| Introduction | p. 3 |
| Electron crystallography provides access to a unique class of problems in structural molecular biology | p. 3 |
| High-resolution crystallography requires averaging of structures that are present in multiple copies | p. 5 |
| Electron crystallography can produce three-dimensional density maps that are interpretable in terms of an atomic model of the structure | p. 7 |
| Electron crystallography has developed from rich intellectual origins in optics, electron microscopy, and x-ray crystallography | p. 11 |
| Objectives of this book | p. 15 |
| Structure Determination as it has Been Developed Through X-Ray Crystallography | p. 17 |
| Introduction | p. 17 |
| Structure analysis by x-ray crystallography requires well-ordered, three-dimensional crystals | p. 18 |
| The practical steps of data collection and data analysis have become very efficient | p. 19 |
| The Fourier transform plays a central role in understanding the analysis of diffraction data | p. 19 |
| The Fourier transform of a crystal represents discrete, regular samples of the continuous Fourier transform of the molecule | p. 26 |
| The disorder that exists in real crystals can result in easily observed changes in the Fourier transform | p. 32 |
| The Ewald sphere: a powerful mental picture that shows what part of the Fourier transform can be measured for every orientation of the specimen | p. 34 |
| Bragg's law relates the measured scattering angle to the size of the repeat-distance for each sinusoidal term in the Fourier transform of the object | p. 36 |
| Information about the relative phase of each sinusoidal term is lost in diffraction patterns | p. 38 |
| The crystallographic phase problem is usually solved by using additional data obtained from heavy-atom derivatives of the original molecular crystals | p. 39 |
| The three-dimensional electron density of the molecule can be calculated from the experimentally measured amplitudes and phases of the Fourier transform | p. 43 |
| The 3-D density map must be interpreted in terms of other available information, to provide a model of the structure | p. 44 |
| A more accurate estimate of the structure can be obtained by further refinement of the model | p. 46 |
| Published structures are made available through a public-domain database | p. 48 |
| Fourier Optics and the Role of Diffraction in Image Formation | p. 49 |
| Introduction | p. 49 |
| Abbe's diffraction theory of images: image formation is the two-dimensional equivalent of the crystallographer's "inverse Fourier transform" | p. 50 |
| Zernike and the invention of phase contrast microscopy | p. 52 |
| The rigorous diffraction theory of image formation describes images in terms of the inverse Fourier transform | p. 54 |
| The lens as a linear system: transfer functions play an important role in Fourier optics | p. 59 |
| The most common applications of Fourier optics in electron crystallography require that the specimen behaves like a weak phase object | p. 63 |
| The image intensity for a weak phase object remains linear in the projected Coulomb potential | p. 64 |
| The concept of a "phase contrast transfer function" is of central importance in the interpretation of high-resolution images | p. 67 |
| Partial coherence imposes an envelope on the phase contrast transfer function | p. 69 |
| Amplitude contrast can also contribute in an important way to images of thin, biological specimens | p. 72 |
| Single side band images: blocking half of the diffraction pattern produces images whose transfer function has unit gain at all spatial frequencies | p. 74 |
| Tilted illumination produces images for which the transfer function includes both phase errors and amplitude modulations | p. 75 |
| Summary: Fourier optics is an important part of the conceptual foundation of electron crystallography | p. 76 |
| Theoretical Foundations Specific to Electron Crystallography | p. 77 |
| Introduction | p. 77 |
| The single-scattering (kinematic scattering) approximation and the weak phase object approximation are mathematically similar but not identical | p. 78 |
| Proof of the projection theorem | p. 81 |
| Two important simplifications of crystallographic structure analysis occur when the specimen is approximated as a weak phase object | p. 82 |
| Three-dimensional Fourier space is sampled by collecting data at many different tilt angles | p. 83 |
| The resolution of a 3-D reconstruction is determined by the spatial frequency limit of the measurements and by the completeness of 3-D data collection | p. 85 |
| Radiation damage represents a much more important experimental constraint in electron crystallography than in x-ray crystallography | p. 93 |
| Images become very noisy at high resolution due to the finite, "low" exposures which are permitted within acceptable limits of radiation damage | p. 101 |
| Spatial averaging must be used in order to overcome the limited statistical definition that is possible when images are recorded with "safe" levels of electron exposure | p. 102 |
| The amount of averaging required is determined by the number of scattered electrons and by the image quality | p. 104 |
| Instrumentation and Experimental Techniques | p. 106 |
| Introduction | p. 106 |
| The basic design of an electron microscope is much like that of a light microscope | p. 107 |
| Technical features that are specific to electron optics | p. 108 |
| Specimen stages | p. 123 |
| Detectors that are suitable for observing and recording images and diffraction patterns | p. 126 |
| Low-dose techniques make it possible to record high-resolution images and diffraction patterns even from easily damaged specimens | p. 131 |
| Spot-scan imaging can minimize beam-induced movement | p. 134 |
| Samples prepared as self-supported specimens within (or over) holes require additional precautions in order to minimize specimen charging | p. 137 |
| Specimen Preparation | p. 139 |
| Introduction | p. 139 |
| Negative staining provides high contrast as well as excellent stability in the electron beam | p. 140 |
| Metal shadowing produces stable samples which reveal surface topography | p. 142 |
| Glucose and other "sustains" can preserve macromolecular structures to high resolution | p. 145 |
| Contrast matching can be manipulated by using embedding media with different densities | p. 147 |
| Embedding in vitreous ice is the preferred alternative for the preparation of unstained, hydrated specimens | p. 150 |
| Charging and mechanical stability vary with details of the specimen preparation method | p. 159 |
| Preparing extremely flat specimens continues to be one of the most important challenges when working with 2-D crystals | p. 161 |
| Symmetry and Order in Two Dimensions | p. 67 |
| Introduction | p. 167 |
| Classes of symmetry in projection | p. 168 |
| Three-dimensional symmetry classes for monolayer crystals | p. 175 |
| The Fourier transform of a 2-D crystal is sampled at discrete points in two dimensions, but it is continuous in the third dimension | p. 182 |
| Disorder and crystalline defects are an important fact of life | p. 187 |
| Two-Dimensional Crystallization Techniques | p. 194 |
| Introduction | p. 194 |
| Integral membrane proteins represent a natural target for 2-D crystallization | p. 195 |
| Many soluble proteins also form very thin crystals | p. 201 |
| Crystallization at interfaces has potential for wide generality | p. 203 |
| Data Processing: Diffraction Patterns of 2-D Crystals | p. 211 |
| Introduction | p. 211 |
| Diffraction intensities are used in a variety of ways in electron crystallography | p. 212 |
| Data that have been recorded on photographic film must be converted to digital form with a scanning microdensitometer | p. 213 |
| Density versus exposure characteristics can be used to convert the film density to the corresponding value of electron intensity | p. 215 |
| Data can also be collected by direct electronic readout rather than on photographic film | p. 217 |
| The digitized diffraction patterns are then indexed and reduced to the final diffraction intensities | p. 219 |
| Intensities from individual diffraction patterns are merged to form a 3-D data set | p. 225 |
| Factors that affect data quality | p. 230 |
| Data Processing: Images of 2-D Crystals | p. 234 |
| Introduction | p. 234 |
| Optical diffraction is an effective tool for the preliminary evaluation of image quality | p. 235 |
| Conversion of the image to a digital form is necessary for computer processing | p. 237 |
| The fast Fourier transform is an efficient algorithm for numerical computation | p. 244 |
| Images of crystals: indexing the Fourier transform is similar to indexing the electron diffraction pattern | p. 246 |
| Extraction of amplitudes and phases from the indexed Fourier transform | p. 247 |
| Establishing a common phase origin allows data from separate crystals to be merged into a 3-D data set | p. 253 |
| Evaluation of data quality is based on the signal-to-noise ratio | p. 257 |
| Quasi-optical filtering reduces the noise in the image | p. 259 |
| Correction for distortions in the image increases the signal quality | p. 263 |
| Corrections are also required for other systematic image defects | p. 270 |
| High-Resolution Density Maps and their Structural Interpretation | p. 277 |
| Introduction | p. 277 |
| Three-dimensional density maps are computed from discrete samples of the complex structure factors | p. 278 |
| Options for the display of 3-D density maps | p. 279 |
| The missing cone of data results in poorer resolution in the direction perpendicular to the plane o{ the 2-D crystal | p. 282 |
| Interpretation of the high-resolution map involves building the known chemical structure into the 3-D density | p. 288 |
| Accurate atomic-resolution models can also be obtained by docking atomic models of individual components into the 3-D density map of a macromolecular complex | p. 291 |
| Refinement of an atomic-resolution model may proceed in a different way for electron crystallography than is traditionally done in x-ray crystallography | p. 293 |
| Difference Fourier maps | p. 300 |
| Electron Crystallography of Helical Structures | p. 304 |
| Introduction | p. 304 |
| Ideal helices and their diffraction patterns | p. 307 |
| Real helices and their diffraction patterns | p. 318 |
| The hardest step: indexing the diffraction pattern | p. 325 |
| Gathering amplitudes and phases is the next step in the reconstruction process | p. 330 |
| Calculating and interpreting three-dimensional maps | p. 336 |
| Helical particles with a seam can be analyzed by extending the method for helical particles | p. 339 |
| Helical structures can be analyzed using single-particle methods | p. 340 |
| The future looks bright | p. 342 |
| Icosahedral Particles | p. 343 |
| Introduction | p. 343 |
| Description of an icosahedron | p. 344 |
| Local symmetries can be present within an asymmetric unit | p. 347 |
| Theory of icosahedral reconstruction | p. 347 |
| Experimental considerations | p. 349 |
| Data evaluation | p. 351 |
| Image restoration | p. 352 |
| Initial model building and structure refinement | p. 354 |
| Resolution evaluation | p. 360 |
| Poststructure analysis | p. 362 |
| Atomic model determination | p. 363 |
| Single Particles | p. 365 |
| Introduction | p. 365 |
| A certain minimum dose is required to align images of single molecules | p. 368 |
| Due to the lack of symmetries, 3-D imaging requires coverage of the entire angular space | p. 369 |
| Conformational variability increases the total number of images needed to achieve higher resolution | p. 370 |
| Alignment of particles is required for averaging and image reconstruction, and its principal tool is the cross-correlation function | p. 371 |
| Classification may be used to divide the projection set according to viewing directions, conformations, and ligand-binding states | p. 374 |
| Variational patterns among images of macromolecules can be found by using multivariate data analysis or self-organized maps | p. 375 |
| Two useful methods of classification in single particle analysis are hierarchical ascendant classification and K-means clustering | p. 385 |
| Real-space reconstruction techniques can deal with the general 3-D projection geometries encountered in single-particle reconstruction | p. 388 |
| Random-conical and common-lines methods can provide angular relationships among the molecule projections, as a way to jump-start a reconstruction project | p. 395 |
| Angular refinement methods are used to proceed from the initial reconstruction to the final reconstruction | p. 399 |
| Single-particle reconstruction in practice | p. 401 |
| What are the prospects of achieving atomic resolution? | p. 413 |
| Special Considerations Encountered with Thick Specimens | p. 415 |
| Introduction | p. 415 |
| Dynamical diffraction can be described by a number of different, but equivalent mathematical formalisms | p. 416 |
| Conditions when kinematic diffraction theory fails | p. 419 |
| Strong dynamical diffraction effects need not interfere with subsequent refinement of an atomic-resolution model of the structure | p. 424 |
| Fresnel diffraction alone can become significant in thick specimens | p. 426 |
| Curvature of the Ewald sphere destroys the appearance of Friedel symmetry at high resolution and at high tilt angles | p. 428 |
| Inelastic scattering becomes an important consideration in thick specimens | p. 430 |
| A final caution: failure of Friedel symmetry for thick specimens can be due to curvature of the Ewald sphere, dynamical diffraction, or inelastic scattering | p. 437 |
| References | p. 441 |
| Index | p. 469 |
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