Fast and reliable alignment and classification of biological macromolecules in electron microscopy images

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Fast and reliable alignment and classification of biological macromolecules in

electron microscopy images

Björn Forsberg

Björn Forsberg Fast and reliable alignment and classification of biological macromolecules in electron microscopy images

Doctoral Thesis in Biochemistry towards Bioinformatics at Stockholm University, Sweden 2020

Department of Biochemistry and Biophysics

ISBN 978-91-7911-050-5

Björn Forsberg

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Fast and reliable alignment and classification of biological macromolecules in electron microscopy images

Björn Forsberg

Academic dissertation for the Degree of Doctor of Philosophy in Biochemistry towards Bioinformatics at Stockholm University to be publicly defended on Friday 24 April 2020 at 13.00 in Magnélisalen, Kemiska övningslaboratoriet, Svante Arrhenius väg 16 B.

Abstract

In the last century, immense progress has been made to charter and understand a wide range of biological phenomena. The origin of genetic inheritance was determined, showing that DNA holds genes that determine the architecture of proteins, utilized by the cell for most functions. Mapping of the human genome eventually revealed around 20000 genes, showing a vast complexity of biology at its most fundamental level.

To study the molecular structure, function and regulation of proteins, spectroscopic techniques and microscopy are employed. Until just over a decade ago, the determination of atomic detail of biomolecules like proteins was limited to those that were small or possible to crystallize. However recent technological advances in cryogenic electron microscopy (cryo-EM) now allows it to routinely reach resolutions where it can provide a wealth of new information on molecular biological phenomena by permitting new targets to be structurally characterized.

In cryo-EM, biological molecules are suspended in thin vitreous sheet of ice and imaged in projection. Collecting millions of such images permits the reconstruction of the original molecular structure, by appropriate alignment and averaging of the particle images. This however requires immense computational effort, which just a few years ago was prohibitive to full use of the image data.

In this thesis, I describe the development of fast algorithms for processing of cryo-EM data, utilizing GPUs by exposing the inherent parallelism of its alignment and classification. The acceleration of this processing has changed how biological research can utilize cryo-EM data. The drastically reduced processing time now allows more extensive processing, development of new and more demanding processing tools, and broader access to cryo-EM as a method for biological investigation. As an example of what is now possible, I show the processing of the fungal pyruvate dehydrogenase complex (PDC), which poses unique processing challenges. Through extensive processing, new biological information can be inferred, reconciling numerous previous findings from biochemical research. The processing of PDC also exemplifies current limitations to established.

Keywords: cryo-EM, electron microscopy, GPU, parallel processing, protein structure.

Stockholm 2020

http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-179802

ISBN 978-91-7911-050-5 ISBN 978-91-7911-051-2

Department of Biochemistry and Biophysics

Stockholm University, 106 91 Stockholm

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FAST AND RELIABLE ALIGNMENT AND CLASSIFICATION OF BIOLOGICAL MACROMOLECULES IN

ELECTRON MICROSCOPY IMAGES

Björn Forsberg

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Fast and reliable alignment and classification of biological macromolecules in

electron microscopy images

Björn Forsberg

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ISBN print 978-91-7911-050-5 ISBN PDF 978-91-7911-051-2

Printed in Sweden by Universitetsservice US-AB, Stockholm 2020

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Eventually, all things

will make sense

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1 | Introduction

1.1 Biomolecules and their structure

Some 3.5 billion years ago, life started as a set of chemical processes which were self- sustaining by storing, repairing and replicating information. The molecules involved are collectively known as biomolecules. DNA, RNA, proteins, lipids and sugars are the most fundamental, found throughout the kingdom of life. Lipids and sugars form sheltered and regulated compartments known as cells, where DNA stores the genetic blueprint of the organism, and proteins perform both its reparation and replication.

Proteins also facilitate a wide range of cellular functions, and are therefore of enormous scientific interest.

Until the 20th century, there was no way for scientific inquiry to determine how molecules like proteins worked. Scientist like August Kekulé had already in the 19th century formulated the notion of a chemical structure through bonds that connected atoms, and Antoine Fourcroy had recognized proteins as a distinct class of molecules. There was however no way to visualize them to understand their mechanistic or catalytic function. One could occasionally infer something about their function, e.g. Louis Jacques Thénard found in 1818 that the enzyme catalase was able to break down hy- drogen peroxide. But it was not until 1923 that the first biomolecules were visualized using X-ray crystallography, starting with fatty acids (lipids). The atomic structure of cholesterol (1937) and penicillin (1946) were next determined, and in 1958 hemoglobin (a oxygen-carrier molecule in blood) became the first protein structure determined, using X-ray crystallography.

The development of X-ray crystallography in the last century has naturally been enormously important for our understanding of biology. By visualizing the atomic structure and interactions that proteins and other biomolecules undergo, we understand

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how biological processes proceed. It has guided drug development by targeting specific biochemistry, and shown us why genetic disorders manifest as they do. In fact, genetics itself was in large part accepted once we were able to visualize DNA in 1953, although its role as a hereditary substance had been suggested earlier. It is becom- ing increasingly clear that nature supplies a near-endless variation in specialization of proteins and other biomolecules. We are continually finding new aspects of molecular structure and their dynamics that inform us on how biological processes work, and what we might do to influence or use them. This is why structural biology has grown so much since its invention, and why the development of tools for the determination of biomolecular structures will continue to be important.

1.2 The challenges of structural biology

1.2.1 The size of biomolecules

The use of X-rays and an arrangement of biomolecules in a crystal-lattice overcomes two fundamental challenges to atomic resolution. First, the physical diffraction limit imposed by the wavelength of the light according to the Rayleigh criterion[1], which states that the smallest resolvable distanceδ in an optical imaging system is

δ = 0.61λ

n sinθ (1.1)

Where n is the refractive index of the material andθ is the half-angle of the optical sys- tems effective aperture. At best, the numerical aperture term n sinθ will be unity, so optical imaging of biomolecules which are 1-100 nm in size will require high-energy X- rays. Second, a single biomolecule would need to interact with an enormous amount of light in order to overcome the noise of a macroscopic measurement and establish a well-resolved image. This would inevitably cause energy to be deposited in the molecule, imparting so-called radiation damage!in crystallography. Collected images would therefore not be possible to interpret.

Figure 1.1:Accessible lengthscales of methods in structural biology. For electron microscopy and tomog- raphy the smallest resolvable detail is much smaller than the samples possible to examine. The discrepancy is indicated in grey. For instance, cryo-electron tomography can resolve details of ∼ 1nm in favorable cases, but is generally only applicable to samples of 40 nm or larger. Illustration by the author.

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The challenges of structural biology 3

A crystal lattice allows many millions of biomolecules to be identically imaged. This multiplies the signal at the same nominal noise level, allowing faithful images to be collected without detrimental radiation damage. But a crystal formed by self-assembly, proceeding slowly enough that the energetically most favorable state is found. Hence, conditions must be found in which the protein of interest is stabilized in a crystal form and the sought state. This can be enormously challenging, but usually confers a well- resolved imaging result when successful. This may also favor states or assemblies of biomolecules that would not spontaneously persist for long enough to be imaged in the natural cellular environment.

1.2.2 The complexity of the cellular environment

The human genome is composed of roughly 3.2 billion base pairs of DNA, the full extent of which could be stored on 760 megabytes of storage media. This code of life contains the blueprint of around 20 000 unique proteins [2]. The transcription of these genes for protein synthesis is strictly regulated by multiple mechanisms, which dictate the overall transcription level. Protein synthesis is additionally and separately regu- lated to ultimately determine the proteins expression level. The relative expression of many genes determines cell identity, function and proliferation. Hence, the function, functional regulation and interactions of the natural protein variations are important to understand the biology of any cell, in addition to the structure of any one protein of interest.

Figure 1.2:Functional association of gene products. Genes within DNA are utilized for various processes, e.g. enzymatic catalysis, DNA replication or acting as structural scaffolds. The number of genes, their distribution among cellular processes and the regulation of their production varies according to species specialization and evolutionary history. Here, so-called proteomaps show the quantitative composition of pro- teomes [3]. Each protein is represented by a polygon: areas reflect protein abundance, and functionally related proteins are arranged in common and similarly colored regions. Illustrations retrieved from bionic- vis.biologie.uni-greifswald.de with permission.

Proteins and other biomolecules are therefore best studied outside the strict confines of a crystal lattice. Methods which attempt to instead visualize individual biomolecules or assemblies are collectively known as single-particle methods . The strength of single- particle image analysis (SPA) compared to crystallographic methods is that variability in the imaged sample is not prohibitive, which allows more native conditions during imaging. Variability or protein heterogeneity can also be computationally detected and

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isolated under favorable circumstances. The complexity of the cellular space can thus be approached, examining natural mixtures of proteins displaying varying interactions and/or composition. While SPA currently relies on extensive biochemical purification to be efficient, they can in theory be applied to the native cellular environment. In order to develop this capacity further, broad progress must be made to better pre- serve samples for imaging, detect and quantify sample heterogeneity, and characterize smaller particle subsets. We will touch upon many of these aspects in the course of this thesis.

1.2.3 The challenge for single-particle analysis

In SPA, the crystal lattice is omitted and one must therefore contend with a much higher level of noise. A sufficiently large number of single-particle images should however be possible to average appropriately so that information becomes significant above the noise. The grand challenge of SPA is precisely the concept of appropriate averaging, because unlike crystallographic methods neither the relative orientation nor the identity of any single particle image is known. Computational methods must instead align and classify single-particle data to allow faithful reconstruction of the true object(s) in the sample.

Such algorithms have long been established [4, 5] to i) align extremely noisy images to a reference object, and ii) faithfully average them, generalizing the images to higher di- mensional spaces. These steps are often combined, so that the averaged object is used to further align images. This results in a looped, or iterated procedure which under favorable conditions successively improves the averaged object(s).

1.3 What this thesis is about

This thesis describes the implementation of fast and reliable SPA algorithms to iter- atively align and classify single-particle images collected by cryogenic transmission electron microscopy (cryo-EM), to produce optimal reconstructions that permit interpretation of image data to further biological knowledge. The algorithms presented are contained within the software suiteRELION, and make use of graphical processing units to accelerate the reconstruction 3D-volumes representing macromolecules from their 2D projections. First, the basis of operation and image formation by the electron microscope will be described. Second, the mathematical formalism behind the optimization of the reconstruction will be detailed. Third, the overall process- ing pipeline will be described. Fourth, our implementation of broad parallelism and its automation will be described. Finally, we provide examples of processing using the established methods, which provides new biological insight. This will also illustrate the benefits of fast processing and currently employed methods for alignment and classification, but also delimit how far reconstructions based on the presented optimization procedure may be interpreted.

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2 | Electron microscopy

Electron microscopy (EM) is magnified imaging using electrons rather than light to illuminate the sample. The following chapter will describe its advantages for biological structure determination, as well as describe the anatomy and functional principle of the electron microscope. This will lay the foundation for our understanding of how EM data is pro- cessed.

2.1 Why electrons?

Just as in light microscopy, there are different imaging methods or modalities. Elec- tron microscopy can be broadly distinguished as imaging by scanning (SEM), or transmission (TEM). SEM utilizes a focused beam that is scanned across the sample, and various detectors then build the image by recording the response of each point, which might be emission of secondary electrons, X-rays, or induced sample charge. TEM on the other hand, illuminates a large area of the sample and forms an image based on

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how the sample modifies the beam. A combination of the two is also possible, known as STEM .

There are a number or reasons why EM and TEM in particular holds significant ap- peal for examination of biological samples. Principally, the (non-relativistic) de Broglie wavelength of electrons is very short, even for moderate energies. Comparing the wavelengthλlof light andλeof electrons

λl= h

cE_l (2.1)

λe= h

p2meE_e (2.2)

one finds that as long as the energy is below ≈1 MeV, electrons have shorter wavelength. This is far below what is conventionally used in electron microscopy. In fact, the diffraction limit for atomic objects requires only about 150 eV for electrons, so the diffraction limit is of no practical consideration in EM, however the wavelength must be precisely known. For TEM energies, relativistic correction as described in appendix A must then be considered.

In addition to being easy to produce, electrons carry electric charge, which makes them easy to manipulate through electrostatic and electromagnetic fields. This permits construction of lenses that focus a beam of electrons used in SEM, and magnify the image formed in TEM. Using X-rays, lenses are difficult to arrange since the refractive index of most materials is close to 1 . Moreover, their focal length is fixed, while the electron lenses can be tuned according to the current applied.

Finally, electrons interact very favorably with matter for biological applications. Elec- trons cause less radiation damage on biological samples then e.g. X-rays, comparing exposures which permits the same nominal information recovery by transmission imaging[6, 7](see figure 2.1C).

2.2 Electron-matter interactions - scattering processes

Imaging is based on collection of particles which have interacted with (or scattered off of ) the sample, thereby containing information about it. The types of scattering utilized in SEM and TEM are different in nature. In SEM, electrons interact with a bulk sample (being thicker than the mean free path of the incident electrons). The detected electrons may thus be so-called secondary electrons, originating from the sample by excitation rather than the beam. Higher energies of incident electrons leads to mul- tiple scattering over a large interaction-volume (compare figure 2.1B which decreases spatial resolution, so the choice of energy is typically low (0.3-4 kV).

Conversely, TEM requires electrons to pass through the sample. The incident electron energy is thus much higher and the sample consequently thin. Primary electrons passing through the sample are further categorized as having preserved or altered energy

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Electron-matter interactions - scattering processes 7

Figure 2.1:Scattering of electrons by matter. a The strong interaction of each electron in the incident beam with the sample may be cause ejection of secondary electrons or emission of radiation, as well as the absorption or hole-pair generation in the sample. These can all be measured in different modes of scanning electron microscopy (SEM). In Transmission EM (TEM), the scattered electrons of the beam are instead collected. b The interaction of 300 keV electrons impacting a 350µm layer of Silicon, where the top 35µm are shown in grey, showing why thin samples are important for TEM. c Atomic interaction cross-sections of biological samples with neutrons, electrons and X-rays. Image attributions: b reproduced from [8] with permission, c Reproduced from [6] with permission.

(i.e. velocity). Small alterations in energy arise from inelastic collisions with the sample atoms or electrons[6], while larger alterations can be ignored for the purposes of TEM, since i) the sample is so thin that a large loss of energy is unlikely[6], and ii) such electrons are typically removed by use of an energy filter.

Electrons which pass through the sample with their energy unchanged are either unscattered or elastically scattered. The distinction is somewhat unclear if we consider electrons as localized particles. We may instead consider the matter-wave property of electrons. Each electron is then delocalized, passing simultaneously through the en- tire sample as one plane wave. In this view, elastic scattering is the alteration of this delocalized wave by he sample, whereas non-scattering reflects the probability of no interaction occurring between the incident electron and the sample. The detection of each electron finally follows from the probability distribution of the superposition of the scattered and unscattered wave function, as described by conventional quantum mechanics. At this point, it should be emphasized that the high velocity of electrons in a typical microscope creates a large spatial separation between electrons in the beam,

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so that at most one electron is likely to be interacting with the sample at any given point in time. The plane wave passing through the full field of illumination is thus a quantum-mechanical statement about each electron individually, and not macroscop- ically literal. This must in fact be that case since the electrostatic repulsion between proximal electrons indeed prohibits a dense beam.

2.3 Anatomy of the microscope

2.3.1 Electron gun (emitter)

The electrons are emitted from a so-called gun, or emitter. A field emission electron gun (FEG) is most commonly used, which is based on extraction of electrons from an atomically sharp tip, using a strong electric field to lower the barrier of electron es- cape[1]. The FEG may also be heated to increase the energy of electrons in the tip and thus lower the barrier further. This however affects the spatial coherence of the beam[1], which may affect the attainable resolution since electrons appear to originate from different positions of the emitter, blurring the subsequent image.

2.3.2 Lens systems

Originally, the lenses employed in electron microscopes were electrostatic in nature[9].

The principle is such a lens is simply one or more electrically charged circular loops[10]

that will act repulsively or attractively upon charged particles passing through it, any- where off-axis, as shown in figure 2.2A. Current electron microscopes however use electromagnetic lenses based on deflection according to the Lorentz force. The reason is that they can be operated at lower voltages and achieve similar aberrations at shorter focal lengths[9].

Multiple lenses are organized in the electron microscope to manipulate the beam.

Conventionally, we enumerate the condenser, objective, intermediate and projector lens systems. We term each a system since each one includes not only the lens itself, but also deflectors, stigmators and apertures. These additionally allow the beam position, direction, symmetry and size to be adjusted.

The role of the condenser lens systems (there are typically at least two, denoted C1 and C2) is to tune the electron beam before it encounters the sample. By adjusting the convergence angle and apertures one typically adjusts the illumination area according to the field of view (by consequence of the magnification), which also affects the beam intensity and coherence.

After interaction with the sample, the objective lens forms an image of the sample from the elastically scattered electron wave function. This rationale and many illustrations depict the objective lens as being situated after the sample, but in reality the sample is immersed within the objective lens to utilize a minimal focal length[9].

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Anatomy of the microscope 9

Figure 2.2:Lenses for electron optics. a Electrostatic lenses are based on Coulomb attraction or repulsion from one or more plates with a circular aperture. As an example, an einzel-lens is shown in cut-trough, showing the path of electrons to be focused, as well as electrostatic field lines in red. b Electromagnetic lenses instead rely on the Lorentz force to bend the path of the electrons as they pass through the magnetic field produced by a current-carrying coil. This also introduces a rotation to the produced image, which is rarely of any practical importance. c Simulation of electron paths through the magnetic field caused by a current-carrying loop shows the path and non-ideal behavior of magnetic lenses far off-axis.

The intermediate and projector lens systems finally magnify the primary image or diffraction plane (depending on the imaging mode) onto the detector. Because the magnification is dependent on the current applied to electromagnetic lenses, the ex- act magnification is not known. Typically, calibration must be performed by reconstruction of a sample with known size or spacing, e.g. a crystalline metal sample like gold.

2.3.3 Energy filter

A critical component of high-resolution TEM (HRTEM) is a filter to remove inelastically scattered electrons, which are incoherent with the unscattered beam. As will become clear later, this does contributes a detectable signal component, but one that is not consistent with the image formation model of the dominant (elastic) scattering. Pro- cessing can to some extent account for this, but it is nonetheless desired to minimize its influence. An energy-filter (EF) is therefore placed between the sample and detec-

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tor, which bends the path of the electrons in proportion to their velocity, which will depend on the energy lost during sample interaction. An aperture can then limit the detected electrons to those within a specific range of energy (loss).

2.3.4 Detectors

Detection of electrons during beam adjustment has historically been a simple phos- phorescent screen which emits light when hit by electrons, followed by photographic film during data collection. Digital detection has entirely replaced both these methods, either by charge-coupled device (CCD) or complementary metal-oxide semicon- ductor (CMOS) sensors. The fundamental distinction between the latter two, is that a CCD detects electrons by proxy of light emitted from a scintillating material, whereas CMOS detectors detect charge separation caused by the electrons themselves[11]. This makes the spatial resolution and sensitivity of CMOS or so-called direct detection much better for applications where high-resolution information has low signal-to-noise ra- tio (SNR)[12, 13]. For this reason, nearly all current research utilizing TEM for single- particle analysis of biological specimens use a CMOS detector.

Typically, the cumulative response of a detector pixel is summed or integrated to yield its measured value, hence known as integration mode image acquisition. The good lo- calization and fast readout of CMOS-detection however allows the detected signal to be converted into single electron detection events. In contrast to integration mode, this counting mode acquisition provides an integer number of electrons for each (sub- ) pixel, increasing the high-resolution information. Consequently, most currently collected data for SPA analysis make use of counting-mode acquisition.

2.4 Image formation

In our description of electron-matter interactions, we briefly stated that the delocalized wave of an elastically scattered electron carries all information about the sample.

The aim is next to somehow extract this information by measurement of each such electron. This amounts to i) understanding how electrons carry the sample informa- tion and ii) devising a method to convert detected electrons into information regarding the sample. We will first devote some attention to understanding how the exit beam behaves following sample interaction, by considering a simplified optical system. This will rationalize image formation in the electron microscope, and the aberrations of particle images we must consider for faithful information recovery.

2.4.1 Huygens principle

Waves are propagations of energy that are caused by a disturbance at a point P , which causes a nearby region Q to alter in response. It is easily seen that the alteration of Q

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Image formation 11

is itself a source of disturbance, which then continues to propagate. The medium in which this disturbance propagates determines its speed, periodicity and dissipation.

In 1690 Christian Huygens formulated a method of analyzing such wave phenomena, based on the principle that the wavefront of a propagating disturbance (i.e. a wave) is a source of spherically propagating waves[11]. In a homogeneous medium, this agrees with a number of experimental observations:

A spherically propagating wave remains spherical.

An infinite plane wave remains planar.

An plane wave appears to bend around the edge of any obstruction.

These cases are also exemplified in figure 2.3A-C. Huygens considered light as his main example of a propagating wave¹, but the propagation of any wave can be understood by utilizing constructions based on Huygens principle. We will e.g. use it to understand the propagation of the electron wave-function in TEM. To this end we say that a spherical wave source at point P causes a disturbance at all other points Q described by

exp(i kr )

r where r = |P −Q| (2.3)

This reflects both the oscillation of the wave amplitude and dissipation of the wave with respect to the distance from P . A continuum of such point emitters throughout space is the foundation of Huygens’ principle, and how we will understand TEM.

Figure 2.3: Huygens principle. Huygens principle states that the instantaneous disturbance of some medium at a point P propagates radially, and that each point outside P causes a similar disturbance in response, modulated by the properties of the medium at that point. This allows us to formulate the expected propagation to all points given the original perturbation, which by a simple construction find that a spheri- cal and b planar waves remain as such. A plane wave incident on a barrier can thus easily be modeled as the sum of contributions from sources in the plane of the barrier. c A plane wave is e.g. bent around an edge, and d emanates spherical waves from a infinitely narrow slit. e Two slits result in an interference pattern de- pendent on the wavelength of the incident wave and slit separation. The angles of constructive interference are indicated.

1Light was shown to be a wave much later by Maxwell, but it had already been suggested by Hooke and Huygens principally relied on Römer observation that it had finite speed.

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2.4.2 Fresnel and Fraunhofer regimes

Consider a plane wave representing a single particle in a coherent beam, incident along z, which then hits an obstructing plane along x = [−∞,∞] at z = 0, as in fig- ure 2.4. In this case, no wave propagates to the region z > 0 (denoted the post-field), hence particle will be detected there. If we introduce a small slit (an opening) of width w in the obstructing plane, then according to Huygens’ principle any wave in the post- field can be found by constructing the sum of point sources situated in the slit. For an infinitely narrow slit, spherical waves are evident in the post-field as in figure 2.4A. For any finite width w , the waves emanating from several point sources located in the slit interfere to create a non-trivial post-field wave. If the extent of the slit is infinite, but only in one direction [−∞,0], then a special case known as a knife-edge diffraction is encountered. In figure 2.4A-C these cases are shown in an overhead view, showing the propagation of waves in the xz-plane.

Figure 2.4:Simulated post-field of plane-wave modulation. All panels were generated by simulation ac- cording to Huygens’ principle - placing equispaced point emitters of spherical waves along the line repre- senting the slit opening, and summing the contributions of these waves to each point in the post-field. a An infinitely narrow slit results in spherical waves from a single point emitter. b The slit is wider so that many point emitters contribute, which produces interference effects due to diffraction of the incoming wave. This interference is even more evident in c, where so called knife edge diffraction displays bending of the wave around the single edge of the infinite half-plane slit. d Two slits are present, and the barrier does not block the incident plane wave but merely delays the wave, altering the phase in the post-field.

Another way to visualize the post-field is to plot the time-averaged magnitude of dis- turbance at a distance z = d into the post-field². Such a plot is denoted an interference pattern, and it will clearly change depending on the distance d into the post-field. The post-field is therefore divided into regions where certain approximations are valid and the theoretical interference pattern is more easily expressed. The Fresnel region occurs in the near-field, specifically under the approximate restriction

w²>> dλ Fresnel (near-field) diffraction limit (2.4)

2you may think of this as the wave amplitude at all points on a screen placed in the post-field

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Image formation 13

Conversely, the Fraunhofer region or far-field diffraction when

w²<< dλ Fraunhofer (far-field) diffraction limit (2.5) To see that it is possible to recover information regarding the slit from the Fraunhofer interference pattern, we will examine a simple example. We follow Huygens’ principle to construct a single-slit diffraction experiment. A continuous set of point emitters are thus defined within a range [−w/2, w/2], defining a slit of width w. The contributions of all such point emitters to a plane z = d forms an integral over the slit. For reasons that will become evident, we will formulate this as an integral over a full plane of emitters multiplied by a window function Tw(x) which defines the extent of the slit. One then simply sums the contribution of the infinitesimal waves according to eq. (2.3), arriving with different relative phases:

Z _∞

−∞

T_w(x)exp (i k(d + δr(x))

d + δr(x) d x, where T_w(x) =

(1, if |x|< w/2.

0, otherwise. (2.6) Here,δr(x) represents path-difference from each point emitter and the point x com- pared to the minimum distance d . Now, in the Fraunhofer regime a line connecting any slit point with a point x in the interference pattern forms approximately the same angle with the optical axis. Making this approximation explicit, we effectively state that the far-field Fraunhofer interference pattern is the infinite post-field pattern (without complete attenuation). In this case we denote it the diffraction pattern of the slit func- tion T_w(x). Mathematically, this approximation makes the path-differenceδr(x) negligible in the denominator. We can also reformulateδr(x) in the exponential, reducing the integral[11] to

Z_∞

−∞

T_w(x) · exp(i kx) d x (2.7)

This integral is the definition of the Fourier transform, barring some constants. Hence, we can conclude that the diffraction pattern is the Fourier transform of the slit function, and that we may potentially recover information regarding the slit by performing an inverse Fourier transform of the measured diffraction pattern.

2.4.3 Limitations to inversion of diffraction patterns

Let us consider the case where we measure a diffraction pattern in the Fraunhofer regime and use it to compute the form of the diffracting object (the slit function T_w(x) in the previous example) by an inverse Fourier transform. The detection measures the amplitude of the wave disturbance, irrespective of its absolute phase. The phase of the wave is necessary for faithful inversion of the Fourier transform, but clearly missing from our measurement. Hence, the diffraction pattern is not a complete description of the object in this sense.

phase problemThis issue is realized in X-ray crystallography, where one collects diffrac- tion patterns and thus face this so-called phase problem. One method to address it, is

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to determine the most likely phases using a prior model of T (x), or at least some as- pect of it. In the context of our single-slit example, we might say that the position and width of the slit are unknown parameters, and search a space covering these to find the most likely combination in view of the observed diffraction pattern. This may predict the phases and allow inversion of the diffraction pattern. In the context of molecular structure determination the parameter space is much larger and the prior belief must be more specific. Typically, prior structures of homologs are required for so-called molecular replacement phasing[14], but in favorable cases simple structures such as α-helices can still provide sufficient conditions to make a highly credible and often legitimate estimation of the true phases[15].

2.4.4 The focusing lens as a Fourier Transform Operator

X-ray scattering methods collect diffraction patterns in the Fraunhofer regime, where it approximates a Fourier transform. This approximation however only becomes strictly true in the infinite post-field, which is impractical for data collection. Fortunately, ba- sic optics tells us that a spherical lens can converge parallel rays into a single point in its plane of focus³. One can therefore recreate the infinite post-field of the propagat- ing wave in the back-focal plane (also known as the diffraction plane), using a lens.

Tracing the paths under these conditions beyond the back-focal plane also shows that all rays emanating from a given point in the sample (or slit in our simple example) are mapped to a single point in the primary image plane. Through the reversibility of optical paths, the immediate post-field d = 0 has thus essentially been re-established.

In the image plane, a measurement is independent of the relative phase, and in this sense complete, circumventing the phase problem. While such lenses are problematic for use with X-rays (as noted previously), they are easily constructed and adjusted for electron illumination.

2.4.5 The optical transfer function of the microscope

The quality of lenses in a microscope are clearly essential for faithful imaging of the sample, but we cannot expect perfect imaging. The (in)ability to recover details of the sample in the image plane is typically described by a response function which represents how the imaging system reproduces a theoretical singular point, including any distortions and blurring. It is also known as the point-spread function (PSF) for this very reason. The PSF can be formulated as a function of spatial resolution, i.e. resolv- able size. It is then known as the optical transfer function (OTF), and is related to the PSF through the Fourier transform.

The OTF acts as a linear filter operation in Fourier space, corresponding to the convolution of the untransformed image with the PSF of the microscope. From this, we directly observe that a binary OTF with a spatial cutoff would blur any higher resolution fea- ture beyond possible retrieval. Incomplete attenuation of information may however

3Equivalently, it converges an incoming plane wave into a single point without relative phase delay

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Image formation 15

still be recovered through appropriate de-convolution (inverting the OTF filter operation). Knowledge of the OTF is thus critical to restore the best possible representation of the true object given the captured images of it.

Abbreviating the OTF to T , we recapitulate that the imperfections of the optical system can be considered a linear filter applied to the post-field wave in the back-focal plane (BFP) (a.k.a. diffraction plane, where the immediate post-field or exit-plane (EP) wave has been Fourier transformed):

ΨBFP(ω) = T (ω)F^[ΨEP(x)] = T (ω)ΨEP(ω) (2.8) whereω denotes spatial resolution. In the back image plane (BIP) we next find the inverse Fourier-transform of this, i.e.

Ψimage(x) = ΨBIP(x) =F⁻¹^{[T (}ω)F^[ΨEP(x)]] = (2.9)

=F⁻¹[T (ω)ΨEP(ω)] (2.10) From eq. (2.9) it would appear that an ideal transfer function T (ω) would permit com- plete recovery the exit plane wave in the back image plane. A non-ideal transfer function still permits retrieval of non-zero-attenuated spatial frequencies by inverse filtering, subject to the additional condition that signal is significant above noise and thus suitably considered. To adequately apply such an inverse filter, we must thus establish the analytical form of the transfer function T .

2.4.6 Phase contrast

In the preceding sections we have found that using a system of lenses will enable the immediate post-field or exit plane waveΨEP(x) to be imaged by utilizing lenses. We have considered a barrier which blocks the incoming plane wave completely apart from within a simple slit, where it does not alter the plane wave at all, according to the definition of T_w(x) in eq. (2.6). Because this slit can be considered as changing the amplitude of the wave by some amount at all points x, it is said to constitute an ampli- tude object. Conversely, a transparent wall which delays the wave instead of blocking it as in figure 2.4D is said to be a phase object, since it alters the phase of the wave but not its amplitude. To generalize these concepts with emphasis on phase objects, we may say that the barrier at each point modifies the incident wave by multiplication with a factor

T (x) = exp(i δ_φ(x)) (2.11)

Whereδφ(x) is the phase delay incurred at position x of the barrier. Amplitude scat- tering of he wave can be considered the imaginary part of a complex phase difference δφ, but for a pure phase objectδφreal-valued. This is the only case we will consider since it reflects the mechanism by which elastic scattering by the sample modifies the incident electron plane wave.

So what image can we expect to collect given a phase object? A measurement of the wave disturbance in the image plane will be independent of relative phase delay, so it

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would appear that a phase object will fail to produce any contrast. In order to acquire phase contrast, we must make an indirect measurement of the relative phase delay at each point in the image plane, e.g. by interference with a wave of known phase. In optical phase contrast microscopy, this is supplied by a reference beam with known phase delay, which differentially alters the amplitude of the image of the exit plane image through interference. Supplying such a reference wave in electron microscopy is more problematic due to the difficulty of introducing a precise phase delay to the un- scattered component of the beam. Electron phase plates do exist and are currently used, but due to their instability and complicated operation one typically relies on other methods of generating differential interference. Primarily, one utilizes an imperfect transfer function to ensure that each point in the image plane is the sum of waves form many points in the object, which thus sum and modulate the amplitude through interference. This generates image contrast, while knowing the analytical form of the transfer function still allows the imperfect imaging to be efficiently de-convoluted.

To establish the mathematical form of the transfer function, we consider only weak phase objects, i.e. whereδ_φin eq. (2.11) is small as defined by the small-angle approximation. In such a case, a truncated Taylor expansion shows us that the immediate post-field wave function modulated in phase by the weak phase object (the thin TEM sample) is

ΨEP(x) = Ψ0(x) exp(iδφ(x)) ≈ Ψ0(x)(1 − i δφ(x)) (2.12) We will denote this as the exit-plane waveΨEPand setΨ0= 1 as the absolute phase of the incident wave is arbitrary. Hence, within the weak phase-object approximation (WPOA),

ΨEP(x) ≈ 1 − i δφ(x) (2.13)

Given the form of our transfer function, the recorded image intensity is therefore I (x) =¯

¯Ψimage(x)¯

¯

2 (2.14)

= |ΨEP(x) ⊗ T (x)|² (2.15)

=¯

¯¡1 − iδφ(x)¢ ⊗ ¡Treal(x) + i Timag(x)¢¯

¯

2 (2.16)

=¯

¯1 + δφ(x) ⊗ Timag(x) − i δφ(x) ⊗ Treal(x)¯

¯

2 (2.17)

=¡1 + δφ(x) ⊗ Timag(x)¢2

+¡

δφ(x) ⊗ Treal(x)¢2

(2.18)

= 1 + 2δφ(x) ⊗ Timag(x) +¡

δφ(x) ⊗ Timag(x)¢2

+¡

δφ(x) ⊗ Treal(x)¢2

(2.19) We truncated the Taylor-expansion ofδφ under the WPOA since its values are small and can be considered negligible for powers of two and above. We may thus truncate eq. (2.19) as well, to find

I (x) = 1 + 2δˆ _φ(x) ⊗ Timag(x) (2.20) Or equivalently in Fourier space

I (ˆω) = δ(0) + 2δ_φ(ω)Timag(ω) (2.21)

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Image formation 17

Hence, in the WPOA employed for phase contrast transmission imaging, the imaginary component T_imag of the transfer function T modulates the frequency response in a resolution-dependent manner, taking the form of a real-valued scalar filter. We now assume the form of T as introducing a phase-shiftχ in addition to δ_φ. The imaginary component of T thus gives the image intensity the following form

I (ˆω) = δ(0) + 2δφ(ω) · Im£exp(iχ(ω))¤ (2.22)

= 2δφ(ω)sin¡χ(ω)¢ (2.23)

This recapitulates that aberration-free phase contrast of a perfect optical system (i.e.

χ = 0) results in no phase contrast. We now only need to establish how each spatial res- olution component is delayed in phase by optical imperfections to completely model how image contrast is formed. Intentionally introduced distortions can thus be used to produce contrast, while still allowing complete image interpretation.

2.4.7 The CTF

The expression

sin¡

χ(ω)¢ (2.24)

established in the previous section is known as the (phase) contrast transfer function (CTF). Anything which introduces a phase-shift of a given componentω in the Fourier- transformed particle image can thus be modeled by this function, within the approximations of its derivation. The two dominating contributions toχ(ω) are spherical aberration C_sand the applied defocus∆f , which are both rotationally symmetric, i.e. they depend only on the spatial resolution |ω|. The phase shift of these factors can be ex- plicitly entered into the CTF as follows;

CTF(ω) = sin¡χdefocus(ω) + χspherical aberration(ω) + χother(ω)¢ (2.25)

= sin µ

π∆f λω²+1

2πCsλ³ω⁴+ χother(ω)

¶

(2.26) From this we can expect the CTF to oscillate with increased frequency as resolution increases, intermittently inverting the contrast of our recorded images. We also expect that a Fourier-transform of a collected image should show this pattern of oscillating component magnitude. Indeed, this is observed, as shown in the below figure. The circular pattern is known as Thon rings[16], which become increasingly elliptical with larger astigmatism (anisotropic defocus). Examples of the expected CTF and observed Thon rings are shown in figure 2.5.

2.4.8 Mathematical formulation of image formation

Given how images of biomolecules in thin sections of vitreous ice are formed through transmission imaging in the electron microscope, we can now formulate a simple mathematical model that will permit us to design algorithms capable of faithfully interpret a

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Figure 2.5:Manifestation of the CTF as Thon rings. The theoretical form of the CTF predicts that an ap- plied defocus should be an oscillating function with respect to spatial frequency. a A simulation using a tool developed by Takanori Nakane shows the predicted appearance of the CTF, under 1µm defocus of 300 keV electrons with spherical aberration Cs=2.7 mm and notable astigmatism. b A typical micrograph, and its c power-spectrum calculated from each quadrant, showing the characteristic Thon-rings. Note that Thon- rings are more well-defined in the presence of holey carbon support film. Defocus is estimated by fitting the theoretical form in collected images such as c.

large set of such images. In mathematical terms, we say that the following linear image formation in Fourier space produces an image Xi, where i = 1... N covers a dataset of N particle images.

X_i= CiI_i+ Ni (2.27)

C_i is given by eq. (2.26) for the estimated defocus∆f applied during collection of im- age i , and the spherical aberration C_sis taken as a fixed constant of the microscope used. N_i is an additive noise component. The ideal projection image I_i is formed by projection

I_i=X

`

P^φ_j_`V_` (2.28)

where V_`is the true object made from components`, projected in orientation φ by an operator P^φto produce a transmission image I_i. The projection operator may be considered a matrix of dimension j × `, such that the image formation eq. (2.27) with explicit Fourier-component index j is

Xi j= Ci j

X

`

P^φ_j_`V_`+ Ni j (2.29)

We express this image formation model in Fourier space for two reasons. First, the aberration C is most easily formulated as a real-valued filter that is applied by pixel- wise multiplication, rather than a convolution in real-space. Second, Fourier space permits a concise formulation of common resolution-dependent quantities. For in- stance, since each component j of the noise N was assumed to originate from a Gaus- sian distribution with a resolution-dependent varianceσ²(ω) colored noise is easily handled.

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3D reconstruction from 2D transmission data 19

2.5 3D reconstruction from 2D transmission data

The main objective is to employ the image formation model eq. (2.29) to understand our image data. The natural way to achieve this is by recovering the object which gave rise to the projection images. One thus desires a method which allows a sufficient number of projection images of an object to be reconciled into its higher-dimensional object through a process denoted reconstruction. In cryo-EM, 3D-volumes are recon- structed from 2D TEM data, but for clarity we will describe the reconstruction of a 2D image from 1D projections.

To this end, we consider the true 2D image as a general 2D field of scalar values f (x, y).

At first we consider a set of projections which are uniformly sampled in orientation and free of noise. In this case, the projection angleθ can be seen as a continuous parameter. Since we find a 1D-projection parameterized by r at each value ofθ, there is a transformed 2D-space parameterized by modified polar coordinates (r,θ) which holds all possible projections of f (x, y). This defines the Radon transform of f (x, y), denoted R_f(r,θ) (see appendix C for details).

Figure 2.6:Examples of Radon transforms of scalar fields. a-c Any scalar field f (x, y) can be projected along a directionθ, so that it is parametrized along the orthogonal direction by a parameter r (see appendix C for details). Consideringθ a second parameter, we define the Radon transform of f (x, y) as this set of projection, denoting it R_f(r,θ). d-f The Radon transforms of panels a-c are shown. Illustrations were produced using a program released under CPOL1.2[17].

The Radon transform maps a single point in f (x, y) as sinusoidal line in the trans- formed space ( as can also be seen in figure 2.6A+D), and is therefore sometimes de- noted a sinogram. The Radon transform can in fact be inverted, recovering f (x, y) ex- actly. However, in practice our measurements are discretely and usually non-uniformly sampled. In this case, the 3D scalar field can not be uniquely determined using an inverse Radon transform[18]. Instead, we must rely on approximate methods to find an estimate bf (x, y) of the true field f (x, y).

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One approximate method known as the algebraic reconstruction technique (ART) [19, 20] considers each measured pixel value as the sum of an equation summing (through projection) discretely sampled points of f (x, y). The set of all projections thus consti- tutes a system of equations for which we can find an optimal solution using conventional optimization methods. We will however not discuss this at any length, as other methods are more relevant for the present considerations.

2.5.1 Backprojection

While the Radon transform is not possible to invert for discretely sampled data, the projection operation at each sampling point (i.e. each projection) can be inverted with incomplete information recovery. This is natural since a projection constitutes a sum along the line of projection, and so the distribution of the projected potential along the line is not measurable. The estimate of each point in f (x, y) provided by a single pro- jection is thus the value of the projection along the line that intersects that point. Con- sequently, the same value is assigned to all points in the estimated field bf (x, y) along that line of integration. One may think of this as smearing each measured projection value along the direction of projection. This simple procedure is known as backprojec- tion P_θ(r ) along the orientationθ, and we will denote this operation B^θ. We might then estimate the original scalar field f (x, y) as the sum of all such smeared projections:

fb_BP(x, y) = X

projections

B^θh P_θ(r )i

(2.30)

As seen in figure 2.7, this results in a reasonable estimate bf_FBP(x, y) that appears to im- prove with sampling density. However, it is noticeably blurred, and this effect does not diminish with increased sampling. Since bf_BP(x, y) does not approach f (x, y) even with infinitely detailed sampling, we say that backprojection is not a consistent estimator of f (x, y). This is intuitively reasonable, as shown in appendix E. It can however be shown that a modification or filter may be applied to each projection prior to backprojection that will produce a consistent estimator of f (x, y) (see appendix E for details). This filter is most easily expressed in Fourier space, where it becomes a ramp filter which amplifies the magnitude of each component in direct proportion to its frequency:

fb_FBP(x, y) = X

projections

B^θh

P_θ(r ) ∗ h(r )i

, where F^h^{h(r )}ⁱ= h(ω) = |ω| (2.31)

While this is theoretically well-founded, it will over-amplify higher spatial frequencies since these universally contain a higher proportion of noise than lower frequencies. In the presence of noise then, some cut-off frequency or low-pass filter is typically en- forced to limit the amplification of noise into reconstructions by filtered backprojec- tion. Some typically used filters are shown in figure 2.8. Depending on how this filter is applied in a noise-dependent fashion, FBP may not be an efficient estimator of f (x, y).

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Figure 2.7:The principle of backprojection. a An image is projected and re-established by backprojection, using b 1, c 2, d 4, e 32 and f 64 uniformly sampled orientations of projection, respectively. The reconstructed estimate bf of the original image f provided by each projection constitutes a smearing along the orientation of projection in the original x, y-space. As back-projection of multiple orientations are added, the estimate f becomes better, however it does not approach f (x, y), as there is a persistent blur. Illustration reprintedb from [21] with permission of Pearson Education, Inc., New York.

Figure 2.8:Conventionally used filters for backprojection. The Ram-Lak filter is the theoretically appro- priate filter in the absence of noise the the collected images, and the one referenced in eq. (2.31). Other filters modulate the amplification at high spatial frequencies, where noise tends to dominate realistic measurements. The parameters of these filters may be adjusted based on heuristics or estimation of the data, additional filters may be suitably applied to the reconstruction.

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2.5.2 Fourier inversion

Another method of reconstruction from discretely sampled projection data relies on a mathematical result known as the Fourier slice theorem. It states that the Fourier transform of a projection represents the values of a central section through the Fourier transform of the higher-dimensional object (see appendix D). This suggests a natural method of reconstruction by considering each measured projection to inform us on part of the transformF£ f (x, y)¤ rather than the entire, untransformed, field f (x, y).

So-called Fourier inversion thus assigns values of each central section ofF£ f (x, y)¤

based on the Fourier-transformed projections. Finally, an inverse Fourier transform recovers bf_FI(x, y). In theory this is in fact equivalent to bf_FBP(x, y) , but it avoids the explicit application of a filter by virtue of not estimating each pixel as the direct average of all projections. This makes it less sensitive to amplification of noise at higher spatial frequencies. It does however require higher dimensional Fourier transforms, which may make an implementation method slower or more computationally demanding than that of e.g. filtered backprojection. This is typically the case where large reconstructions are made, such as in electron tomography. For the purposes of SPA however, Fourier inversion offers significant benefits with negligible performance impact, and is therefore conventionally used.

2.5.3 Limits of reconstruction fidelity

While consistent estimators of the inverse Radon transform clearly exist even in the presence of noise, we must also consider the reality of data collection, where the projections are neither uniformly sampled nor perfectly known. This begs the question, under what circumstances can we expect a faithful reconstruction of f (x, y)? Consider- ing Fourier inversion, it is clear that each orientation of projection supplies estimations for a unique set of points inF£ f (x, y)¤. Covering 180^◦sampling at increasing sampling density provides arbitrarily good estimates of the full space, allowing the subsequent inversion to recover f (x, y).

Conversely, if less than 180^◦is covered, the transformed space contains a wedge of missing information, for which no estimate is provided by any measurement. This missing wedge will introduce artifacts following inversion, since it contains zeros in lieu of accurate measurements. It is arguably incorrect to let the value zero represent unknown measurements as it may in fact be the correct value and does not signify an unknown component in a mathematical sense. However no procedure exists to assign confidence to each value that in a way the can be accounted for during inversion.

Generalizing the notion of a missing wedge to a non-uniform distribution of orientations, one may formulate an anisotropic and resolution-dependent confidence measure. This can be considered an anisotropic point-spread function (PSF), reflecting the smallest interpretable length scale in an orientation-dependent manner[22]. From this, it is clear that we might expect the reconstruction to be stretched in the direction of lesser confidence. This is particularly noticeable in cases of single-particle pro-

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cessing where the data displays the so-called preferred orientation problem, where a single orientation predominates due to e.g. a strong interaction of the particle with the air-water interface prior to vitrification. Features within reconstructions displaying anisotropic resolution are thus difficult to interpret correctly[23].

Alternatively, we might have complete (and even uniform) but sparse or low sampling covering 180^◦. In this case we find that the projections supply a more complete sampling of lower spatial frequencies, akin to the sampling density in the untransformed space during conventional backprojection (see figure 2.7). Hence, higher spatial fre- quencies are measured less frequently, reflecting a limitation imposed on resolution by the limited sampling⁴. Compared to non-uniform sampling, this limitation is much less problematic since it will be reflected in subsequent resolution estimates of the reconstruction, and can be addressed by simply extending data collection.

4This is intuitively reasonable, reflecting that more measurements are required to deconvolute a volume considered to consist of smaller elements.

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3 | Hypothesis optimization of cryo-EM data

3.1 Likelihood optimization . . . 25 3.1.1 The likelihood . . . 25 3.1.2 The probability of observing a dataset: P (X|Θ) . . . . 27 3.1.3 Optimization ofΘ . . . 27 3.2 Extensions . . . 30 3.2.1 The posterior probability . . . 30 3.2.2 Marginalization . . . 31 3.2.3 Classification . . . 33 3.2.4 Symmetry . . . 33 3.2.5 Resolution estimation and filtering . . . 36

We have established that electron microscopy is able to image the projection of biological particles by phase-contrast, through introduction of known distortions, and that a sufficiently complete dataset of such images allow the reconstruction of a 3D object representing these particle images. We will now describe the consistent procedure to achieve this as implemented inRELION, but the methods are largely similar for other currently used software.

3.1 Likelihood optimization

Most, if not all currently utilized SPA processing tools, optimize the likelihood of the data or reconstruction thereof. We will begin by briefly explaining what the likelihood is, to then derive a simple form of it in the context of cryo-EM SPA reconstruction.

Importantly, we will understand what aspects of the reconstruction are optimized, and how to interpret the behavior and output of the optimization procedure.

3.1.1 The likelihood

A simple example will be used to conceptualize the likelihood. Consider the case where we seek the true value x of some quantity, which has been repeatedly measured. A set of n measurements form a set X of individual measurements xi. It stands to reason that the best estimatex of x would be the average value of all xb _i:

25

Fast and reliable alignment and classification of biological macromolecules in electron microscopy images

Fast and reliable alignment and classification of biological macromolecules in

electron microscopy images

Björn Forsberg

Department of Biochemistry and Biophysics

Fast and reliable alignment and classification of biological macromolecules in electron microscopy images

Björn Forsberg

Department of Biochemistry and Biophysics

FAST AND RELIABLE ALIGNMENT AND CLASSIFICATION OF BIOLOGICAL MACROMOLECULES IN

ELECTRON MICROSCOPY IMAGES

Björn Forsberg

Fast and reliable alignment and classification of biological macromolecules in

electron microscopy images

Björn Forsberg

Eventually, all things

will make sense

Contents

1 | Introduction

Contents

1.1 Biomolecules and their structure

1.2 The challenges of structural biology

1.2.1 The size of biomolecules

1.2.2 The complexity of the cellular environment

1.2.3 The challenge for single-particle analysis

1.3 What this thesis is about

2 | Electron microscopy

Contents

2.1 Why electrons?

2.2 Electron-matter interactions - scattering processes

2.3 Anatomy of the microscope

2.3.1 Electron gun (emitter)

2.3.2 Lens systems

2.3.3 Energy filter

2.3.4 Detectors

2.4 Image formation

2.4.1 Huygens principle

2.4.2 Fresnel and Fraunhofer regimes

2.4.3 Limitations to inversion of diffraction patterns

2.4.4 The focusing lens as a Fourier Transform Operator

2.4.5 The optical transfer function of the microscope

2.4.6 Phase contrast

2.4.7 The CTF

2.4.8 Mathematical formulation of image formation

2.5 3D reconstruction from 2D transmission data

2.5.1 Backprojection

2.5.2 Fourier inversion

2.5.3 Limits of reconstruction fidelity

3 | Hypothesis optimization of cryo-EM data

Contents

3.1 Likelihood optimization

3.1.1 The likelihood