3D reconstruction from the Fourier transform of a single superlattice image of an oblique section

Ultramicroscopy 41 (1992) 153-167 North-Holland

3D reconstruction from the Fourier transform of a single superlattice image of an oblique section K.A. Taylor Cell Biology Department, Box 3011, Duke Unit~ersity Medical Center, Durham, NC 27710, USA

and

R.A. Crowther MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, UK

Received 16 October 1991

An image of a thin oblique section through a 3D crystal exhibits superlattice periods much larger than the unit cell dimensions of the crystal. Within a superlattice period the contents of the unit cell of the 3D crystal are sampled at different levels, so that a 2D image of the section contains 3D information about the crystal. The 2D Fourier transform of an electron micrograph of such an oblique section thus exhibits superlattice spots, which provide an estimate of the 3D transform of the original crystal. The strengths of the observed spots are reduced from their true values by convolution with a weighting function that depends on section thickness. A method is described that uses phase relationships among symmetry-related structure factors to determine the section thickness and hence the weighting function. Wiener filter deconvolution of the section thickness is performed, in which the filter level is set by the ratio of diffraction spot intensity to background intensity. From the deconvoluted set of structure factors a 3D map of the unit cell can be computed by a standard crystallographic Fourier program. The approach is illustrated with images of oblique sections through rigor insect flight muscle.

I. Introduction F r o m a single image of a n o b l i q u e section t h r o u g h a two- or t h r e e - d i m e n s i o n a l crystalline s p e c i m e n it is possible to p r o d u c e a t h r e e - d i m e n sional m a p of the s p e c i m e n [1,2]. Provided the section is thin c o m p a r e d with the length of the unit cell axis r u n n i n g approximately n o r m a l to the section, successive areas in the image systematically sample the t h r e e - d i m e n s i o n a l c o n t e n t s of the unit cell of the crystal. O b l i q u e section reconstruction ( O S R ) exploits this systematic s a m p l i n g to r e - b u i l d a t h r e e - d i m e n s i o n a l m a p of the d e n sity in the unit cell. T h e m a n i p u l a t i o n s n e e d e d to m a k e the m a p c a n be carried out with the image densities directly or with their F o u r i e r transform. D e p e n d i n g o n the n a t u r e of the s p e c i m e n a n d o n the g e o m e t r y of sectioning, o n e of these ap-

p r o a c h e s may be m o r e a p p r o p r i a t e t h a n the other. W e have previously described in detail a protocol b a s e d on image densities [3], which we t e r m e d crystallographic serial s e c t i o n r e c o n s t r u c t i o n (CSSR) by analogy with c o n v e n t i o n a l serial section r e c o n s t r u c t i o n . I n this p a p e r we give details of a protocol based on F o u r i e r transforms, which exploits the superlattice periodicities p r e s e n t in an image of an oblique section t h r o u g h a threed i m e n s i o n a l crystal a n d which we therefore call superlattice r e c o n s t r u c t i o n (SLR). W e illustrate the m e t h o d with images of insect flight muscle ( I F M ) in rigor. T h e image of an oblique section t h r o u g h a t h r e e - d i m e n s i o n a l crystal exhibits a period m u c h larger t h a n the u n i t cell of the crystal, b e c a u s e the density cut out of a p a r t i c u l a r u n i t cell by the section will be e q u i v a l e n t to that cut out of other

0304-3991/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

154

K.A. Taylor, R..4. Crowther / O b l i q u e section 3D reconstruction

unit cells further along the section. The period of the observed superlattice depends on the slope of the section, with shallower slopes giving larger superlattice spacings, as the section then has to cut through more unit cells before reaching an equivalent density. In the two-dimensional Fourier transform of the image the superlattice manifests itself as sets of spots with spacings much finer than those that arise from the true crystal lattice. Consider the case of IFM, which consists of a crystalline hexagonal array of parallel thick and thin filaments. A section cut exactly tranverse to the filaments would give an image displaying a hexagonal unit cell of side equal to the thick filament spacing. The Fourier transform of the image would show a hexagonally related set of spots, equivalent to the so-called equatorial or (h, k, 0) terms of the three-dimensional transform. In an image of a slightly oblique section, however, although the hexagonal pattern of thick filament profiles is still seen, a superlattice appears because the contents of neighbouring unit cells are cut at slightly different heights. In the two-dimensional transform of such an image each prominent (h, k, 0) spot is therefore accompanied by a set of satellite spots ("superlattice spots"). These superlattice spots represent spots from the three-dimensional transform of the crystal mapped by convolution into the two-dimensional transform of the image. Thus a three-dimensional set of structure factors can be recovered from a single image. The thickness of the section reduces the weights with which the higher order spots are m a p p e d into the two-dimensional transform. If the section thickness is known, the weighting down of the structure factors can be corrected and a three-dimensional map with improved resolution computed. SLR has the advantage over CSSR that it can be applied to a far wider range of section geometries.

2. Methods

The electron micrographs of sections of IFM in rigor were obtained from muscle tissue pre-

pared as described [4] on a Philips EM300 or EM301 electron microscope. All of the micrographs processed were obtained from 120-250 sections cut approximately transverse to the illament axis. A superlattice image therefore represents sampling of object density in successive unit cells along the filament or c axis. The transform of the image can then be interpreted as an (h, k, 0) zone with additional superlattice spots representing structure factors from upper level planes along c* (i.e. h, k, l zones). The (h, k, 0) zone of the IFM transform is commonly referred to as the equator and the c* axis as the meridian. Micrographs were digitized on a computerlinked film scanner [5] or on a Perkin-Elmer PDS 1010 M microdensitometer at a step size corresponding to 20 ,~ on the specimen. Usually the entire myofibril was digitized to facilitate windowing of the largest, most coherent area from which to calculate the Fourier transform of the image after all the preprocessing steps had been completed. Each image was processed as follows. The initial step was to correct for distortions of the lattice using correlation and unbending [6]. A cross-correlation map was obtained between the image and a circular patch windowed from its center. The patch had an area equivalent to four unit cells and was centered on a thick filament. In calculating the correlation map, a circular mask with a radius equal to the reciprocal diameter of the patch was applied to each of the equatorial spots in the image transform. A lattice was fit to the peaks in the resulting filtered correlation map and one cycle of unbending [6] carried out. A second correlation map was then calculated from the unbent image, but in this case the amplitudes of the [1,0], [2,0] and hexagonally related spots were weighted down by a factor of 0.1 in the grid-filtering step. This procedure sharpened the correlation peaks considerably and improved the visibility of the superlattice. Correlation peaks were also fit in this second unbent image and a second cycle of unbending was done if warranted from the size of the residual positional errors. More often than not an additional unbending cycle was not warranted and the second correlation map was instead utilized to define the largest coherent area sampled by the section.

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

A coherent area that included as close to an integral number of superlattice periods as possible was then widowed from the unbent image. The windowed image was preprocessed to remove outlying densities and density gradients [3], Fourier-transformed, and a reciprocal lattice fitted to the superlattice spots. After determining the thickness of the section, the structure factors were appropriately corrected and used to compute a three-dimensional map. Further details are given in section 3.5 (Results). The performance of the program for determining section thickness (see section 3.4) was tested on model data generated by taking random structure factors within a reciprocal asymmetric unit and then generating a full 3D data set with symmetry operators using the E X P A N D program in the CCP crystallographic computing programs. Model data sets corresponding to those from oblique sections were then generated (see appendix). Section gradients ranged from 4 ° to 16°, the gradient axis was constrained to 15 ° relative to a * and the thickness was set to 150 or 300 ,~. R a n d o m noise was added to each complex structure factor after sinc function convolution. To simulate as closely as possible the actual structure under investigation, only those layer planes that have significant intensity in native I F M were used in the calculation and the data were low-passfiltered using an ellipsoidally shaped filter with semiaxes of length 0.02, 0.02 and 0.0091 A-1. All of the software used in these studies was developed on a V A X 11/750 computer (Digital Equipment Corporation, Maynard, MA) running under the VMS operating system.

3. Results

3.1. Theoretical considerations The relationship between an image of an oblique section and its parent 3D crystal can be conveniently formulated in Fourier space, as a convolution of the transform of a rectangular windowing function with that of the crystal. The discussion that follows uses this formulation to develop methods for determination and deconvo-

155

lution of section thickness. The relationships for specifying the orientation of the oblique section with respect to the lattice of the crystal and for orthogonalization of the reciprocal lattice are described in the accompanying appendix. In the following description, script will be used to denote coordinate frames of the transforms of oblique section and parent crystal, sans serif to denote transformation matrices and bold italics to denote vectors with the symbol to denote unit length. Coordinate frames are denoted in lower case for real space and upper case for Fourier space. Let tr(x, y, z) represent the continuous object density of the oblique section in an orthogonal coordinate space, x. The periodic object density of the parent crystal, p(u, v, w), is conveniently described in another orthogonal coordinate space, t~ or by the geometrically related function p'(x, y, z) in the space of the oblique section. The transforms of or(x, y, z), p(u, v, w) and p'(x, y, z) are d e n o t e d as S(X, Y, Z), R(U, V, W) and R'(X, Y, Z), respectively. In general both the reciprocal and real space unit cells will have non-orthogonal cell axes. Thus, an orthogonalization step will be required to convert an index (h, k, l) in the reciprocal space of the crystal to (U, V, W) before calculating distances from the plane of the oblique section transform. There are a number of possible orthogonalization schemes but the equations are most easily derived if we adopt the convention that c* is coincident with 1~ and a* lies within the U-W plane (see appendix 1). The relationship between an oblique section and its parent 3D crystal is shown in fig. 1. Mathematically, ~r(x, y, z) can be described as the product of the periodic object density, p'(x, y, z), with a rectangular windowing function, w(z), assumed to be of infinite extent in x and y and have width t along z equal to the section thickness (fig. la). That is to say ^

o~(x, y, z) = w ( z ) p ' ( x , y, z), where w(z) has the form w(z) =rect(z/t) =

{10 i f l z l < - t / 2 , if I z l >t/2.

(1)

K.A. Taylor, R.,4. Crowther / Oblique section 3D reconstruction

156

A

cell 1 I

III

cell 2

.......

oblique section

cell i

i iIIIlillllllllllllllllll]llllllllllh

where

\

t

cellj

cell 1

]i[1111111111111111111111111]11;111111!1111111111 Pa J

B h= 3 1=2 •

20

-2

-1

0

2

3

•

•

•

•

1

•

•

•

•

•

•

•--

a" - - O I

Fig. 1. (A) Diagram illustrating the relationship between an

oblique section and a 3D periodic lattice. The oblique section cuts through the 3D lattice at an angle 0 so that the section gradient samples the contents of successive unit cells at different levels along the c axis. The physical process of sectioning is represented mathematically as multiplication of the crystal with the windowing function. The micrograph displays the sampled cell contents in the form of a projected image, %. (B) The relationship between the Fourier transform of the oblique section image and the 3D transform of the parent crystal. The oblique section image is a projection and therefore its transform is a central section in the Fourier transform of the parent crystal. Convolution of the 3D transform of the crystal with the transform of the windowing function extends each spot in a direction perpendicular to the plane of the oblique section transform. The intersection of these extended spots with the central section gives rise to the set of superlattice spots observed in the 2D transform of the oblique section.

T h e image, ~p(x, z of o-(x, y, z )

•(x, y ) =

y),

*

denotes

W(Z) = t s i n c ( Z t ) (TrZt).

the c o n v o l u t i o n o p e r a t i o n , a n d s i n c ( Z t ) = sin(~'Zt)/

T h e convolution o f R'(X, Y, Z) with W(Z) extends F o u r i e r t e r m s o f R ' in the Z direction, i.e. p e r p e n d i c u l a r to the p l a n e of the o b l i q u e section, a n d m o d u l a t e s t h e i r a m p l i t u d e by the factor t s i n c ( Z t ) . B e c a u s e the i m a g e of the oblique section r e p r e s e n t s a projection, its 2D transform, S(X, Y, 0), is a c e n t r a l section t h r o u g h S(X, Y, Z) that intersects the e x t e n d e d spots from the various layer p l a n e s (fig. lb). T h u s the finely s p a c e d s u p e r l a t t i c e spots seen in the transform o f the oblique section i m a g e r e p r e s e n t F o u r i e r t e r m s from different layer p l a n e s of R ' m a p p e d by W(Z) o n t o the 2D t r a n s f o r m . T h e m o d u l a t i o n of these F o u r i e r terms by W(Z) will d e p e n d on t h e i r d i s t a n c e from the p l a n e of the central section a n d on the section thickness. D e p e n d i n g u p o n the section g e o m e t r y , the convolution will m a p s y m m e t r y - r e l a t e d 3D s t r u c t u r e factors into the 2D t r a n s f o r m with different weights and different signs, t h e r e b y f o r m i n g the basis of a m e t h o d for d e t e r m i n i n g the section thickness.

3.2. Deconvolution of the section thickness If the section g e o m e t r y and unit cell are such that the h i g h e r - l a y e r spots do not o v e r l a p w h e n m a p p e d into the 2D t r a n s f o r m , as in the e x a m p l e of I F M c o n s i d e r e d here, the convolution (2) m e r e l y weights down the o b s e r v e d values of individual structure factors b u t d o e s not mix d i f f e r e n t structure factors. In this situation the true values of the structure factors R'(X, Y, Z) can, in principle, be r e c o v e r e d by a simple division of the form:

r e c o r d s the p r o j e c t i o n down S ( X , Y, O)

R ' ( X , Y, Z ) -

y, z ) d z .

t

sinc(Zt)

(3)

F o u r i e r t r a n s f o r m a t i o n o f eq. (1) converts the m u l t i p l i c a t i o n o p e r a t i o n into a convolution

S ( X , Y, Z) = W ( Z ) * R ' ( X , Y, Z),

W(Z )

1 S ( X , Y, O)

(2)

However, W(Z) is an oscillating function with zeros at Z = j / t , w h e r e j is any integer, so division by W(Z) in the n e i g h b o r h o o d of the zeros is not practical. T o o v e r c o m e this a n d to m i n i m i z e

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

possible noise amplification, we have utilized a Wiener filter [7] of the form: sinc(Zt)

Q ( Z ) = {sincZ(Zt) + y } ,

(4)

where the filter level, y, is a function of the signal-to-noise ratio in the data. Because correction is carried out on individual structure factors, signal-to-noise ratio should be treated at the level of the individual diffraction spots. Thus the filter level, 7h~t, is assigned for each structure factor according to the relationship

lated structure factors after deconvolution over a range of plausible section thickness. Each spot in a symmetry-related set in the oblique section transform represents an independent measurement of a particular complex structure factor in the asymmetric unit of the crystal. For all biologically relevant space groups except triclinic P1, there will be at least two such measurements. After deconvolution an estimate of the complex structure factor, Fh~~, within the asymmetric unit is obtained from the weighted average: 1

ln( hkl ) Yhkl-- is( hkl ) ,

(5)

where In(hkl) is the background intensity and Is(hkl) is the intensity of the diffraction spot. The filter, Qhkt used here to restore Fhk t is thus: sinc(Zt)

Qhk, = {sincZ(Zt) + Yhk,}

(6)

and Fh~l = Sh~tQhkl.

The factor 1/t in eq. (3) acts as a scale factor to relate oblique sections of different thickness. However, since the individual micrographs must be scaled to one another anyway, 1/t can be combined with the overall scale factor.

3.3. Determining the thickness from the superlattice diffraction spots In principle, the section thickness can be obtained from the extracted 3D set of structure factors by using the symmetry of the crystal (see fig. lb). In the mapping of the 3D transform of the crystal onto the 2D transform of the oblique section image the individual weights applied to a set of symmetry-related structure factors by W ( Z ) will differ in magnitude and possibly in sign, because the spots lie at different distances from the central section. For example, in space groups with an even-ordered rotation axis parallel to c, ] F(hkl)l = ] F(hki)[ and a(hkl) = - a ( h k i ) . The section thickness can thus be determined by calculating the agreement between symmetry-re-

157

Fhkl = _ ~ F~,k,rQh,k, J J r

ei~,k,t,,

nj=l

where F~,k, r represents a set of symmetry-related structure factors from the 2D transform, Q~'k'r is the filter function (from eq. (6)) and 6~,k, r is an additive phase term for relating the individual structure factors to the asymmetric unit. 6],,k,r depends upon the particular space group. The summation is taken over the n symmetry-related structure factors whose observed intensity in the 2D transform is above background. We have tried several measures of fit including unweighted phase residuals, amplitude R factors and the RMS vector difference. On actual IFM data, only the unweighted phase residuals proved useful for determining section thickness. The unweighted phase residual was calculated for each symmetry-related set of structure factors with respect to the average for that set according to the equation: res

=

__

--

OLhkl -- 6Jh'k'l' [,

(7)

nj=l

where a~,~,t, is the phase after deconvolution of an individual structure factor, ~hkl ~asym is the phase of the corresponding average structure factor in the asymmetric unit. The summation over all Fourier terms in the asymmetric unit gives an overall phase residual.

3.4. Model calculations for determining section thickness Model calculations were made to investigate the determination of section thickness from sym-

K.A. Taylor, R.A. Crowther /Oblique section 3D reconstruction

158

metry-related Fourier terms for different section geometries and with noise-corrupted data (fig. 2). The range of section gradients investigated corresponded closely to those encountered in our IFM work. The results of the phase residual plots show that a clear minimum is produced at the correct value of section thickness. They also reveal that the depth of this minimum depends upon two factors: (1) the gradient of the oblique section and (2) the amount of data from innerlayer planes (i.e. for small l) that are unmodified in phase by W(Z). For a 150 A section with a section gradient as low as 4 °, there is ambiguity in the section thickness since minima are observed at 150 A but also at 250 ,& (fig. 2A) and 300 A (fig. 2C). This ambiguity occurs because the node of the sinc function can pass entirely between layer planes for several values of section thickness without

AoGO

altering the phases of any symmetry-related structure factors. If the actual section thickness corresponds to a spacing very different from the spacing of upper-layer planes, then W(Z) may have no effect on structure factors within a symmetryrelated group other than to change the sign of all members of the group together. Thus there can be several values of section thickness that have the same effect on the phases, only one of which gives the correct change. At higher section gradients this ambiguity does not occur because the node of W(Z) will cross several layer planes simultaneously so that only one value of section thickness will correct all of the altered phases. The second effect observed in these model calculations occurs when the section thickness is close to the resolution limit of the data. In IFM, the X-ray diagram contains little sampled diffraction for axial spacings beyond 129 ,~ [8]. For low

B

8

O ~

816°

8' 40

2° OO,.I

o o,,I o

o

260 t(A)

O

360

13 o

150

Co_

t(A)

45o 'sso

D o.

~ !

o

.

e

o

16o

260 t(A)

aoo

460

150

250

16o 4°

!

t(A)

Fig. 2. Model calculations for section thickness determination. The model data correspond to section thickness 150 A (A and C) or 300 ,& (B and D) and to section gradients of 4°, 8 °, 12° and 16°. The data were low pass filtered in 3D using an elliptical filter with semiaxes of 0.02 ,~-l, 0.02 , ~ - i in the a * - b * plane and, 0.0091 A i along c* and noise was added to the remaining structure factors. The phase residual calculation used only those structure factors with a signal to noise ratio of > 4. • is the phase residual calculated from eq. (7) and t is the section thickness used in deconvolution of the model data. Curves (A) and (B) were obtained using layer planes l = 0, 2, 4, 6. Curves (C) and (D) were obtained using just the l = 4 and l = 6 layer planes.

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

section gradients and section thickness around 150 A, only a few Fourier terms are therefore affected by the sinc function and thus the depth

159

of the minimum is not very great (fig. 2A). The sensitivity can be improved by eliminating those layer planes with structure factors whose mea-

Fig. 3. Electron micrograph of an oblique section through rigor IFM. Six superlattice periods are observed in this micrograph. The area processed has been outlined. The transform of this image is shown in fig. 5,

160

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

sured phases are unmodified by W(Z) (fig. 2C) or by processing images with a higher section gradient. For the thicker section, where the data were distributed more equally about the first node of W(Z), the minimum was deeper for all four gradients modelled. It is worth pointing out that while eliminating the inner-layer planes increases the sensitivity, those same inner-layer planes are important for excluding higher values of section thickness. The simulation therefore showed that initial attempts should use all the data and that only after some indication of the probable range of section thickness has been obtained should inner-layer planes be excluded to increase the

Select Micrographs and Digitize

Correct Lattice Disorder

sensitivity of the calculation. The more data substantially unaffected by W(Z) that are included in the phase residual calculation, the less sensitive the phase residual is to the section thickness, because the lower resolution data effectively produce a constant background.

3.5. Application to insect flight muscle Thin, ~ 150 ,~ transverse sections of rigor IFM reveal a hexagonal lattice with a unit cell consisting of one thick filament surrounded by six thin filaments placed midway between pairs of thick filaments. At successive 129 A levels along oblique section should step uniformly through the 3-D or 2-D lattice

use a spline fitting method, i.e. unbending

Window Coherent Superlattice and Transform Fit Reciprocal Lattice

the reciprocal net must encompass vectors for a*, b* as well as c*

Extract Amplitudes and Phases

Assign Signal/Noise Ratio

Determine Section Thickness

Deconvolute and Average Symmetry Related Structure Factors to the Asymmetric Unit

Reconstructed 3-D Image

from ratio of spot intensity to background intensity

by fitting phase relationships between spots depending upon the space group

use a Wiener filter for the deconvolution

crystallographic Fourier synthesis

Fig. 4. Flow chart for 3D reconstruction from an oblique section through a 3D crystal.

K.A. Taylor, R.A. Crowther /Oblique section 3D reconstruction

the filament axis, crossbridges connect the thick filament to four of these thin filaments producing a structure that has been described as the flared-X [4]. Oblique transverse sections through rigor I F M (fig. 3) show the flared-Xs rotating clockwise for positive section gradients due to the lefthanded helix of actin target zones [9] which define the locations where geometry and steric constraints permit crossbridge formation. Two descriptions of the rigor filament lattice of I F M as a 3D crystal have been proposed, one in space group P64 [8], the other in space group P65 [10]. The two models are related, for in the P65 case the symmetry at the thin filament positions is 2~ while for P64 the symmetry is two-fold. The native unit cell is hexagonal with dimensions o o a=b-560 A, c---770 A, a - - / 3 = 9 0 °, y = 1 2 0 ° given by X-ray diffraction. The fixed and embedo ded muscle has dimensions a = b = 520 A, c = o 770 A. The experimentally measured transform

161

of native IFM [8] shows that the innermost l = 0, 2 layer planes have most of the reflections while the I = 4, 6 planes have fewer and weaker reflections. For the purpose of this demonstration the P6 5 lattice can be used without loss of generality, if it is realized that restricting the data in P6 5 to layer planes for which l = 2n results in a nearly equivalent description to that for P6 4. The protocol for SLR can be divided into 9 steps (fig. 4). Micrographs with the largest possible number of superlattice repeats should be selected. The micrographs can be preprocessed to remove section gradients and outliers, but this is not as critical in this method as it is in the real-space reconstruction protocol [3]. Correction of lattice disorder is r e c o m m e n d e d as the image will cover a large area if it contains a large number of superlattice repeats. In such cases, instrumental image distortions will reduce the coherence between unit cells that are widely sep-

Fig. 5. Transform of the oblique section image (fig. 3) before and after correction for lattice disorder. (A) Original image transform shows clear evidence of a superlattice by the presence of satellite spots which run along the vertical axis in the figure. However, the spots are not sharp and fade rapidly at high resolution. (B) After correction for lattice disorder, the spots are clearly resolved and sharp diffraction spots are observed to higher resolution.

162

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

arated. In addition, the lattice of IFM shows local filament disorder that it has proven advantageous to correct before extracting the 3D structure factors. We used the unbending procedure [6] to correct the image for filament disorder and instrumental distortions. Comparison of the transform of the original image (fig. 5a) with the transform of the distortion corrected image (fig. 5b) shows that considerable sharpening of the diffraction spots is produced by unbending. The procedures used to obtain structure factors from the 2D transform are conventional except that three rather than the normal two vectors are used to describe the reciprocal lattice, the third vector indexing the separation of superlattice spots in each set. One of the interesting consequences of the mapping of upper-layer structure factors onto the transform of the oblique section image is their shift to relatively low resolution. For instance, the 129 A meridional term in IFM occurs close to the origin of the 2D transform. Unbending proved to have some limitations. In all the images analyzed in this study, unbending produced dramatic improvements in the sharpness of all spots, particularly those from the equatorial plane. However, unbending was less effective at improving the sharpness of the superlattice spots, which sometimes had unacceptable phase gradients across the peaks indicative of a lack of coherence across the image. The sharpness of the superlattice spots represents the regularity with which the section gradient samples the 3D density, which depends on the sectioning process rather than instrumental image distortions. We thus found that once the equatorial spots were sharpened, further unbending did not improve the superlattice spots. The quality of the unbending was therefore judged solely on the basis of the equatorial data and the most regular superlattice was obtained by windowing different areas of the micrograph. We did not investigate the minimum number of superlattice repeats needed to obtain a reconstruction. In no case did we process images with less than four repeats of the 3 8 7 , ~ spacing. The determination of the section thickness involved first using the equatorial spots to place the phase origin onto the 65 axis which is coincident

with the thick filament axis. This could be done without prior deconvolution because the section gradient and thickness were sufficiently small for the phases of the equatorial structure factors to be unmodified by W(Z), although their amplitudes were slightly changed. Thus only small local refinements for the phase origin were needed when testing values for the section thickness. In nearly all cases, the final phase origin was identical to that found using just the equatorial data. i t, for space The phase correction term, 6hk group P65 is given by the relationship

- ~ - ( j - 1)t ~l~kl = -

6

where j is the hexant of the spot numbered counterclockwise starting with the asymmetric unit and l is the Miller index. The asymmetric unit is defined as h >_ 0, k > 0, l > 0. Tested values of section thickness ranged from 50 to 400 A and @ was calculated according to eq. (7). The calculation was done using various combinations of layer planes such as l = 0, 2, 4, 6; l = 2, 4, 6; or just l = 4, 6. This was done to test the utility of the l = 0, 2 data which lie well within the first node of W(Z) for estimates of section thickness from 200-100 A and section gradients less than 9 °. The plots obtained (fig. 6) indicate that sensitivity is clearly increased by omitting the inner layer plane data but at the cost of a significant increase in phase residual. In addition, we repeated the calculation for different thresholds of signal-to-noise ratio. The calculation of section thickness using the symmetry-related Fourier terms depends upon having a measurement of amplitude and phase for at least two spots in each symmetry-related set. Using just the data with high signal-to-noise ratio sometimes does not produce a clear minimum, because there are too few structure factors to compare among upper-layer plane data. When a high signal-tonoise threshold is used, the phase residual calculation usually suggests that the actual thickness lies within a certain range. Lowering the signalto-noise threshold results in more comparisons and usually a clearer minimum. Though this procedure sometimes means that the minimum is

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

determined by the weaker data, repeating the calculation for different thresholds insures that the minimum is compatible with the stronger data. From the residual curves plotted in fig. 6, the thicknesses of the two obliqueo sections were determined to be 134 and 194 A, respectively. The width of the minima in the plots of fig. 6 suggests that section thickness can be rather accurately determined by this method. In addition, it is clear that the accuracy is better for the thicker section because the reciprocal section thickness was closer to the spacing of the l = 4 layer plane. Thus more of the data were affected by the sinc function weighting. For the thinner section, the reciprocal section thickness was close to the spacing of the l = 6 layer plane which is only lightly populated with reflections that are generally rather weak. The value of the section thickness obtained with the thinner section is thus based

A

upon data with relatively low signal-to-noise ratio. Despite the unfavorable distribution of sampled diffraction in IFM, this method of determining section thickness is reproducible (fig. 7). We were able to obtain three data sets from different areas of one section, that included two myofibrils. The section gradients were 7.5 °, 7.1 ° and 4.4 ° and the calculated values of section thicknesso for these three data sets were 134, 120 and 128 A, respectively. The curves in fig. 7c illustrate the difficulty, already seen in model calculations, of finding the thickness of sections with small gradient. Nevertheless, this small sample suggests a precision of better than 10% for this method of determining section thickness. We analyzed eight oblique section micrographs in this study. The general agreement among data from these different micrographs can best be illustrated by plotting the amplitude and phase of

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K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

164

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60-

data for each structure factor are not spread equally over Zt. All the l = 0 and l = 2 data fall within the first node of W(Z) at Zt = 1 and only the generally weaker l = 4 and l = 6 data are distributed across the first node of W(Z).

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and (B) are from the same myofibril while (C) is from a second myofibril. These curves were calculated using only the l = 4 and l = 6 layer planes. (A) Section gradient 7.5°, (B) section gradient 7.1° and (C) section gradient 4.4°.

selected spots determined from different images against the product Zt, the argument of the sinc function (fig. 8). For this comparison we used data that had not been corrected for section thickness so that the variations in amplitude and phase should reveal the effects of W(Z). The plots indicate that phases are more accurately recovered than amplitudes in the structure factors from oblique sections. Because the range of section thicknesses and gradients was limited, the

In this paper we have described superlattice reconstruction (SLR) for computing a 3D image from the superlattice pattern in the Fourier transform of a single micrograph of an oblique section. The section thickness can also be determined from the single image using phase relations among symmetry-related structure factors. Once the thickness and geometry of the section are known, the effects of the section thickness on the observed structure factors can be deconvoluted. The set of corrected structure factors can then be used to compute the 3D image. The method is generally applicable to thin oblique sections through 3D crystals. Previously we described a real-space method of reconstruction from oblique sections, which we called crystallographic serial section reconstruction (CSSR) [3]. It is worth at this point comparing the two methods. CSSR is particularly limited in the type of section geometry that can be conveniently reconstructed and is most easily attempted when one of the cell axes is parallel with the gradient axis. In SLR the restriction on section geometry is to avoid overlap of superlattice spots, which is most likely to occur when the X axis is parallel to a*. Thus the SLR is best obtained from sections with general orientations substantially different from the special orientations used in CSSR. We have shown that section thickness can be reliably determined in both CSSR and SLR. In CSSR the section thickness is determined by comparison with an image of a thick section cut in a direction perpendicular to the orientation of the thin oblique section. The precision of section thickness determination in CSSR depends upon the resolution of the data common to both images. In SLR, thick section data could also be used for section thickness determination but we

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

section. If the gradient is too steep, the overlap between samples will be poor and deconvolution cannot be attempted. The gradient also directly determines the sampling and therefore the Nyquist limit on the resolution. In SLR the section gradient also affects resolution but in a slightly different manner. Inspection of fig. lb indicates that the higher the section gradient, the more readily can signal be recovered from higher-order layer planes. This is because with steeper gradients the plane of the central section

have shown here that for section gradients greater than 4 °, the oblique section image itself contains sufficient information to determine the thickness, provided the crystal has high symmetry. Thus CSSR can be made without reference to 3D space group symmetry, whereas SLR uses crystal symmetry to determine section thickness. The section gradient poses slightly different problems in CSSR and SLR. In CSSR the obtainable resolution in a direction normal to the section plane depends directly on the gradient of the

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Fig. 8. Data for four structure factors from different layer planes are plotted as a function of Zt, the argument to the sinc function, expressed as multiples of ~-. The indices in each amplitude panel indicate (h, k, l) for the particular spot. These data have not been deconvoluted but have been folded into the asymmetric unit for hexagonal space group P6 5.

166

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction

will lie closer to some of the upper-level structure factors. Thus, a steeper gradient in SLR can result in higher potential resolution along the c axis. However, a steep gradient causes some difficulty with the section thickness determination because for every upper-level spot brought closer to the central section using a steep gradient, a symmetry-related spot will be moved further away resulting in a much lower sinc function weighting and signal-to-noise ratio. Because the section thickness determination requires that at least two symmetry-related copies of any included structure factor be recovered with reasonable signal-tonoise ratio, a steep section gradient may complicate section thickness determination, especially when the crystal symmetry is low. It may be possible to overcome this problem by combining data from sections with shallow gradients to form a data base for cross correlation with sections having steeper gradients. Correction for section thickness in oblique sectioning is mathematically similar to defocus correction in electron microscopy. Knowledge of the defocus permits correction of the reversals in the phase contrast transfer function; knowledge of the section thickness allows for correction of contrast reversals in the sinc function weighting. Just as images of different defocus can be combined to span the zeros in the contrast transfer function, so oblique sections of different thickness can be used to fill in zeros in the sinc function. Although it is remarkable that a single image can be used to generate 3D information, the quality of the reconstruction can be improved by combining oblique sections of different thickness and different gradient.

Acknowledgements This research was supported by N I H grant G M 30598. The authors are grateful to Mary Reedy for permission to reproduce the micrograph in fig. 3. The authors are grateful to Drs. J.F. Deatherage and J. Frank for their comments on the manuscript.

Appendix A.1. Orthogonalization of reciprocal unit cell The orthogonalization matrix for the theoretical development above is based upon the following assumptions (fig. 9). In reciprocal space, let the c* axis be parallel with Z, and the a* axis lie in the X - Z plane; h, k, l are the Miller indices for the transform spots. The orthogonalization matrix thus defined is given as:

= [a21

a22

a23

[_a31

a32

a33

(A.I.1)

where (U, V, W) is the orthogonal coordinate. The coefficients aij are given as follows: all =a*

sin

fl*,

a21 = 0, a31 ~ a *

cos l~*, COS OZ* COS /~* -- COS 3/*

a12 = b *

sin /3 * W

-.%

7

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IJ

Fig. 9. Diagram of the coordinate frames of crystal and oblique section in Fourier space. The reciprocal unit cell axes a*, b* and c* are contained in an orthogonal coordinate space with axes UVW. c* is coincident with W and a* lies in the U - W plane. These vectors are related to the coordinate space of the oblique section transform, XYZ, by a gradient axis ~ and a gradient angle 0. The projections of a*, b* and c* onto the X - Y plane are at, bt, and ct.

K.A. Taylor, R.A. Crowther / Oblique section 3D reconstruction a22 = b*(1 - cos20l * - c0s2/~ * - c o s 2 ' y *

+ 2 cos a * cos /3" c o s ' ) / * ) l / 2 [ s i n / ~ , ] - 1 , a32=b

* c o s oz*,

a13 : 0, a23 = 0, a33 = c * .

A.2. The oblique section orientation Interpretation of the oblique section transform depends on a description of the orientation of the central section with respect to the transform of the unit cell. This coordinate transformation is most easily derived in Fourier space. In the discussion that follows, it is assumed that the image is scanned so that the gradient of the oblique section is parallel to the y axis in real space and that the gradient axis is parallel with the x axis. The normal to the oblique section is z. In the image transform, the corresponding coordinate frame • has axes X, Y and Z. Let ~ be the coordinate frame of the crystal transform, with axes U, V, and W. ~9 is related to g/ by the transformation S=

P~',

(A.2.1)

where the matrix P is given by

P=PoP~=

[, 0 01 0 0

×

cos 0 -sin0

sin 4' 0

sin 0 cos0

cos 4' 0

,

(A.2.2)

where Pu, is a transformation describing rotation about W by the angle 4' and Po is a transformation describing rotation about the transform X axis by the tilt angle 0. The signs of the angles 4' and 0 are positive for a counterclockwise rotation from X to U in the case of 4' and about X from c* to Z in the case of 0.

167

The Fourier transform of the oblique section image will contain an array, of spots whose positions are defined in terms of the reciprocal lattice vectors ate, bt~, and ct~,. Usually the gradient of the oblique section will be too small to cause appreciable differences in l at,~, I and I bt~ ] with respect to I a* I and I b* I thereby making it necessary to determine the gradient of the oblique section using ct~. That gradient is given by the equation sin 0 =

ct~/c*

(A.2.3)

Note that the sign of the tilt angle is ambiguous and must be derived from the known hand of the structure. The tilt axis angle, 4', is obtained from tan 4' =

at~ - a * c o s / 3 " sin 0 at~c cos 0

(A.2.4)

The distance from the coordinate (U, V, W) to the plane of the oblique section transform is Z obtained from eq. (A.2.1).

References [1] R.A. Crowther, Ultramicroscopy 13 (1984) 295. [2] R.A. Crowther, P.K. Luther and K.A. Taylor, Electron Microsc. Rev. 3 (1990) 29. [3] K.A. Taylor and R,A. Crowther, Ultramicroscopy 38 (1991) 85. [4] M.K. Reedy and M.C. Reedy, J. Mol. Biol. 185 (1985) 145. [5] U.W. Arndt, J. Barrington-Leigh, J.F.W. Mallett and K.E. Twinn, J. Phys. E Ser. 2, 2 (1969) 385. [6] R. Henderson, J.M. Baldwin, K.H. Downing, J. Lepault and F. Zemlin, Ultramicroscopy 19 (1986) 147. [7] N. Wiener, Extrapolation, Interpretation, and Smoothing of Stationary Time Series (Wiley, New York, 1949). [8] K.C. Holmes, R.T. Tregear and J. Barrington-Leigh, Proc. Roy. Soc. (London) B 207 (1980) 13. [9] M.K. Reedy, J. Mol. Biol. 31 (1968) 155. [10] J. Wray, P. Vibert and C. Cohen, J. Mol. Biol. 124 (1978) 501.