1. Introduction

In recent years, cryo-electron microscopy (cryo-EM, [1]) combined with single-particle analysis (SPA) has emerged as an important technique for the reconstruction of the three dimensional (3D) structure of biologically relevant macromolecules [2,3]. In cryo-EM, a monolayer of randomly oriented macromolecules embedded in an amorphous ice layer is imaged using transmission electron microscopy (TEM), resulting in the collection of a large number of randomly oriented projections of the same macromolecular structure. SPA makes use of image processing techniques to combine the information from the two dimensional (2D), often very noisy, projection images, into a 3D density map of the molecule [4–7].

The cryo-EM technique relies on using the native phase contrast generated by the macromolecules themselves rather than on the use of staining to enhance contrast. Due to the lack of reliable phase-plates for electron microscopy, the standard method for converting phase contrast to intensity contrast has been by defocusing the specimen - a practice that has its roots in the early days of light microscopy [8]. Defocusing the specimen leads to a parabolically varying (as a function of radial spatial frequency) phase shift (defocus aberration) in the exit pupil plane [9,10]. The transfer of contrast in this case is not ideal and is described by the contrast transfer function [11,12]:

where λ is the electron wavelength, v, the spatial frequency, Δz, the defocus and C_s, the spherical aberration coefficient ([12], p. 230). That a term describing the influence of spherical aberration has been included in Eq. (1) is due to the fact that it is the most significant aberration apart from the defocus term [9]. Perfectly coherent illumination is assumed in Eq. (1) though the partially-coherent illumination case can be described by a convolution with the appropriate illumination shape [13–17]. The form of the CTF implies that contrast will be reversed whenever the sinusoidal changes sign and that there will be spatial frequencies at which the transfer of contrast is equal to zero (the sinusoidal is equal to zero).

Owing to the fact that electron dose has to be severely limited in order to prevent specimen damage, one is forced to contend with poor signal-to-noise ratios (SNRs) in cryo EM images. As a consequence, the inversion of the CTF is difficult, especially in regions close to the CTF zeroes. Conventionally, this problem has been dealt with by taking images at different defoci, such that the CTF zeroes within the spatial frequency region of interest occur along different loci. By averaging “like” projection images taken at different defoci, it is thus possible to reduce the amount of missing information ([18], section 9.7).

Real-space averaging presupposes that the individual images taken at different defoci have the same magnification. The variation of magnification with defocus is often assumed to be negligible, or of a random nature. In the latter case, it may possible to measure and correct for differences in magnification using image processing techniques [18–20]. It has however recently been shown from a theoretical standpoint that for a coherent system, the magnification varies systematically with the specimen defocus and that this effect is strongly dependent on the curvature of the beam incident on the specimen [21,22]. To explain this effect, one needs to keep in mind that the diffraction plane is conjugate to the source. If the source is at infinity, then the diffraction plane coincides with the focal plane, but as the source is brought closer to the specimen, so does the diffraction plane move closer to the image plane. The image position and magnification are related to the object, but are independent of the source position. Consider now a sinusoidally varying object pattern. The various diffraction orders will be brought into focus at points in the diffraction plane. With continued propagation, the diffraction orders will diverge and combine, eventually forming the image at the image plane. The divergence of the diffraction orders will necessarily be stronger the closer the diffraction plane is to the image plane. Therefore, the variation in magnification with the defocus of the image will also be stronger. In simple terms, if a line is drawn from the centre of the diffraction plane to an image point, the intersection of this line with a defocused image plane gives us the height of the corresponding defocused image point in that plane.

For the simple model imaging system depicted in Fig. 1, the magnification is given by:

where M_o is the ratio of the image height to the specimen height, assuming that the specimen is positioned at the conjugate object distance, d_o, for an image plane located at d_i and a lens with a focal length f, Δf is the defocus effected by changing the focal length of the objective lens and d_c is the position of the condenser cross-over in relation to the objective lens. It should be noted that although the defocus is effected by changing the focal length on the microscope used in this study, it is specified in terms of the equivalent physical defocus of the object, Δd , which is related to the former by:

If we assume that the object distance and focal length are approximately equal, d_o ≈ f, and that the defocus is small compared with the focal length, Δf ≪ f, we may approximate Eq. (2) by:

where the change in focal length has been replaced by the equivalent physical defocus using Eq. (3). It should here be pointed out that if one uses the equation for the magnification variation for the case where it is effected by physically defocusing the object [21] as the starting point, one still arrives at the approximation given in Eq. (4). The key features of Eq. (4) are that a) the magnification varies linearly with defocus, b) the variation in magnification changes sign when we go from overfocused to underfocused illumination and c) the magnitude of the variation is exacerbated with increasing magnitude of curvature of illumination.

In this study, we have sought to experimentally verify these key features as well as to ascertain whether the magnification variation under typical imaging conditions is of such magnitude as to warrant any concern. In order to do so, we must be able to vary Δd and d_c - d_o in a controlled manner. Varying the defocus is relatively simple as it can be set to a certain value using a microscope control knob or via the microscope control software.

Although there is often a large discrepancy between the actual defocus and the nominal value, in our experience, the relationship between the two is repeatable and highly linear and can be compensated for once calibration has been performed. The position of the condenser crossover is controlled by the “Intensity” knob which in fact controls the focal length of the C2 condenser lens. However although the microscope provides a read-out of the lens currents, it does not provide a read-out of this focal length, or any other relevant length or distance for that matter. Rather than try and convert lens currents to relevant distances ourselves, we have opted for a simpler method which allows us to specify the variation in d_c - d_o in a relative sense which is adequate for the purposes of this study. The real microscope illumination system consists of several lenses, but as in [21,22] we replace the condenser system by a single thin lens coincident with the aperture stop as depicted in Fig. 1. The diameter of the aperture stop, w_c, can be related to the diameter of its projection onto the specimen plane, w_s, from basic geometric principles as

The approximation follows from the fact that for most work we use a C2 condenser aperture with a diameter that is greater than 50μm while working with a condenser projection that is 1-2μm in diameter; if we assume that the aperture stop of our model condenser system has approximately the same diameter as the C2 condenser aperture, it is reasonable to assume that |d_c - d_o| will be much smaller than d ₂₃. If w_c and d ₂₃ are fixed then changes in |d_c - d_o| are directly proportional to changes in the diameter of the condenser projection. The sign of d_c - d_o can be obtained by noting the direction in which the Intensity knob has been turned from the point at which the condenser is focused onto the specimen - clockwise for an overfocused condenser. There is no direct read-out for the diameter of the illumination projection, w_s, but it can be specified in terms of the fraction covered of the viewing screen, α, and M_o, if for a reference magnification, M_ref, the diameter of an object covering the viewing screen, w_ref, is known:

where the approximation is valid as long as we do not consider second order effects in subsequent steps. The viewing screen is designed to correspond to a specimen diameter of 1μm at a magnification of 100kX ([12], p. 93) and these values could be used for w_ref and M_ref respectively, although the experiments performed in this study have been designed to be independent of these parameters. By combining Eqs (4–6), the relative change in magnification can be expressed as:

where sgn(d_c - d_o) is equal to one for an overfocused condenser. M_ref, w_ref and d ₂₃ are constant throughout the measurements, the rest of the parameters can be quantified and varied.

In order to measure magnifications with high enough precision we have used a 2D crystalline specimen, catalase, and measured the distances between Fourier transform peaks corresponding to lattice periodicities with sub-pixel precision. In order to rule out the possibility that any variations that are observed are an artifact of the periodicity of the specimen, we have also performed magnification measurements by measuring the real-space distances between viral particles. The precision of the latter measurements is poorer than the former due to SNR issues.

 

Fig. 1. Schematic of the model system used to describe the imaging properties of the transmission electron microscope in this study. A defocused image of the point source is imaged onto the specimen via a condenser lens (rays in cyan). The specimen is imaged onto the image plane by the objective lens (rays in magenta). The conjugate object and image distances, for an objective lens with a focal length f, are d_o and d_i respectively. The specimen is defocused by changing the focal length of the objective lens by Δf. d_c is the distance between the objective lens and the illumination cross-over and d₂₃ is the distance between the specimen and the condenser lens. w_c is the diameter of the condenser aperture and w_s is the diameter of its projection onto the specimen. The naming convention used here has been adopted from [21,22].

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2. Materials and methods

2.1 Specimens

The high precision magnification measurements were performed using a specimen that consisted of negatively stained catalase crystals on an amorphous carbon support film (Agar Scientific Ltd, Essex, UK). Catalase has lattice plane spacings of approximately 8.75nm and 6.85nm [23]. A negatively stained (2% uranyl acetate) tomato bushy stunt virus (TBSV; kindly supplied by Dr Martin Svenda, Dept. of Cell and Molecular Biology, Uppsala University, Sweden) specimen was used for corroborating the experiments performed on the catalase crystal.

2.2 Electron microscopy

All experiments were performed on a CM300 electron microscope (FEI, Eindhoven, The Netherlands) equipped with a 2048² pixels TemCam-F224 CCD camera system (TVIPS GmbH, Gauting, Germany). Experiments were performed at room temperature but in low-dose mode using an acceleration voltage of 300kV, a gun lens setting of 4 and a spot size setting of 4. Three different magnifications were used with nominal values of 33kX, 51kX and 70kX, though they were found to correspond to 32kX, 62kX and 82kX, respectively, after calibration. In this paper, we have used the nominal values when categorizing the magnifications, but have used the calibrated values in all calculations. Defocus series, series of images taken at a fixed location but with the defocus incremented/decremented in fixed steps between images, were acquired using the EM-MENU 3.0 image processing software (TVIPS) that accompanied the CCD camera for different nominal magnifications, C2 condenser settings and C2 aperture settings. Four categories of the parameter α were examined: full screen - condenser aperture projection covering the full viewing screen, half screen - condenser aperture projection covering approximately half the viewing screen, double screen - condenser aperture projection spread so that the dose is approximately half that obtained with full screen and inner circle - condenser aperture projection filling the marked circle on the viewing screen, which is smaller than the half screen category. Most measurements were performed using the full screen and half screen categories. Note that for conventional cryo-EM work we use a magnification of approximately 50kX and illumination settings between the full screen and half screen categories depending on the thickness of the specimen and the desired dose.

In order to rule out the possibility that the effects that we were seeing were due to the accumulation of specimen damage, we a) repeated the experiments at a number of different locations in the specimen, b) changed the order in which they were performed, e.g. changing from first using overfocused illumination to first using underfocused and c) collected both, series where the defocus was incremented as well as series where it was decremented.

2.3 Image processing

Initial rough measurements of magnification were performed using EM-MENU. This involved manually measuring the distance between points in real or Fourier space. For the TBSV specimen, distances were measured between the centers of several viral particles on each image. More precise measurements on the catalase crystal specimen were performed using the Imagic image processing system (Image Science GmbH, Berlin, Germany). The acquired TIFF images were first converted to native Imagic format using the EM2EM command. A central 1024² pixel patch was then cut out from each catalase crystal image. A circular mask with a fractional diameter that was 0.9 times the size of the image was then applied to the cut patches, which were then normalized so that the average density in each was zero and the standard deviation of the densities equal to a fixed value (10). These images were then Fourier transformed to obtain images of their amplitude spectra. The PEAK-SEARCH-ALL command was then used to locate peaks corresponding to lattice periodicities with sub-pixel precision. This command finds the peak position by fitting a paraboloid to the peak density distribution. The magnification in real space is reciprocally related to scalings obtained in Fourier space.

Although nominal defocus values were available for all images, it has been our experience that there can be a significant discrepancy between these and those estimated from the micrographs. To overcome this difficulty we used the EM-MENU software to fit a CTF to the two extremes of each defocus series using the known catalase lattice period to determine the real magnifications. Since the automated focus steps used to collect the series have proven to be very consistent we estimated the intermediate defocus values by interpolation. It should noted that the measured defocus values will be affected by the beam curvature as pointed out in [21]. As this effect amounted to a less than 5% change in defocus values in the most severe cases presented, we have opted to present defocus values without compensating for it.

3. Results

According to Eq. (7), the relative change in magnification should vary linearly with defocus, with a positive gradient for an overfocused condenser and vice versa for an underfocused condenser. All the measurements presented in Fig. 2 display an essentially linear relationship between the relative change in magnification and the defocus. The dependency of the sign of the gradient on whether the condenser is overfocused or underfocused is confirmed if we compare the graphs on the left-hand side (overfocused condenser; Figs. 2(a), 2(c) and 2(e)) with those on the right-hand side (underfocused condenser; Figs. 2(b), 2(d) and 2(f)).

The ratio between the gradients of the curves for half screen and full screen illumination cases, given that all other parameters are fixed, should be equal to two according to Eq. (7). If we take the ratio between the gradients of half screen and full screen curves presented in each graph in Fig. 2, the mean value and standard deviation are 2.1 and 0.3 respectively. In Fig. 3(a) one curve corresponding to a more planar illumination setting than full screen (double screen) and one corresponding to a more divergent illumination that half screen (inner circle) are shown with both measurements taken at 51kX magnification. As expected, if these curves are compared with those shown in Fig. 2(c), the gradient is smaller for the double screen case than the full screen case and greater for the inner circle case than half screen case.

The ratio between the gradients for two different magnifications, given that all other parameters are constant, is equal to the ratio of the magnifications. In Fig. 2, the mean and standard deviation for the ratio, between the gradients of the curves at 51kX and corresponding curves at 33kX are 2.2 and 0.4 compared with an expected value of 1.9, between the curves at 70kX and 51kX are 1.6 and 0.2 compared with 1.4 and between the curves at 70kX and 33kX are 3.4 and 0.7 compared with 2.6.

Finally, the ratio between the gradients for two different condenser aperture diameters should be equal to the ratio of the diameters. The ratio between the curves for a C2 aperture diameter of 150μm to 50μm, as shown in Fig. 3(b), is 2.1. The measurements performed using the TBSV specimen (data not shown) confirm all the trends in magnification variation that have been mentioned above. The measurements performed thus confirm in their entirety the general features of the relationship between the relative change in magnification and defocus as predicted by Eq. (7).

4. Conclusion

The magnitude of the relative change in defocus under conditions that are typically used for cryo-EM, e.g. 51kX, C2 aperture diameter between 50 and 150 μm and illumination between the full and half screen settings, ranges from 0.4% to 0.9% per micron defocus. Considering a defocus range of ∼ 3μm, this amounts to a magnification variation of between 1% and 3% among the particle images collected. For a particle with a diameter of 20nm, this would translate into a variation in the diameter of between 0.2 and 0.6nm. Thus whereas this effect can be essentially ignored in low resolution projects aiming for no better than, say, 1nm resolution, it can be a limiting factor in projects aimed at resolving the secondary structure, or perhaps even the atomic resolution structure of the molecule.

On a practical level, the main take-home message of this study is to work with a small C2 aperture diameter and a large condenser projection on the specimen. However in cryo-EM, the freedom with which these parameters can be selected is severely restricted by the dose required to achieve an adequate SNR, specimen drift as well as beam-damage. Additionally, the use of other microscope parameters, such as the spot size, to compensate for the reduction in dosage may lead to an increase in CTF damping.

The use of image processing techniques for the a-posteriori compensation of magnification variation is limited by practical issues such as the SNR of the specimen [19] or the number of particle images available [20]. To achieve a sufficiently high precision, it may be necessary to mix the specimen of interest with a macromolecule that is easier to use for calibration purposes, such as an icosahedral virus. In our experience, a-posteriori compensation is tedious and time consuming and it would be preferable if the magnification variation could be compensated at the microscope level so that the images taken were virtually free from this effect.

 

Fig. 2. Magnification variation as a function of defocus. Each graph consists of two data sets, one corresponding to full screen illumination (red triangles) and one corresponding to half screen illumination (blue circles). Linear fits to these data sets, along with the equations of best fit and regression coefficients are shown in corresponding colors. The title of each graph specifies whether overfocused or underfocused illumination was used and the magnification. Comparing the graphs on the left (overfocused illumination) with those on the right (underfocused illumination); the sign of the variation can be seen to change. Comparing graphs row-by-row from top to bottom, the magnitude of the variation can be seen to increase with magnification. The magnitude of the variation can also be seen to be consistently higher for the half screen case than the full screen one in each graph.

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Fig. 3. (a) Magnification variation with defocus for two, more extreme cases of illumination than those shown in Fig. 2. The red triangles correspond to a case where the illumination is much more convergent than the half screen case, i.e., the inner circle case, and the blue circles correspond to the case where the condenser projection onto the specimen is approximately twice as large as the full screen case. (b) Comparison of the magnification variation for two different condenser aperture diameters and half screen illumination. The variation is stronger for the larger aperture diameter.

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The ideal solution to the problem of magnification variation is an illumination system that provides planar illumination over a useful range of illumination-area diameters, i.e., a Köhler illumination system. On the CM series and related microscopes, this could be achieved by coupling the change in the C2 condenser lens focal length to a change in the focal length of the “minicondenser” lens. An alternative, if illumination planarity cannot be achieved, would be to tweak the magnification of a non-critical post-objective projector lens to compensate for the magnification variation.