Handout for MBB710b/CMP710b Spring 2005 Electron Crystallography

Transcription

1 Handout for MBB710b/CMP710b Spring 2005 Electron Crystallography Topics What are 2D-crystals, and what are the advantages of using 2D-crystals? Electron imaging the properties of electron micrographs How can we retrieve structural information from a single image of a 2D-crystal? How do we combine information from different images? How do we create a three-dimensional data set? 4/18/05 & 4/20/05: V. Unger

2 Definition: Two-dimensional crystals (2D-crystals) are single layers of molecules (complexes) ordered within the x,y-plane. A second layer is permissible, but only if the two layers are related by a (non)crystallographic two-fold axis. In this particular case, the double layer is described as a single object. 2-layered 2D-crystals The twofold axis is marked by a black ellipse. Note, that the twofold axis can fall inbetween molecules. Symmetries consistent with the shown examples are p622 (left example) and p321(right). Types of 2D-crystals There are two very different types of 2D-crystals. Type I crystals are for membrane proteins only. Type II crystals are mostly for soluble proteins and macromolecular complexes because a regular layer of hydrocarbon based lipids is not stable in the presence of the detergents needed to keep membrane proteins in solution. TYPE I 2D-CRYSTALS TYPE II 2D-CRYSTALS TYPE I TYPE II Reconstitution of purified protein into lipids by dialysis or detergent removal with bio-beads In situ - removal of excess lipid from membranes enriched with target protein Crystallization at interfaces Lipid monolayer (with and without specific lipid ligands such as Ni-lipids) Water-Air Water-Carbon 2

3 Advantages of working with 2D-crystals: All molecules on the crystal lattice have identical orientations. This largely improves the signal-tonoise-ratio (S/N-ratio), which is proportional to SQRT(N), where N is the number of particles averaged. The Fourier Transform (FT) of a 2D-crystal is discrete, i.e. the information about the structure of the molecule is contained in defined spots (reflections). This is because in real space, a 2D-crystal can be described as the result of the convolution of a molecule with a lattice (i.e. place a copy of the molecule in each of the points of a lattice). In reciprocal space: the FT of the two-dimensional lattice with spacings a and b is the same lattice rotated through 90 and with spacings 1/a and 1/b (this does not hold for three-dimensional lattices!). In contrast, the FT of the molecule is a continuous variation of amplitudes and phases. The convolution in real space corresponds to a multiplication in reciprocal space the FT of a two-dimensional crystal is the product of the FT of the lattice with the FT of the molecule. In other words, the FT of a 2Dcrystal is the FT of the molecule sampled at the positions of the reciprocal lattice points (because zero wins in multiplication) we observe a discreet patterns of spots. FT calculated from images provide amplitude and phase for each reflection. The discrete nature of the FT makes it easy to correct for image distortions such as astigmatism or the contrast function (CTF) of the objective lens. The contrast transfer function reflects the fact, that the objective lens does not faithfully transfer all spatial frequencies with the same fidelity. Depending on the spherical aberration constant of the lens, the wavelength of the electrons and the amount of defocus, the CTF appears as a pattern of alternating bright and dark zones in the FT of the image. Apart from modulating the diffraction intensities, the CTF also introduces alternating contrast reversals, i.e phases of the reflections on adjacent bands are shifted by 180 with respect to each other. This needs to be corrected if one wants to obtain a consistent representation of the structure. Direct electron diffraction data can be collected if the crystals are large enough ( 1µm x 1µm). Note: Electron diffraction patterns are not affected by the contrast transfer function! However, much the same way as X-ray diffraction data, electron diffraction patterns do not contain any phase information. Disadvantage of working with 2D-crystals: Working with 2D-crystals has only 2 disadvantages. First, one needs to grow a crystal, and second, each image provides only a single view of the molecule. Therefore, we need pictures of tilted crystals that have random orientations with respect to the tilt axis in order to be able to reconstruct the structure. As a rule of thumb: to get sufficient data for a reconstruction to ~6Å resolution one needs ~2-3 images per degree of specimen tilt. How to make 2D-crystals The Reconstitution Approach This technique uses the controlled reconstitution of purified protein into complex or defined lipid bilayers at high protein to lipid ratios. The reconstitution technique is the traditional approach to 2D-crystallization and, to date, has produced the largest number of crystalline specimen. Examples: bovine rhodopsin (Schertler et al., 1993, Krebs et al., 1998), plant light harvesting complex II (Kühlbrandt et al., 1994), aquaporin-1 (Mitra et al., 1995, Jap and Li, 1995), bacterial light harvesting complex 1 (Karrasch et al., 1995), bacterial light harvesting complex 2 (Walz et al., 1998), photosystem II (Rhee et al., 1998), H + -ATPase (Auer et al., 1998), mannitol transporter enzyme II (Koning et al., 1999), glutathione transferase (Schmidt-Krey et al. 1998), NhaA (Williams et al, 1999), ClC-chloride channel (Mindel et al 2001), EmrE (Tate et al, 2001), OxlT (Heymann et al, 2001). 3

4 Advantages From all approaches, offers the most control about crystallization conditions. This may result in a higher degree of reproducibility. Allows crystallographic symmetries with screw-axes because molecules may insert into the membrane with both possible orientations. Disadvantages Needs purified protein. Purification may remove trace lipids that are required for function or crystallization. Since reconstitution results in formation of vesicles or tubes, adding substrates for time resolved studies does not work well if the protein is arranged on a lattice that contains screw-axes. The ratio of crystalline to non-crystalline material may be higher. The crystals may be large enough to collect electron diffraction data. Protocol: a typical crystallization experiment will use µl of a solution containing detergent purified protein at final concentrations of 0.5-2mg/ml, and lipids at molar protein-to-lipid ratios ranging from Lipids should be prepared freshly for each experiment to minimize the negative impact of oxidized lipids on protein crystallization. To this end, aliquots of lipid stock are dried under a stream of argon, and the resulting lipid film is solubilized in detergent. After mixing of detergent solubilized lipids with the protein, the detergent are removed by dialysis or the use of hydrophobic BioBeads. The latter approach is mandatory for detergents with small critical micelle concentrations because in these cases the detergent s monomer concentration in solution is too low to allow efficient removal by dialysis (Rigaud et al., 1997). Parameters such as lipid (e.g. E.coli polar lipid extract, di-myristoyl-phosphatidylcholine, di-oleoyl-phosphatidylcholine), ph, temperature, divalent cations, and substrates should be taken into account in the screening procedure. The In Situ Approach The principle of the in situ approach is to raise the concentration of the protein above the critical threshold required for lattice formation by extracting an excess of bulk lipid from the membranes. Another way of looking at this technique is that the removal of excess lipid induces a phase separation that segregates the protein into a crystalline domain. There are no restrictions as to what detergents are suitable provided the detergent does not solubilize the membrane protein of interest under the conditions being used. The in situ approach is feasible if the target protein constitutes the major fraction of the total membrane protein in a particular membrane fraction. Examples: Frog rhodopsin, constituting ~90% of the total membrane protein in the discs of rod outer segments, was crystallized by treating a preparation of disc membranes with Tween 80 and mixtures of Tween20/Tween 80, respectively (Schertler and Hargrave, 1995). Note that rhodopsin has been crystallized in situ and by reconstitution. Similarly, loosely packed arrays of gap junction intercellular channels were converted into tightly packed crystals by treatment with Tween 20 and di-heptanoyl-sn-glycero-phophatidylcholine. This structure could be solved from sub-microgram amounts of recombinant material (Unger et al., 1997a, Unger et al., 1999) because gap junction channels naturally segregate into specialized lipid domains, which can be enriched on sucrose gradients. Advantages Least disruptive because protein is never removed from original membrane. Absence of crystallographic screw-axes makes time resolved studies easier because substrates can reach all target sites on the lattice. Absence of crystallographic screw-axes allows determination of the absolute orientation of the molecule within the membrane. Disadvantages Size of the unit cell may vary between different preparations. Restricted number of crystallographic symmetries that are possible (p1, p2, p3, p4 and p6). Crystals may be small and (heavily) contaminated with non-crystalline membrane material. Protocol: typical experiments use about 0.5-1ml of a membrane suspension at a final protein concentration of ~1mg/ml. Extraction of lipids is achieved by incubation with detergents, usually for several hours and at 4

5 temperatures ranging from 4-40 C. After treatment the detergent is removed by passing the membranes over a sucrose cushion/gradient, followed by dialysis to remove monomeric detergent that still resides inside the membranes. If successful, then an equivalent of about 100µl of the extraction reaction provides sufficient material for the structure determination to an intermediate resolution. However, more material is needed to solve the structure to near atomic resolution. The Surface Crystallization Approach (for membrane proteins) The surface crystallization technique exploits the tendency of membrane proteins to spontaneously form twodimensional arrays at the interface between two phases such as the air-water or a carbon-water interface. The latter can be exploited to grow 2D-crystals directly on the carbon support film of the electron microscope grid. This has been demonstrated for the plasma membrane H + -ATPase, which was the first, and so far only, protein crystallized by this approach (Auer et al., 1999). A different way of looking at this approach is that it represents a substrate-induced 3D-crystallization, which is interrupted at the stage where the crystal is only 1-2 layers thick. Advantages May result in large arrays, suitable for electron diffraction. Arrays form within minutes to periods of a few hours. Crystals are formed directly on the grid. The presence of substances such as polyehtylene glycols and glycerin allows cryo-preservation without intermediate steps. Needs very little protein. Disadvantages While large, crystals may not be coherent across large areas. The presence of precipitants such as polyehtylene glycols in the crystallization mix causes high background in the diffraction patterns. The crystals are held together by detergent. Any changes to the environment are likely to disorder the crystals. This makes them unsuitable for timeresolved studies and observation in negative stain. Protocol: ~5-10µl of protein containing crystallization solution will be placed on a siliconized cover slip. A gold plated, carbon coated electron microscope grid is placed onto the drop immediately. In analogy to a hanging drop experiment, the coverslip is then transferred to a Linbro plate. The reservoir is filled with water and the coverslip assembly is sealed with immersion oil. Various temperatures and incubation times should be tested to establish whether suitable conditions can be found. Parameters to be optimized for the crystallization solution are: protein and detergent concentrations, ph, salt, precipitants such as PEGs, cryoprotectants (e.g. trehalose, glucose, glycerol) as well as amphiphilic additives (e.g benzamidine, heptanetriol). At the end of the incubation, grids are removed from the surface of the droplet, excess crystallization solution is blotted with filter paper and the grids are vitrified in liquid ethane. Specimen have to be observed in a frozen hydrated state because negative staining will destroy lattices that are held together mostly by the detergent present in the crystallization solution. Lipid Monolayer Crystallization for Soluble Proteins This approach exploits the enrichment of proteins at a lipid monolayer interface. The high protein concentration in vicinity of the lipid layer can induce the formation of 2D-crystals. Several tricks have been used to immobilize the protein at the lipid-buffer interface. For instance, small amounts of charged lipids can be used to bind the protein by electrostatic interactions. More specific interactions can be enforced by using lipids with defined affinity-labels such as Ni-binding headgroups. Such bait-lipids can specifically bind to any histidine-tagged protein. Examples: HIV-1 reverse transcriptase (Kubalek, EW et al, 1994), RNA polymerases (Edwards, AM et al, 1990, Asturias, FJ and Kornberg, 1995) + many others (Brisson A. et al., 1994). Advantages Disadvantages 5

6 May result in large arrays, suitable for electron diffraction. Arrays form within minutes to periods of a few hours. Needs very little protein. While large, crystals may not be coherent across large areas. The crystals are extremely fragile, and transfer to the electron microscope grid is tricky Protocol: Protein solution (concentration 0.1-1mg/ml) is placed in a reaction well (10-35µl depending on design). Using a mixture of an inert filler lipid and a so-called bait lipid (e.g with Ni-bindign head group) in organic solvents, a lipid monolayer is generated by adding 1µl of the lipid mixture to the well. The crystallization is then incubated at 100% relative humidity and temperatures ranging from 4-40 C. Incubation times must be established empirically (it takes ~20-30 minutes for the protein to be concentrated at the lipid layer crystallization will take >30 minutes). At the end of the incubation, an electron microscope grid is placed on top of the solution in the well. With a bit of luck, the crystal should adhere to the surface of the EM-grid when the latter is withdrawn. After removal from the crystallization solution, the grid is either negatively stained, or prepared for vitrification. Difficulties with transfer of the crystals may be overcome by using holey carbon films instead of continuous carbon (use of continuous carbon mimics the use a platinum wire loop ). However, in this case negative stain will only work if the protein forms a very stable lattice. Electron Imaging the properties of electron micrographs Before anything else, lets contemplate the nature of our images for most applications, they are still recorded on EM-film (although digital imaging is becoming more common, especially for single particle applications). EM-film is a piece of plastic backing covered with silver halide grains ~3-5µm in size. Notably, it only takes a single electron to reduce a silver ion. This is different from light photography where multiple photons are required to develop the emulsion. The 1:1 stoichiometry (1 electron hit = 1 silver ion reduced) is a great advantage because it directly links the number of scattering events (including not being scattered at all) with the optical density of the film after developing. Unfortunately, there is no way to picture how a particular grain being hit (multiple times in some cases) correlates with the actual scattering events at the specimen. However, intuitively we know that there must be a relationship between the physical grain size of the emulsion and what structural information we can retrieve from the image. In other words, what is the scale imprinted in the grains? This depends on the magnification. For instance imaging a 2D-crystal at 10,000x and 50,000x magnifications will result in different numbers of grains representing a unit cell. Using more grains to describe the molecules on the lattice goes in parallel with recording a lager amount of structural detail (i.e. higher resolution). Why? At 10,000x magnification, a 5µm grain represents a 5Å detail at the level of the specimen (5µm/10,000). At 50,000x magnification, the same grain would represent a 1Å detail at the level of the specimen (5µm/50,000). Consequently, if our unit cell was 100x100Å, then we d use 20x20=400 grains in the first case, while expending 100x100=10,000 grains in the second. Note, if the same total number of electrons is used for imaging at both magnifications then we get a weaker exposure at the higher magnification because the same number of scattering events is captured on a larger number of grains (i.e. some grains will not be developed). And consequently, as we increase magnification we also need to increase the total electron dose delivered per Å 2 if we want to maintain the same overall counting efficiency (= optical density). Recalling that electrons interact with matter rather strongly, we immediately see that radiation damage will increase as we try to image detail at higher resolution. At the same time, a smaller number of unit cells is imaged onto a piece of film compared to a picture taken at low magnification. Decreasing the number of unit cells in turn will reduce the signal-to-noise ratio because the latter is directly proportional to SQRT(N), where N is the number of molecules in the crystal. Consequently, the first conclusion to be drawn is that we need to find a suitable compromise between magnification and resolution to be obtained. Lower magnification = less electrons = less damage, but also less detail. Higher magnification records more detail at the expense of higher radiation damage (primarily affecting the fine detail in our molecules) and a reduction of the number of molecules observed in a single image. In practice, if the aim is to recover data at near atomic resolution then images need to be collected at magnifications of 6

7 60,000-70,000x (corresponding to a grain resolution of ~0.8 and 0.7Å respectively). Why do we need that good a grain resolution? Part of the answer is the need to sample an image in order to be able to work with it on the computer. Sampling of the image means that we replace the random pattern of grains by an xy-array of evenly spaced pixel. Each pixel represents the averaged optical density of the corresponding area of the micrograph. The smallest pixel size that can reliably be recorded by high-end flatbed scanning densitometers is 5µmx5µm, i.e. about the size of individual silver grains. However, accuracy is usually better if a slightly larger sampling aperture is used. In practice, people scan their images at stepsizes of 7-14µm. In particular, the introduction of fast CCD based film scanners allows routine use of 7µmx7µm pixel sizes. With this in mind, we can reevaluate what magnification to use for high-resolution imaging. Assuming the image will be sampled in 7µm pixel, then a magnification of 70,000 fold corresponds to a pixel resolution of 1Å at the level of the specimen. This still sounds like an overkill.. why would one want to sample at 1Å intervals? The answer is that the best resolution obtainable from an image sampled at 1Å intervals is about 2-3Å.. This is also known as Nyquist limit stating that an image needs to be sampled at a frequency of at least twice the frequency of the highest resolution term to be retrieved. In other words: to achieve a final resolution equal to 5Å in a structure reconstruction one needs to sample the image at least at 2.5Å (preferentially better than that). Applied to near atomic resolution, i.e. ~3Å, this means we need a 1-1.5Å sampling at the level of the specimen. Why? Because, we cannot distinguish a wave whose period corresponds to the cutoff from a wave that is in phase but has half the period (= double the resolution). For instance, if sampling is in 5Å intervals then we cannot tell whether any particular 5Å modulation stems from waves corresponding to reflections at 5Å or waves representing 2.5Å reflections that are in phase with the 5Å waves. Consequently, we need 2.5Å or higher frequency of sampling to unambiguously describe a 5Å detail. The practical information limit that can be achieved with a good microscope is somewhere around 1.5Å, i.e. in theory we would need to sample at ~0.5Å increments at the level of the specimen to make sure that all frequency contributions can be described appropriately. The Nyquist limit Pixelsize/magnification = sampling distance at the level of the specimen For instance: pixel is 10µm, magnification was 50,000-fold ==> 2Å/pixel A plain sine-curve is defined by any pair of values taken from within one period ==> if the sampling in the image 2Å per pixel, and 2 pixel values are needed to define frequencies, then the highest frequency that can be reliably described is 2*2=4Å This relation is referred to as the Nyquist limit and says that an image needs to be sampled at at least twice the frequency of the highest resolution to be obtained RESOLUTION-and pixelsize 2.6Å/pixel 80, , , Å/pixel 240, Å/pixel 1.65Å/pixel The two figures to the right are meant to illustrate the above: Graphitized carbon - lattice spacing is 3.4Å In summary: to prepare images for digital processing, we need to sample the image, i.e. break it down into evenly spaced pixel. This process is called digitization. The pixel size depends on the magnification of the micrograph and the resolution that is targeted. Example: Let the magnification be 55,000x. What is the best possible resolution that can be obtained if the image was digitized with a 7µm pixel size? Answer: ~3Å ([7/55,000]*2=2.5Å frequencies at better than 2.5Å cannot be retrieved, and description of frequencies exactly at 2.5Å is not absolutely reliable) 7

8 Obtaining structures from images of 2D crystals Optical diffraction is the first step to identify good images of (hopefully) good crystals. Moving the image through a highly coherent laser beam results in a diffraction pattern if any part of the image represents a crystalline array. The number of orders of spots that we can see, the size of the spots, their shape and sharpness immediately tells us how well ordered the crystals is. We also will see whether the underfocus was chosen correctly, whether the image is astigmatic and whether specimen movement occurred during the exposure. At this stage, we also can get a first impression how large the coherently scattering area is, making it easy to identify which images are worthwile to process any further. Once a suitable crystal has been identified, the image is digitized, i.e. the crystal is represented by an array of evenly spaced pixel of optical densities. After digitization, a Fourier Transform is calculated. This results in a pattern of reflections similar to diffraction patterns observed in X-ray crystallography. Compared to X-ray diffraction patterns, a large difference and great advantage of transforms calculated from images is that both the amplitude and phase information can be calculated from the complex numbers representing the transform pixel! The availability of phase information allows several things to be accomplished: We can directly go forth and back between reciprocal space and real space allowing us to use whichever is more suitable for certain types of operations required to analyze the images. The lattice imperfections are encoded in the phase information of the transform this we can use for a computational improvement of the crystallinity (= correct for certain lattice distortions) through correlation methods. Phase information allows us to combine the data from different images. The combined data from different images in turn allow us to refine image parameter such as astigmatism, and the tilt geometry where applicable. To understand the general philosophy of processing information from images of 2D-crystals, we need to recall/ familiarize ourselves with some basic facts. First, the FT of the crystals is discrete. If we had an infinitely large and perfect crystal, we should observe a diffraction pattern where each diffraction spot takes up just a single pixel. As shown in the figure, the transform of a raw image tells us that this is far from being true part of raw image calculated Fourier Transform 8

9 Original transform masked transform r=7 pixel masked transform r=1 pixel Principle of Digital Filtering entire FT raw image enlarged area of FT filtered image low local averaging = large maskhole circular maskholes applied (FT has now non-zero values only within maskholes filtered image high local averaging =small maskhole Although spots are visible, they are not always perfect (=sharp). Furthermore, the FT of the raw image shows very significant intensity everywhere and the oscillating pattern of dark and light bands indicates the modulation of the image by the contrast transfer function (CTF). Knowing that the structural information about the crystalline material is constrained to the discrete diffraction spots, we know that all other non-zero values in the transform are noise. This allows us to use a computational trick known as digital image filtering for an immediate improvement of the quality of the image. The principle of image filtering is to construct a mask for the original transform that will eliminate all but the part of the information we are interested in. A sharp edged hole is the most commonly used type of mask. An example how this affects the image is shown in the figure. The two panels in the middle illustrate the effect of placing circular maskholes with a 7-pixel radius around each of the diffraction spots out to ~9Å. With the exception of the transform values inside the maskholes, all other transform densities are set to zero. Inverse transformation gives the filtered image showing the periodic information of the crystal lattice much more clearly. However, we can still detect considerable variation of the periodic motif. This is not the case if we only allow the information from the predicted and precise lattice position to contribute to the filtered image. Using a maskhole radius of 1 (right two panels in previous figure) it is almost impossible to detect the individual spots without magnification of the panel. However, the filtered 9

10 image below illustrates an important aspect of this step. By choosing a maskhole radius of one, we simulate an infinite and perfect crystal. Consequently, the inverse transform shows a uniform representation of the crystalline object, and the noise is largely reduced. While this looks nice, it is not the ultimate result. Using the information contained in both these filtered images, we can further improve our image data. To understand how this improvement can be made lets first consider the hallmark of a perfect crystal. In the latter, all unit cells will be arranged on a precise lattice, without any deviation + all molecules in all unit cells will look identical. In practice, this is never the case. First, taking low-dose images means that we will image different detail from each of the unit cells, i.e the molecules in different unit cells will appear (very) similar but not identical. Second, the unit cells will never be on a perfect lattice. Several types of lattice disorder may be present in a 2D-crystal. The easiest one is translational disorder meaning that the xy coordinates of consecutive unit cells are shifted away from their expected lattice position by some small amount. As will be introduced later, this type of lattice distortion can be corrected because of the availability of phase information. The second type of lattice disorder is rotational disorder. This type of disorder could be dealt with, but current software will not correct for this type of disorder. Lastly, unit cells may be slightly tilted out of plane. This type of defect cannot be corrected without having information about the complete 3D-structure of the molecule, and even then, it would be computationally quite challenging. From this brief outline it becomes clearer that the 2D-image processing largely deals with the correction of translational lattice disorder. How does translational disorder impact on the FT of the 2D-crystal, and how can we correct for translational disorder? Translational disorder can best be understood as having a collective of small crystals with slightly different unit cell vectors instead of one coherent crystalline area with constant cell parameter. Because the length of the unit cell vectors determines where in the FT the diffraction intensity will appear, a mosaic of very similar yet slightly different vectors will result in blurred diffraction spots. This is true especially at high resolution. For instance, suppose our unit cell vectors deviate by 0.5 pixel over one order and that the unit cell is square with overall dimensions of 100x100Å. Consequently a (10,0) or (0,10) reflection will describe a 10Å detail, the (20,0) and (0,20) reflections describe 5Å and about 36 orders of reflections are required to describe the structure at the ~2.8Å resolution limit obtainable by electron diffraction from biological specimen. If the different unit cell vectors in the mosaic crystal are off by ~0.5 pixel, this will distribute the diffraction intensities at ~2.8Å over a circular area with ~18 pixel radius centered about the predicted peak position. Such a spread of the signal will make it undetectable over the autocorrelation two copies of same object crosscorrelation two similar objects two identical objects with translational offset background noise. Yet, the example also hints at how we can correct for translational disorder. Because the blurring of the diffraction spots contains information about the translational disorder, we can use this information to bring all unit cells into perfect register with an arbitrarily chosen reference. In praxis, we use the difference in the information content of a tightly and 10 Δx 0 origin w x c(x) -w 0 w c(x) -w w Δx 1 1 c(x) -w w 1 x x x

11 loosely filtered image to achieve this goal. Referring back to the figure illustrating the effects of digital filtering, the image created by using a small maskhole radius serves to define a reference area. While looking pretty, this image does not contain any significant information about the lattice disorder. However, the loosely filtered image (maskhole radius of 7) does include the information from the blurred diffraction spots. Since each of the non-zero transform pixel of the masked transform contains phase information, we can determine the relative positioning of each unit cell in the loosely filtered image with respect to the reference picked from the tightly filtered image. Consequently, correction of translational lattice disorder comes down to shifting all unit cells in the crystal into precise register with the reference area. By doing this, the collective of slightly different lattice vectors will vanish and the diffraction contribution of each unit cell will be directed to the transform positions dictated by the unit cell vectors of the straightened (= unbent /corrected) lattice. In practice, this goal can easily be achieved by correlation methods. To understand how these methods work lets consider a pictorial representation of the two types of correlation approaches, autocorrelation and crosscorrelation. Autocorrelation involves correlating an object with itself. The corresponding correlation function c(x) is maximal if both copies of the object are in register. For any displacement of the two copies against each other, c(x) will be smaller, falling off to c(x)=0 if the copies are shifted by +/- their width (= no overlap). Note, the shape of the autocorrelation peak does not represent any particular structural detail but reflects the sampling of the object along the correlation axis (=x axis in the one-dimensional example). In other words, if an object has an undefined shape showing only little modulation along x, then the autocorrelation will be a broad peak. If on the other hand the object shows a high degree of modulation, then the autocorrelation will produce a narrow and well-defined peak. This feature can be exploited to evaluate lattice distortions. How? The answer comes from looking at the outcome of crosscorrelation events. In the case of a crosscorrelation, one of the objects serves as reference, the other copy as target. Let us consider two extreme cases. First, let two nonidentical but generally similar objects be centered at equivalent origins. In this case, c(x) will still be maximal for the zero-displacement position. However, as the copies are not identical, the overall value of c(x) will be less than that obtained in the autocorrelation. The other extreme is shown in the bottom panel where two identical copies of the object are shifted out of register by Δx. In this case, c(x) will have the same shape and maximum as that obtained in an autocorrelation yet the corresponding crosscorrelation peak will be displaced by the off-set Δx. Consequently, crosscorrelation will provide two parameters: similarity and translational offset. Naturally, in most real cases we will have a mixture between the two extremes, i.e. we will deal with two objects that are not identical but similar and are offset. It is now easier to appreciate how we can correct for translational lattice disorder. The first step involves a crosscorrelation between a reference area and the entire Autocorrelation Map image. This will give a socalled crosscorrelation map consisting of an array of crosscorrelation peaks across the area of the image. To mine this information, we need a search object that serves as a model for what we are looking for. In our case, an autocorrelation map calculated from a small area within the reference serves this purpose. The similar shapes of the autocorrelation and crosscorrelation peaks allow the autocorrelation peak to be used as a model to Part of Crosscorrelation Map Note that the shape of the central peak in the autocorrelation map is very similar to the shape of the cross-correlation peaks. 11

12 determine the position (=defining translational disorder) and height (=similarity with reference) of the crosscorrelation peaks within the crosscorrelation map. Starting at the center of the reference area, we move along the lattice and compare the height of the crosscorrelation peak with that of the autocorrelation peak for each unit cell. This comparison tells us how similar the inspected unit cell is with respect to the reference. An example is shown in the figure. The information contained in this part of the cross-correlation map is an easy way for us to identify and select the best area of the crystal. For instance, based on the similarity alone, we know that the area in the right lower corner does not belong to the crystal. In fact, in this particular image this part was occupied by plain carbon film. The carbon film however does not contribute to the information we are seeking all it does is adding background noise. Consequently, being able to locate the best area of the crystal in the image allows us to box this area, i.e. replace all but the best part of the crystal by a uniformly gray area. Calculating amplitudes and phases from the FT of such a boxed image greatly improves data in cases where not all parts of the image are of the same quality, or where some parts are non-crystalline at all. Regardless of this trick, now lets have a look at the part of the cross-correlation map that describes the translational lattice distortions. On a perfect lattice, we would always find the crosscorrelation peaks in their predicted position (i.e. spaced away from the center of the reference by integral multiples of the lattice vectors). This is not the case if the lattice suffers from translational disorder. In this case the crosscorrelation peak of the unit cells are often offset from the positions predicted by a perfect lattice. However, this is no longer a problem because we can locate the crosscorrelation peak by searching for it using the autocorrelation peak, calculated from the reference area, as search model. This step of the entire process provides us with a list of exact xy-coordinates for the positions of the crosscorrelation peaks and allows us to calculate a displacement vector for each unit cell. An example is shown in the figure. To correct for the translational lattice disorder, all we need to do is to reinterpolate the image, i.e shift all pixel of translationally disordered unit cells by the negative amount of the displacement vector. This procedure aligns all unit cells with the reference area, thereby restoring the high-resolution information in the image. To illustrate this point, the information retrieved from the calculated transforms of an image before and after correction for lattice disorder is shown below. The xy-offset vectors (x20) are shown for each of the unit cell positions across the image area. Blank areas indicate that no crosscorrelation peak was found in this position. The random pattern of vectors towards the right of the crosscorrelation map indicate that this area is not crystalline. 12 The data shown to the left were retrieved from a calculated FT of an untreated raw image. In this case, the data are not statistically significant beyond ~15Å resolution. However, after correction for translational lattice

13 At this point we are ready to summarize the correction for translational lattice disorder by the following flowchart. In this diagram, gray boxes represent Fourier Transforms that are calculated at some intermediate stage of the processing. Bold arrows indicate a direct input output relationship. In some cases, we use information from a processing intermediate to manipulate another file. For instance, we use the list of crosscorrelation peak positions to reinterpolate the original digitized raw image. Such a relation is indicated by a dotted arrow. The flowchart empahsizes how we jump forth and back between real space and reciprocal space. Having phase information in the calculated diffraction patterns makes this a straightforward process: image FT of image image (= real space reciprocal space real space) is achieved by calculating consecutive Forward and Backward Transforms. Note that the crosscorrelation itself is done in reciprocal space because in this domain, it comes down to a multiplication of two Fourier Transforms (which is fast and easy to do). If the crosscorrelation were done in real space, we would have to Autocorrelation Map Digitized Raw Image Digitally Filtered Image (small radius) Reference Area Crosscorrelation Map List Positions of Crosscorrelation Peaks Reinterpolated Unbend Image Boxed and Unbend Image FT of Reference Area LIST OF AMPLITUDES AND PHASES FT of Digitized Raw Image Masked FT of Digitized Raw Image (large radius) FT of Crosscorrelation Map FT of Boxed and Unbend Image 13

14 sin φ(α) perform a convolution, i.e. calculate correlation values for each possible shift of the reference area against the entire image. This would be very cumbersome + suffers from the noisy background in the image. By using Fourier methods, we can largely circumvent the noise because the digital filtering allows us to create a noise free reference. This is a huge advantage over the real space approach. However, the crosscorrelation-transform that we obtain does not directly reveal the positions of the crosscorrelation peaks!! To retrieve this information, we first need to translate the information of the crosscorrelation transform back into real space. We can then use the real space crosscorrelation-map and the real space autocorrelation peak to determine the distortion parameters of the lattice. The last paragraphs discussed the principles of how we can use the phase information in the calculated Fourier transforms of images of 2D-crystals to correct translational lattice disorder, and how to identify the best area of an image for computation of a final set of amplitudes and phases. However, in dealing with each of the images this is not the final result. We still have to correct for the impact of the contrast transfer function, CTF, on the phase data. To illustrate this fact, the following figure shows again part of the tightly filtered image. The second panel shows an equivalent part of the image after correction of the data obtained from this image for the effects of the CTF. Tightly filtered image data from same image (r=1) after correction for CTF For the unbending procedures, this modulation caused by the CTF does not matter because all parts of the image are affected in the same way if the specimen is untilted with respect to the incident electron beam. However, if we want to calculate the actual density distribution in the molecules then we need to correct for the CTF. Knowing the wavelength, magnification, spherical aberration constant of the objective lens, the level of defocus and pixel size of the digitized array, this can be achieved by a simulation of the CTF. As shown in the figure, this simulation allows us to determine which reflections require adjustments of their phase by 180 (= make all reflections contribute with the same contrast; in the table below, all reflections with negative values for the CTF had their phases adjusted by 180 ) first zero At this stage we have reduced our image to a list of data containing the indices (h,k) of the individual reflections, amplitude, phase, K 14 H

15 quality (IQ), the rms-background (bck) around the diffraction spot and the numerical value of the CTF at the location of the reflection. H K Amp Phase IQ Bck CTF How do we generate a structure? Again, the key to success is the availability of direct phases! This allows us to determine the symmetry of the crystal lattice, to combine data of different images, and to refine image parameter. The first step is to determine the crystal symmetry. There are 17 possible two sided plane plane groups for 2D-crystals. Most of these are described by primitive unit cells (indicated by a p ). The nomenclature for these plane groups sometimes differs from the crystallographic conventions in that the first symmetry element, following the cell type, describes the symmetry along the c axis. This is followed by additional symmetry elements along the a- and b-axes (in that order) where applicable. For instance, p4 symmetry of a 2D-crystal means that there are four molecules in the unit cell which are related by a simple 4-fold axis of rotation perpendicular to the plane of the crystal. Similarly: p1, p2, p3, and p6 describe the remaining symmetries that only involve simple rotations about the c-axis. However, there are many cases, where the symmetry of the crystal is higher. For instance, p22121 symmetry indicates twofold symmetry about the cell s c-axis, and twofold screwaxes along both a and b. Having symmetry means that the information in the Fourier Transform is redundant as long as the crystal is not tilted with respect to the incident electron beam. If this applies then the information of so called symmetry-related reflections is the same, and provides more than one independent measurement of the same structural detail from the same image. This behavior can be exploited to determine the symmetry of the crystal because symmetry related reflections obey specific phase relationships in projection (i.e. if no crystal tilt is present). For instance, a twofold axis about c constrains the phases of all reflections to adopt values of either 0 or provided that the two related molecules are centered at the right position within the unit cells. As illustrated in the following figure, we can use this fact to test for the presence of symmetry elements. To this end, the molecules of a unit cell are placed into all possible locations within the cell and conformity with applicable phase constraints is expressed as a phase error. The calculated phase error will be smallest for positions within the unit cell that relate the Principle of Symmetry Determination (0,0) a 15 b twofold-axis boundary of unit cell

16 molecules by permissible symmetry operations (twofold, threefold, fourfold, sixfold, twofold screw). For instance, the representation of the calculated diffraction pattern shown on the previous page immediately suggests that the molecules are arranged on a hexagonal lattice (a*=b*, γ*=60 ). However, such packing of the molecules is compatible with more than one potential symmetry: p1, p2, p3, p321, p6, p622. We can differentiate between these possibilities because each of these two sided plane groups imposes a characteristic set of phase constraints. For instance, in p1 no constraints apply and positioning of the phase origin is arbitrary. In p2, p6 and p622 all phases are constrained to 0 and 180. Furthermore, in p3, p321, p6 and p622 triplets of symmetry related reflections have identical phases (and amplitudes). However, the lack of twofold symmetry in p3 and p321 causes the loss of the THE MATRIX BELOW IS CENTRED ABOUT AN ORIGIN WITH A PHASESHIFT OF 0.00 FOR THE (1,0) REFLECTION 0.00 FOR THE (0,1) REFLECTION STEP SIZE BEST PHASE SHIFTS ARE FOR THE (1,0) REFLECTION FOR THE (0,1) REFLECTION NOTE THAT THESE SHIFTS INCLUDE THE INITIAL SHIFTS AS WELL AS THE ADDITIONAL REFINED SHIFTS only one position for sixfold symmetry axis SPACEGROUP Phase resid(no) Phase resid(no) OX OY v.other spots v.theoretical (90 random) (45 random) 1 p p2 25.4! b p12_b a p12_a b p121_b a p121_a b c12_b a c12_a p b p2221b a p2221a p c p p p p3 13.5* p p p6 15.4* p * = acceptable! = should be considered ` = possibility c e n t r o s y m m e t r i c condition (i.e. all phases are real = 0 / 180 ). In other words, we can d e t e r m i n e t h e symmetry of the crystal by testing the phase relationships of all symmetry related reflections for all possible locations within the unit cell, and for all 17 twosided plane groups: PHASE ERROR AT MINIMUM IS 18.5 DEGREES in p6 The example shown above demonstrates that in this case the phase relationships are compatible with a unique sixfold axis of symmetry or in other word, the crystallographic symmetry of the crystal is p6. The following table illustrates even more explicitly what it means to bring symmetry related reflections to their common phase origin:

17 Reflection Phase extracted from Phase after shifting to (h,k) calculated FT proper phase origin (12) (2,-3) (-3,1) In p6 triplets of symmetry reflections all adopt the same phase when centered at the correct sixfold origin. Brought to this phase origin, the phase of each of the symmetry related reflections has to be close to either 0 or 180 because the inherent twofold symmetry (p6 symmetry implies p3 and p2 symmetry) constrains phases to be real in projection. Once the symmetry has been determined, we can then align the data from independent images based on the same principle. This is illustrated in the following figure. Alignment of Data from Independent Images Find common phase originby comparison with data from single untilted crystal Once a number of images have been brought to the proper phase origin, the independent measurements can then be averaged to yield a final list of (h,k,l (=0 in projection), amplitude, phase, and a figure of merit (FOM)). Note: if crystallographic symmetry is present, then only the unique set of reflections is generated from the data, applicable phase constraints are enforced, and the appropriate symmetry operators are applied during calculation of potential maps. Up to this point, we only considered the treatment of images from untilted 2D-crystals. These data allow us to calculate a so-called projection density map, such at that shown in the figure below: Iteratively merge and refine phase origins of each image against average of all others Assume that the rectangle represents the schematic outline of a fictiuous unit cell, and that in an initial step the data of the first image (marked ref) where brought to the proper crystallographic phase origin. If compared to data from other images (which at the end of the initial image processing will have an arbitrary phase origin), there will be no agreement between the phase data. However, this can be changed by calculating the phase shifts necessary to superimpose the images. Initially, only the data of the very first image are available as reference. However, once other images have been brought to a common phase origin, the combined data can be used to carry out further refinements until the best possible overall agreement has been reached. B e i n g a t r u e p r o j e c t i o n, t h i s p o t e n t i a l m a p represents the sum of all atoms within the structure that are ordered. In this case, Projection Density Map and some of the Corresponding Structure Factors 17 (H,K,L) amplitude phase FOM

18 the projection is of a gap-junction intercellular channel as seen at ~6Å resolution. Protein is represented by continuous lines, while lipid and the aqueous channel are contoured with dotted lines. In this particular case, both the amplitude data as well as the phase data were obtained from the images, illustrating that structures can be obtained from images alone (as long as the resolution is moderate). For various reasons, this changes as near atomic resolution is approached. Yet discussion of the reasons is beyond the scope of this introductory course. Step 1 embed & freeze Step 2 search & shoot Step 4 Use phases to align images and calculate average Projection density map Step 3 Use amplitude and phase information from images for correction of lattice distortions (= computational improvement of crystal quality) SUMMARY: FROM SPECIMEN TO PROJECTION STRUCTURE As you may notice, the projection map does not directly reveal information about the three-dimensional structure of the molecule. Because of this, one needs to be extremely careful about any conclusions drawn from density maps such as those shown above. Only in very special cases is it possible to assign density features in the projection map to secondary structure elements such as α-helices. To unambiguously resolve secondary structure and/or side chains, one needs to collect a complete three-dimensional data set. How can this be done? Lets consider again what the calculated Fourier Transform of an image represents. In technical language, the calculated Fourier Transform of an image represents a central section through the three dimensional molecular transform. In the case of projection data, this section corresponds to the equatorial plane within the sphere of reciprocal space, and describes an end-on view of the molecule. To retrieve the full three-dimensional information we need to collect images representing different views of the structure. This can easily be achieved by tilting the crystals with respect to the incident electron beam. Remember that the molecular transform will rotate with the crystal. However, the imaging plane does not rotate. As a result, the projections of the tilted crystals correspond to different sections through the molecular transform. Since the crystals are randomly oriented on the electron microscope grid, this procedure provides us with the data necessary to reconstruct the three-dimensional structure if we are able to work out the geometrical relationship that relates one view to another. Fortunately, this is possible, and hence we can assign the 18

19 phases and amplitudes from the calculated transforms of the different projections to their correct place in the molecular transform. This leaves the question, how does the molecular transform of a 2D-crystal looks like? The answer comes from contemplating the characteristics of Fourier Transforms. As outlined in the beginning, the Fourier Transform of an image from a 2D-crystal is discrete, ie displays spots. This we explained by the fact, that the observed transform of the crystal comes about by multiplying the transform of a 2D-lattice with the molecular transform of the molecule (= sampling the molecular transform at the reciprocal lattice points). Contrary to 3D-crystals, 2D-crystals lack unit cell repeats along the third dimension. Consequently, while being discrete in the xy-plane, the Fourier Transform of a 2Dcrystal is continuous along the z-direction. In other words, instead of determining a set of discrete reflections along a*, b*, and c*, we need to determine the amplitude and phase variation along so-called lattice lines. As sketched in the figure, this is achieved by collecting central sections through the transform by recording images at different tilt angles of the specimen. Concept of Lattice Lines and Principle of Sampling their Data In practice, lattice line data look like those shown in this figure: Once sufficient independent measurements have been taken along each of the lattice lines, structure factors are generated by fitting curves to the experimental data, and sampling of the amplitude and phases at intervals of no less than [1/thickness of the molecule]. This sampling generates a set of discrete structure factors with indices (H,K,L). The three-dimensional structure can then be calculated by simple Fourier Summation. Note that the maximum z* value shown in the lattice line is small. This is due to an extremely limited amount of tilt data that were available to compile this data set. In general, the maximum tilt that can be achieved is about 60. Still, even at this amount of tilt a cone of missing data will be present causing a distortion of the structure. This distortion will be more or less pronounced depending on how much data are missing. In all cases, however, the distortion manifests itself as a stretch along the direction of c in the real space potential map of the molecule. This anisotropy in resolution 19 Example for a lattice line This figure shows the variation of phase (top panel) and amplitude (bottom) of the (2,5)- reflection of a gap-junction 2D-crystal as function of z*. The amplitudes were obtained from the image transforms and therefore are very noisy (=large scatter).

20 cannot be overcome unless edge-on views of the crystal (providing data for the 00L lattice line) are available, which rarely is the case. Nevertheless, if 60 of tilt have been covered by the data set then the distortion is small e n o u g h t o a l l o w a n unambiguous interpretation of the potential map. This is true in particular for maps at near atomic resolution. In cases with limited in plane resolution (5-10Å) the situation becomes worse and vertical resolutions (= resolution along the direction of c) of 10Å or worse are not uncommon. Note that the projected views obtained from the tilted crystals are no longer 6-fold symmetric as was the case for the projection of the untilted crystal. While lacking the defined relationships between symmetry related reflections of untilted views, the alignment of images from tilted specimen is still carried out by phase comparisons. This is possible because the transforms of all images share the same (0,0) origin in reciprocal space, i.e. each pair of images shares certain phase values (= common lines) allowing the common phase origin to be calculated for this pair. Moreover, each pair of images will have its own characteristic common line because the random orientation of the crystals results in projections showing the same object from different directions. For instance, suppose you have three images A, B and C. While images and A and B may share certain phase values, the phase values shared between images A and C or B and C are different. However, since all the projections (A,B, and C) are derived from the same structure each of them must obey the phase constraints imposed by all the different common lines. This particular constraint will only be met if one has found the correct geometrical relationship between all the images and has used this knowledge to bring all images to their shared phase origin. An example for an intermediate resolution structure (~6Å) is shown in the last figure. The contoured 3D-volume represents a longitudinal section through the 3D-map of a gap junction channel. Instead of presenting the structure as 3D-volume one also can obtain individual crosssections along the z-direction. Albeit limited in what it shows, the cell-to-cell channel is clearly visible in both representations. Similarly, individual transmembrane α- M 20