrons

conventional CCD detector

Falcon-II direct electron detector

50 µm

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

Scintillator

Fiber optics

CCD chip

Peltier coolerSeparate vacuum

-35°C

4000 x 4000 14 µm pixelsread out 17 times per second

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Matteo Allegretti, Janet Vonck, Deryck Mills

Fast readout overcomes drift

Frame displacement in Å

3.36 Å

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Science, 28 March 2014Science, 28 March 2014

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Science, 28 March 2014Science, 28 March 2014

Donnerstag, 3. Juli 14