Ultramicroscopy 46 (1992) 263-285 North-Holland


Processing images of helical structures: a new twist David Gene Morgan and David DeRosier Rosenstiel Basic Medical Sciences Research Center, Brandeis University, Waltham, MA 02254, USA Received at Editorial Office 9 April 1992

Helical macromolecular assemblies are particularly difficult to study by X-ray diffraction but are quite well suited t o analysis by electron microscopy. Most of our information about helical macromolecular assemblies has come from the electron microscope but has been limited to about 25 A resolution. With the use of low-dose electron cryomicroscopy, one can obtain structural data to near atomic resolution on two-dimensional crystals, but the problem is to extract the information from the noise. In this paper we present methods to extract signal from low-dose electron cryomicrographs of helically symmetric structures. We apply these methods to extract 10 ,~ data from the bacterial flagellar filament.

1. Introduction

The recent successes of electron microscopy in solving the structures of membrane proteins to near atomic resolution [1,2] push us to consider whether other difficult classes of structure such as helical assemblies might also be studied at high resolution by electron microscopy. Structures with helical symmetry fall in a category of those which are difficult to work with by X-ray diffraction; only one helical macromolecular assembly, tobacco mosaic virus (TMV), has been solved to atomic resolution using conventional methods such as multiple isomorphous replacement [3]. Helical assemblies at low resolution have been examined using electron microscopy. Indeed, the first three-dimensional reconstruction from electron micrographs was carried out on the helical tail of the T4 bacteriophage [4]. More recently, there have been some encouraging results on studies of T M V to about 9 A resolution by electron microscopy and image processing [5]. In this p a p e r we will discuss the difficulties and advantages of examining helical structures at high resolution by electron microscopy. We will present new methods we have been developing to process images in order to extract high-resolution data.

2. Helical structures are weak diffractors

The Fourier transform of a two-dimensional array of atoms and the Fourier transform of the same array rolled into a helix of radius r 0 are related to one another [6]. If the transform of the two-dimensional array is

T,,I= Y'~f/ e 2~('xl/'+tz,/C), J


where f/ is the scattering factor of the jth atom, n and l are the coordinates of the Fourier coefficients, x~ and z / a r e the coordinates of the atom, a and c are the lengths of the edges of the helical unit cell (a 0304-3991/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved


D. Morgan, D. DeRosier / Processing images


of helical structures:


a new twist


.2 .1 0 .1 .2









- - ? ?


2-D 3 D

















i ~I~1















Fig. 1. Diffracted amplitudes from helices versus planar crystals. (a) The diffraction amplitude of a helix having one atom in the asymmetric unit is a set of Bessel functions. The plot shows the amplitude of an order 10 Bessel function. The maximum amplitude is about 0.3. The amplitude of the corresponding reflection from a two-dimensional crystal also having one atom in the unit cell is 1. Thus the diffraction from the helix would be about three times weaker than that of a two-dimensional crystal. (b) Ratio of the peak amplitudes on layer lines of different order, n, relative to amplitudes for the corresponding reflection from a planar crystal. The smooth curve shows the ratio of amplitudes (helix/planar crystal) if the asymmetric unit in both cases is a single atom. The dashed curve shows the ratio of amplitudes if the asymmetric unit is a three-dimensional distribution of atoms. The result depends on the details of the distribution. We chose an asymmetric unit consisting of 21 point atoms. In the helix, these lie evenly spaced along a radial row which starts at radial pixel 25 and ends at radial pixel 120. In the planar crystal, the 21 atoms lie in a line perpendicular to the plane of the crystal.

is the length of the azimuthal edge 2rrr 0 and c is the length of the axial edge, i.e., the axial repeat); then the transform of the helix is

F( R, tb, I/c) = T,,.tJ,,(2rcRro) e in('/'+rr/2)


1( R, ~, l/c)






J,, is an integral Bessel function of order n,

In,l = ]T,,,/I 2,

R, & and 1/c are the cylindrical coordinates of a point in reciprocal (Fourier) space. The Bessel function has an oscillatory behavior that decreases in magnitude with increasing argument (fig. la). It reaches its maximum amplitude at an argument that is not far from n. The effect of the Bessel function on a reflection T,,,~ is to stretch it out from a discrete point into a radial line called a layer Fine. The amplitude maximum on a layer line relative to the corresponding reflection in the crystal is simply the value for the maximum of J~ which depends on n. These amplitudes of the maxima begin at

D. Morgan, D. DeRosier / Processing images o f hefical structures: a new twist


1.0 and decrease as a function of n as shown in fig. lb. The amplitudes of the maxima for low-order layer lines (e.g., n = 5) are about three times less than the corresponding crystalline reflections. The situation worsens by another factor of three when one considers realistic helices that are not derived from two-dimensional sheets but rather from three-dimensional molecules. These helices have structural features that extend radially. In fig. lb, the dashed curve shows the ratio of the peak layer-line amplitude for a helix having a three-dimensional subunil relative to the amplitude of the corresponding reflection for a single-layer crystal made of the same subunit. Thus, the maximum amplitude on a low-order layer line is at best a factor of three times weaker than the corresponding reflection from a planar crystal. More realistically, they are ten times weaker. To solve the structure of the helix, we need the amplitudes from the whole layer line, not just from the strongest peak on it.

3. Collection of three-dimensional data is easier

The weakness in amplitude arises because the helical particle is providing three-dimensional data in a single view. In studying a m e m b r a n e protein having planar symmetry, one gets only one view of the subunit, albeit many times over. One has to get the three-dimensional data set by obtaining a series of images in which the planar crystal is tilted. The number and angular increments of the tilts determine the resolution to which one can solve the three-dimensional structure [4]. Helical structures, by their nature, present the subunit structure in different orientations. If the axial repeat of a helical structure occurs after 101 subunits, then one has 101 different views of the subunit just as though one had carried out 101 tilts. This sets the useable resolution one can obtain from a single view. The diffraction pattern conveniently separates out the three-dimensional data until one reaches a resolution at which there is an overlap of reflections (i.e., layer-line overlap at which juncture the observed Fourier coefficient, F, is a sum of G,.I(R) [6] where values of n are different). To get beyond this resolution one has to obtain different views of the helix. With a set of different views, one can extract the individual values Gn,t(R) from the observed values of F in each view. Unlike the case for planar crystals, collecting a set of different views does not usually require tilting of the specimen in the microscope. The reason is that the helices, by their cylindrical shape, tend to sit with varying azimuthal orientation. Thus a set of ten particles on a support film will all have their axes in the plane of the support film but will pick rotational orientations around that axis at random. These different orientations will provide the data needed to extract the overlapping terms. The procedures to carry out the extraction are described by Jeng et al. [5], Crowther et al. [7], and Crowther et al. [8]. Given the set of G,,I(R) to some given resolution, the three-dimensional density map is generated: 1

p(r, 6, z) = 7 Z fG.j(R)R

dR e i'6 e -2~iz'z.


In studies of planar crystals such as bR, the reflections are strong and generally well resolved from one another. In helical structures, however, where the reflection is, in effect, streaked out into a layer line, it often happens that layer lines will overlap or at least be separated by a small axial distance. For example, in the case of the bacterial flagellar filament, from Salmonella typhimurium strain SJW1660, the symmetry is such that even low-resolution layer lines are close together, in this case separated by only about 0.001 .&-1 (fig. 2). If one can resolve such layer lines, there is no actual overlap for this structure out to a resolution of more than 8 ,~. The separation of such layer lines poses a problem for studies by X-ray diffraction where misalignment of the filaments in the sample results in widening of the layer lines in the diffraction pattern. The widening due to misalignment, if not severe, can be sorted out by deconvolution [9], but even in the most favorable cases the problem of layer-line overlap is more severe


D. Morgan, D. DeRosier / Processing images of helical structures." a new twist 1/7






xXXX'^ 1

+++~ xxXX~" xXXX'~

+++~ xXXX'"

~xxxXX ++++





++++-{-,~. ~ace.xXX X X ~-+++ xxXX^ o


,+T+t~ -40

,,.,xxxX, 0



80 n

Fig. 2. Plot of n versus z for the flagellar filament of

Salmonella typhimurium SJW1660. The plus symbol indicates the value of n for the flagellar filament. The cross is put in as the mirror symmetric point to simulate the mirror symmetry of the diffraction pattern. The plot extends to a resolution of 7 A. The vertical scale is given in reciprocal ~ngstr6ms. Note that to this resolution there is no layer-line overlap. The potential for overlap will become a problem when row lines of plus symbols meet the row line of cross symbols at the right or left edges of the plot.

in X-ray diffraction studies than in electron microscopical work. With X-ray diffraction the way to sort out layer-line overlap is with isomorphous derivatives. One derivative is required for every Gn,t(R) contributing to a layer line. Thus layer-line overlap puts heavy demands on the generation of isomorphous derivatives. In electron microscopical studies, on the other hand, we can manipulate images of single particles and align them in the computer. We can thus align particles perfectly and avoid this problem. In general, the better one's ability to resolve layer lines the higher the resolution to which one can work without facing layer-line overlap (remembering, of course, that the spacing of the layer lines is set by the symmetry of the object).

4. Helical particles are naturally disordered In particles with perfect symmetry, the width of the layer line is inversely proportional to the length of particle being examined. In practice, disorder in each particle determines the minimum layer-line width. As is the case with planar crystals such as bR, the disorder reduces the amplitudes of the reflections resulting in a loss of signal-to-noise and, consequently, in a loss of reliability of the data. As is also the case with planar crystals, we can correct some of the disorder. Helical filaments tend to flex in solution. Images of the filaments attached to electron microscope grids exhibit the same curvature seen in solution. Egelman [10] has applied a spline algorithm for straightening images of curved filaments and thereby removing a source of disorder. Such flexing also occurs in a plane perpendicular to the plane of the support film. This is usually eliminated by the attachment of the particle to the flat support film. There are cases, however, where such curvature has been seen [11]. Two other kinds of disorder are variable stretch (axial disorder) and variable twist (angular disorder). We have built the measurement of these two forms of disorder into our new procedures but have yet to implement their correction. A real-space algorithm to do so has been successfully used to correct for these disorders in sickle cell fibers [12].

D. Morgan, D. DeRosier / Processing images of helical structures: a new twist


Flattening of the particle on the support film is also a distortion. Moody [13] has an extensive analysis of the effects of flattening. One often sees in the transforms of images a left-right (or near-far) asymmetry in the layer-line amplitudes. This arises because the particle deviates from helical symmetry. For unstained specimens, it is usually the side in contact with the support film that is weaker than that away from the support film. It seems likely that the support film may disrupt the side of the structure binding to it. The effect is even noted somewhat in ice-embedded specimens viewed over holes. It may be that the particles interact with the water meniscus. The seriousness of these flattening and binding distortions depends on the structure being studied and on the conditions used in preparing the specimen. They can be monitored to some extent by comparing the data obtained from the near and the far sides of the particles, as we do in our new procedures which we describe below.

5. Electron diffraction

Electron diffraction provides a way to assess the degree of preservation of the specimens being used for microscopy. If the specimen diffracts to, say, 4 to 5 A, as is the case for the flagellar filaments, then, given good imaging conditions, we can expect, at best, data to that resolution from the images. We can thus set an upper limit on the resolution we can obtain from the images. In addition to this qualitative use, the electron diffraction pattern can also provide a set of accurate amplitudes with which to combine phase information obtained from the images. In this regard the procedure parallels the work done on m e m b r a n e crystals.

6. Automating the collection of data

Given the inherently weak nature of the diffraction, it is essential to average together many images and thereby extract the signal from the noise. In the past one typically used about ten images to generate a 20 _A map. However, with the inherently weaker diffraction caused by the use of lower electron dose and the gathering of higher-resolution data, many more than ten or twenty images will be needed. To make the collection of data a practical matter, we need to implement computer programs to carry out automatically the procedures that have been done traditionally by hand. These procedures are: (1) straightening the curvature of the particle, (2) locating the positions of the layer lines for each particle transform, (3) locating the particle axis and determining any tilt, (4) aligning the different particles prior to averaging, (5) carrying out C T F weighted averaging and calculating the statistics. We have recently automated all of these steps and used them to produce a preliminary 10 .~ data set for the flagellar filament.

7. Use of a reference data set

In order to automate the procedures, we use a reference data set which allows us to locate the helix axis, determine the positions of the layer lines, and align particles for averaging. After one pass of alignment and averaging, we replace our reference data set by the new average and begin a second round of alignment and averaging. To get started, we need a reference data set. We have tried two strategies. The first is to use the old interactive procedures to generate a low-resolution average from a few images. This works quite well. The second is to select one image from the set to serve as reference. This


D. Morgan, D. DeRosier / Processing images ~>['helical structures: a new twist

single-filament reference works but not as well as an average of several images. In the first round of analysis using the single-image reference, the corrections and alignments are all worse than those attained using an average, but the results (for the flagellar filaments) were good enough to include about a fourth of the images in the first-round average. This, then, was quite adequate to proceed with the rest of the analysis. Subsequent rounds of averaging removed the bias of the starting reference data set.

8. Strategies for choosing the lengths of helices to be analyzed From images of flagellar filaments, we select segments having lengths of 2500 to 5000 ,~. We need lengths above 2000 A to be sure we can resolve the closely spaced axial layer lines (1/87(I A). We break these lengths into shorter segments of 200 or 800 A, depending on the kind of analysis. By comparing each segment to the reference, we can detect deviations of the particle symmetry from that of the reference. Given a segment-by-segment analysis of a particle, there are two possible methods of proceeding. One is to extract the data from each segment as if each segment is an independent view of some fixed segment of the structure. One would then use real-space reconstruction to compute a three-dimensional map. See, for example, Bluemke et al. [12]. The second method is to correct the segments, piece them back together into a more perfect filament, and then treat the restored filament as a single particle. We have chosen the second method because it allows us to carry along the noise in a convenient form and to assess the improvement in signal-to-noise much the way it is done for planar crystals. When done, we can display the averaged data and the averaged noise in the form of a diffraction pattern of the averaged image. The noise is seen in the regions of the transform where there are no layer-line data. We use lengths up to about 5000 A in order to keep down the execution times of the Fourier transforms. Moreover, there is no gain in using longer lengths since, for an 8 ,& map, all the layer lines are well resolved. We extract data from the layer lines by interpolating values from the transform. There is some controversy over the best way to interpolate data so as not to reduce the amplitudes extracted from the transforms. If one oversamples the image, then one is better off interpolating in real space. If one oversamples the transforms, then one is better off interpolating in the transform. We routinely oversample the transforms, by a factor of three to four, by padding the images with borders of constant density. o

9. Reordering of the procedures We have found it beneficial to reorder some of the steps set out by DeRosier and Moore [14]. In their work, they extractcd the layer-line data and used the expected symmetry to determine the position of the helix axis and to measure and correct tilt of the axis out of the plane of the image. The layer-line data were then split into near- and far-side sets. Alignment and averaging ensued. We find it advantageous to extract the entire layer line and to carry out the alignment and averaging without selecting the helix axis or splitting the data sets into near and far sides. This has the advantage of comparing only near sides with near sides and far with far. Due to the near/far-side asymmetry, this strategy ensures we carry out alignment using only equivalent data. We also delay the fixing of the position of the axis until the final step when the signal is considerably strengthened relative to the noise. The gain is a factor of two in signal-to-noise. This increases the reliability of the determination. What we actually do is determine the tilt of each particle against that of the current average. Only at the end of each round of averaging do we then determine the tilt of the axis out of the plane of the image. Thus we do as little altering of the data as possible, making as few assumptions as possible, until the averaging is complete.

D. Morgan, D. DeRosier / Processing images of helical structures." a new twist


10. Initial estimate of particle curvature Fig. 3 shows a flow chart of the steps we use in image analysis. The first stage is the initial estimate of particle curvature, which is obtained from a cross-correlation of. the particle's projected radial density with that of the reference. The particle, assumed to have its axis in a near vertical orientation, is divided into small segments. The size of each segment will depend on the structure being studied. For the flagellar filaments we find that boxes 50 pixels long (corresponding to about 200 A or 40 subunits having a total mass of 1.6 million daltons) work well. The equatorial data from the Fourier transform of each segment are multiplied with those of the reference and the product back-transformed to generate a cross-correlation map. The position of the peak in the map fixes the position of the helix axis of the segment relative to that of the reference. This initial estimate is not highly accurate, but it is important in constraining the search we use subsequently to precisely locate the axis. The reason is that this initial measurement uses equatorial data which may contain a strong contribution from the matter surrounding the particle. The cross-correlation peak can therefore be perturbed if the distribution of matter around the particle is different from that surrounding the reference. We keep track of the peak position, and if it deviates too widely from the estimates derived from other segments or if it is too close to the edge of the image, we try the second-highest peak and, if necessary, the third, fourth or fifth. If none of the positions are satisfactory, we skip this segment and go on to the next. Only occasionally do we ever need to look farther than the second peak, and this is when we have noisy images. From the positions of the correlation peaks, we have a set of points marking the path of the helix across the image. Due to the presence of noise in the image, the points do not follow a smooth curve but,


dlgltlzecl~Images ~RV[ AND ~LT ,.e. etralghtsed



axl.tions posi

layedine posi tions


~("~YERUNEDATA la)~rllne data nta

awraged layurllne data

near elde reconltruction

far side reconstruction

Fig. 3. Flow chart for the analysis of images. The cross-correlation operation is assumed to contain both the initial estimate of axis position based on equatorial data only as well as the analysis based on non-equatorial data.


D. Morgan, D. DeRosier / Processing images of helical structures: a new twist

rather, they show some jitter. We fit the set of points with a smooth spline that does not follow the points exactly but instead smooths out the errors in the positions [10]. This smoothed curve defines our initial estimate of the path of the helix axis in the image.

11. Helical cross-correlation The off-equatorial data is less sensitive to variations in the particle's surroundings. We use it to determine more accurately the path of the axis of each particle. We also use it to predict the positions of the layer lines. The math covering this is contained in the appendix, but the idea is as follows: If the reference and a particle have exactly the same helical symmetry, then having determined the orientation of one segment of a particle, we can predict exactly the orientation of each neighboring segment. We can do this simply from the helical symmetry operator consisting of a rotation by a constant angle and a translation by a constant distance. If the helical symmetry of the particle is different from that of the reference, then the orientations of neighboring segments deviate in a systematic way from that of the reference. The systematic variations can be used to determine the exact helical parameters for each particle relative to the reference. The non-systematic variations provide a measure of the disorder in the structure. To carry out this procedure, we again divide each particle into segments. From the Fourier transform of each segment, we collect layer-line data. The use of short segments increases layer-line width. Thus we need only know the approximate positions of the layer lines to collect layer-line data for each segment. The use of segments that are shorter than the repeat generates layer-line overlap. Thus when we collect layer-line data for short segments, it is biased by the data from the neighboring layer lines. This makes no difference in the cross-correlation, which is defined in real space but carried out in reciprocal space. The answer is the same as if we had carried out a real-space correlation of that segment against an infinitely long reference helix. The cross-correlation assumes we have correctly chosen the helix axis and correctly aligned the polarity of the reference and unknown segment. We can refine the axis position and tilt and choose the orientation by varying both until the height of the cross-correlation peak is maximized. In varying the position of the axis, we cannot search over more than about 10 to 15 ,~. If we stray too far, we obtain another strong peak which corresponds to displacing the lattice laterally and axially to the next unit cell. We therefore require a relatively good initial estimate of the position of the axis at any point along each particle. The smoothed spline obtained as described in the preceding section provides that estimate. The position of the cross-correlation peak in the helical unit cell gives us the orientation of a segment relative to the reference. We repeat the process for all the segments of a single particle. For each segment we obtain a refined estimate of the position of the axis, the polarity of the axis (up versus down), and an orientation in the form of a position in the helical unit cell. The two coordinates of positions in the helical unit cell are separately plotted as a function of the segment number (see eq. (A.38)). The best straight lines through these two sets of points determine the precise helical symmetry for the particle. Deviations from the straight lines provide estimates of the axial and angular disorder present in that particle. The procedure works well as long as the predicted and observed positions are within 0.5 of a unit cell dimension, since all positions are plotted relative to the nearest unit cell. The longer the segment used, the more accurate must be the initial estimates of the helical symmetry.

12. Estimates of image quality at this stage Following the helical cross-correlation, we have a collection of statistics that can be used to assess the images:

D. Morgan, D. DeRosier / Processing images of helical structures: a new twist

(1) (2) (3) (4) (5)


correlation coefficient for each particle derived from the averages for each segment, average difference in up versus down correlation coefficients, amount of angular and axial disorder, amount of particle curvature, measured helical parameters.

13. Extraction of layer-line data Having determined the curvature for each particle, we now use a spline algorithm [15] to straighten each of the particles and tighten the mask (i.e., we rebox them with an apodized box if desired). The spline algorithm is different from the one used in our first estimate for the particle axis in that it draws a smooth curve that goes through every point. The reason that we do not do additional smoothing is that we have fewer points defining the axis and the points are more accurate. We have no information as to the deviation of the points from the true axis. If we further smooth the curve, we will necessarily miss going exactly through the points and may therefore not correct the data as well as we could. No doubt, further work and tests may indicate a better plan of approach. Each straightened image is transformed and the positions of all desired layer lines to some specified resolution are calculated from the helical symmetry determined for that particular image. In addition to these layer lines, we collect a set of background lines; that is, we collect lines of data between the layer lines. For the flagellar filament, we have been collecting all layer lines to 8 A resolution. There are 83 such layer lines. We fill in the gaps between these with background lines which are as evenly spaced as possible and with at least one background line between neighboring layer lines. The spacing of the background lines is about 1/1740 A. In all we collected 255 lines of data plus background. The lines of data and background when assembled into a single frame represent a 256 by 512 two-dimensional transform of that particular image. Since the actual positions of the layer lines vary slightly from one image transform to another due to slight variations in helical symmetry, the layer-line positions have been adjusted axially so that the assembled frames of data and noise from different images all have the same layer lines at the same axial positions. This is necessary to carry out the remaining steps of alignment and averaging.

14. Alignment of data sets and initial averaging Prior to averaging the collected layer-line data and background, we aligned the data sets to the reference set. To effect the alignment, we used a subset of the layer lines which we thought had significant data on them and where there is no need to correct for the CTF or curvature of the Ewald sphere. This could be changed. The alignment was carried out as given by DeRosier and Moore [14]. Successful alignment is judged as the position at which the phase residual is a minimum. The phase residual was calculated from reference amplitudes and squared phase differences as done by Amos and Klug [16]. At the end of each round we had determined the azimuthal rotation, axial shift, lateral shift, tilt, and polarity needed to align each image to the reference. Then we computed an average of the subset of layer lines to be used in the next round of alignment and then repeated the alignment using the new average as the reference. The cycles continue until there is no further gain. Using as reference an average derived from about 10 filaments, we found that 4 or 5 cycles of alignment and averaging were required. Using a noisy data set and one filament, we found that 8 cycles were required. Using an average derived from about 60 filaments using the new procedures, we found that additional images aligned in only one cycle.


D. Morgan, D. DeRosier / Processing images of helical structures: a new twist

The quality of the individual particles can be judged from the residual and the up-versus-down difference (i.e., the change in the residual calculated using the opposite polarity of the axis). Together with the measures of quality generated during the helical cross-correlation analysis, there is sufficient information to begin looking for correlations among the various measures. We have not yet done this, but it may provide the information one needs to weight the contribution of each particle to the final average.

15. Use of weights and filters

In both helical cross-correlation analysis and alignment using phase residual, we selected the data used. If all layer lines are used, even ones where there is little or no signal, one is adding more noise than signal, and the quality of the fit suffers. Alternatively, if one aligns using only the strongest few layer lines, one reduces the effect of the weaker, higher-resolution layer lines, which potentially can give a more accurate result. To allow for adjusting the way the data are used, we introduced layer-line filters and weights which select a n d / o r weight the data being compared. Each layer line has a weight which determines how data on that line will be weighted into the correlation coefficient or the residual determining the best fit. These values can be different for each round of fitting and averaging. In the early rounds, one might have only the strongest layer lines contributing, and in later rounds, one can weight these down and add in weaker, but higher-resolution, layer lines. In practice we have not found this makes an appreciable difference in the result, but perhaps it will for other structures. In any layer line there are regions where the amplitudes are known to be nearly zero, in particular, the meridian for G,,.I(R) where n is different from zero. We exclude data in this region by applying a "Bessel" filter. With an estimate of the outermost radius, r 0, of the particle and the order, n, of the layer line, we can determine the relative contribution of the meridional part of the layer line. We determine the largest value of R 0 for which J,,(2rrRor o) is less than some preset fraction (say 10%) of its maximum value. We then construct a filter that is 0 for values of R < R 0 and is 1 otherwise. A second filter we apply is a resolution-cutoff filter. Positions in the filter corresponding to data exceeding some resolution are set to 0; the remainder of the filter is set to 1. Thus all data exceeding the resolution limit will be set to zero. We also construct a filter based on the relative strengths of the amplitudes of the reference. We usually take the top 10% to 20% of the amplitudes on each layer line. To do so we order all amplitudes on a layer line according to strength. We exclude amplitudes that would be excluded by the Bessel and resolution filters, of course. We then pick the strongest 10% or 20% and construct a filter which is 1 if the amplitude at that position is in the selected percentage and is 0 elsewhere. This selects the regions of each layer line that have the strongest signal (based on the current average or reference). It increases the signal-to-noise in the alignment. Another filter we have instituted is one based on the statistical significance of each amplitude in the reference. At the end of a round of averaging, we compute the average amplitude and standard error of the average for every reciprocal pixel. Using a t-test, we can determine at some confidence level (say 5%) whether an amplitude is significantly different from zero. We can then construct a filter like that in the preceding paragraph. The filter contains a 1 at every position at which the averaged amplitude was significantly different from zero and a 0 elsewhere. In practice so far, this filter did not work better than the amplitude-cutoff filter, although in principle it should. The composite filter we apply to each layer line is the product of all the individual filters. The filters are then applied to the ensuing round of alignment and serve to transmit into the residual or cross-correlation calculation only those pixels that are thought or known to be carrying some signal.

D. Morgan, D. DeRosier / Processing images of helical structures: a new twist


16. Corrections and calculating the final average In averaging together data sets from different images, we want to take into account the differences in the contrast transfer function. In particular, we do not want to average with full weight the data in a region where the contrast function is near zero, since at best one will reduce the amplitude of the signal and at worst will add in extra noise where no signal is expected. We therefore use a Wiener filter of the following form [17]: CTF,, .l,j ( n )


= ~ [CTF.j.k(R)] 2 + N / S '



where CTF.,t.j(R) is the contrast transfer function for the jth particle at some position R along the layer line n, I, the sum over k in the denominator is taken over the entire set of particles, and N / S is the noise-to-signal ratio. The positions of the zeros and the amount and direction of astigmatism are determined from transforms of large areas of each micrograph. We then calculate the CTF using the formula in Erickson and Klug [18]. Having calculated all the CTFs, we then calculate the Wiener filter for each transform and apply it. The averaged value for each G . j ( R ) is:

(G.j( R)) = Y'~ W.,,,j( R)G.,Li( R ).



Since we are carrying not only layer lines but also lines of noise, we can get an estimate of the noise as passed through the averaging procedure. We treat background lines as if they are n = 0 layer lines. We can use this to estimate the significance of the amplitudes seen on each of the layer lines. In particular, we can assume that the standard error of each average amplitude is simply the averaged background that surrounds that pixel. This allows us to estimate the resolution of our data. We could also use the local noise to determine the reliability of our data and to weight each data point on its inclusion into the inverse Fourier-Bessel transform. 17. Materials The micrographs we used were taken of flagellar filaments from the strain SJW1660. These are straight, left-handed filaments which were embedded in a g l u c o s e - P T A mixture (equal weights). The microscopy was carried out on a Philips 420 at the Laboratory of Molecular Biology, Cambridge, England, and on a Philips CM12 in the Structural Biology Laboratory at Brandeis University. In the former location we used a cold stage especially designed by Richard Henderson [19] and, at Brandeis, a Gatan cryoholder. The specimens were photographed between - 1 5 0 and - 1 7 5 ° C depending on the stage. The magnification was about 60K x and the dose about 10 electrons per ,~2. We used three sets of images taken at three different defocus settings ( - 2 5 0 0 , - 4 0 0 0 , and - 7 0 0 0 A). The number of images in each set were 8, 64, and 92, respectively. Film scanning was done on a flat-bed scanner or an Eikonix CCD scanner at a 20 ~ m sampling raster. Processing was done on a cluster of VAX computers using programs written in F O R T R A N . 18. Results The following shows the results of carrying out these procedures on the flagellar filaments. Table 1 gives the relevant parameters used to carry out the helical cross-correlation. The only filter used was the Bessel filter which eliminates the meridional intensities resulting from the masking


D. Morgan, D. DeRosier / Processing images of hefical structures: a new twist

Table 1 Parameters for helical cross-correlation Bessel filter: Resolution filter: Amplitude filter: Significance filter:

Radial cutoff starts at 20% Off Off Off

Layer lines included and weights Z


Weight (%)

1/870 1/52 1/49 1/26 1/13

- 11 -5 +6 + 1 +2

50 50 1(10 100 100

procedure. Of the five layer lines included in the calculations, the two strongest layer lines were weighted down so that the higher-resolution, weaker layer lines could more strongly influence the precise determination of orientation. Fig. 4 shows a page of output for one of the images following helical cross-correlation. The leftmost box shows the particle curvature obtained by cross-correlation using only equatorial data (smaller points). The curve corresponds to the spline-fitted line through the points. The larger points are those found using only non-equatorial data. These points are also shown in the box to the right. In this box the curve represents the spline fitted through the large points. Alongside the two plots of curvature are listed the best orientation of each segment (up versus down) relative to the reference and with it the difference between the up versus down correlation coefficients for that segment. The data for the unit cell position versus segment number are shown in the smaller boxes on the right. From the best straight-line plots through the points, the exact helical parameters and disorders are estimated. At the bottom is given the image name, image size and the axial positions (in transform pixels) of the n = - 11 and n = - 5 layer lines which we use to define the helical symmetry of the particle. From these two values, we can generate the positions of all the other layer lines.














alpha vs box number

beta vs box number


Fig. 4. Sample output from the helical cross-correlation program. See text tier details. File name: 4181H.BOX. The dimensions are 87 by 972. This particle is oriented down. The height of the 11 Bessel is 16.8933 and the height of the 5 Bessel is 276.0080.


D. Morgan, D. DeRosier / Processing images of helical structures: a new twist

Table 2 Results of helical cross-correlation Defocus of data set (A) Number of images

- 7000

- 4000

- 2500




Av. correlation coef.

0.95 (0.01)

0.90 (0.03)

0.93 (0.01)

Up/down difference

0.19 (0.04)

0.32 (0.08)

0.16 (0.04)

RMS axial disorder

0.22% (0.22)

0.27% (0.23)

0.23% (0.11)

RMS angular disorder (degrees/subunit)

0.0078 (0.0045)

0.0085 (0.0067)

0.0091 (0.0040)

Z(n = - 11) (I/A)

0.00108 (0.00005) 0.01925 (0.00009)

0.00108 (0.00008) 0.01925 (0.00008)

0.00108 (0.00004) 0.01918 (0.00012)

Z(n = - 5) (1/,~)

The first value is the average and the second is the standard deviation.

T a b l e 2 shows the results of applying helical c r o s s - c o r r e l a t i o n to the t h r e e sets of d a t a e a c h t a k e n at a different defocus. It is c l e a r that in some aspects the t h r e e sets are indistinguishable. T h e a v e r a g e d p o s i t i o n s of t h e n = - 11 a n d n = - 5 layer lines a r e essentially identical. T h e rms axial d i s o r d e r s are all a b o u t the same. T h e value of ~ 0.2% m e a n s that on a v e r a g e a n e i g h b o r i n g s e g m e n t was within 1.6 A of its e x p e c t e d location. T h e rms a n g u l a r d i s o r d e r s are also all a b o u t the same. A value of 0.008 ° p e r subunit m e a n s t h a t a n e i g h b o r i n g s e g m e n t was within 1.3 ° of its e x p e c t e d a n g u l a r o r i e n t a t i o n . T h e t h r e e d a t a sets d o show s o m e significant differences. T h e set at - 4 0 0 0 ( t a k e n in C a m b r i d g e , E n g l a n d ) shows a lower a v e r a g e c r o s s - c o r r e l a t i o n a n d a g r e a t e r u p - v e r s u s - d o w n difference. This m e a n s that the d a t a a r e not as g o o d but show g r e a t e r polarity. T h e r e are several possible e x p l a n a t i o n s for why a p o o r e r d a t a set might show i n c r e a s e d polarity. F o r e x a m p l e , the d a t a are of h i g h e r r e s o l u t i o n (albeit noisier), a n d these d a t a b e t t e r reflect the i n h e r e n t polarity. T h e e x p l a n a t i o n we favor is that the images d o not differ simply d u e to defocus b u t that t h e r e m u s t be d i f f e r e n c e s in the particles p e r h a p s p r o d u c e d by the e m b e d d i n g m e d i u m . T a b l e 3 gives the r e l e v a n t p a r a m e t e r s used in four r o u n d s of a l i g n m e n t a n d averaging. T h e values for p h a s e r e s i d u a l a n d u p - v e r s u s - d o w n d i f f e r e n c e s are the limits set for including an i m a g e in the average. T h u s the p h a s e residual must be less than 90 ° a n d the u p - v e r s u s - d o w n d i f f e r e n c e must be g r e a t e r t h a n 10 ° (i.e., the p a r t i c l e m u s t show a r e s p e c t a b l e polarity). O n each s u b s e q u e n t r o u n d of a l i g n m e n t a n d averaging, we i n c l u d e d m o r e layer lines a n d thus h i g h e r - r e s o l u t i o n data. W e p u t in all layer lines with full weight. T h e m a t t e r of w e i g h t i n g m e r i t s f u r t h e r e x p e r i m e n t a t i o n . T a b l e 4 shows t h e o u t c o m e of the a l i g n m e n t for the - 7 0 0 0 A defocus images, the - 4 0 0 0 ,~ set a n d finally for all t h r e e sets at once. F o r the last set, we a p p l i e d the W i e n e r filter p r i o r to alignment. Fig. 5 shows t h e sums of the s q u a r e d C T F s for all the i m a g e s in all t h r e e d a t a sets. This sum is an i m p o r t a n t t e r m in the W i e n e r filter. T h e sum varies in m a g n i t u d e from a b o u t 15 at its lowest to a b o u t 140 at its highest. R e c o v e r y of the d a t a will be g o o d in regions w h e r e the sum differs significantly from 0. R e c o v e r y will be p o o r w h e r e the sum is n e a r zero, but t h e r e are very few such positions b e c a u s e of the large n u m b e r of images. A s an aside, a s t i g m a t i s m in the i m a g e s h e l p s b e c a u s e it r e d u c e s the n u m b e r of small values in t h e sum by effectively a d d i n g in a g r e a t e r r a n g e of defocus to t h e set of images. T h u s the o p t i m a l way to collect d a t a m a y be to use a mix of various levels of defocus with a m o d e s t a m o u n t of a s t i g m a t i s m ( ~ 1000 ,~). Since we have not d e t e r m i n e d the noise-to-signal ratio for the set of images, we u s e d a value of 1 in the filter. Fig. 6 shows a typical p a r t i c l e f r o m the - 7 0 0 0 .& d a t a set, its t r a n s f o r m , a n d its t r a n s f o r m after the W i e n e r filter h a d b e e n a p p l i e d . B e c a u s e we u s e d the W i e n e r filter as a w e i g h t i n g function in the

D. Morgan, D. DeRosier / Processing images of helical structures: a new twist


Table 3 P a r a m e t e r s for particle a l i g n m e n t Bessel filter: R e s o l u t i o n filter: A m p l i t u d e filter: Significance filter:

R a d i a l cutoff starts at 10% Off Only top 10% a d m i t t e d Off

M a x i m u m a c c e p t a b l e p h a s e residual: M i n i m u m a c c e p t a b l e up-versus-down difference:

90 ° 10°

Layer lines included and weights (in %) for each r o u n d Z


R o u n d No. 1





- 11 22



100 10()

10(J 100

- 33


+ 17 +6


10() 10()

100 100

- 5

1 O0


1 O0

I ()0





10() 1()()

100 I(JO







+ 12 + 1 -





+ 18 + 7 4 - 15

1O0 100 100

l()(J 1(JO 1O0 100

lO0 1O0 l O0

+ 13 +2


100 100

I00 100 lO0

- 9


- 2O




+ 19 + 8



- 3

1 O0

I ()0


10() I()0

14 + 3 - 8

l()() I 0()

l()i) l(JO

average, we did not calculate the statistically significant amplitudes. The - 7 0 0 0 A set compared to the 4000 A set shows a lower rms phase residual and less rapid rise with increasing resolution (i.e., in each succeeding round). The polarity of the - 7000 .A set is again found to be less than that of the - 4000 ,~ set at least in the first rounds. In both cases, the number of images included in the average at the end of each round remains quite stable. The number of pairs of amplitudes that were symmetrically disposed about the meridian and were statistically significant was about the same for both sets of data in the early rounds (i.e., about 9% of the maximum number of pairs possible), but the - 7 0 0 0 A, data set had more when higher-resolution data were included in the later rounds. The percentages in both cases decrease since the number of possible pairs increases with increasing resolution. The difference between the two data sets again points to the higher-resolution data being in the - 7 0 0 0 * set.


D. Morgan, D. DeRosier / Processing images o f helical structures." a new twist

Table 4 Results of particle alignment for two of the data sets Round 1




Defocus = - 7000 ~, only No. images accepted out of 92 RMS phase residual (deg) Up/down difference (deg) % significant amplitudes

88 25 24 9

89 31 24 7

89 35 23 7

89 36 23 7

Defocus = - 4000 ~, only No. images accepted out of 64 RMS phase residual (deg) Up/down difference (deg) % significant amplitudes

62 36 34 9

62 48 29 7

62 58 24 5

62 59 24 5

131 41 27

139 39 21

lib 38 21

ll6 39 21

Defocus = - 7000, - 4000, - 2500 ,~ No. images accepted out of 164 RMS phase residual (deg) Up/down difference (deg)

W h e n t h e t w o sets a r e r u n t o g e t h e r , w e find an e s s e n t i a l d i f f e r e n c e b e t w e e n t h e m . T h e r e f e r e n c e d a t a set in t h e a l i g n m e n t was t h e - 4 0 0 0 ,~ a v e r a g e . In t h e first r o u n d , t h e p r o g r a m a c c e p t e d 131 i m a g e s w h i c h i n c l u d e d a l m o s t all t h o s e f r o m t h e - 4 0 0 0 A set b u t left o u t m a n y f r o m t h e - 7 0 0 0 ]~ set. In s u b s e q u e n t r o u n d s , t h e - 7 0 0 0 A set t o o k t h e u p p e r h a n d so t h a t m o r e o f t h e s e i m a g e s w e r e i n c l u d e d a n d m a n y o f t h e - 4000 ,~ set w e r e e x c l u d e d f r o m t h e a v e r a g e . T h e - 2500 A set, a l t h o u g h h a v i n g only 8 i m a g e s , fit v e r y w e l l w i t h t h e - 7000 ,~ d a t a set. T h e s e t w o sets w e r e b o t h t a k e n at B r a n d e i s w h e r e a s t h e -4000 ,~ set was t a k e n in C a m b r i d g e . T h i s a g a i n c o n f i r m s t h a t it is n o t t h e d i f f e r e n c e in d e f o c u s but, r a t h e r , s o m e d i f f e r e n c e in t h e p r e p a r a t i o n . W e h a v e n o t yet d i s c o v e r e d t h e c a u s e . D e s p i t e t h e s e d i f f e r e n c e s w e r e c o v e r e d h i g h - r e s o l u t i o n d a t a t h a t is b e t t e r t h a n t h e t h r e e sets t a k e n i n d i v i d u a l l y . Fig. 7 s h o w s t h e r e s u l t a n t a v e r a g e set o f data. W e h a v e w e i g h t e d - u p t h e h i g h - r e s o l u t i o n d a t a

Fig. 5. Grey-scale image of the sum of the squares of all the CTFs. The values range from a high of 140 (white regions) to a low of about 15 (dark areas). The top of the figure corresponds to a radius of 1/7 A.


D. Morgan, D. DeRosier / Processing images of helical structures." a new twist

Fig. 6. Transforms of a single image. (a) Image from thhe -7000 A, underfocus set. The magnification is about 150000×. (b) Transform of image in (a). The highest-resolution layer is the near meridional (n = 1) at 1/26 A. The transform has been radially scaled to increase the amplitude of the higher-resolution data. (c) Transform as in (a) but with the Wiener filter applied.

which is o t h e r w i s e quite weak. T h e t r a n s f o r m shows a striking gain in signal-to-noise. D a t a out to 10 are clearly visible, a n d we even t h i n k t h e r e are d a t a o n e zone f u r t h e r out (at 8.7 ,~ resolution). It is also clear from the two rings of noise in fig. 7 that o u r W i e n e r filter can be i m p r o v e d . This we leave for the future. T h e most i m p o r t a n t result, however, is that we b e g a n with sets of images whose t r a n s f o r m s showed strong layer lines to only 25 A resolution, at best, and that, by careful a l i g n m e n t and averaging, we e x t r a c t e d d a t a to at least 10 ,~ resolution. T h e s e investigations are still in t h e i r early stages, and we expect f u r t h e r i m p r o v e m e n t s in resolution.

19. C o n c l u s i o n W e d e s c r i b e a set of a l g o r i t h m s that can b e used to extract the signal from low-dose e l e c t r o n c r y o m i c r o g r a p h s of helical particles. T h e a d v a n t a g e o f these a l g o r i t h m s is that they can be used with a m i n i m u m of h u m a n i n t e r v e n t i o n a n d thus can s p e e d up the analysis so that h u n d r e d s of i m a g e s can be

D. Morgan, D. DeRosier / Processingimages of helical structures: a new twist


Fig. 7. Transform of the final averaged data set. The position of the set of layer lines at about 10 ,~ resolution is marked. The transforms have been radially scaled to increase the visibility of the weak, high-resolution terms. Compare this figure to fig. 6c to see the increase in signal-to-noise upon averaging. analyzed in a short space of time. Large numbers of images are needed to extract the weak signal from the noise. Beginning with a set of images that show strong layer-line diffraction to only 25 ,~ resolution, we extracted reliable data to 10 A resolution.

Acknowledgements We would like to show our gratitude to Dr. Cam Owen whose wonderful subroutines to handle images made our job easier. We also thank him for his considerable help in getting the programs running. We also owe a debt of gratitude to Linda Melanson, who took many of the images used in this work. The appendix was prepared with the help of Ms. Elaine Ames and the photographic work was done by Marie Craig. We thank them. We also acknowledge the hospitality of Dr. Richard Henderson when one of us (D.J.D.) was on sabbatical leave. We acknowledge the support of the John Simon Guggenheim Foundation for a fellowship (to D.J.D.) supporting the work in Dr. Henderson's lab. This work was also supported by grants GM26357 and GM35433 from the N I H to D.J.D.

Appendix A. 1. Helical cross-correlation Assume we have a reference helix and an unaligned helix. The reference helix is pr(r, &, z) and the unaligned helix is pu(r, oh, z). Recall that for a helix, p(r, ~b, z), F ( R , ~b, Z ) = ~ Gn( R , Z,,) e i'~'+~/2),




G.( R, Zn) = f p(r, See Klug et al. [6].


gr) e

e2=izZ,,rdr d4, dz.



D. Morgan, D. DeRosier / Processing images of helical structures: a new tw&t

Construct a correlation coefficient

c(,o, ~) = f po(r, 49, Z)pr(r, 49-o), z-l~)r

dr d49 dz.


Both Pu and Or can be written in terms of their Fourier-Bessel coefficients:

pu(r, 49, z ) = ~_,[[Gj,,(R, Zi)Ji(2"n'Rr)R dR e ij~b e-2CrizZ,], jr--.



pr(r, 49-w,z-,C,)= ~._,[fGk,u(R',Zk)Jk(2rrR'r)R' dR'eik(4"-~°)e-2~i("-~'zk].



Substitute (A.4) and (A.5) into (A.3):

c¢o),~j)=f{~[fGj,u(R, Zj)Jj¢2~'Rr)RdR]



×{~[fGk,dR',Z~)J~(2crR'r)R'dR'] e'k¢~'-"'e 2~i¢:-~'Zk}rdrd49dz.


Rewriting the terms:

c(w, ~) = 2 Y'~ •


Jj(2wRr)Jk(2~R'r)r dr Gj,,(R, Zj)Gk,~(R', Zi)RR' dR dR'


) (A.7a)




where z o is the length of an axial repeat. Evaluate the terms of (1.7) one at a time. The middle integral term (A.7b) is zero unless k = - j whereupon it is 27re ij~°. Since k = - j , the bottom integral, (1.7c) is f~"e-2~i"(Zi+Z i). But ( Z j = - Z _ v) and thus the bottom term, (A.7c), equals z o e 2rriZ~ Since k = - j , the innermost integral in eq. (A.7a) becomes




The integral in (A.8) equals [20]

6(2rrR- ZrrR')(- l) i. (A.7a) then is

~-" Y" fGi'u(R' Zi)( fGL-- J r ( R " j k

Z ,)(-1)'6(2~rR-27rR')R'

dR'JR dR,


which becomes 2~'(-

1)Jf G,,u(R, Zj)G_j,r( R, Z_i)R



D. Morgan, D. DeRosier / Processing images of helical structures: a new twist


but G j,r(R, Z_j)=



z)J_j(2rrrR)R dR

e -i(-j)'b e 2~i'-z,,z r d r d~b d z ,

= ( - 1)JGj*r(R, Zj),


w h e r e G * denotes the complex conjugate of G. T h e n (A.10) b e c o m e s 2~fCj,o(R, and, letting

ZS)Cj*r(R, Z s ) R d R ;

Gs,,(R, Zj)


be d e n o t e d by Gj, u,

c ( w , ~) = ~ (2'n')2Z0 e ij

Processing images of helical structures: a new twist.

Helical macromolecular assemblies are particularly difficult to study by X-ray diffraction but are quite well suited to analysis by electron microscop...
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