Electron microscopy in molecular cell biology I

Electron microscopy in molecular cell biology I Electron optics and image formation Werner Kühlbrandt Max Planck Institute of Biophysics

chemistry biology Objects of interest Galaxy 10 6 light years 10 22 m Solar system light minutes 10 10 m Planet 20,000 km 10 7 m Man 1.7 m 10 0 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? Resolution limit with visible light Galaxy 10 6 light years 10 22 m Solar system light minutes 10 10 m Planet 20,000 km 10 7 m Man 1.7 m 10 0 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

Structural methods in molecular cell biology light microscopy AFM NMR x-ray crystallography electron microscopy 0.1nm=1Å 1nm 10nm 100nm 1µm 10µm 100µm 1mm 10mm atoms proteins bacteria organisms small molecules viruses cells

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 microscopes 300 kv 120 kv

The transmission electron microscope electron gun condenser objective lens specimen projector lenses vacuum pump projection chamber viewing screen camera

The transmission electron microscope electron source specimen objective lens diffraction plane image plane focus 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 (LaB 6 ) 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 LaB6 FEG

Field emission tip

Coherence LaB6 filament field emission tip incoherent electron beam coherent electron beam 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)

Electrons: particles or waves? 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.

Electron parameters Rest mass Charge Kinetic energy Rest energy Velocity of li ght Planck s constant m0 = 9.1091x10 31 kg Q = e = 1.602x10 19 C E = eu,1ev = 1.602x10 19 Nm E0 = m0c 2 = 511keV = 0.511MeV c = 2.9979x10 8 ms 1 h = 6.6256x10 34 Nms Non-relativistic (E << E0) Relativistic (E ~ E0) Mass m = m 0 m = m0( 1 v 2 / c 2 ) 1 / 2 Energy E = eu = 1 2 m0v 2 mc 2 = m0c 2 + eu = E 0 + E m = m0 ( 1 + E / E0 ) Velocity Momentum de Brogli e wavelength v = ( 2E / m0) 1 / 2 v = c [1 1 ( 1+ E/ E 0 ) 2 p = m 0v = ( 2m 0E) 1/ 2 p = mv = [ 2m0E ( 1+ E /2E0)] 1 / 2 = 1 c 2EE0 + E 2 ( ) 1 / 2 λ = h p = h( 2m0E ) 1 / 2 λ = h[ 2m0 E( 1+ E/ 2E0 )] 1 / 2 ] 1 / 2 =hc 2EE0 + E 2 ( ) 1 / 2

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 10 8 m/s 3.7 pm = 0.037 Å 200 kv 2.5 pm = 0.025 Å 300 kv 2.0 pm = 0.02 Å

Resolution limit in Reciprocal space! crystallography Highest possible resolution: 1/d = 2/λ d = λ/2 2 sin θ = (1/d)/(1/nλ) θ = 90

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

The transmission electron microscope electron source specimen objective lens diffraction plane image plane focus plane

Electron- specimen interactions Incident beam of energy E hν E-ΔE E-ΔE Z = 0 Z = t E E E E -ΔE E -ΔE elastic inelastic

Scattering of electrons, X-rays and neutrons carbon 4 x 20 ev scattering cross section radiation damage 10 x 8000 ev ~10 5 R. Henderson, Quart. Rev. Biophys. 1995

Large elastic cross section ~10 6 higher than for x- rays high signal per scattering event but even larger inelastic cross section σ inel /σ el 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 10 x 8000 ev 4 x 20 ev ~10 5

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 Effective energy deposited per elastic event 20 ev 8000 ev 2000 ev 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/å 2 for single- particle cryoem Low electron dose means low image contrast

The transmission electron microscope electron source specimen objective lens diffraction plane image plane focus 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 iron concentrate the magnetic field

Electromagnetic Lens The magnetic field is rotationally symmetric and changes gradually

Refraction in light optics 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.

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 lens diagram looks the same for glass and magnetic lenses The light microscope is focussed by varying the object distance. The electron microscope is focussed by varying the focal length of the lens f.

Magnification steps 60x 600x 6,000x 60,000x EM grid grid square with holey carbon film hole FAS in vitreous buffer 2mm x 2mm 20 µm x 20 µm 2µm x 2µm 0.2 µm x 0.2 µm

The transmission electron microscope electron source specimen objective lens diffraction plane image plane focus plane

Electron diffraction d 2θ diffraction angle Bragg s law: nλ = 2d sinθ

Elastic scattering by atoms (Rayleigh scattering) d θ path length difference nλ x 2 (sin θ = nλ/d) 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 Bragg s law: nλ = 2d sinθ

Interference constructive interference destructive interference superposition 1 st wave 2 nd wave

Resolution limit in Reciprocal space! crystallography Highest possible resolution: 1/d = 2/λ d = λ/2 2 sin θ = (1/d)/(1/nλ) θ = 90

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

The transmission electron microscope electron source specimen objective lens diffraction plane image plane focus plane (UNDERfocus, i.e. weaker objective lens)

Electron- specimen interactions Incident beam of energy E hν E-ΔE E-ΔE Z = 0 Z = t E E E E -ΔE E -ΔE elastic inelastic

Contrast mechanisms Image no interaction electron wave amplitude object modified amplitude, unchanged phase amplitude contrast phase contrast phase object shorter wavelength within object modified phase modified phase, unchanged amplitude

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 ψ incident V t (x, y) = t 0 V (x, y,z)dz V(x,y,z) ψ exit The vacuum wavelength λ of the electron changes to λ` within the sample: h h λ = 2meE λ " = 2me(E +V (x,y,z)) When passing through a sample of thickness dz, the electron experiences a phase shift of: dϕ = 2π( dz λ $ dz λ ) dϕ = π V (x,y,z)dz = σv (x, y,z)dz λe (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 weak phase approximation As we have seen, the phase shift is proportional to the projected potential of the specimen: ϕ = σ V (x,y,z)dz = σv t (x, y) The potential V of the specimen shift modifies the phase of the incident plane wave ψ incident to: ψ incident Ψ exit = Ψ incident e iϕ = Ψ incident e iσv t For V t <<1 and thus φ<<2π as for thin biological specimens: V(x,y,z) Ψ exit = Ψ incident e iϕ i = e i π 2 = cos( π 2) + isin( π 2) Ψ incident (1 iϕ) = Ψ incident iψ incident ϕ = Ψ incident iψ scattered ψ exit à For thin biological specimens there is a linear relation between the transmitted exit wave function and the projected potential of the sample: Ψ exit Ψ incident! " # (1 iϕ) =1 iϕ =1 iσv t 1(background )

Taylor series Any function can be approximated by a finite number of polynomial terms à n =0 f (n ) (a) n! (x a) n à à à sin(x) x x 3 e x 1+ x + x 2 3! + x 5 5! x 7 2! + x 3 3! + O(x n ) 7! + O(x n ) e iϕ 1 iϕ + ϕ2 2! i ϕ3 3! + O(ϕn ) 1 iϕ if φ << 1! taken from Wikipedia...

Phase contrast exit wave = scattered wave + unscattered wave imaginary exit wave Scattered wave is 90 out of phase with unscattered wave. Ψ exit = ψ unscattered + iψ scattered real Vector notation. Vector length shows amplitude (not drawn to scale) Cosine wave notation (amplitudes not drawn to scale)

Phase contrast exit wave = scattered wave + unscattered wave A perfect lens in focus gives minimal phase contrast à Positive phase contrast Since ψ scattered = φψ unscattered with φ << 1: ψ scattered << ψ unscattered I image = ψ exit 2 = ψ unscattered + iψ scattered 2 ψ unscattered 2 = I incident

Phase contrast exit wave = scattered wave + unscattered wave exit wave Ψ exit = ψ unscattered + iψ scattered Minimal contrast Maximal contrast Defocus phase contrast Minimal contrast Lens aberrations cause an additional phase shift that generates phase contrast

Lens aberrations Spherical aberration Chromatic aberration

Defocus generates an additional phase shift Perfect lens Image 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) Lens with spherical aberration Image plane Spherical aberration causes a path length difference χ which depends on the scattering angle f 2 f 1 χ = e.g. 3λ/2

Depending on its phase shift, the scattered wave reinforces or weakens the exit wave exit wave = scattered wave + unscattered wave Ψ exit = ψ unscattered + iψ scattered scattered and unscattered wave are in phase: exit wave gets stronger 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 scattered and unscattered wave are out of phase by π = 180 (half a wavelength): exit wave gets weaker

Optimal (Scherzer) focus The phase shift χ(θ) introduced by the spherical aberration C s of the objective lens depends on the scattering angle θ and on the amount of defocus Δf. χ(θ) = (2π/λ) [Δf θ 2 /2 - C s θ 4 /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 R 2 λ - C s R 4 λ 3 ] (Scherzer formula) At Scherzer focus, the terms containing Δf and C s nearly add up to 1, and a positive phase shift of χ +π/2 applies to a wide resolution range.

Defocus and spherical aberration have similar but opposite effects

Contrast transfer in exact focus Scherzer focus high underfocus (Erikson and Klug, 1971)

The contrast transfer function The phase Contrast Transfer Function B(θ) of the electron microscope is defined as B(θ) = -2 sin χ(θ) (CTF) 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 C s is constant, Δf can be adjusted to produce maximum phase contrast of ±π/2 for a particular range of scattering angles.

Fourier transforms of amorphous carbon film images underfocus underfocus FT FT Contrast Transfer Function, CTF

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: E damping (k) = E spatial (Z,C s,α,k) E temporal (C c,δe,k) E specimen E detector. C S is the spherical aberration coefficient, C C is the coefficient for chromatical aberration Z=100 nm, C S = 3 mm, C C = 3 mm Z=100 nm, C S = 0 mm, C C = 3 mm Z=100 nm, C S = 0 mm, C C = 0 mm spatial frequency [1/nm]

Chromatic aberration (C c ) corrector Max Haider

The transmission electron microscope electron source specimen objective lens diffraction plane image plane focus plane

Film image Janet Vonck, Deryck Mills

CCD vs. direct detector 300 kv electrons -35 C Separate vacuum Scintillator Fiber optics CCD chip Peltier cooler CMOS active pixel sensor 5-15 µm pixels radiation-hard back-thinned rapid readout (17-400 Hz) 50 µm conventional CCD detector Direct electron detector Richard Henderson, Greg McMullan (MRC-LMB Cambridge, UK)

DQE of electron detectors (300 kv) DQE = SNR 2 o/snr 2 i counting mode, CCD McMullan, Faruqi and Henderson http://arxiv.org/pdf/1406.1389.pdf Greg McMullan, MRC LMB Cambridge

Fast readout overcomes drift 3.36 Å Frame displacement in Å Matteo Allegretti, Janet Vonck, Deryck Mills

Science, 28 March 2014 Science, 28 March 2014