- Email: [email protected]
The role of multivariate image analysis in solving the architecture of the limulus polyphemus hemocyanin molecule
Ultramicroscopy 13 (1984) 153-164 North-Holland, Amsterdam
153
T H E R O L E O F MULTIVARIATE IMAGE ANALYSIS IN SOLVING T H E A R C H I T E C T U R E O F T H E L I M U L U S P O L Y P H E M U S HEMOCYANIN M O L E C U L E Joachim F R A N K Center for Laboratories and Research, New York State Department of Health, Albany, New York 12201, USA
Received 1 November 1983; presented at EMSA SymposiumAugust 1983
Multivariate analysis facilitates the study of the structure of complexbiomolecularassembliesof which Limulus polyphemus hemocyaninis an example. Both the analysis of classes of moleculeviewsand the investigationof continuous structure-related variations have provided new informationon the architectureof hemocyanin.
I. Introduction
2. Levels of structural analysis of single molecules
Hemocyanins are respiratory proteins of invertebrates. The hemocyanins of arthropods are built from subunits with molecular weight 75,000 and occur as 6-, 12-, 24-, and 48-meric aggregates [1]. Each of these stages of aggregation can be observed after partial dissociation of the 48-meric hemocyanin of Limulus polyphemus [2]. While it has been possible to solve the structure of Panulirus interruptus which consists of a single hexamer by X-ray crystallography [3], the aggregated forms have not been available in crystal form amenable to this approach. The study of the architecture of the aggregated forms therefore involves electron microscopy of single molecules, aided by immunolabeling, image processing and model building [4-7]. The different stages of association of the hexameric building block, and the different positions assumed by each of the association forms on the specimen grid produces a bewildering variety of views (fig. 1). While it is very difficult to infer the architecture of the 48-mer of Limulus from its various views (pentagonal, ring, cross, bowtie; figs. l d - l h ) , the additional information contained in the views of the 24-meric (figs. l a - l c ) and 12-meric building blocks helps in unscrambling this puzzle.
In this review, the hemocyanin molecule serves an an example for large biomolecular assemblies that cannot be analyzed in crystalline form using classical methods of X-ray crystallography or electron crystallography. When analyzing electron micrographs of such a molecule, we may distinguish three different levels of structural information, each requiring different tools of analysis: (A) The level of gross morphology, characterized by a description of the typical molecule shape as it emerges after visual analysis of a large number of molecules. Many molecules assume only a few preferential positions on the specimen grid, positions which maximize the areas of contact between molecule surface and specimen grid. Each such position leads to a preferential view which must be described on the level of gross morphology. As an example, we take van Bruggen's observation about the shape of the 24-meric halfmolecule of Limulus polyphemus hemocyanin in its most frequently occurring top view (see figs. l a and lb): " T h e molecules are observed as large square profiles with sides of about 24 nm and a 2 nm wide gap parallel to one of the sides" [1]. Indeed, the eye is most efficient in distinguishing molecule views on this level. Since there is no ambiguity in classification, little would be gained by using a
0304-3991/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
154
J. Frank / Multivariate image anal)sis
a
b
c
g
h
Fig. 1. Differenl views of the hemocyanin molecule in two states of association: (a-c) the two top views and the side view of the (4 x 6) halfmolecule; ( d - h ) five different views of the complete (8 x 6) molecule: pentagonal, ring, cross, and two different forms of the bowtie view, respectively. (Reprinted with permission from ref. [4]. Copyright 1982 American Chemical Society.)
fully computer-automated recognition procedure. (B) The level of fine structural detail common to all images of one kind (i.e., relating to the same view of the molecule). After the molecule images have been sorted, on the crude level, into their different classes, the retrieval of information on the fine structure common to all images in each
class would require a tedious visual scanning through all images, if visual evaluation on this level were attempted. Through repeated pairwise comparison of images, features that are reproducible among a majority of images are eventually recognized, and those that are not are rejected as untypical. The result of such a procedure would be
J. Frank / Multivariate image analysis
a mental image which is difficult to quantify and communicate from one observer to another. Computer averaging combined with correlation alignment [8-10] is obviously far superior to a visual analysis in the task of extracting common features from a set of images. In the above example of the Limulus hemocyanin halfmolecule, the presence and exact location of crossbridges between the four constituent hexamers can thus be ascertained, and a distinctly rhombic outline can be defined (see below for details). (C) The level of systematic variations of structural details, either due to the coexistence of closely similar views of a molecule possessing quasi-symmetries (e.g., the appearance of the halfmolecule in flip/flop related positions [5]), or due to a continuous range of positions of a molecule supported in an unstable way (e.g., the observed rocking movement of the halfmolecule in either of its top view positions [5]). It is on this level of description where the visual analysis fails completely. It is no surprise that observations on the variability of molecule images and its possible structure-related causes are rare in the literature.
3. Rationale and method of multivariate analysis The reason why visual analysis is insufficient for the task of identifying classes or systematic patterns of variation is the large number of image elements (up to 4000) occurring in multiple measurements (e.g. 100 images) of the molecule view that require simultaneous comparison. The visual system, often praised for its efficiency in pattern recognition problems, appears to be overtaxed by such a task. The method of multivariate analysis of computer-aligned images [5,11] (specifically, correspondence analysis because of its special properties [12]) accomplishes this task in a very effective way, providing a wealth of information previously thought hidden in noise. In this account we can only sketch the principle of the method, referring to more rigorous treatments elsewhere [11,12]. Aligned images of more or less similar molecule views may be represented as a point cloud in a multi-dimensional coordinate system ("hyperspace") where each axis represents a single pixel value. The finite size of the cloud reflects the
THE £2GEN VALUES VAU(1)=O,99999982 .......................................................................
NUN I I T [ ~
* £lGgJl VtL
........................
| 2 l ) l 4 l 5 l 6 l 7 I 0 I 9 I 10 111 l 12 I I$ l 14 I 15 1 16 I 17 l 18 I 19 l 20 l 21 l 22 1 23 l 24 I 25 I 26 I 27 I 20 l 29 I ]0
0
I ~E~CE~T ......
0 00019269
20 647
0.00006040 0°00004570 0,00004252 0.0000$788 0.00002842
6.472 4,~Q? 4.557
0.00002761 0,0000246~ 0.0000230* 0.0000222~ 0,00001990 0,000019?7
0.00001922 0.0000173~ 0,00001644 0.00001577 0,00001498 0.00001105 0,00001301 0,00001215 0,00001195 0.00001175
4*059 3.046 2.950 2.639 2.468
2,384 2,133 2.118 2,059
1,059 1.762 1,690 1.605 1.484
1.J94 1,302 1.281 1,259
0,00001144
1,225
0,00001054
1,130 1.10J 1.002 1.012 0.991 0j9~4
0.00001030 0.00001010 0.00000944 0.00000Q25 0.00000~72
I CUflOU
.............
l*l ............
. ..................
HISTDGRAM o r ........
155
....
£10[~
. ....
...................................
VEkUES
...............................................................
20 647 27o119 32.015 36.572
I****************'1'*****$*******'2"**************|***********$*** *********************** I**************** l*l************* 40.63! Iw************* 4~.677 I'It******** 4e.e]b l,I********* 49.]74 LtI*$****** 51.742 [ * l t * * * * * * 54.126 I*It*****t 5e.258 I*l**e*** 58.3?6 l*ltee*** b0,435 lel*****e b2.294 I'It**** b4.056 I******* b5.746 I*l***** 67.J50 L'i***** 68.834 [,1,*** 70.220 I*l**** 71.5)0 Ie***** 72.811 I'I**** 74.071 leI**** 75.29b l * l * * * * 76.426 I*I*** 77*529 I'I*** 78.611 /*It** 79*62] IeI*** 80,613 I ' I * * * el.sis 1,****
Fig. 2. Example for a histogram of eigenvalues (reproduction of the original printout) resulting from correspondence analysis of pentagonal views). The histogram represents the relative contributions of highest ranking independent factors to the total interimage variance. In our example, the first factor (associated with the rocking to be explained below) is by far the largest: it presents 21% of the total interimage variance. The second factor follows with 6.5%.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qi . . . . . . .
G
•
............ ~.......................................... • . . . . . . . . . . . . . . . . . . .
ib
1
•
S
¢
!
1 !
l
d,
e
Fig. 3. Analysis of dodecamer views using correspondence analysis. Seventy molecules were selected from the micrographs (a) and aligned. Some of the aligned molecules are shown in (b). The m a p of the first versus second factor (c) reveals a division of the data into two clusters: averages (d,e) using the encircled images on the right and left of the map show the particle in two mirror-related projections. (Reprinted with permission from ref. [14]. Copyright 1981, North-Holland Publishing Company. On the correspondence analysis map, numbers designating images were replaced by dots.)
J. Frank / Multivariate image analysis
l
157
180 °
I
J
I
90 °
90 °
90 °
Fig. 4. Hemocyanin dodecamer in four stable positions (a-d) related by 90° rotations. The four views corresponding to these positions are pairwise identical, related by 180° in-plane rotations (a = d; b = c). Alignment algorithms that bring the roughly rectangular outline of the molecule (indicated in a) into register without distinguishing the hexagonal shape of the topview (a, top) from the rectangular shape of the sideview (a, bottom) of the hexamer will produce four classes of images in the multivariate analysis.
v a r i a t i o n a m o n g the images, either due to noise or d u e to s o m e s y s t e m a t i c v a r i a t i o n of their s t r u c t u r e - r e l a t e d contents. However, the cloud m a y be structured, either consisting of separate subc l o u d s ( c o r r e s p o n d i n g to different classes of images) or peculiarly shaped, deviating in some w a y from the shape of a c o n s t a n t " s i g n a l " image s u p e r p o s e d with different realizations of totally r a n d o m noise. T o describe the m a j o r c o m p o n e n t s of variation, a new c o o r d i n a t e system is introduced, whose m a i n ( m u t u a l l y o r t h o g o n a l ) axes ( " f a c t o r s " ) run parallel to the p r o m i n e n t directions into which the cloud extends. F o r a given d a t a set a n d a given d i s t a n c e m e a s u r e (metric) in hyperspace, the new c o o r d i n a t e system of factors is uniquely defined (see below the e x a m p l e of the p e n t a g o n a l view). The relative i m p o r t a n c e of the factors in r e p r e s e n t i n g the i n t e r i m a g e variation is seen from a h i s t o g r a m of variance c o m p o n e n t s (eigenvalues of a m a t r i x associated with this p r o b lem [11,12]). A n e x a m p l e of such a h i s t o g r a m is shown in fig. 2; this is the d i s t r i b u t i o n of interi m a g e variance c o m p o n e n t s f o u n d in the analysis o f the p e n t a g o n a l view of the full (48-mer) Limulus m o l e c u l e to be discussed below. It is often the case that a few h i g h e s t - r a n k i n g factors are sufficient to represent the m a j o r variance c o m p o n e n t s . F o r convenience, t w o - d i m e n sional m a p s involving different c o m b i n a t i o n s of
two factors (i.e., 1 versus 2, 1 versus 3, etc.) are used to p r e s e n t p r o j e c t i o n s of the p o i n t cloud.
Fig. 5. Composition of the halfmolecule from two dodecamers according to ref. [4]. The three-dimensional representation was done by Dr. R. Feldmann with computer graphical methods. The labels show the positions of the eight subunits that make up the molecule, as localized by immunological methods. The model incorporates the slight shift and skewing between the two dodecamers as postulated by Van Heel and Frank [5]. The resulting halfmolecule which has a slightly rhombic outline is able to rock between two stable positions on the support grid. (Reprinted with permission from ref. [16]. Copyright 1983 American Chemical Society.)
.+,, . . . . . . . . . .
°.+°
.9,
om£° *~t,
*~z*
....
i~i:
*z1*
*z~*
*gz*
:~:
.... °z~*
£z
*~ge
t~
°~t**6{~"
:
¢I:,
.... :%: :
.~F*
tgn~*
,~tt°
:o1.
U6
J. Frank / Multivariate image analysis
These maps show in one glance whether important trends of data divisions exist (fig. 3). 4. Separation of classes Examples for the existence of sharply defined classes are found in the analysis of images of 2 x 6-mer (dodecamer) and the 4 x 6-mer. The dodecamer (insets in fig. 3) is formed by a sideby-side association of two hexamer building blocks (see fig. 4) where a 90 ° rotation is applied to one of the hexamers around the axis of dimeric association. Micrographs of this dissociation product were first analyzed by Van Heel [13,14] with the method of multivariate analysis. Any of the four stable side views (fig. 4) shows a clear projection of one hexamer down its axis of three-fold symmetry next to a square-profiled side view of the hexamer. If the images seen in a micrograph would be aligned such that the rectangles roughly inscribing the projections (indicated in fig. 4a) coincide, without any further discrimination, then four classes of images would be generated which show the molecule in all four stable positions. However, the four views are pairwise identical (figs. 4a, 4d and 4b, 4c), related by 180 ° rotation in the image plane due to the symmetry of the molecule (i.e., at a level of resolution where individual variations among the eight subunits involved in building the dodecamer are not considered). Any alignment that distinguishes the squarish from the hexagonal profile within the rectangle fuses each pair of classes into a single one [13,14] as evident in fig. 3. The two remaining, genuinely different views originate from flip/flop related positions of the molecule. Thus these views are mirror-related unless the staining is so shallow that preferential visualization of one side of the molecule occurs. However, all evidence suggests that
159
fig. 3 shows more or less " t r u e " projections of the entire molecule. We also take note, in view of the topic discussed in the next section, that the four original classes or the two fused classes exist without transitions, indicating that the energetically disfavored positions where the dodecamer molecule rests on two edges of the two constituent hexamers are in fact never assumed. The 4 x 6-mer is formed by side-by-side association of two dodecamers, each having been tilted by 45 ° around its long axis (fig. 5). Since the association involves a slight shift, the top view of the resulting molecule ("halfmolecule" relative to the fully associated 8 x 6-mer) shows a rhombic outline (figs. la and lb). However, since both flip and flop related positions of the halfmolecule occur, this top view exists in two mirror-related versions (fig. l a versus fig. lb) which are distinguished by the multivariate analysis [5]. We reproduce here Bijholt et al.'s results (fig. 6) because the larger number of images used by these authors makes it easier to recognize the sharp delineation of two clusters (a + c) and (b + d) on the correspondence analysis map than in the original study [5] (we ignore the more or less artificial divisions within the two main classes). The results obtained, in terms of the grouping of images on the map and the appearance of the class averages, are exactly the same in both studies [5,6]. In addition, another interesting phenomenon is observed: the two clusters of points on the map (fig. 6) representing the two classes of images appear elongate, suggesting the existence of systematic differences within each class. 5. Continuous variations First some general remarks on this phenomenon. Continuous variations of molecule views on
Fig. 6. Analysis of the top view of the halfmolecule using correspondence analysis [6]. The division of the data into two major clusters a + c and b + d reflects the difference between images of molecules lying in flip and flop related positions. Each of the main clusters is elongate: in the direction of this elongation the molecule views appear sorted according to stain differences between the two hexamers not in contact with the support (always the lower left and the upper right hexamer in inserts labeled (a) through (d)). In the comparison of averages (a) versus (c) and (b) versus (d), the two supporting hexamers, on the upper left and the lower right of the halfmolecule, have equal staining while the two others are unequal. Since the averages are taken over large ranges of the cluster elongation the stain imbalance is somewhat blurred out. However, for each cluster a gallery of particles picked at roughly equal intervals along the direction of elongation, such as done in [5], clearly shows a large range of stain variation. (Reprinted with permission from ref. [6]. Copyright 1982 Academic Press Inc. (London) Ltd.)
J. Frank / Multivariate image analysis
160
ROCKING
b
a
ROCKING
Fig. 7. Schematic drawing showing the halfmolecule in four different positions (a-d). Because of the non-coplanar arrangement of the four constituent hexamers (indicated by spheres), only two hexamers (indicated by stars) are in permanent contact with the support film while the other two assume varying positions in a see-saw manner, between close contact with the support film and m a x i m u m distance from it. Views (a) and (b) are related by rocking of the molecule, and so are (c) and (d); the hexamer lifted "toward us" into m a x i m u m distance from the plane of support is drawn as an enlarged sphere. Note that (a) and (b) as well as (c) and (d) are respectively related by 180 ° in-plane rotation. Molecules in positions (a) or (b) are brought into positions (c) or (d) by flipping. Since the arrangement of the hexamers is rhombic (exaggerated in these drawings), the two hexamers having permanent contact are either joined by a short (c, d) or by a long axis (a, b).
the scale we are observing have to do with the gradual changes in the molecule's position and concurrent changes in the pattern of staining. If a molecule has a uniquely defined position on the specimen grid (as a result of a stable minimum of the surface energy) we may expect to find no more than small random variations in its appearance, leading to a diffuse, globular appearance of the cluster on the map. However, if the molecule is able to rock between two or more positions, due to the topology and binding properties of the surface in contact with the support film, then the cluster will be elongated, branched, or otherwise structured, consisting of images in all intermediate
positions between the two or more extremes. Local map averages, i.e., averages of images within subdomains of the map show the molecule in its different "frozen" positions. The analysis of such local averages shows that the elongation of the two clusters (a + c) and (c + d) in fig. 6 arises from a see-saw redistribution of stain between diagonally juxtaposed subunits whereas the other two are relatively constant in stain intensity. Since upon going from the flip to the flop position, the varying hexamers become constant and vice-versa, the halfmolecule behaves very much like a noncoplanar structure (fig. 7). (Since the two extremes of each cluster contain images that are related by 180 ° rotation, the argument is somewhat more complicated; important, however, is the existence of a range of seesaw stain redistribution from a completely imbalanced to a completely balanced state. See ref. [13] where the analysis was repeated after the 180 ° ambiguity was removed.) This is indeed the interpretation put forth by Van Heel and Frank [5] which was later verified by direct observation of side views of the halfmolecule (Van Heel, personal communication): the four hexamers are in a slightly tetragonal arrangement which allows three hexamers at most to touch the support simultaneously; in addition to the slight shift, a slight skewing has to be introduced when forming the halfmolecule from the two dodecamers (fig. 4). Another example for a rocking effect associated with a continuous change in the molecule view is obtained in the analysis of the pentagonal view of the complete (8 × 6) molecule (fig. ld). The model of the complete molecule [4] emerges * when two halfmolecules (fig. 5) are stacked above each other such that their pseudo dyad axes (perpendicular to the plane of fig. 5) align; rotation by approximately 45 ° produces a closest fit (the most likely association involves the flipping over of one of the halfmolecules before stacking, see ref. [4]). The resulting model exhibits all views known from electron microscopy (figs. l d - l h ) . The pentagonal * A reevaluation of structural data and recent immunological findings (Lamy, personal communication) makes it necessary to revise the model by Lamy et al. [4]. However, this revision does not affect the building principle outlined in ref. [41 and the interpretation of interimage variations described here.
-I-
•i°
°o~
w
t
•
°
• FACTOR 2
d
"
b
Fig. 8. Factor map (1 versus 2) obtained by correspondence analysis of molecules presenting the pentagonal view. The scale of the factor 2 axis is exaggerated by a factor of two; there are no large changes between images falling onto the right and those on the left of the map as can be seen by comparing the upper two or the lower two averages. The main changes occur along factor 1: The pentagonal view changes from a quasi-symmetric (top averages) to an asymmetric appearance (bottom averages). At the same time, the pronounced central structure which is similar to the projection of a single dodecamer (fig. 3) shifts with respect to the outline of the molecule in horizontal direction (see fig. 9). (Reprinted with permission from ref. [4]. Copyright 1982 American Chemical Society.)
a
t~ %-.
e,
2
t~
;ili~i
C
Fig. 9. (a-e) Five "local" averages of pentagonal views obtained by averaging over horizontal strips of the map shown in fig. 8. The averages show a gradual change between quasisymmetric and asymmetric appearance, and a shift of the central dodecamer projection from right to left. (f-i) Difference images (b a), ( c - a ) , etc. showing the gradual shift of the dodecamer.
!~i,'~'~.. !!!~!~ .........
J. Frank / Multivariate image analysis
view (fig. ld) is seen to be dominated by a central structure which closely resembles the projections of the dodecamer (fig. 3): this view is in fact generated when the model of the full molecule is put to rest on one of its constituent dodecamers. However, such a demonstration at once reveals that in this position the molecule is able to pivot around one of the edges of the supporting dodecamer, up to a limiting position defined by contact with the tip of the neighboring dodecamer. (Pivoting around the other edge leads to no stable position.) Within that one-sided rocking range, the molecule's outline remains pentagonal, but the apparent position of the supporting dodecamer shifts relative to the entire molecule's outline, because the structures constituting the outline are farther away from the pivoting axis than the supporting dodecamer. It may come as a surprise to many electron microscopists that such small movements can be reliably measured by analyzing noisy images of the kind shown in fig. ld. However, the analysis of 100 pentagonal views [3] (fig. 8) provides a most detailed confirmation of the behavior expected for the model: (a) By far the largest component of interimage variation (21%) is due to the rocking (fig. 2). (b) The rocking (figs. 9a-9e) is one-sided, from a quasi-symmetric appearance (fig. 9a) to an asymmetric appearance (fig. 9e). (c) The central dodecamer projection moves relative to the molecule's outline in a direction perpendicular to the dodecamer edge (horizontal movement in fig. 9). This can be verified from the difference images (b - a), (c - a), (d - a), and (e a) in fig. 9. (d) The movement is continuous, i.e., all intermediates occur with the same frequency. The pentagonal view may serve to illustrate the high degree of reproducibility of findings obtained with correlation alignment and multivariate image analysis: the same analysis of L i m u l u s hemocyanin molecules showing the pentagonal view was independently done by Van Heel (personal communi-
163
cation). The detailed agreement (fig. 10) in the relative size of the rocking factor and in the averaged molecule views obtained from the "symmetric" and the "asymmetric" sides of the factor map confirms that we are dealing with universal properties of this molecule, not with laboratory artifacts.
6. Conclusions The direct observation of such subtleties as rocking [5,6] and effects due to the rise of the peripheral stain level around a molecule [15] introduces new and exciting possibilities into electron microscopic structure research. Reevaluation of micrographs whose structural information had been thought exhausted by visual examination may produce some unexpected results.
Acknowledgements Grateful acknowledgement is made of many stimulating discussions with J. Lamy and M. van Heel on the subject of hemocyanin architecture. I would like to thank J. L a m y for permitting me to use some unpublished figures which were obtained during a collaborative visit of P.-Y. Sizaret in Albany. I would also like to thank M. van Heel for making his averages of pentagonal views available. Special thanks are due to Adriana Verschoor for helping me with the preparation of the figures. This work was supported by N I H grant No. 1 R01 G M 29169.
References [1] E.F.J. van Bruggen, in: Proc. 9th Intern. Congr. on Electron Microscopy, Toronto, 1978, Ed. J. Sturgess. Vol. III, p. 450. [2] W.G. Schutter, E.F.J. van Bruggen, T. Bonaventura, C. Bonaventura and B. Sullivan, in: Structure and Function of Haemocyanin, Ed. J.V. Bannister (Springer, Berlin, 1977) pp. 13-21.
Fig. 10. Reproducibilityof results obtained with multivariate image analysis: Upper two averages of pentagonal views obtained by M. van Heel (unpublished), lower two obtained by Lamy et al. [4]. The averageson the left show the quasi-symmetricform, those on the right the asymmetric form of the pentagonal view. Detailed agreement is found in the distribution of stain.
164
J. Frank / Multivariate image analysis
[3] E.J.M. van Schaick, W.G. Schutter, W.P.J. Gaykema, A.M.H. Schepman and W.G.J. Hol, J. Mol. Biol. 158 (1982) 457. [4] J. Lamy, P.-Y. Sizaret, J. Frank, A. Verschoor, R. Feldmann and J. Bonaventura, Biochemistry 21 (1982) 6825. [5] M. van Heel and J. Frank, Ultramicroscopy 6 (1981) 187. [6] M.M.C. Bijholt, M.G. van Heel and E.F.J. van Bruggen, J. Mol. Biol. 161 (1982) 147. [7] M. van Heel, W. Keegstra, W. Schutter and E.F.J. van Bruggen, in: Structure and Function of Invertebrate Respiratory Proteins, Ed. E.J. Wood (Harwood, 1983) pp. 69-73. [8] J. Frank, W. Goldfarb, D. Eisenberg and T.S. Baker, Ultramicroscopy 3 (1978) 283. [9] J. Frank, A. Verschoor and M. Boublik, Science 214 (1981) 1353.
[10] J. Frank, J. Microscopy 117 (1979)25. [11] J. Frank and M. van Heel, J. Mol. Biol. 161 (1982) 134. [12] L. Lebart, A. Morineau and N. Tabard, Techniques de la Description Statistique (Dunod, Paris. 1977). [13] M. van Heel, PhD Thesis, R.ijksuniversiteit Groningen (1981). [14] M. van Heel and W. Keegstra, Ultramicroscopy 7 (1981) 113. [15] J. Frank, A. Verschoor and M, Boublik, J. Mol, Biol. 161 (1982) 107. [16] J. Lamy, P.-Y. Sizaret, P. Billiard, P. Jolles, J. Joll~s, R.J. Feldmann and J. Bonaventura, Biochemistry 22 (1983) 5573.
153
T H E R O L E O F MULTIVARIATE IMAGE ANALYSIS IN SOLVING T H E A R C H I T E C T U R E O F T H E L I M U L U S P O L Y P H E M U S HEMOCYANIN M O L E C U L E Joachim F R A N K Center for Laboratories and Research, New York State Department of Health, Albany, New York 12201, USA
Received 1 November 1983; presented at EMSA SymposiumAugust 1983
Multivariate analysis facilitates the study of the structure of complexbiomolecularassembliesof which Limulus polyphemus hemocyaninis an example. Both the analysis of classes of moleculeviewsand the investigationof continuous structure-related variations have provided new informationon the architectureof hemocyanin.
I. Introduction
2. Levels of structural analysis of single molecules
Hemocyanins are respiratory proteins of invertebrates. The hemocyanins of arthropods are built from subunits with molecular weight 75,000 and occur as 6-, 12-, 24-, and 48-meric aggregates [1]. Each of these stages of aggregation can be observed after partial dissociation of the 48-meric hemocyanin of Limulus polyphemus [2]. While it has been possible to solve the structure of Panulirus interruptus which consists of a single hexamer by X-ray crystallography [3], the aggregated forms have not been available in crystal form amenable to this approach. The study of the architecture of the aggregated forms therefore involves electron microscopy of single molecules, aided by immunolabeling, image processing and model building [4-7]. The different stages of association of the hexameric building block, and the different positions assumed by each of the association forms on the specimen grid produces a bewildering variety of views (fig. 1). While it is very difficult to infer the architecture of the 48-mer of Limulus from its various views (pentagonal, ring, cross, bowtie; figs. l d - l h ) , the additional information contained in the views of the 24-meric (figs. l a - l c ) and 12-meric building blocks helps in unscrambling this puzzle.
In this review, the hemocyanin molecule serves an an example for large biomolecular assemblies that cannot be analyzed in crystalline form using classical methods of X-ray crystallography or electron crystallography. When analyzing electron micrographs of such a molecule, we may distinguish three different levels of structural information, each requiring different tools of analysis: (A) The level of gross morphology, characterized by a description of the typical molecule shape as it emerges after visual analysis of a large number of molecules. Many molecules assume only a few preferential positions on the specimen grid, positions which maximize the areas of contact between molecule surface and specimen grid. Each such position leads to a preferential view which must be described on the level of gross morphology. As an example, we take van Bruggen's observation about the shape of the 24-meric halfmolecule of Limulus polyphemus hemocyanin in its most frequently occurring top view (see figs. l a and lb): " T h e molecules are observed as large square profiles with sides of about 24 nm and a 2 nm wide gap parallel to one of the sides" [1]. Indeed, the eye is most efficient in distinguishing molecule views on this level. Since there is no ambiguity in classification, little would be gained by using a
0304-3991/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
154
J. Frank / Multivariate image anal)sis
a
b
c
g
h
Fig. 1. Differenl views of the hemocyanin molecule in two states of association: (a-c) the two top views and the side view of the (4 x 6) halfmolecule; ( d - h ) five different views of the complete (8 x 6) molecule: pentagonal, ring, cross, and two different forms of the bowtie view, respectively. (Reprinted with permission from ref. [4]. Copyright 1982 American Chemical Society.)
fully computer-automated recognition procedure. (B) The level of fine structural detail common to all images of one kind (i.e., relating to the same view of the molecule). After the molecule images have been sorted, on the crude level, into their different classes, the retrieval of information on the fine structure common to all images in each
class would require a tedious visual scanning through all images, if visual evaluation on this level were attempted. Through repeated pairwise comparison of images, features that are reproducible among a majority of images are eventually recognized, and those that are not are rejected as untypical. The result of such a procedure would be
J. Frank / Multivariate image analysis
a mental image which is difficult to quantify and communicate from one observer to another. Computer averaging combined with correlation alignment [8-10] is obviously far superior to a visual analysis in the task of extracting common features from a set of images. In the above example of the Limulus hemocyanin halfmolecule, the presence and exact location of crossbridges between the four constituent hexamers can thus be ascertained, and a distinctly rhombic outline can be defined (see below for details). (C) The level of systematic variations of structural details, either due to the coexistence of closely similar views of a molecule possessing quasi-symmetries (e.g., the appearance of the halfmolecule in flip/flop related positions [5]), or due to a continuous range of positions of a molecule supported in an unstable way (e.g., the observed rocking movement of the halfmolecule in either of its top view positions [5]). It is on this level of description where the visual analysis fails completely. It is no surprise that observations on the variability of molecule images and its possible structure-related causes are rare in the literature.
3. Rationale and method of multivariate analysis The reason why visual analysis is insufficient for the task of identifying classes or systematic patterns of variation is the large number of image elements (up to 4000) occurring in multiple measurements (e.g. 100 images) of the molecule view that require simultaneous comparison. The visual system, often praised for its efficiency in pattern recognition problems, appears to be overtaxed by such a task. The method of multivariate analysis of computer-aligned images [5,11] (specifically, correspondence analysis because of its special properties [12]) accomplishes this task in a very effective way, providing a wealth of information previously thought hidden in noise. In this account we can only sketch the principle of the method, referring to more rigorous treatments elsewhere [11,12]. Aligned images of more or less similar molecule views may be represented as a point cloud in a multi-dimensional coordinate system ("hyperspace") where each axis represents a single pixel value. The finite size of the cloud reflects the
THE £2GEN VALUES VAU(1)=O,99999982 .......................................................................
NUN I I T [ ~
* £lGgJl VtL
........................
| 2 l ) l 4 l 5 l 6 l 7 I 0 I 9 I 10 111 l 12 I I$ l 14 I 15 1 16 I 17 l 18 I 19 l 20 l 21 l 22 1 23 l 24 I 25 I 26 I 27 I 20 l 29 I ]0
0
I ~E~CE~T ......
0 00019269
20 647
0.00006040 0°00004570 0,00004252 0.0000$788 0.00002842
6.472 4,~Q? 4.557
0.00002761 0,0000246~ 0.0000230* 0.0000222~ 0,00001990 0,000019?7
0.00001922 0.0000173~ 0,00001644 0.00001577 0,00001498 0.00001105 0,00001301 0,00001215 0,00001195 0.00001175
4*059 3.046 2.950 2.639 2.468
2,384 2,133 2.118 2,059
1,059 1.762 1,690 1.605 1.484
1.J94 1,302 1.281 1,259
0,00001144
1,225
0,00001054
1,130 1.10J 1.002 1.012 0.991 0j9~4
0.00001030 0.00001010 0.00000944 0.00000Q25 0.00000~72
I CUflOU
.............
l*l ............
. ..................
HISTDGRAM o r ........
155
....
£10[~
. ....
...................................
VEkUES
...............................................................
20 647 27o119 32.015 36.572
I****************'1'*****$*******'2"**************|***********$*** *********************** I**************** l*l************* 40.63! Iw************* 4~.677 I'It******** 4e.e]b l,I********* 49.]74 LtI*$****** 51.742 [ * l t * * * * * * 54.126 I*It*****t 5e.258 I*l**e*** 58.3?6 l*ltee*** b0,435 lel*****e b2.294 I'It**** b4.056 I******* b5.746 I*l***** 67.J50 L'i***** 68.834 [,1,*** 70.220 I*l**** 71.5)0 Ie***** 72.811 I'I**** 74.071 leI**** 75.29b l * l * * * * 76.426 I*I*** 77*529 I'I*** 78.611 /*It** 79*62] IeI*** 80,613 I ' I * * * el.sis 1,****
Fig. 2. Example for a histogram of eigenvalues (reproduction of the original printout) resulting from correspondence analysis of pentagonal views). The histogram represents the relative contributions of highest ranking independent factors to the total interimage variance. In our example, the first factor (associated with the rocking to be explained below) is by far the largest: it presents 21% of the total interimage variance. The second factor follows with 6.5%.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qi . . . . . . .
G
•
............ ~.......................................... • . . . . . . . . . . . . . . . . . . .
ib
1
•
S
¢
!
1 !
l
d,
e
Fig. 3. Analysis of dodecamer views using correspondence analysis. Seventy molecules were selected from the micrographs (a) and aligned. Some of the aligned molecules are shown in (b). The m a p of the first versus second factor (c) reveals a division of the data into two clusters: averages (d,e) using the encircled images on the right and left of the map show the particle in two mirror-related projections. (Reprinted with permission from ref. [14]. Copyright 1981, North-Holland Publishing Company. On the correspondence analysis map, numbers designating images were replaced by dots.)
J. Frank / Multivariate image analysis
l
157
180 °
I
J
I
90 °
90 °
90 °
Fig. 4. Hemocyanin dodecamer in four stable positions (a-d) related by 90° rotations. The four views corresponding to these positions are pairwise identical, related by 180° in-plane rotations (a = d; b = c). Alignment algorithms that bring the roughly rectangular outline of the molecule (indicated in a) into register without distinguishing the hexagonal shape of the topview (a, top) from the rectangular shape of the sideview (a, bottom) of the hexamer will produce four classes of images in the multivariate analysis.
v a r i a t i o n a m o n g the images, either due to noise or d u e to s o m e s y s t e m a t i c v a r i a t i o n of their s t r u c t u r e - r e l a t e d contents. However, the cloud m a y be structured, either consisting of separate subc l o u d s ( c o r r e s p o n d i n g to different classes of images) or peculiarly shaped, deviating in some w a y from the shape of a c o n s t a n t " s i g n a l " image s u p e r p o s e d with different realizations of totally r a n d o m noise. T o describe the m a j o r c o m p o n e n t s of variation, a new c o o r d i n a t e system is introduced, whose m a i n ( m u t u a l l y o r t h o g o n a l ) axes ( " f a c t o r s " ) run parallel to the p r o m i n e n t directions into which the cloud extends. F o r a given d a t a set a n d a given d i s t a n c e m e a s u r e (metric) in hyperspace, the new c o o r d i n a t e system of factors is uniquely defined (see below the e x a m p l e of the p e n t a g o n a l view). The relative i m p o r t a n c e of the factors in r e p r e s e n t i n g the i n t e r i m a g e variation is seen from a h i s t o g r a m of variance c o m p o n e n t s (eigenvalues of a m a t r i x associated with this p r o b lem [11,12]). A n e x a m p l e of such a h i s t o g r a m is shown in fig. 2; this is the d i s t r i b u t i o n of interi m a g e variance c o m p o n e n t s f o u n d in the analysis o f the p e n t a g o n a l view of the full (48-mer) Limulus m o l e c u l e to be discussed below. It is often the case that a few h i g h e s t - r a n k i n g factors are sufficient to represent the m a j o r variance c o m p o n e n t s . F o r convenience, t w o - d i m e n sional m a p s involving different c o m b i n a t i o n s of
two factors (i.e., 1 versus 2, 1 versus 3, etc.) are used to p r e s e n t p r o j e c t i o n s of the p o i n t cloud.
Fig. 5. Composition of the halfmolecule from two dodecamers according to ref. [4]. The three-dimensional representation was done by Dr. R. Feldmann with computer graphical methods. The labels show the positions of the eight subunits that make up the molecule, as localized by immunological methods. The model incorporates the slight shift and skewing between the two dodecamers as postulated by Van Heel and Frank [5]. The resulting halfmolecule which has a slightly rhombic outline is able to rock between two stable positions on the support grid. (Reprinted with permission from ref. [16]. Copyright 1983 American Chemical Society.)
.+,, . . . . . . . . . .
°.+°
.9,
om£° *~t,
*~z*
....
i~i:
*z1*
*z~*
*gz*
:~:
.... °z~*
£z
*~ge
t~
°~t**6{~"
:
¢I:,
.... :%: :
.~F*
tgn~*
,~tt°
:o1.
U6
J. Frank / Multivariate image analysis
These maps show in one glance whether important trends of data divisions exist (fig. 3). 4. Separation of classes Examples for the existence of sharply defined classes are found in the analysis of images of 2 x 6-mer (dodecamer) and the 4 x 6-mer. The dodecamer (insets in fig. 3) is formed by a sideby-side association of two hexamer building blocks (see fig. 4) where a 90 ° rotation is applied to one of the hexamers around the axis of dimeric association. Micrographs of this dissociation product were first analyzed by Van Heel [13,14] with the method of multivariate analysis. Any of the four stable side views (fig. 4) shows a clear projection of one hexamer down its axis of three-fold symmetry next to a square-profiled side view of the hexamer. If the images seen in a micrograph would be aligned such that the rectangles roughly inscribing the projections (indicated in fig. 4a) coincide, without any further discrimination, then four classes of images would be generated which show the molecule in all four stable positions. However, the four views are pairwise identical (figs. 4a, 4d and 4b, 4c), related by 180 ° rotation in the image plane due to the symmetry of the molecule (i.e., at a level of resolution where individual variations among the eight subunits involved in building the dodecamer are not considered). Any alignment that distinguishes the squarish from the hexagonal profile within the rectangle fuses each pair of classes into a single one [13,14] as evident in fig. 3. The two remaining, genuinely different views originate from flip/flop related positions of the molecule. Thus these views are mirror-related unless the staining is so shallow that preferential visualization of one side of the molecule occurs. However, all evidence suggests that
159
fig. 3 shows more or less " t r u e " projections of the entire molecule. We also take note, in view of the topic discussed in the next section, that the four original classes or the two fused classes exist without transitions, indicating that the energetically disfavored positions where the dodecamer molecule rests on two edges of the two constituent hexamers are in fact never assumed. The 4 x 6-mer is formed by side-by-side association of two dodecamers, each having been tilted by 45 ° around its long axis (fig. 5). Since the association involves a slight shift, the top view of the resulting molecule ("halfmolecule" relative to the fully associated 8 x 6-mer) shows a rhombic outline (figs. la and lb). However, since both flip and flop related positions of the halfmolecule occur, this top view exists in two mirror-related versions (fig. l a versus fig. lb) which are distinguished by the multivariate analysis [5]. We reproduce here Bijholt et al.'s results (fig. 6) because the larger number of images used by these authors makes it easier to recognize the sharp delineation of two clusters (a + c) and (b + d) on the correspondence analysis map than in the original study [5] (we ignore the more or less artificial divisions within the two main classes). The results obtained, in terms of the grouping of images on the map and the appearance of the class averages, are exactly the same in both studies [5,6]. In addition, another interesting phenomenon is observed: the two clusters of points on the map (fig. 6) representing the two classes of images appear elongate, suggesting the existence of systematic differences within each class. 5. Continuous variations First some general remarks on this phenomenon. Continuous variations of molecule views on
Fig. 6. Analysis of the top view of the halfmolecule using correspondence analysis [6]. The division of the data into two major clusters a + c and b + d reflects the difference between images of molecules lying in flip and flop related positions. Each of the main clusters is elongate: in the direction of this elongation the molecule views appear sorted according to stain differences between the two hexamers not in contact with the support (always the lower left and the upper right hexamer in inserts labeled (a) through (d)). In the comparison of averages (a) versus (c) and (b) versus (d), the two supporting hexamers, on the upper left and the lower right of the halfmolecule, have equal staining while the two others are unequal. Since the averages are taken over large ranges of the cluster elongation the stain imbalance is somewhat blurred out. However, for each cluster a gallery of particles picked at roughly equal intervals along the direction of elongation, such as done in [5], clearly shows a large range of stain variation. (Reprinted with permission from ref. [6]. Copyright 1982 Academic Press Inc. (London) Ltd.)
J. Frank / Multivariate image analysis
160
ROCKING
b
a
ROCKING
Fig. 7. Schematic drawing showing the halfmolecule in four different positions (a-d). Because of the non-coplanar arrangement of the four constituent hexamers (indicated by spheres), only two hexamers (indicated by stars) are in permanent contact with the support film while the other two assume varying positions in a see-saw manner, between close contact with the support film and m a x i m u m distance from it. Views (a) and (b) are related by rocking of the molecule, and so are (c) and (d); the hexamer lifted "toward us" into m a x i m u m distance from the plane of support is drawn as an enlarged sphere. Note that (a) and (b) as well as (c) and (d) are respectively related by 180 ° in-plane rotation. Molecules in positions (a) or (b) are brought into positions (c) or (d) by flipping. Since the arrangement of the hexamers is rhombic (exaggerated in these drawings), the two hexamers having permanent contact are either joined by a short (c, d) or by a long axis (a, b).
the scale we are observing have to do with the gradual changes in the molecule's position and concurrent changes in the pattern of staining. If a molecule has a uniquely defined position on the specimen grid (as a result of a stable minimum of the surface energy) we may expect to find no more than small random variations in its appearance, leading to a diffuse, globular appearance of the cluster on the map. However, if the molecule is able to rock between two or more positions, due to the topology and binding properties of the surface in contact with the support film, then the cluster will be elongated, branched, or otherwise structured, consisting of images in all intermediate
positions between the two or more extremes. Local map averages, i.e., averages of images within subdomains of the map show the molecule in its different "frozen" positions. The analysis of such local averages shows that the elongation of the two clusters (a + c) and (c + d) in fig. 6 arises from a see-saw redistribution of stain between diagonally juxtaposed subunits whereas the other two are relatively constant in stain intensity. Since upon going from the flip to the flop position, the varying hexamers become constant and vice-versa, the halfmolecule behaves very much like a noncoplanar structure (fig. 7). (Since the two extremes of each cluster contain images that are related by 180 ° rotation, the argument is somewhat more complicated; important, however, is the existence of a range of seesaw stain redistribution from a completely imbalanced to a completely balanced state. See ref. [13] where the analysis was repeated after the 180 ° ambiguity was removed.) This is indeed the interpretation put forth by Van Heel and Frank [5] which was later verified by direct observation of side views of the halfmolecule (Van Heel, personal communication): the four hexamers are in a slightly tetragonal arrangement which allows three hexamers at most to touch the support simultaneously; in addition to the slight shift, a slight skewing has to be introduced when forming the halfmolecule from the two dodecamers (fig. 4). Another example for a rocking effect associated with a continuous change in the molecule view is obtained in the analysis of the pentagonal view of the complete (8 × 6) molecule (fig. ld). The model of the complete molecule [4] emerges * when two halfmolecules (fig. 5) are stacked above each other such that their pseudo dyad axes (perpendicular to the plane of fig. 5) align; rotation by approximately 45 ° produces a closest fit (the most likely association involves the flipping over of one of the halfmolecules before stacking, see ref. [4]). The resulting model exhibits all views known from electron microscopy (figs. l d - l h ) . The pentagonal * A reevaluation of structural data and recent immunological findings (Lamy, personal communication) makes it necessary to revise the model by Lamy et al. [4]. However, this revision does not affect the building principle outlined in ref. [41 and the interpretation of interimage variations described here.
-I-
•i°
°o~
w
t
•
°
• FACTOR 2
d
"
b
Fig. 8. Factor map (1 versus 2) obtained by correspondence analysis of molecules presenting the pentagonal view. The scale of the factor 2 axis is exaggerated by a factor of two; there are no large changes between images falling onto the right and those on the left of the map as can be seen by comparing the upper two or the lower two averages. The main changes occur along factor 1: The pentagonal view changes from a quasi-symmetric (top averages) to an asymmetric appearance (bottom averages). At the same time, the pronounced central structure which is similar to the projection of a single dodecamer (fig. 3) shifts with respect to the outline of the molecule in horizontal direction (see fig. 9). (Reprinted with permission from ref. [4]. Copyright 1982 American Chemical Society.)
a
t~ %-.
e,
2
t~
;ili~i
C
Fig. 9. (a-e) Five "local" averages of pentagonal views obtained by averaging over horizontal strips of the map shown in fig. 8. The averages show a gradual change between quasisymmetric and asymmetric appearance, and a shift of the central dodecamer projection from right to left. (f-i) Difference images (b a), ( c - a ) , etc. showing the gradual shift of the dodecamer.
!~i,'~'~.. !!!~!~ .........
J. Frank / Multivariate image analysis
view (fig. ld) is seen to be dominated by a central structure which closely resembles the projections of the dodecamer (fig. 3): this view is in fact generated when the model of the full molecule is put to rest on one of its constituent dodecamers. However, such a demonstration at once reveals that in this position the molecule is able to pivot around one of the edges of the supporting dodecamer, up to a limiting position defined by contact with the tip of the neighboring dodecamer. (Pivoting around the other edge leads to no stable position.) Within that one-sided rocking range, the molecule's outline remains pentagonal, but the apparent position of the supporting dodecamer shifts relative to the entire molecule's outline, because the structures constituting the outline are farther away from the pivoting axis than the supporting dodecamer. It may come as a surprise to many electron microscopists that such small movements can be reliably measured by analyzing noisy images of the kind shown in fig. ld. However, the analysis of 100 pentagonal views [3] (fig. 8) provides a most detailed confirmation of the behavior expected for the model: (a) By far the largest component of interimage variation (21%) is due to the rocking (fig. 2). (b) The rocking (figs. 9a-9e) is one-sided, from a quasi-symmetric appearance (fig. 9a) to an asymmetric appearance (fig. 9e). (c) The central dodecamer projection moves relative to the molecule's outline in a direction perpendicular to the dodecamer edge (horizontal movement in fig. 9). This can be verified from the difference images (b - a), (c - a), (d - a), and (e a) in fig. 9. (d) The movement is continuous, i.e., all intermediates occur with the same frequency. The pentagonal view may serve to illustrate the high degree of reproducibility of findings obtained with correlation alignment and multivariate image analysis: the same analysis of L i m u l u s hemocyanin molecules showing the pentagonal view was independently done by Van Heel (personal communi-
163
cation). The detailed agreement (fig. 10) in the relative size of the rocking factor and in the averaged molecule views obtained from the "symmetric" and the "asymmetric" sides of the factor map confirms that we are dealing with universal properties of this molecule, not with laboratory artifacts.
6. Conclusions The direct observation of such subtleties as rocking [5,6] and effects due to the rise of the peripheral stain level around a molecule [15] introduces new and exciting possibilities into electron microscopic structure research. Reevaluation of micrographs whose structural information had been thought exhausted by visual examination may produce some unexpected results.
Acknowledgements Grateful acknowledgement is made of many stimulating discussions with J. Lamy and M. van Heel on the subject of hemocyanin architecture. I would like to thank J. L a m y for permitting me to use some unpublished figures which were obtained during a collaborative visit of P.-Y. Sizaret in Albany. I would also like to thank M. van Heel for making his averages of pentagonal views available. Special thanks are due to Adriana Verschoor for helping me with the preparation of the figures. This work was supported by N I H grant No. 1 R01 G M 29169.
References [1] E.F.J. van Bruggen, in: Proc. 9th Intern. Congr. on Electron Microscopy, Toronto, 1978, Ed. J. Sturgess. Vol. III, p. 450. [2] W.G. Schutter, E.F.J. van Bruggen, T. Bonaventura, C. Bonaventura and B. Sullivan, in: Structure and Function of Haemocyanin, Ed. J.V. Bannister (Springer, Berlin, 1977) pp. 13-21.
Fig. 10. Reproducibilityof results obtained with multivariate image analysis: Upper two averages of pentagonal views obtained by M. van Heel (unpublished), lower two obtained by Lamy et al. [4]. The averageson the left show the quasi-symmetricform, those on the right the asymmetric form of the pentagonal view. Detailed agreement is found in the distribution of stain.
164
J. Frank / Multivariate image analysis
[3] E.J.M. van Schaick, W.G. Schutter, W.P.J. Gaykema, A.M.H. Schepman and W.G.J. Hol, J. Mol. Biol. 158 (1982) 457. [4] J. Lamy, P.-Y. Sizaret, J. Frank, A. Verschoor, R. Feldmann and J. Bonaventura, Biochemistry 21 (1982) 6825. [5] M. van Heel and J. Frank, Ultramicroscopy 6 (1981) 187. [6] M.M.C. Bijholt, M.G. van Heel and E.F.J. van Bruggen, J. Mol. Biol. 161 (1982) 147. [7] M. van Heel, W. Keegstra, W. Schutter and E.F.J. van Bruggen, in: Structure and Function of Invertebrate Respiratory Proteins, Ed. E.J. Wood (Harwood, 1983) pp. 69-73. [8] J. Frank, W. Goldfarb, D. Eisenberg and T.S. Baker, Ultramicroscopy 3 (1978) 283. [9] J. Frank, A. Verschoor and M. Boublik, Science 214 (1981) 1353.
[10] J. Frank, J. Microscopy 117 (1979)25. [11] J. Frank and M. van Heel, J. Mol. Biol. 161 (1982) 134. [12] L. Lebart, A. Morineau and N. Tabard, Techniques de la Description Statistique (Dunod, Paris. 1977). [13] M. van Heel, PhD Thesis, R.ijksuniversiteit Groningen (1981). [14] M. van Heel and W. Keegstra, Ultramicroscopy 7 (1981) 113. [15] J. Frank, A. Verschoor and M, Boublik, J. Mol, Biol. 161 (1982) 107. [16] J. Lamy, P.-Y. Sizaret, P. Billiard, P. Jolles, J. Joll~s, R.J. Feldmann and J. Bonaventura, Biochemistry 22 (1983) 5573.