Accurate determination of local defocus and specimen tilt in electron microscopy

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Abstract

Accurate knowledge of defocus and tilt parameters is essential for the determination of three-dimensional protein structures at high resolution using electron microscopy. We present two computer programs, CTFFIND3 and CTFTILT, which determine defocus parameters from images of untilted specimens, as well as defocus and tilt parameters from images of tilted specimens, respectively. Both programs use a simple algorithm that fits the amplitude modulations visible in a power spectrum with a calculated contrast transfer function (CTF). The background present in the power spectrum is calculated using a low-pass filter. The background is then subtracted from the original power spectrum, allowing the fitting of only the oscillatory component of the CTF. CTFTILT determines specimen tilt parameters by measuring the defocus at a series of locations on the image while constraining them to a single plane. We tested the algorithm on images of two-dimensional crystals by comparing the results with those obtained using crystallographic methods. The images also contained contrast from carbon support film that added to the visibility of the CTF oscillations. The tests suggest that the fitting procedure is able to determine the image defocus with an error of about 10 nm, whereas tilt axis and tilt angle are determined with an error of about 2° and 1°, respectively. Further tests were performed on images of single protein particles embedded in ice that were recorded from untilted or slightly tilted specimens. The visibility of the CTF oscillations from these images was reduced due to the lack of a carbon support film. Nevertheless, the test results suggest that the fitting procedure is able to determine image defocus and tilt angle with errors of about 100 nm and 6°, respectively.

Introduction

Electron cryo-microscopy is a versatile technique to determine the three-dimensional (3D) structure of proteins and protein complexes. It can be applied to samples of different symmetry and geometry, such as two-dimensional (2D) crystals (Henderson et al., 1990; Kühlbrandt and Wang, 1991; Mindell et al., 2001; Murata et al., 2000; Nogales et al., 1998), helical particles (Morgan et al., 1995; Wells et al., 1999; Wendt et al., 2002), highly symmetrical viruses (Böttcher et al., 1997; Kuhn et al., 2002; Zhou et al., 2001), and other particles with lower or no symmetry (Frank and Agrawal, 2000; Grigorieff, 1998; Ranson et al., 2001). The processing of an image obtained from any of these samples usually includes the determination of lens defocus and astigmatism needed to correct the measured data for the contrast transfer function (CTF) of the electron microscope (Erickson and Klug, 1971; Wade, 1992). The correction is necessary because the CTF causes resolution-dependent amplitude modulations (Thon rings; Thon, 1966) and phase reversals in the image (Fig. 1). The Thon rings can be observed in a power spectrum of an image, and they are particularly strong when the image contains additional contrast from a carbon film supporting the sample. Towards higher resolution, as the CTF oscillations become more rapid, accurate knowledge of defocus and astigmatism are essential for correct determination of the image phases. When ordered samples are imaged, such as 2D crystals, the diffraction patterns calculated from the images can be used to determine image defocus and astigmatism accurately (for example, see Henderson et al., 1986). Yet, methods based on diffraction patterns are not applicable to samples that contain aperiodic structures, such as single protein molecules and complexes. In those cases, image defocus and astigmatism can still be determined from the shape and position of the Thon rings. Often, samples of single protein molecules are prepared for electron microscopy using a carbon film containing approximately 1 μm wide holes. This has the advantage that images of molecules suspended in a thin layer of ice inside the holes show reduced background due to the absence of the carbon support film. However, the lack of contrast from a carbon film also means that the Thon rings are reduced in their visibility, making the determination of defocus parameters more difficult. An additional complication arises with tilted specimens because the amount of defocus is then also a function of the position on the specimen. Tilt can be introduced deliberately to obtain 3D information of a sample that exhibits a preferred orientation on the carbon support film, such as a 2D crystal. Often, however, a small but significant tilt is introduced unintentionally due to undulations in the carbon support film. These undulations can sometimes be enhanced when the specimen is frozen, due to the difference in thermal expansion coefficients between the carbon support film and the metal grid the film is deposited on. This effect is sometimes referred to as ‘cryo-crinkling’ and can be reduced by using molybdenum grids (Vonck, 2000).

Image formation in the electron microscope results from a combination of sample-induced elastic and inelastic electron scattering. In general, inelastic scattering produces an almost featureless background in the image power spectrum that is high at low resolution and falls off towards higher resolution (Zhu et al., 1997). Amplitude contrast in an image is produced by high-angle scattering when the electrons are scattered outside the objective aperture. The elastically scattered electrons that pass through the objective aperture produce the phase contrast that contains most of the structural information in cryo-EM of unstained specimens; the amount of amplitude contrast from these specimens is usually very small. The amplitude and phase contrast are modulated by the CTF of the microscope which is a function of defocus, astigmatism, lens errors, electron wave length, as well as temporal and spatial coherence of the electron beam (Zhu et al., 1997). The image recording process on film or CCD also contributes to the CTF (Koeck, 2000). The inelastic component of the contrast in an image giving rise to the smooth background is not used in current structural analyses.

Precise determination of the CTF based solely of the shape and position of Thon rings is difficult; it would require knowledge of all the parameters listed above as well as a detailed understanding of inelastic and multiple scattering. Comprehensive determination of the CTF from images of single molecules has been successful with energy-filtered images, where the bulk of the background due to inelastic scattering was removed (Zhu et al., 1997). Energy filters are not, however, routinely available on electron microscopes used in structural biology. Even without a detailed understanding of all the scattering components contributing to the contrast in an unfiltered image, the oscillations due to the CTF can usually be interpreted in a straight-forward manner. The periodic phase reversals corresponding to these oscillations are the only effect of the CTF on image phases, and these in turn are the predominant determinants of molecular structure (Ramachandran and Srinivasan, 1961). The phase reversals can be roughly described by a simple oscillating function of constant amplitude, determined by defocus, astigmatism, and spherical aberration of the objective lens. In the comprehensive description, this in turn is multiplied by an envelope function which monotonically attenuates the CTF towards higher spatial resolution, and which captures the effects of spatial and temporal beam incoherence (Fig. 1; Wade, 1992). We show here that defocus and astigmatism, the parameters which determine the oscillatory component of the CTF, can be determined in images of untilted specimens with sufficient accuracy to obtain corrected phases at 0.3-nm resolution without knowing the shape of the envelope. The procedure of defocus determination for images of untilted specimens has been implemented in a computer program called CTFFIND3. We further show that the defocus information across an image of a tilted specimen can be used to accurately determine specimen tilt axis and tilt angle. The entire process of defocus and tilt determination has been automated in a new computer program called CTFTILT. Both programs are written in FORTRAN 77 code and are available from the corresponding author. The programs were tested on a 2.4-GHz Pentium 4 PC running Linux. For moderately sized images of about 6000 × 6000 pixels, the analysis performed by CTFFIND3 is completed in about 2 min, whereas the analysis performed by CTFTILT is completed about 15 min.

Section snippets

Determination of defocus from power spectra

To determine the defocus and astigmatism for an image, our objective is to fit the oscillatory component of the measured power spectrum in two dimensions. However, to do so on a full sized image (>10,000 × 10,000 pixels) would be computationally expensive. Furthermore, such a fit would be contaminated by portions of the image containing unexposed areas of film or other aberrations. Thus, we subdivide the image into small square tiles, then calculate the power spectrum of each tile, summing the

Results

Because the CTF is a complex function of many variables, we have used a largely empirical approach to extract the defocus and tilt parameters which strongly influence the final image phases. The two major features of the algorithm for untilted images are the background subtraction and the full, two-dimensional fit of the power spectrum to an unattenuated CTF.

The fitting algorithm implemented in CTFFIND3 was tested on high-resolution electron micrographs of 2D crystals used to calculate a

Discussion

Precise defocus and tilt information is essential for high-resolution protein structure analysis using electron microscopy. The oscillations visible in the power spectrum of an electron micrograph (Thon rings; Thon, 1966) have been used before to determine defocus parameters for untilted specimens (for example, Conway and Steven, 1999; Frank, 1972; Henderson et al., 1986; Ludtke et al., 1999; Morgan et al., 1995; Tani et al., 1996; Zhu et al., 1997), and to obtain tilt information (Henderson

Conclusions

We have developed two computer programs, CTFFIND3 and CTFTILT, that determine defocus and astigmatism in images of untilted specimens, and defocus, astigmatism tilt angle, and tilt axis direction in images of tilted specimens, respectively. The programs run fully automatically and can determine defocus and tilt parameters with high accuracy. The accuracy of the algorithms were tested on untilted and tilted images of two-dimensional crystals with known defocus and tilt values, and on images of

Acknowledgements

The authors thank Dr. Thomas Walz for providing images of aquaporin crystals to test our algorithms. The authors are grateful to Dr. David DeRosier, Dr. Thomas Walz, and Dr. Brian Andrews for critical comments on the manuscript. This work was supported in part by NIH Grants GM-62580 (N.G.) and DK-2801 (J.M.).

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