Image formation

One of the main differences cryo-EM to other structural determination methods is that the output of the TEM is a projection image. The micrograph is a true projection of the Coulomb potential of the protein complexes [66]. The TEM like any other imaging instru-ment needs to generate image contrast to make objects distinguishable. Image contrast is the difference in intensities in an image. In a TEM this image contrast is formed by the wave interference of the incident beam and the electrons scattered by the protein complex in the specimen plane. Therefore, the mathematical representation of the single emitted electron and its detected properties as well as the interaction with other electrons are im-portant. The detected interference pattern describe the signal of the protein complex which is read out to a digital 2D representation of the sample. This image is called phase contrast image.

The electron gun of a TEM (see Figure 2.4) emits electrons with a specific magnitude and phase. The electron in the incident electron beam in Figure 2.5 is expressed by the

wave . The wave describes the movements of the electron.

0 =Â0·exp (i„), (2.26)

where the amplitude is expressed byÂ₀ and„is the phase of the wave . It is an oscillatory function. An equivalent representation is the sinusoidal as seen in 2.1. The wavelength is defined by the smallest distance of two points on the wave with the same phase. Within the column of the TEM the single electron travels through the column, where its path is described by the wave . In a perfect microscope, the properties of the electrons waves emitted do not change until the sample plane. On the sample plane there are two main possible effects visible. Some electrons pass through the specimen without interaction as the black solid arrow in Figure 2.5. Their wave representation properties do not change. Other electrons interact with parts of the specimen. These electrons are scattered by atoms. In general, electron scattering underlies different scattering processes. The elastically scattered electrons are electrons which undergo a change in the direction of propagation. This results in a phase shift in the electron wave Equation 2.26. The image contrast in the TEM is based on the interference of the unscattered electrons with these elastically scattered electrons. The exit wave ex (in Figure 2.5) at a position^ær = (x, y, z) in the specimen plane is described by

ex(^ær) =Â0·exp (i„+ifl P tpr(^ær))

¸ ˚˙ ˝

=Â0·exp (i„)·exp(ifl P tpr(^ær))

(2.27)

ex(^ær) = 0·exp (ifl P tpr(^ær)), (2.28) where 0 is the incident wave emitted from the electron gun in Figure 2.5. The variable fl= _h^m2/2fi^e⁄ is defined by the Max Planck constanth, the wavelength⁄and the electron mass.

P tpr is the integral of inner potential of the specimen P tpr(^ær) =^{⁄ +t/2}

≠t/2 P t(^ær)dz, (2.29)

where it is integrated over the thickness t of the sample along the optical axis z. The phase shift (^ær)the exit wave is equal to (^ær) = fl^s_≠t/2^+t/2P t(^ær)dz. The combination of the exit wave of the scattered electrons and the wave of the incident electron beam describe the interference between the electrons. Under the assumption of the weak phase approximation, meaning the changes of the phase are close to zero, the interference pattern is described by the following exit wave [67]

ex(^ær)¥ 0· (1 +i (^ær))

¸ ˚˙ ˝

with the Taylor series

, (2.30)

The first part of this Equation 2.30 is defined by the unscattered electrons (see the black arrows in Figure 2.5). The scattered electron wave (represented by blue arrows in Figure 2.5) are considered by the second additive term in the Equation 2.30.

Figure 2.5: Image formation in a TEM An incident electron beam is emitted from the electron gun and passes through the condenser lens system. This wave reaches the specimen sample plane with specific in sync wave properties. The solid black arrows is the direct beam. These electrons were transmitted without an interaction with the particles.

The dotted arrows represent the electrons which underwent a shift in their phase due to deflection. The objective lens focuses all waves contained in the beam in the back focal plane. Electrons scatter with too high phase angles as the purple arrows are absorbed by the objective aperture. The back focal plane bundles all electrons with the same wave function properties to a single point. The bundled points equal the diffraction pattern. In the image plane the image intensity is measured.

In Figure 2.5 all electrons with the same scattering angle are bundled on back-focal plane by the objective lens. Each spot on the back-focal plane corresponds to a spatial frequency in diffraction plane. Higher (resp. lower) spatial frequencies correspond to higher (resp. lower) scattering angles. The higher the scattering angle the further distanced from the center is the diffraction spot on the back-focal plane. Too high scattering angles are absorbed by the objective aperture as seen in Figure 2.5. Besides elastic scattering, other scattering processes such as inelastic scattering occur. The electron interacts with the electron shell of the atom in the protein complex and undergoes a phase shift and additionally a change in energy. Inelastic scattered electrons contribute to noise [41].

The position probability density function describes the likelihood to find an electron at a given position. The function is given by the multiplication of the wave function Equation 2.30 with its complex conjugate (see (2.17)). The measurable image intensity [67]

based on the position probability density function for Equation 2.30 is I(^ær) = 1≠(i (^ær))²

¸ ˚˙ ˝

ex· ^úex

(2.31)

Keeping the characteristics of the exponential function in mind, thin organic samples such as proteins will be imaged with weak contrast. The phase shift is not measurable.

Protein complexes are not visible within the projection image. Therefore, the scattered wave is shifted by an additional phase shift of 90 degrees in order to convert an initially small phase shift into a large change in amplitude [67]. This phase shift is introduced by defocusing the objective lens. The digitized image equals the position probability of the electron. The output of the TEM is called micrograph. It contains hundreds of imaged single particles. During image processing these particles are identified and cut-out for the image processing. The i-th projection image Ii is the integral over the Coulomb potential of the particles. The convolution of PSF andenvelope function with the effective potential ensures the correction of some aberrations. The mathematical real space representation of the detected signal of a single particle is given as

Ii(x) =P SFn(x)úen(x)ú

3⁄

P T(Ti(^ær))dz+m^S_i

4+m^B_i , (2.32)

where Ti describes the unknown rotation angles (–,—,“) and translation (x, y) of the single particle shown in the i-th image. The termm^B_i is the colored background noise and m^S_n describes the noise due to scattering by the support film. They describe the amount of noise present in the projection image. Both, m^B_i and m^S_i are assumed to be Gaussian distributed with zero mean, statistically uncorrelated and independent.[66]

A simpler real space representation of the cryo-EM projection image is

I_i =f +m_i m_i ≥N(0,‡²) (2.33) and its corresponding Fourier representation is

Gi =F +Mi, (2.34)

wheref is the signal in real space and F is the Fourier transformed signal. mi and Mi are the corresponding zero-mean, uncorrelated Gaussian noise components in the i-th image in real and Fourier space. These two equivalent representations are the foundation of the image processing tools. The algorithm used to refine the data assumes the image to be a model of the protein signal disturbed by an additive random process (see subsection 1.3.2).