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Reconstruction of symmetry deviations: a procedure to analyze partially decorated F-actin and other incomplete structures
Ultramicroscopy 72 (1998) 187—197
Reconstruction of symmetry deviations: a procedure to analyze partially decorated F-actin and other incomplete structures Linda E. Rost, Dorit Hanein, David J. DeRosier* W.M. Keck Institute for Cellular Visualization and the Rosenstiel Basic Medical Sciences Research Center, Department of Biology, MS 029, Brandeis University, Waltham, MA 02254, USA Received 29 October 1997; received in revised form 3 February 1998
Abstract The absolute value of individual differences (AVID) procedure is a method to map variations within images arising from deviations in symmetry. We devised this procedure to analyze images of actin filaments decorated with actinbinding proteins (ABPs). In three-dimensional maps of such actin complexes, ABPs often appear weak (i.e. they have low density) relative to actin. Because the 3D map represents an average taken over equivalent positions in the helix, the final density at the position of the ABP represents an average of the densities at all ABP sites. If there is either incomplete binding or a conformational variability of the bound ABP, the average density will be lowered. By the same argument, the variation of density at these sites will be increased. The aim of the AVID procedure is to calculate the density variations within partially decorated filaments and thereby attempt to locate the bound protein. We tested the AVID procedure with model data and then applied it to electron micrographs of F-actin decorated with an actin-binding domain of fimbrin known as N375 [Hanein et al., J. Cell Biol. 139 (1997) 387—396]. The AVID maps have peaks at the site where N375 binds. Because it excludes the layer line data, the AVID procedure uses data that are independent of the data used for 3D reconstruction and difference mapping. It therefore provides an independent way to localize the bound subunit without the need for a map of undecorated actin. Moreover, the difficulties of scaling maps are minimized. This procedure could also be applied to structures with non-helical symmetry. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Image analysis; Partial occupancy; Electron microscopy; Actin; Actin binding proteins
* Corresponding author. Tel.: #1 781 736 2494; fax: #1 781 736 2405; e-mail [email protected]. 0304-3991/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 9 1 ( 9 8 ) 0 0 0 1 7 - 5
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1. Introduction 1.1. The problem In three-dimensional (3D1) maps of decorated actin filaments, the features corresponding to the ABP often appear weak relative to actin [1—4]. The cause for the weakening of the density corresponding to ABP is uncertain but loss of bound ABP during blotting and plunging of grids is one possible explanation. It is also possible that the ABP exhibits conformational variability. In either case the average densities corresponding to the ABP will be weaker than the density corresponding to the actin subunit. The layer line data extracted for image analysis correspond to an average in which densities from occupied ABP sites are combined with those from unoccupied sites. While the reduction in occupancy decreases or weakens the density at the binding site, it concomitantly increases the variance. This increased variability can betray the location of the ABP and can be used to verify or even determine the location of an ABP. 1.2. The solution The possibility of locating sites for ABPs motivated us to develop a procedure for visualizing regions of high variation. In this procedure, called absolute value of individual differences (AVID), we subtract the symmetry-averaged data from each individual image and thereby produce an image of the density variations arising from incomplete occupancy and from other sources of variations such as noise or disorder. The variations are both positive and negative. We take the absolute value of the differences, extract the symmetric or repeating features, and then average over an ensemble of images. Fig. 1 shows the one-dimensional analog of the procedure.
1 Abbreviations 3D — three dimensional; ABP — actin-binding proteins; AVID — absolute value of individual differences; N375 — N-terminal actin binding domain of human T-fimbrin (1-375aa).
In the one-dimensional analogue, we represent the partial occupancy of a decorated filament as a comb of density peaks with some of the teeth missing (Fig. 1a). Each tooth of the comb has the shape of the positive half of a sine wave. The particular example used here has about 50% “occupancy”. The transform (Fig. 1b) reveals the expected set of Bragg maxima as indicated by the arrows. Significant values of its transform lie between the Bragg maxima because in the image, the comb lacks some of its teeth and is therefore not perfectly regular. If we apply a filter mask to block off all but the Bragg maxima (Fig. 1d) and compute the inverse Fourier transform, we generate the averaged or filtered image which consists of peaks all having amplitudes of about 50%. This filtered image is the analogue of 2D or 3D reconstructions of symmetric structures. At this point, the AVID procedure departs from other, existing procedures. We subtract the filtered image (Fig. 1c) from the starting image (Fig. 1a) to obtain a difference image having both positive and negative peaks (Fig. 1e). (As expected, the transform of the difference image as seen in Fig. 1f lacks the Bragg maxima, which were removed when we subtracted the filtered image. It does however have all the other parts of the transform shown in Fig. 1b. This transform is not part of the AVID procedure but demonstrates that the data corresponding to the Bragg maxima have really been removed.) The next step in the AVID procedure is to take the absolute value of the data in Fig. 1e which gives Fig. 1g. We call this the “AVID difference image”. The peaks in the AVID difference image are all positive although not all are of equal height. In the Fourier transform of Fig. 1g, we find maxima once again at the Bragg positions (Fig. 1h). Although there is still some intensity remaining off the Bragg positions since the peaks in the AVID difference image in Fig. 1g are not all the same height, the bulk of the Fourier data is now “restored” to the Bragg positions. (Subtracting images and subtracting their transforms are equivalent. Therefore, it may seem counter-intuitive that there is any layer line data in the transform of an AVID difference image since the layer line data has been subtracted from the original image. However, taking the absolute value of the difference is a
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nonlinear operation, which therefore makes the layer lines non-zero as illustrated in Fig. 1h.) We again apply the filter mask, which blocks out all but the Bragg maxima, and generate an averaged image (the AVID image) as seen in Fig. 1i. The AVID
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image reveals the regular array of peaks corresponding to the density fluctuations; it is different from the filtered image in Fig. 1c which reveals the average density. Note that the filtered image and the AVID image are generated from different and independent sets of data extracted from the starting image. 1.3. Comparison to existing procedures Other, existing procedures have been used to measure density variations between filaments (e.g. Ref. [5]). In these existing procedures, which differ from the AVID procedure, data corresponding to the periodic features (e.g., layer-line data) are generally used to determine the variance. Layer line data correspond to an average over equivalent subunits in the image, and hence data taken along layer lines correspond to a filtered or averaged image. In existing procedures, the filtered images of filaments are compared, and the density variations between them are calculated. Since each filtered image is itself an average of the contribution of all protomers within a filament, as the filament length increases the differences between the individual filtered images decrease. This is because each filtered image more closely approximates the true average of the population. If all filaments in a data set were decorated
Q &&&&&&&&&&&&&&&&&&&&&&& Fig. 1. Test of the AVID procedure using a computer-generated, one-dimensional model. The model consists of 256 regularly spaced sites at which a peak can be placed. Sites to be filled were selected using a random number generator which filled about 50% of the sites. The peak placed at each site has the form of the positive peak from a sine wave. (a) One-dimensional model with approximately 50% of the sites filled at random. The figure shows 16 of the 256 sites in the model. (b) Fourier transform of (a). (c) Filtered image of (a) produced by backtransforming the Bragg peaks shown in (d). (e) Difference image in which the image in (c) is subtracted from that in (a). (f ) Fourier transform of (e) which shows as expected, that the Bragg peaks are missing. This step is not a part of the AVID procedure but is included for didactic purposes. (g) Absolute value of the densities in (e). This is the AVID difference image. (h) Fourier transform of (g). Note that most of the Fourier coefficients scattered around the transform in (f ) have been restored to the Bragg positions. (i) Filtered image of (g) made by back transforming only the Bragg peaks shown in ( j). This is the AVID image.
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at exactly the same percentage, existing procedures for mapping variation would generate null images whereas AVID would reveal the variations. On the other hand, when there is cooperative binding in which some filaments are fully decorated and some are undecorated, the existing procedure would reveal the variability whereas the AVID procedure, which looks at variation within each filament, would not. Therefore, these two procedures can provide complimentary information.
2. Methods 2.1. Eight steps in the AVID algorithm 0. The original images are transformed and layer line data from the transforms are used to align a set of images as described by DeRosier and Moore [6]. The alignment parameters are saved. (Note: This is part of the procedure for a standard helical reconstruction.) 1. Next, each image (e.g. 457 pixels long by 45 pixels wide) is floated as described in DeRosier and Moore (1970) and the value of the subtracted constant density is saved. To calculate this constant density, the densities are averaged over the (2]457#2]43"1000) pixels making up the perimeter of each image. The floated image is obtained by subtracting the perimeter average from all the pixels in the image. The image is then put into a much larger array (4096 pixels long by 512 pixels wide). The pixels outside of those containing the image are set to zero. 2. The floated image is transformed and all reciprocal pixels except those on the single rows corresponding to the layer lines are set to zero. The amplitudes along the layer lines are divided by the width of the array (512) and by the length of the original particle (457). This ensures that the densities in the averaged image are scaled to the values in the original image. (Note that the width of the transform not that of the image is used in one direction and the length of the image not that of the transform is used in the other. This is because the width of layerline data extracted is one reciprocal pixel, whereas
all the reciprocal pixels along the layer lines are used.)2 3. The back transform or filtered image is calculated, and the saved, perimeter-averaged density is added back to every pixel. 4. The image resulting from step three is subtracted from the original image, and the average value of the difference is determined and subtracted from every pixel. (The average is usually very close to zero as expected so this last step is a minor correction.) 5. The absolute value of the image from step four is computed. This is the key step that restores data to the original layer lines positions. Using the absolute value (instead of, for example, the square of the differences) keeps the densities in the AVID map on the same scale as those of the 3D maps of the structures. 6. The AVID image is loaded into a 4096]512 array (padded with zeros) and transformed. Data along the same layer line positions as in step two are collected. 7. The alignment parameters determined in step zero are used to phase shift the AVID layer line data sets prior to averaging them over the set of images.
2 The correctness of the weighting scheme can be seen by considering an image n]m pixels. The transform will be an array of Fourier coefficients of amplitude"1. If we back transform all coefficients (m lines each having n pixels), the pixel that had a density of one in the image will now have a density of n*m in the reconstructed image. Thus, a weighting of 1/(n*m) would be needed to scale these densities to the original image. We have embedded, however, our image array of n]m densities in a bigger array that is N]M. This oversamples the Fourier transform by a factor of N/n in the equatorial direction and M/m in the axial or meridional direction. Thus, layer lines (which would have been one pixel wide had we transformed the original array of dimension m) are of now of width M/m pixels. If we backtransformed the sets of M/m layerlines, we would obtain a filtered image whose densities had been multiplied by M]N. Before we backtransform, however, we select from each set of M/m layer lines, a single line of Fourier coefficients at its center; i.e., a line of width 1. We are thus ignoring the other M/m!1 neighboring lines of coefficients. It is as if we had calculated the transform of an array having an axial dimension of m not M. Thus the weighting in the axial direction is m not M. In the equatorial dimension, however, we include all N points along this selected line of width 1. Thus the corresponding weighting is N and the total weighting factor therefore is 1/(Nm) not 1/(NM).
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8. The averaged layer line data are used to generate a 3D AVID map. We used the helical reconstruction algorithms to carry out various steps in this procedure [7]. 2.2. Two important aspects of the process As an alternative to subtracting the filtered image from each image, we tried subtracting the average of all filtered images from each filament image. When we tested this variation, we obtained a very strong “ghost” image of the structure. We believe this occurred because the variability between images of different filaments is larger than that within the image of a single filament. The subtraction of individual filtered images rather than their average avoids ghosting problems associated with interfilament variations such as one-sidedness, variations in amplitudes on a layer line, deviations in helical symmetry among filaments, and imperfect alignment during step zero of the AVID procedure. The second important aspect concerns the alignment parameters used in averaging AVID layer line data from different filaments images. We used the alignment parameters determined from the original layer line data collected from the starting images rather than trying to determine the alignment parameters from the AVID data. If a standard map of the average has already been generated, the alignment parameters are already available, which simplifies the application of the AVID method. Use of these parameters also ensures that the AVID map is aligned to the average and to standard difference maps. Moreover, the AVID layer lines, which tend to have a lower signal-to-noise ratio than those from original images, may be too noisy to obtain accurate alignment parameters. 2.3. The expected density for AVID peaks Suppose the fraction of sites p, occupied on actin by the ABP is 1/3"0.33. Suppose that the electron scattering density for an actin-binding protein subunit is 1.0. Because of partial occupancy, the density corresponding to the ABP in the averaged map would be 0.33]1.0"0.33. When the averaged
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map is subtracted from the image, there will be a positive peak where there was an ABP bound to a site on actin and a negative peak where there was an unoccupied site. The positive peaks will have a density of 1.00!0.33"0.67 and the negative peaks will have a density of 0.00!0.33"!0.33. When we take the absolute value of this difference image, all peaks will become positive but some peaks will be twice the density of others (i.e., 0.67 vs. 0.33). The final AVID image will be an average of the peaks in the AVID difference map. Since the filament was 1/3 decorated, 1/3 of the peaks in the AVID difference images will have a density of 0.67 and the remaining 2/3, a density of 0.33. The density in the AVID image will be an average of the two sets of peaks weighted by relative proportions (i.e., 1]0.67#2]0.33"0.44). Note 3 3 that the density in the AVID image is greater than that in the filtered images (0.44 vs. 0.33). In general, the relative density in the AVID map will be 2p(1!p), whereas the filtered-averaged image will have a density of p. If p(0.5, the AVID peak will be stronger than the peak in the filtered image. An AVID peak will also be stronger than that obtained by the existing methods used to calculate variations (i.e. a map of the standard deviation between filtered images). If the expected fraction of subunits bound is p, then the expected number of occupied sites on a filament with N sites is pN. From the binomial distribution, the variance of the number of occupied sites is p(1!p)N. Thus if we average many filaments with an average of pN bound subunits, we expect the relative density to be p and the standard deviation of that density to be Jp(1!p)/N. The peak value in a standard deviation map is always less than the value obtained by the AVID procedure. Table 1 shows the expected and observed peak heights for AVID and for existing (standard deviation) methods. The data are obtained using computer-generated onedimensional models identical to that in Fig. 1. The one-dimensional filament had 256 possible sites for “ABPs”. The agreement between theory and observation using model data is good. The point is to show that the AVID peak is much stronger than that generated using existing methods.
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Table 1 Peak heights Difference Map
Standard deviation map
AVID Map
SpT
Expected
Observed
Expected
Observed
Expected
Observed
0.20 0.40 0.60 0.80
0.20 0.40 0.60 0.80
0.22 0.38 0.60 0.82
0.025 0.031 0.031 0.025
0.022 0.033 0.027 0.020
0.32 0.48 0.48 0.32
0.34 0.47 0.48 0.30
2.4. Caveat in adapting AVID to 3D structures An electron micrograph is a projection of the three-dimensional structure. The AVID procedure does not rigorously extract the 3-D distribution of variations from 2-D projections. It is possible that in projection the binding sites for ABPs will overlap. If so, the difference density at these sites may not be correct. To see this, suppose that p"0.5 and that only one of the “overlapping” two sites is occupied. Since the sites overlap in projection, the relative occupancy is one (corresponding to one bound subunit) instead of two (subunits on both sites). Since p"0.5, a 3D, averaged map should have a relative density of 0.5 (i.e., half a subunit) at each site. Since the sites overlap in projection, the filtered image will have a relative density of 2]0.5"1. The starting image also has one of the two sites occupied so it too will have a density of one. When we subtract the averaged image from the starting image, we will get a value of 1!1"0. If we took the differences and their absolute values in three-dimensions and then projected the result, we would get a value of 1 instead of 0. To see this, we subtract the average value (0.5) from the actual values at the two sites; namely, at the occupied site (which has a density of 1.0), we get 1.0!0.5"0.5 and at the unoccupied site (which has a density of 0.0), we get 0.0!0.5"!0.5. If we take the absolute values of these and then project them (i.e., add them) we get a value of D 0.5 D # D!0.5D"1. Thus, with images that are projections, we can get incorrect AVID difference densities when sites overlap. The error changes the apparent fraction of occupancy, which is not a serious error. If the ABP is extended, however, overlap may often be partial.
Partial superposition may result in an AVID peak with a shape that does not accurately represent the region of variance. 2.5. Computer-generated model data In order to test the AVID algorithm, we created models based on an actin-like filament [8]. The model has 2.152 units per turn of the left-handed 1-start helix. The rise per subunit is 27.5 A_ . The sampling is 9 A_ /pixel. The actin subunit consisted of a two ball model where a pair of balls approximates the actin protomer. The inner ball (38 A_ diameter) is located at a radius of 12 A_ and the outer ball (38 A_ diameter) is at a radius of 31 A_ and is shifted axially by 10 A_ and rotated by #70° (ccw when viewed from the top). We modeled partial occupancy of an ABP by adding a third ball to the model but at about half the sites on actin. This ball (20 A_ diameter) is at radius of 55 A_ and at the same angular and axial coordinates as the outer ball of actin. 2.6. F-actin decorated with fimbrin (N375) The 3D structure of F-actin filaments decorated with the N-terminal actin binding domain of human T-fimbrin (N375) was determined through the use of electron cryo-microscopy and existing image processing procedures [1]. A 1 : 1 stoichiometry of actin:N375 was determined by centrifugation and subsequent analysis of the supernatants and pellets by SDS-PAGE. Low-dose images of frozen-hydrated preparations were recorded using 120 kV electrons, at a nominal magnification of 60 000],
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and &2.2 lm defocus (electron dose &10e~/A_ 2). Selected electron micrographs were digitized with an Eikonix 1412 densitometer at a scanning raster of 25 lm (&4.3 A_ /pixel). Image processing and three dimensional reconstruction procedures employed the Brandeis Helical Package [7].
3. Results We tested the algorithm using a projection (Fig. 2a) of a computer-generated actin-like model. The “actin” portion of the model is a helical arrangement of subunits, each of which consists of two spheres. The ABP is a third, smaller sphere located at the outside of the outermost sphere of the “actin” subunit. About 50% of the sites are occupied by the ABP. Fig. 2b shows the filtered image, which was generated using layer line data extracted from Fig. 2a. Fig. 2c shows the difference image we calculated by subtracting the filtered image from the original image. To show the negative and the positive peaks, we shifted the background value so that the negative peaks appear as black and the positive as white. The background grey surrounding the filament corresponds to a density of zero. Note that where there is an ABP subunit in Fig. 2a, there is a bright spot (positive difference) in Fig. 2c, and where there should be but is not an ABP, there is dark spot (negative difference). Fig. 2d presents the AVID difference image that we generated by taking the absolute value of the densities presented in Fig. 2c. Note that the ABP spheres appear in a fairly regular helical array of bright spots. A slight ghosting of actin remains in Fig. 2c because only the first 10 layer lines were used to make the filtered image. This leaves higher resolution details corresponding to the higher order layer lines in the original image that are not removed by subtracting the filtered image. We found that the high resolution ghost reduced somewhat the amplitude of the AVID peak but not its shape or location. To produce the 3D map of the density variations, we transformed the image shown in Fig. 2d, collected layer-line data, and computed an inverse Fourier—Bessel transform in the same way we would calculate a map of the structure itself. Fig. 3a
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shows a section of the conventional 3D map of the partially decorated model. Fig. 3c shows a section through the 3D AVID map on the section shown in Fig. 3a. For comparison, Fig. 3b shows a section from the conventional, 3D difference map made by subtracting a reconstruction of the actin model from that of the decorated actin model. The AVID map and the difference map both have a single strong peak located at the position of the third sphere (or ABP of the model) as expected. The deviation of the peaks from a circular shape arises because the resolution radially is greater than that azimuthally (i.e., only layer-lines with small values of the order n are included). The AVID procedure was next applied to electron micrographs of frozen hydrated preparations of F-actin decorated with N375, the N-terminal actin binding domain of fimbrin [1]. In this structure, the location of N375 was determined using conventional difference map procedures. The difference map shows two positive peaks (bold contours in Fig. 4a). Peak 1, which is interpreted to be the N375 domain, is located outside the radius of actin as expected for a bound domain. The variance corresponding to this peak is low. Peak 2 is interpreted as a conformational change in actin [1]. Note that the peaks in the map of standard deviation (Fig. 4c) do not coincide with peak 1 or peak 2. Hanein et al. indicate that in their reconstruction of actin decorated with N375, the density ascribable to N375 is much weaker than that ascribable to actin and that N375 is much larger in the difference map than the N375-actin complex map. This can be seen in Fig. 4a which is taken from Figs. 3 and 4 of Hanein et al., [1]. The cause for the weakening of the density corresponding to N375 appears to be due to the loss of bound N375 during blotting and plunging of grids. The N375-actin complex therefore seemed a good candidate on which to test the AVID procedure. We carried out the procedure and averaged layer line data from 32 filaments which Hanein et al. used to generate the real-space average in their paper. Fig. 4b shows a section through the 3D AVID map. It reveals one main peak which overlaps with peak 1 (the N375 peak) seen in the difference map (Fig. 4a). The peak does not overlap with either of
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Fig. 2. Test of AVID procedure using a computer-generated 3D model: the generation of the AVID difference image. The filament is a simple model of actin, that is a helical array of subunits each of which consists of two spheres corresponding to the two domains of the actin subunit. The analog of the actin-binding protein is a third sphere added to a particular site on the model actin. As in the model in Fig. 1, a random number generator is used to decide which sites to decorate. The images shown here are projections of the 3D models. (a) Model filament decorated at about 50% of the equivalent sites. (b) Filtered image of (a). To produce the filtered image, the strongest ten layer lines from the transform of the projection shown in Fig. 1a are selected and back transformed. (c) Difference image obtained by subtracting the projection in (b) from that in (a). (d) AVID difference image formed by taking the absolute values of the densities in Fig. 1c. Bar"10 A_ .
the peaks detected in the standard deviation map (Fig. 4c). The AVID peak extends axially relative to peak 1 of the difference map; a consequence of this
elongation is the appearance of the peak marked 1’ in the section of the AVID map. The reason that the shape of the AVID peak is different from the
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Fig. 3. Test of AVID procedure using computer generated models: the generation of the AVID map. (a) Transverse section through the 3D map generated from Fig. 2b. To calculate the map, layer lines were selected from the Fourier transform of the image shown in Fig. 2a. An inverse Fourier—Bessel transform of the layer line data generated the 3D map. (b) A transverse section through the difference map generated by subtracting a 3D map of the undecorated actin model (not shown) from the map described in (a). The difference peak is superimposed on the section shown in (a). (c) A transverse section through the 3D AVID map. The AVID peak is superimposed on the section shown in (a). Bar"10 A_ .
Fig. 4. Application of AVID procedure to electron micrographs of the actin—N375 complex. (a) A transverse section of the difference map generated by subtracting a map of actin from a map of actin decorated with N375. The peaks are superimposed on the corresponding section through the actin—N375 map. The data for this figure is taken from the work of Hanein et al. [1]. (b) A transverse section through the 3D AVID map superimposed on the corresponding section of the actin—N375 map. (c) A transverse section through the standard deviation map of actin decorated with N375. The map is obtained from an analysis of the set of filtered images of the actin—N375 complex. The peaks are superimposed on the corresponding section through the actin—N375 map that has been contoured to represent correctly the dimensions of the actin portion. Note that the peaks do not lie at the location of the N375 domain and do not coincide with those obtained from the AVID procedure. Bar"10 A_ .
difference peak in Fig. 4a is not clear. The AVID procedure does not detect a peak of variation corresponding to peak 2, perhaps because it is intrinsically too weak or perhaps because the conformational changes in actin produced by N375 binding are cooperative.
4. Discussion The idea behind the AVID procedure is to map the variations in density that arise from partial occupancy. We must keep in mind, however, that the AVID procedure will map all variations within
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the filaments. Because we subtract the regular features from the starting images in calculating the AVID difference images, we use data that are independent of those used in calculating the standard 3D reconstructions and difference maps. The AVID procedure therefore provides an independent way to locate an actin-binding domain. Because we subtract the filtered image from the original image, we avoid problems associated with scaling. With a simple one-dimensional model, we illustrated the principle behind the AVID procedure. We then applied it to projections of a model helical structure. In this example, we make use of the approximation that the absolute value of the difference between the two projections is equal to the projection of the absolute values of the difference between the two structures. We have shown with a helical model and an actual filament that this is approximately true at least for some classes of decorated actin filaments. We applied the AVID procedure to actin filaments decorated with the known actin-binding domain of fimbrin, N375. This structure had already been solved by Hanein et al. [1], who noted that the portion of the reconstruction corresponding to N375 appeared as a relatively small extension from the portion of the map corresponding to actin. Only 50% of the density is seen relative to actin; this despite the fact that N375 is approximately the same mass as the actin subunit. When Hanein et al. subtracted a map of undecorated actin from the N375-decorated actin however, they observed a statistically significant peak that corresponds in size and shape to the N375 structure. Based on the weakness of the N375 density relative to that of actin, they suggested that there is either incomplete decoration or the N375 domain is floppy. The AVID procedure as expected shows a strong peak at the N375 binding site, and little variation elsewhere in the map. These results show that the AVID procedure is effective in finding the bound N375 domain, without the need for a map of undecorated actin. In an ideal situation (i.e. when applied to a computer-generated model), the AVID procedure produced a peak having the expected location and size of the bound “ABP”. When the AVID procedure
was applied to actin-N375, the location of the peak occurred at the expected position. The existence of the AVID peak suggests that we were not dealing with a mixed population of filaments in which some were fully decorated while the rest were undecorated. On the contrary, N375 was either partially decorating the filaments or it was exhibiting variable conformations or both. The shape of the peak, however, was not exactly the same as that in the difference map. The densities in the peak were also weaker than those in the difference peak. We do not know if partial overlap of binding sites in projection and extraction of an incomplete set of layer lines account for these deviations from the expected results. We need more examples to address this issue.
5. Conclusions The AVID procedure is designed to map all deviations from symmetry. It is an algorithm therefore that can confirm or detect an ABP through deviations in symmetry arising from partial occupancy or domain mobility. Unlike difference mapping, the AVID procedure does not require a map of the undecorated filament. Since the procedure makes use of data that are independent of those used in difference mapping, it provides additional information about the structure. Moreover, when a difference map already exists, the calculation of an AVID map requires little additional work. We used the AVID procedure to look at structures with an underlying helical symmetry. With suitable modification, the AVID procedure should be applicable to non helically symmetric structures. Application of the algorithm to other examples may provide new insights into structural variations.
Acknowledgements This work was supported by grants from the NIHGM26357 and GM07596. The authors also acknowledge funds from the W.M. Keck Foundation.
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[5] S. Trachtenberg, D. DeRosier, J. Mol. Biol. 195 (1987) 581—601. [6] D.J. DeRosier, P.B. Moore, J. Mol. Biol. 52 (1970) 355—369. [7] C.H. Owen, D.G. Morgan, D.J. DeRosier, J. Struct. Biol. 116 (1996) 167—175. [8] E.H. Egelman, N. Francis, D.J. DeRosier, Nature 298 (1982) 131—135.
Reconstruction of symmetry deviations: a procedure to analyze partially decorated F-actin and other incomplete structures Linda E. Rost, Dorit Hanein, David J. DeRosier* W.M. Keck Institute for Cellular Visualization and the Rosenstiel Basic Medical Sciences Research Center, Department of Biology, MS 029, Brandeis University, Waltham, MA 02254, USA Received 29 October 1997; received in revised form 3 February 1998
Abstract The absolute value of individual differences (AVID) procedure is a method to map variations within images arising from deviations in symmetry. We devised this procedure to analyze images of actin filaments decorated with actinbinding proteins (ABPs). In three-dimensional maps of such actin complexes, ABPs often appear weak (i.e. they have low density) relative to actin. Because the 3D map represents an average taken over equivalent positions in the helix, the final density at the position of the ABP represents an average of the densities at all ABP sites. If there is either incomplete binding or a conformational variability of the bound ABP, the average density will be lowered. By the same argument, the variation of density at these sites will be increased. The aim of the AVID procedure is to calculate the density variations within partially decorated filaments and thereby attempt to locate the bound protein. We tested the AVID procedure with model data and then applied it to electron micrographs of F-actin decorated with an actin-binding domain of fimbrin known as N375 [Hanein et al., J. Cell Biol. 139 (1997) 387—396]. The AVID maps have peaks at the site where N375 binds. Because it excludes the layer line data, the AVID procedure uses data that are independent of the data used for 3D reconstruction and difference mapping. It therefore provides an independent way to localize the bound subunit without the need for a map of undecorated actin. Moreover, the difficulties of scaling maps are minimized. This procedure could also be applied to structures with non-helical symmetry. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Image analysis; Partial occupancy; Electron microscopy; Actin; Actin binding proteins
* Corresponding author. Tel.: #1 781 736 2494; fax: #1 781 736 2405; e-mail [email protected]. 0304-3991/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 9 1 ( 9 8 ) 0 0 0 1 7 - 5
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1. Introduction 1.1. The problem In three-dimensional (3D1) maps of decorated actin filaments, the features corresponding to the ABP often appear weak relative to actin [1—4]. The cause for the weakening of the density corresponding to ABP is uncertain but loss of bound ABP during blotting and plunging of grids is one possible explanation. It is also possible that the ABP exhibits conformational variability. In either case the average densities corresponding to the ABP will be weaker than the density corresponding to the actin subunit. The layer line data extracted for image analysis correspond to an average in which densities from occupied ABP sites are combined with those from unoccupied sites. While the reduction in occupancy decreases or weakens the density at the binding site, it concomitantly increases the variance. This increased variability can betray the location of the ABP and can be used to verify or even determine the location of an ABP. 1.2. The solution The possibility of locating sites for ABPs motivated us to develop a procedure for visualizing regions of high variation. In this procedure, called absolute value of individual differences (AVID), we subtract the symmetry-averaged data from each individual image and thereby produce an image of the density variations arising from incomplete occupancy and from other sources of variations such as noise or disorder. The variations are both positive and negative. We take the absolute value of the differences, extract the symmetric or repeating features, and then average over an ensemble of images. Fig. 1 shows the one-dimensional analog of the procedure.
1 Abbreviations 3D — three dimensional; ABP — actin-binding proteins; AVID — absolute value of individual differences; N375 — N-terminal actin binding domain of human T-fimbrin (1-375aa).
In the one-dimensional analogue, we represent the partial occupancy of a decorated filament as a comb of density peaks with some of the teeth missing (Fig. 1a). Each tooth of the comb has the shape of the positive half of a sine wave. The particular example used here has about 50% “occupancy”. The transform (Fig. 1b) reveals the expected set of Bragg maxima as indicated by the arrows. Significant values of its transform lie between the Bragg maxima because in the image, the comb lacks some of its teeth and is therefore not perfectly regular. If we apply a filter mask to block off all but the Bragg maxima (Fig. 1d) and compute the inverse Fourier transform, we generate the averaged or filtered image which consists of peaks all having amplitudes of about 50%. This filtered image is the analogue of 2D or 3D reconstructions of symmetric structures. At this point, the AVID procedure departs from other, existing procedures. We subtract the filtered image (Fig. 1c) from the starting image (Fig. 1a) to obtain a difference image having both positive and negative peaks (Fig. 1e). (As expected, the transform of the difference image as seen in Fig. 1f lacks the Bragg maxima, which were removed when we subtracted the filtered image. It does however have all the other parts of the transform shown in Fig. 1b. This transform is not part of the AVID procedure but demonstrates that the data corresponding to the Bragg maxima have really been removed.) The next step in the AVID procedure is to take the absolute value of the data in Fig. 1e which gives Fig. 1g. We call this the “AVID difference image”. The peaks in the AVID difference image are all positive although not all are of equal height. In the Fourier transform of Fig. 1g, we find maxima once again at the Bragg positions (Fig. 1h). Although there is still some intensity remaining off the Bragg positions since the peaks in the AVID difference image in Fig. 1g are not all the same height, the bulk of the Fourier data is now “restored” to the Bragg positions. (Subtracting images and subtracting their transforms are equivalent. Therefore, it may seem counter-intuitive that there is any layer line data in the transform of an AVID difference image since the layer line data has been subtracted from the original image. However, taking the absolute value of the difference is a
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nonlinear operation, which therefore makes the layer lines non-zero as illustrated in Fig. 1h.) We again apply the filter mask, which blocks out all but the Bragg maxima, and generate an averaged image (the AVID image) as seen in Fig. 1i. The AVID
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image reveals the regular array of peaks corresponding to the density fluctuations; it is different from the filtered image in Fig. 1c which reveals the average density. Note that the filtered image and the AVID image are generated from different and independent sets of data extracted from the starting image. 1.3. Comparison to existing procedures Other, existing procedures have been used to measure density variations between filaments (e.g. Ref. [5]). In these existing procedures, which differ from the AVID procedure, data corresponding to the periodic features (e.g., layer-line data) are generally used to determine the variance. Layer line data correspond to an average over equivalent subunits in the image, and hence data taken along layer lines correspond to a filtered or averaged image. In existing procedures, the filtered images of filaments are compared, and the density variations between them are calculated. Since each filtered image is itself an average of the contribution of all protomers within a filament, as the filament length increases the differences between the individual filtered images decrease. This is because each filtered image more closely approximates the true average of the population. If all filaments in a data set were decorated
Q &&&&&&&&&&&&&&&&&&&&&&& Fig. 1. Test of the AVID procedure using a computer-generated, one-dimensional model. The model consists of 256 regularly spaced sites at which a peak can be placed. Sites to be filled were selected using a random number generator which filled about 50% of the sites. The peak placed at each site has the form of the positive peak from a sine wave. (a) One-dimensional model with approximately 50% of the sites filled at random. The figure shows 16 of the 256 sites in the model. (b) Fourier transform of (a). (c) Filtered image of (a) produced by backtransforming the Bragg peaks shown in (d). (e) Difference image in which the image in (c) is subtracted from that in (a). (f ) Fourier transform of (e) which shows as expected, that the Bragg peaks are missing. This step is not a part of the AVID procedure but is included for didactic purposes. (g) Absolute value of the densities in (e). This is the AVID difference image. (h) Fourier transform of (g). Note that most of the Fourier coefficients scattered around the transform in (f ) have been restored to the Bragg positions. (i) Filtered image of (g) made by back transforming only the Bragg peaks shown in ( j). This is the AVID image.
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at exactly the same percentage, existing procedures for mapping variation would generate null images whereas AVID would reveal the variations. On the other hand, when there is cooperative binding in which some filaments are fully decorated and some are undecorated, the existing procedure would reveal the variability whereas the AVID procedure, which looks at variation within each filament, would not. Therefore, these two procedures can provide complimentary information.
2. Methods 2.1. Eight steps in the AVID algorithm 0. The original images are transformed and layer line data from the transforms are used to align a set of images as described by DeRosier and Moore [6]. The alignment parameters are saved. (Note: This is part of the procedure for a standard helical reconstruction.) 1. Next, each image (e.g. 457 pixels long by 45 pixels wide) is floated as described in DeRosier and Moore (1970) and the value of the subtracted constant density is saved. To calculate this constant density, the densities are averaged over the (2]457#2]43"1000) pixels making up the perimeter of each image. The floated image is obtained by subtracting the perimeter average from all the pixels in the image. The image is then put into a much larger array (4096 pixels long by 512 pixels wide). The pixels outside of those containing the image are set to zero. 2. The floated image is transformed and all reciprocal pixels except those on the single rows corresponding to the layer lines are set to zero. The amplitudes along the layer lines are divided by the width of the array (512) and by the length of the original particle (457). This ensures that the densities in the averaged image are scaled to the values in the original image. (Note that the width of the transform not that of the image is used in one direction and the length of the image not that of the transform is used in the other. This is because the width of layerline data extracted is one reciprocal pixel, whereas
all the reciprocal pixels along the layer lines are used.)2 3. The back transform or filtered image is calculated, and the saved, perimeter-averaged density is added back to every pixel. 4. The image resulting from step three is subtracted from the original image, and the average value of the difference is determined and subtracted from every pixel. (The average is usually very close to zero as expected so this last step is a minor correction.) 5. The absolute value of the image from step four is computed. This is the key step that restores data to the original layer lines positions. Using the absolute value (instead of, for example, the square of the differences) keeps the densities in the AVID map on the same scale as those of the 3D maps of the structures. 6. The AVID image is loaded into a 4096]512 array (padded with zeros) and transformed. Data along the same layer line positions as in step two are collected. 7. The alignment parameters determined in step zero are used to phase shift the AVID layer line data sets prior to averaging them over the set of images.
2 The correctness of the weighting scheme can be seen by considering an image n]m pixels. The transform will be an array of Fourier coefficients of amplitude"1. If we back transform all coefficients (m lines each having n pixels), the pixel that had a density of one in the image will now have a density of n*m in the reconstructed image. Thus, a weighting of 1/(n*m) would be needed to scale these densities to the original image. We have embedded, however, our image array of n]m densities in a bigger array that is N]M. This oversamples the Fourier transform by a factor of N/n in the equatorial direction and M/m in the axial or meridional direction. Thus, layer lines (which would have been one pixel wide had we transformed the original array of dimension m) are of now of width M/m pixels. If we backtransformed the sets of M/m layerlines, we would obtain a filtered image whose densities had been multiplied by M]N. Before we backtransform, however, we select from each set of M/m layer lines, a single line of Fourier coefficients at its center; i.e., a line of width 1. We are thus ignoring the other M/m!1 neighboring lines of coefficients. It is as if we had calculated the transform of an array having an axial dimension of m not M. Thus the weighting in the axial direction is m not M. In the equatorial dimension, however, we include all N points along this selected line of width 1. Thus the corresponding weighting is N and the total weighting factor therefore is 1/(Nm) not 1/(NM).
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8. The averaged layer line data are used to generate a 3D AVID map. We used the helical reconstruction algorithms to carry out various steps in this procedure [7]. 2.2. Two important aspects of the process As an alternative to subtracting the filtered image from each image, we tried subtracting the average of all filtered images from each filament image. When we tested this variation, we obtained a very strong “ghost” image of the structure. We believe this occurred because the variability between images of different filaments is larger than that within the image of a single filament. The subtraction of individual filtered images rather than their average avoids ghosting problems associated with interfilament variations such as one-sidedness, variations in amplitudes on a layer line, deviations in helical symmetry among filaments, and imperfect alignment during step zero of the AVID procedure. The second important aspect concerns the alignment parameters used in averaging AVID layer line data from different filaments images. We used the alignment parameters determined from the original layer line data collected from the starting images rather than trying to determine the alignment parameters from the AVID data. If a standard map of the average has already been generated, the alignment parameters are already available, which simplifies the application of the AVID method. Use of these parameters also ensures that the AVID map is aligned to the average and to standard difference maps. Moreover, the AVID layer lines, which tend to have a lower signal-to-noise ratio than those from original images, may be too noisy to obtain accurate alignment parameters. 2.3. The expected density for AVID peaks Suppose the fraction of sites p, occupied on actin by the ABP is 1/3"0.33. Suppose that the electron scattering density for an actin-binding protein subunit is 1.0. Because of partial occupancy, the density corresponding to the ABP in the averaged map would be 0.33]1.0"0.33. When the averaged
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map is subtracted from the image, there will be a positive peak where there was an ABP bound to a site on actin and a negative peak where there was an unoccupied site. The positive peaks will have a density of 1.00!0.33"0.67 and the negative peaks will have a density of 0.00!0.33"!0.33. When we take the absolute value of this difference image, all peaks will become positive but some peaks will be twice the density of others (i.e., 0.67 vs. 0.33). The final AVID image will be an average of the peaks in the AVID difference map. Since the filament was 1/3 decorated, 1/3 of the peaks in the AVID difference images will have a density of 0.67 and the remaining 2/3, a density of 0.33. The density in the AVID image will be an average of the two sets of peaks weighted by relative proportions (i.e., 1]0.67#2]0.33"0.44). Note 3 3 that the density in the AVID image is greater than that in the filtered images (0.44 vs. 0.33). In general, the relative density in the AVID map will be 2p(1!p), whereas the filtered-averaged image will have a density of p. If p(0.5, the AVID peak will be stronger than the peak in the filtered image. An AVID peak will also be stronger than that obtained by the existing methods used to calculate variations (i.e. a map of the standard deviation between filtered images). If the expected fraction of subunits bound is p, then the expected number of occupied sites on a filament with N sites is pN. From the binomial distribution, the variance of the number of occupied sites is p(1!p)N. Thus if we average many filaments with an average of pN bound subunits, we expect the relative density to be p and the standard deviation of that density to be Jp(1!p)/N. The peak value in a standard deviation map is always less than the value obtained by the AVID procedure. Table 1 shows the expected and observed peak heights for AVID and for existing (standard deviation) methods. The data are obtained using computer-generated onedimensional models identical to that in Fig. 1. The one-dimensional filament had 256 possible sites for “ABPs”. The agreement between theory and observation using model data is good. The point is to show that the AVID peak is much stronger than that generated using existing methods.
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Table 1 Peak heights Difference Map
Standard deviation map
AVID Map
SpT
Expected
Observed
Expected
Observed
Expected
Observed
0.20 0.40 0.60 0.80
0.20 0.40 0.60 0.80
0.22 0.38 0.60 0.82
0.025 0.031 0.031 0.025
0.022 0.033 0.027 0.020
0.32 0.48 0.48 0.32
0.34 0.47 0.48 0.30
2.4. Caveat in adapting AVID to 3D structures An electron micrograph is a projection of the three-dimensional structure. The AVID procedure does not rigorously extract the 3-D distribution of variations from 2-D projections. It is possible that in projection the binding sites for ABPs will overlap. If so, the difference density at these sites may not be correct. To see this, suppose that p"0.5 and that only one of the “overlapping” two sites is occupied. Since the sites overlap in projection, the relative occupancy is one (corresponding to one bound subunit) instead of two (subunits on both sites). Since p"0.5, a 3D, averaged map should have a relative density of 0.5 (i.e., half a subunit) at each site. Since the sites overlap in projection, the filtered image will have a relative density of 2]0.5"1. The starting image also has one of the two sites occupied so it too will have a density of one. When we subtract the averaged image from the starting image, we will get a value of 1!1"0. If we took the differences and their absolute values in three-dimensions and then projected the result, we would get a value of 1 instead of 0. To see this, we subtract the average value (0.5) from the actual values at the two sites; namely, at the occupied site (which has a density of 1.0), we get 1.0!0.5"0.5 and at the unoccupied site (which has a density of 0.0), we get 0.0!0.5"!0.5. If we take the absolute values of these and then project them (i.e., add them) we get a value of D 0.5 D # D!0.5D"1. Thus, with images that are projections, we can get incorrect AVID difference densities when sites overlap. The error changes the apparent fraction of occupancy, which is not a serious error. If the ABP is extended, however, overlap may often be partial.
Partial superposition may result in an AVID peak with a shape that does not accurately represent the region of variance. 2.5. Computer-generated model data In order to test the AVID algorithm, we created models based on an actin-like filament [8]. The model has 2.152 units per turn of the left-handed 1-start helix. The rise per subunit is 27.5 A_ . The sampling is 9 A_ /pixel. The actin subunit consisted of a two ball model where a pair of balls approximates the actin protomer. The inner ball (38 A_ diameter) is located at a radius of 12 A_ and the outer ball (38 A_ diameter) is at a radius of 31 A_ and is shifted axially by 10 A_ and rotated by #70° (ccw when viewed from the top). We modeled partial occupancy of an ABP by adding a third ball to the model but at about half the sites on actin. This ball (20 A_ diameter) is at radius of 55 A_ and at the same angular and axial coordinates as the outer ball of actin. 2.6. F-actin decorated with fimbrin (N375) The 3D structure of F-actin filaments decorated with the N-terminal actin binding domain of human T-fimbrin (N375) was determined through the use of electron cryo-microscopy and existing image processing procedures [1]. A 1 : 1 stoichiometry of actin:N375 was determined by centrifugation and subsequent analysis of the supernatants and pellets by SDS-PAGE. Low-dose images of frozen-hydrated preparations were recorded using 120 kV electrons, at a nominal magnification of 60 000],
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and &2.2 lm defocus (electron dose &10e~/A_ 2). Selected electron micrographs were digitized with an Eikonix 1412 densitometer at a scanning raster of 25 lm (&4.3 A_ /pixel). Image processing and three dimensional reconstruction procedures employed the Brandeis Helical Package [7].
3. Results We tested the algorithm using a projection (Fig. 2a) of a computer-generated actin-like model. The “actin” portion of the model is a helical arrangement of subunits, each of which consists of two spheres. The ABP is a third, smaller sphere located at the outside of the outermost sphere of the “actin” subunit. About 50% of the sites are occupied by the ABP. Fig. 2b shows the filtered image, which was generated using layer line data extracted from Fig. 2a. Fig. 2c shows the difference image we calculated by subtracting the filtered image from the original image. To show the negative and the positive peaks, we shifted the background value so that the negative peaks appear as black and the positive as white. The background grey surrounding the filament corresponds to a density of zero. Note that where there is an ABP subunit in Fig. 2a, there is a bright spot (positive difference) in Fig. 2c, and where there should be but is not an ABP, there is dark spot (negative difference). Fig. 2d presents the AVID difference image that we generated by taking the absolute value of the densities presented in Fig. 2c. Note that the ABP spheres appear in a fairly regular helical array of bright spots. A slight ghosting of actin remains in Fig. 2c because only the first 10 layer lines were used to make the filtered image. This leaves higher resolution details corresponding to the higher order layer lines in the original image that are not removed by subtracting the filtered image. We found that the high resolution ghost reduced somewhat the amplitude of the AVID peak but not its shape or location. To produce the 3D map of the density variations, we transformed the image shown in Fig. 2d, collected layer-line data, and computed an inverse Fourier—Bessel transform in the same way we would calculate a map of the structure itself. Fig. 3a
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shows a section of the conventional 3D map of the partially decorated model. Fig. 3c shows a section through the 3D AVID map on the section shown in Fig. 3a. For comparison, Fig. 3b shows a section from the conventional, 3D difference map made by subtracting a reconstruction of the actin model from that of the decorated actin model. The AVID map and the difference map both have a single strong peak located at the position of the third sphere (or ABP of the model) as expected. The deviation of the peaks from a circular shape arises because the resolution radially is greater than that azimuthally (i.e., only layer-lines with small values of the order n are included). The AVID procedure was next applied to electron micrographs of frozen hydrated preparations of F-actin decorated with N375, the N-terminal actin binding domain of fimbrin [1]. In this structure, the location of N375 was determined using conventional difference map procedures. The difference map shows two positive peaks (bold contours in Fig. 4a). Peak 1, which is interpreted to be the N375 domain, is located outside the radius of actin as expected for a bound domain. The variance corresponding to this peak is low. Peak 2 is interpreted as a conformational change in actin [1]. Note that the peaks in the map of standard deviation (Fig. 4c) do not coincide with peak 1 or peak 2. Hanein et al. indicate that in their reconstruction of actin decorated with N375, the density ascribable to N375 is much weaker than that ascribable to actin and that N375 is much larger in the difference map than the N375-actin complex map. This can be seen in Fig. 4a which is taken from Figs. 3 and 4 of Hanein et al., [1]. The cause for the weakening of the density corresponding to N375 appears to be due to the loss of bound N375 during blotting and plunging of grids. The N375-actin complex therefore seemed a good candidate on which to test the AVID procedure. We carried out the procedure and averaged layer line data from 32 filaments which Hanein et al. used to generate the real-space average in their paper. Fig. 4b shows a section through the 3D AVID map. It reveals one main peak which overlaps with peak 1 (the N375 peak) seen in the difference map (Fig. 4a). The peak does not overlap with either of
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Fig. 2. Test of AVID procedure using a computer-generated 3D model: the generation of the AVID difference image. The filament is a simple model of actin, that is a helical array of subunits each of which consists of two spheres corresponding to the two domains of the actin subunit. The analog of the actin-binding protein is a third sphere added to a particular site on the model actin. As in the model in Fig. 1, a random number generator is used to decide which sites to decorate. The images shown here are projections of the 3D models. (a) Model filament decorated at about 50% of the equivalent sites. (b) Filtered image of (a). To produce the filtered image, the strongest ten layer lines from the transform of the projection shown in Fig. 1a are selected and back transformed. (c) Difference image obtained by subtracting the projection in (b) from that in (a). (d) AVID difference image formed by taking the absolute values of the densities in Fig. 1c. Bar"10 A_ .
the peaks detected in the standard deviation map (Fig. 4c). The AVID peak extends axially relative to peak 1 of the difference map; a consequence of this
elongation is the appearance of the peak marked 1’ in the section of the AVID map. The reason that the shape of the AVID peak is different from the
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Fig. 3. Test of AVID procedure using computer generated models: the generation of the AVID map. (a) Transverse section through the 3D map generated from Fig. 2b. To calculate the map, layer lines were selected from the Fourier transform of the image shown in Fig. 2a. An inverse Fourier—Bessel transform of the layer line data generated the 3D map. (b) A transverse section through the difference map generated by subtracting a 3D map of the undecorated actin model (not shown) from the map described in (a). The difference peak is superimposed on the section shown in (a). (c) A transverse section through the 3D AVID map. The AVID peak is superimposed on the section shown in (a). Bar"10 A_ .
Fig. 4. Application of AVID procedure to electron micrographs of the actin—N375 complex. (a) A transverse section of the difference map generated by subtracting a map of actin from a map of actin decorated with N375. The peaks are superimposed on the corresponding section through the actin—N375 map. The data for this figure is taken from the work of Hanein et al. [1]. (b) A transverse section through the 3D AVID map superimposed on the corresponding section of the actin—N375 map. (c) A transverse section through the standard deviation map of actin decorated with N375. The map is obtained from an analysis of the set of filtered images of the actin—N375 complex. The peaks are superimposed on the corresponding section through the actin—N375 map that has been contoured to represent correctly the dimensions of the actin portion. Note that the peaks do not lie at the location of the N375 domain and do not coincide with those obtained from the AVID procedure. Bar"10 A_ .
difference peak in Fig. 4a is not clear. The AVID procedure does not detect a peak of variation corresponding to peak 2, perhaps because it is intrinsically too weak or perhaps because the conformational changes in actin produced by N375 binding are cooperative.
4. Discussion The idea behind the AVID procedure is to map the variations in density that arise from partial occupancy. We must keep in mind, however, that the AVID procedure will map all variations within
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the filaments. Because we subtract the regular features from the starting images in calculating the AVID difference images, we use data that are independent of those used in calculating the standard 3D reconstructions and difference maps. The AVID procedure therefore provides an independent way to locate an actin-binding domain. Because we subtract the filtered image from the original image, we avoid problems associated with scaling. With a simple one-dimensional model, we illustrated the principle behind the AVID procedure. We then applied it to projections of a model helical structure. In this example, we make use of the approximation that the absolute value of the difference between the two projections is equal to the projection of the absolute values of the difference between the two structures. We have shown with a helical model and an actual filament that this is approximately true at least for some classes of decorated actin filaments. We applied the AVID procedure to actin filaments decorated with the known actin-binding domain of fimbrin, N375. This structure had already been solved by Hanein et al. [1], who noted that the portion of the reconstruction corresponding to N375 appeared as a relatively small extension from the portion of the map corresponding to actin. Only 50% of the density is seen relative to actin; this despite the fact that N375 is approximately the same mass as the actin subunit. When Hanein et al. subtracted a map of undecorated actin from the N375-decorated actin however, they observed a statistically significant peak that corresponds in size and shape to the N375 structure. Based on the weakness of the N375 density relative to that of actin, they suggested that there is either incomplete decoration or the N375 domain is floppy. The AVID procedure as expected shows a strong peak at the N375 binding site, and little variation elsewhere in the map. These results show that the AVID procedure is effective in finding the bound N375 domain, without the need for a map of undecorated actin. In an ideal situation (i.e. when applied to a computer-generated model), the AVID procedure produced a peak having the expected location and size of the bound “ABP”. When the AVID procedure
was applied to actin-N375, the location of the peak occurred at the expected position. The existence of the AVID peak suggests that we were not dealing with a mixed population of filaments in which some were fully decorated while the rest were undecorated. On the contrary, N375 was either partially decorating the filaments or it was exhibiting variable conformations or both. The shape of the peak, however, was not exactly the same as that in the difference map. The densities in the peak were also weaker than those in the difference peak. We do not know if partial overlap of binding sites in projection and extraction of an incomplete set of layer lines account for these deviations from the expected results. We need more examples to address this issue.
5. Conclusions The AVID procedure is designed to map all deviations from symmetry. It is an algorithm therefore that can confirm or detect an ABP through deviations in symmetry arising from partial occupancy or domain mobility. Unlike difference mapping, the AVID procedure does not require a map of the undecorated filament. Since the procedure makes use of data that are independent of those used in difference mapping, it provides additional information about the structure. Moreover, when a difference map already exists, the calculation of an AVID map requires little additional work. We used the AVID procedure to look at structures with an underlying helical symmetry. With suitable modification, the AVID procedure should be applicable to non helically symmetric structures. Application of the algorithm to other examples may provide new insights into structural variations.
Acknowledgements This work was supported by grants from the NIHGM26357 and GM07596. The authors also acknowledge funds from the W.M. Keck Foundation.
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[5] S. Trachtenberg, D. DeRosier, J. Mol. Biol. 195 (1987) 581—601. [6] D.J. DeRosier, P.B. Moore, J. Mol. Biol. 52 (1970) 355—369. [7] C.H. Owen, D.G. Morgan, D.J. DeRosier, J. Struct. Biol. 116 (1996) 167—175. [8] E.H. Egelman, N. Francis, D.J. DeRosier, Nature 298 (1982) 131—135.