III.E.3. Particles with Rotational Symmetry

Specimens with only rotational symmetry, such as individual oligomeric proteins, spherical viruses, and bacteriophage baseplates (Fig.III.E.3.1), have been studied by the rotational photographic-superposition, digital rotational filtering, and 3D reconstruction techniques. Digital rotational-filtering and photographic superposition techniques produce qualitatively similar results, but the photographic methods should be used with caution and usually with specimens displaying obvious or well-established symmetry. Crowther and Amos (1971) and Misell (1978) compare the real-space and Fourier-space methods.

FIg.III.E.3.1 A negatively stained preparation of base plates from bacteriophage T4. Rotationally filtered images of particles A and B are shown in Fig.III.E.3.6. (From Crowther and Amos, 1971, Plate I)

a. Rotational photographic superposition method

Fig.III.E..3.2. Apparatus for integrating detail in micrographs with radial symmetry. (a) Enlarger unit containing micrograph with region to be studied accurately centered in relation to the lens. (b) Movable board containing rotary disc. (c) The center of rotation must be carefully aligned to the selected region of the micrograph shown at (a). (From Horne and Markham, p.413)

Markham et al. (1963) devised a simple, real-space method for analyzing images of particles with rotational symmetry. The basic apparatus used (Fig.III.E.3.2) consists of a photographic enlarger and a movable board to which a card is attached that can be rotated about an axis. The specimen image is projected onto a photographic print attached to the card and a series of n images are exposed onto the print, with the print rotated by 360°/n after each of n exposures. For example, if the object of interest had 6-fold rotational symmetry, and a normal, straight photographic print required a 12 second exposure, then a total of six 2-second exposures would be required to produce the rotational superposition photographic image. Presumably, maximum reinforcement of detail is given when n is the true periodicity in the image of the object. This presumes, of course, that the rotation axis of the print can be positioned accurately at the center of symmetry in the image. If they do not coincide, the true symmetry of the object may be missed and details will certainly get smeared out in the reconstruction. Thus, the correct centering of the photographic print with respect to the rotational symmetry axis in the image and the correct choice of n are important components in successful application of this method. A mistake in either choice leads to erroneous results.

The main disadvantage of the optical superposition method is that it requires a visual assessment step which may be more influenced by what is eye-catching rather than what is correct. The computational, rotational filtering method that is described in the next section provides quantitative information in the form of the rotational power spectrum (RPS). The RPS allows quantitative assessment of the presence of a particular rotational symmetry. Once the symmetry is known, a filtered image can be resynthesized from only those components that obey the chosen rotational symmetry.

A second type of apparatus for producing rotationally symmetrized images by the photographic superposition method is shown below (Fig.III.E.3.3).

Fig.III.E.3.3. The arrangement of an apparatus for analyzing electron micrographs by rotational integration with the aid of a strobe illuminator. (From Horne and Markham, p.415)

b. 2D digital processing - Power spectrum and rotational filtering analysis

Fig.III.E.3.4. Cartesian (x,y) and polar (r,f) coordinate systems in real (left) and reciprocal (right) space.

The digitization, boxing, and floating of the specimen image is performed as usual and subsequent computations are conveniently performed in polar coordinates (r,f) (Fig.III.E.3.4). Thus, a Cartesian image, r(x,y) is converted to a polar image, r(r,f) by subdividing the Cartesian image into a series of equally spaced annuli (Fig.III.E.3.5) and interpolating the densities within each annulus. The polar density function can be expanded into a series of circular waves much as a Cartesian image is conveniently expanded into a series of plane waves (Fig.III.E.3.5).

(1)

In equation (1),each gn(r) represents the weight of the n-fold azimuthal component of the image at a radius r. The phase term, exp(inf), positions the peak of each circular wave with respect to an origin (usually the x axis) so that all gn(r) are properly summed.

Each gn(r) is integrated over the radius of the particle, a, to obtain a measure of the total n-fold rotational component of the image. Power in the image is defined as:

(2)

en = 2 accounts for the fact that Pn has equal contributions from gn and g-n for n>0.

The rotational power spectrum is a plot of Pn as a function of n . This is a compact way to represent the rotational symmetry components in the image. P0 is usually normalized to 1.0 and the spectrum is displayed with the Pn on a logarithmic scale (Figs.III.E.3.6 and E.3.7).

Fig.III.E.3.5. The Fourier method for finding rotational symmetry. The image (a) is divided into a series of concentric, equal-spaced annuli, of which one (b) is expressed as the sum of two of its rotational Fourier components ((c), the zero-fold and (d) the eight-fold symmetric component). The Fourier transforms of (c) and (d) are shown in (e) and (f), respectively. (From Moody, 7.66, p.239)

FIg.III.E.3.6 (Left) A logarithmic plot of the rotational power spectrum of a T4 bacteriophage base plate (particle A in Fig.IIII.E.3.1), showing the strong 6-fold symmetry of the image. The curve is normalized with Po = 1 and the power associated with rotational frequencies higher than n = 36 is less than 0.001. (From Crowther and Amos, 1971, Fig.2, p.126). (Top) 6-fold rotationally-symetrized images of negatively stained base plates from bacteriophage T4 (particles labeled A and B in Fig.IIII.E.3.1). Although the two filtered images have rather different appearances, because they have been plotted at different density levels, the main features of each are very similar. (From Crowther and Amos, 1971, Plate I)

Fig.III.E.3.7 Logarithmic plots of the rotational power spectra of two images of discs of tobacco mosaic virus protein. (a) A well-preserfved disc (Fig.III.E.3.8(a)) in which no one component is dominant. In each case the solid curve (-l-) refers to a choice of origin which maximizes the 17-fold component. In (a) the triangles (s) refer to a choice of origin which simultaneously maximizes the 16- and 18-fold components, while in (b) the triangles (s) and squares (n) refer to choices of origin which maximize respectively the 16- and 18-fold components. The curves are normalized with Po = 1 and the power associated with rotational frequencies higher than n = 26 is less than 0·001. (From Crowther and Amos, 1971, Fig.3, p.127)

Fig.III.E.3.8. (a) and (c) Images of negatively stained discs of tobacco mosaic virus protein. (b) and (d) Results of 17-fold filtering of the images shown in (a) and (c), respectively. These images are well preserved as judged by the dominance of a single rotational symmetry in the power spectrum (Fig.III.E.3.7(a)). Note that the two filtered images, which have been processed in an identifical manner, are of opposite hand, thus confirming the polar nature of the disc. The density level in plotting has been chosen to be rather high in order to emphasize the azimuthally varying component, thereby accentuating the hole at centre of the particle. (From Crowther and Amos, 1971, Plate II)

Fig.III.E.3.9. (a) Image of a negatively stained disc of tobacco mosaic virus protein. This is a poorly preserved particle as judged by the rotational power spectrum (Fig.III.E.3.7(b)). (b), (c), and (d) show respectively the results of 16-, 17- and 18-fold filtering. 16-fold filtering produces the most eye-catching image, although it is not the strongest harmonic. (From Crowther and Amos, 1971, Plate III)

As with other types of specimens, it is convenient with rotationally-symmetric specimens to perform computations in Fourier space rather than real space. The polar Fourier transform coordinates are R and F. The transform is expanded in the following way:

(3)

(4)

Jn(X) is a Bessel function of order n. Each Jn is a circularly-symmetric, oscillatory function (Fig.III.E.3.10). The first maximum of Jn(X) for large n (i.e. n > about 5) appears at about X+2.

Fig.III.E.3.10. (Left) The behavior of the Bessel function Jn(X) for various values of n. (From Sherwood, 16.7, p. 565). (Right) Amplitude variations of the Bessel function Jn(X) for n = 0 to 3. (From Misell, 3.26, p.98)

The transform of a ring of radius a is given by 2paJ0(2paR) (Fig.III.E.3.11). Such a ring can be considered to be generated from a pair of points, separated by the distance 2a, that are rotated through the angle p. A single pair of points at opposite ends of a diameter give rise to cosine fringes When rotationally averaged, the fringes reinforce at the origin but tend to cancel away from the origin. This gives rise to a Bessel function of zero order (J0).

d(r-a) 2paJ0(2paR)

Fig.III.E.3.11. A ring (left) and its transform (right). (Taken from Crowther unpublished course notes, 1973)

The expansion of the Fourier Transform (eqn. 4) is analagous to the expansion of the polar image densities as given in equation (1). Thus the Gn (R) are the coefficients (weights) of each azimuthal component in the Fourier transform. The two sets of coefficients, Gn(R) and gn(r), are connected by what is called the Fourier-Bessel transform.

(5)

In practice, the above integral would normally only be evaluated out to some resolution limit (i.e. with R < _).

The inverse relationship also holds:

(6)

where r0 is the radial limit of the object.

Examples of the relationship between objects with n-fold sinusoidal variations in azimuth (gn(r)) and the corresponding Fourier-Bessel transforms (Gn(R)) are illustrated in Fig.III.E.3.12.

Fig.III.E.3.12. (Left two columns) Density functions with 2-, 3-, and 4-fold azimuthal variations and the corresponding Fourier-Bessel transforms. (Right two columns) Two-fold azimuthal density functions of different radii and orientation and the respective Fourier-Bessel transforms. (Taken from Crowther unpublished course notes, 1973)

Fig.III.E.3.13. An object consisting of two rings of radii a and 2a , each with an n-fold azimuthal variation (left) gives rise to overlapping Bessel functions which tend to canncel apart from theri major peaks (right). (Taken from Crowther unpublished course notes, 1973)

It is essential that the origin of the polar coordinate system lie on the symmetry axis of the image. Initially, the origin chosen by eye during the boxing procedure necessarily becomes the phase origin of the Fourier transform. The origin point is then shifted to get the best Pn for the assumed symmetry. By changing the assumed symmetry, m, one gets a series of origins and computes for each of these separate origins a series of rotational power spectra. These are compared to look for the dominant symmetry.

Fig.II.E.3.14. A plot of the residual function obtained when determining the best position for the origin of a T4 bacteriophage base plate (particle A, Fig.III.E.3.1), based on the assumption of 6-fold symmetry. The origin is shifted by steps Dx, Dy of approximately 2.5 Å from the initial approximate position chosen by eye. For a particle with perfect 6-fold symmetry the residual should be zero when the origin coincides with the 6-fold axis. It is the sharpness of the minimum which is important for accurate determination of the position of the origin, and it can be seen that, in this case, the residual approximately doubles for a shift of origin of about 10 Å from the position corresponding to the minimum. (From Crowther and Amos, 1971, Fig.1, p.125)

One typically examines the rotational power spectra computed from several different particle images to get an idea of the relative preservation of the particles. Those images which show the highest Pn are used to synthesize rotationally-filtered images. Equation (5) is used to convert each Gn to a corresponding gn and only those gn for which n is a multiple of m are computed, thereby omitting all other components considered to be noise. Noise may arise from several sources such as:

1) the particle may not be viewed directly along an axis of symmetry

2) the particle may be distorted or may be non-uniformly stained, shadowed, etc.

3) the other usual forms of noise (e.g. support film, electron optical effects, etc.) may be present.

The Gn(R) are computed from the Fourier transform by the inverse of equation (4):

(7)

In using equation (5) to compute the gn(r), n is set to some limit since Pn is effectively zero beyond certain n (resolution limit) Setting an upper limit for n, thus limits the fineness of detail that can be seen in the reconstructed image.

Note that the computation of Gn(R) from F(R,F) (eqn. 7) allows the Pn to be computed either from densities directly (eqn. 2) or from the Fourier transform as follows:

(8)

Once the gn(r) are computed, equation (1) is used to resynthesize the density function, r(r,f). This polar image is then reconverted back to a Cartesian format, r(x,y) and displayed (e.g. Figs. III.E.3.6, III.E.3.8, III.E.3.9).

Recall the analogy of optical filtering and translational photographic superposition with crystalline (translationally symmetric) specimens (Sec.III.D.2.g, p.226). The optical diffraction pattern of such a specimen is basically a translational power spectrum of the specimen image. This diffraction pattern makes possible the objective analysis of the periodicities and preservation of the object's translational symmetry.

It is not possible to perform optical filtering of rotationally symmetric objects because the wanted and unwanted Fourier components are not spatially separated in the diffraction plane. Thus, it is necessary to compute the power spectrum and filter the image numerically. This procedure, like with translationally-symmetric specimens, involves two steps:

1. Analysis (Fourier analysis) to separate the image into Fourier components.

2. Synthesis (Fourier synthesis) to recombine just those components that satisfy the symmetry.

Photographic superposition methods attempt to determine the symmetry AND produce an average at the same time. The digital filtering approach is more reliable and powerful because of the separation of these two steps. An additional benefit of the digital processing procedure is that it provides quantitative assessment. Other advantages are that more complex operations can be performed on numerical data (e.g. CTF corrections), and it is easy to combine data from a number of different images and compute difference images.

References on Rotational Filtering: see page 7 of reading list. The following are the "best" sources for information.

Crowther, R. A. and L. A. Amos (1971) "Harmonic Analysis of Electron Microscope Images with Rotational Symmetry" J. Mol. Biol. 60:123-130.

Misell, D. L. (1978) "Image Analysis, Enhancement and Interpretation" Pract. Meth. Elec. Microsc. (Ed. A. M. Glauert) 4:129-139. (502.8 P881: on reserve; also TSB)

Moody, M. F. (1990) "Image Analysis of Electron Micrographs" In Biophysical Electron Microscopy: Basic Concepts and Modern Techniques (Eds. P. W. Hawkes and U Valdre) Academic Press, New York pp. 238-242 (578.45 B524: on reserve; also TSB).

c. 3D reconstruction of particles with icosahedral (532) point group symmetry

Three-dimensional reconstructions of spherical viruses are calculated by combining several unique views of the particle. Details of the procedure are outlined by Crowther (1971), Crowther and Amos (1972), and Crowther et al. (1970). These methods provide quantitative measures of the agreement between individual views and also indicate the resolution to which the 532 icosahedral point group symmetry is obeyed.

Examples of image processing studies of rotationally symmetric particles are given in Tables 3, 1.III.B.10, and IV.A.2 of Baker (1981). Except for the spherical viruses, no 3D reconstructions of rotationally symmetric particles have been computed with Fourier methods. This probably reflects the difficulty in reconstructing particles much smaller and of lower symmetry than the spherical viruses. Some 3D reconstructions of isolated enzyme molecules have been calculated using real-space, back-projection methods (Kiselev (1978) Proc. Ninth Intl. Cong. Elec. Microsc. (Toronto) 3:94-106; Samonidze, et al. (1978) Dokl. Biophys. 240:95-99; Tsuprun, et al. (1978) Sov. Phys. Cryst. 23:417-420; Vainshtein (1973) Sov. Phys.-Usp. 16:185-206). These methods will not be discussed in class.

The "common-lines" method for determining the orientation of a particle's rotation axes. ABCD is a section of the F.T., calculated from one image. The three-fold axis (along the Z-axis, pointing at the reader) generates from this two symmetry-related planes, of which only one (A'B'C'D') is shown. Both planes have common values along the line (1, 2, 3) of their intersection. (From Moody, 7.68, p.245)

The planes ABCD (a) and A'B'C'D' (b) of Fig. 7.67 laid flat. The common values lie along two lines, shown in (c). (From Moody, 7.69, p.246)