InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 378
## Section 2.5.7.4. Assessment of the data quality and estimation of the image formation parameters P. A. Penczek
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The initial assessment of the quality of the micrographs is usually performed during the data collection and in most cases before the micrographs are digitized. The micrographs are examined visually and those that have noticeable drift, astigmatisms, noticeable contamination or simply too low a number of particles to justify further analysis are simply discarded. After digitization of the accepted micrographs, the first step is estimation of the power spectrum, which will be examined for the presence of Thon rings (thus confirming that the micrograph is indeed usable) and astigmatism.

The method of averaged overlapping periodograms (Welch, 1967) is commonly used in EM to calculate the power spectrum. It is designed to improve the statistical properties of the estimate by taking advantage of the fact that when *K* identically distributed independent measurements are averaged, the variance of the average is decreased with respect to the individual variance by the ratio 1/*K*. Thus, instead of calculating a periodogram (squared moduli of the discrete Fourier transform) of the entire micrograph field, one subdivides it into much smaller windows, calculates their periodograms and averages them. Typically, one would chose a window size of 512 × 512 pixels and an overlap of 50%, which will result in the reduction of the variance of the estimate to few percent with respect to the variance of the periodogram of the entire field (Fernandez *et al.*, 1997; Zhu *et al.*, 1997). Further reduction of the variance is achieved by rotational averaging of the 2D power-spectrum estimate. The resulting one-dimensional (1D) profile is finally used in the third step of our procedure.

For a set of micrographs the power spectra can be evaluated either visually or computationally in an automated fashion. Of main concern are the presence of Thon rings, the astigmatism and the extent to which Thon rings can be detected. Although in principle astigmatic data could be used in subsequent analysis (in fact, astigmatism could be considered advantageous, as particles from the same micrograph would contain complementary information in Fourier space), in practice they are discarded as currently there is no software that can process astigmatic data efficiently. The extent of Thon rings indicates the `resolution' of the data, *i.e.*, the maximum frequency to which information in the data can be present.

A number of well established programs can assist the user in the calculation of power spectra and automated estimation of defocus and astigmatism (Huang *et al.*, 2003; Mindell & Grigorieff, 2003; Sander *et al.*, 2003; Mallick *et al.*, 2005). Given the analytical form of the CTF [(2.5.7.4)], the problem is solved by a robust fitting of the CTF parameters such that the analytical form of the CTF matches the power spectrum of the micrograph. Usually, the steps employed are: (1) robust estimation of the power spectrum; (2) calculation of the rotational average of the power spectrum; (3) subtraction from this rotational average of the slowly decreasing background [roughly corresponding to *P*_{B} in (2.5.7.5)]; (4) fitting of the defocus value using known settings of the microscope (voltage, spherical aberration constant, …) and usually assuming a constant and known value of the amplitude contrast ratio *q* (for cryo-EM data, *q* should be in the range 0.02–0.10); and (5) using the established defocus value , analysis of the 2D power spectrum and fitting of the astigmatism amplitude and angle while refining the defocus. As long as the defocus value is not too small and there are at least two detectable zeros of the CTF, all available programs give very good and comparable results.

In some single-particle packages, the automated calculation of defocus is integrated with the estimation of additional characteristics of the image-formation parameters that are required for advanced application of a Wiener filter [(2.5.7.18)] (Saad *et al.*, 2001; Huang *et al.*, 2003), *i.e.*, the power spectra of two noise distributions *P*_{S} and *P*_{B} and the envelope function of the microscope for each micrograph. A possible approach is to select slowly varying functions and fit their parameters to match the estimates of *P*_{S}, *P*_{B} and *P*_{d} obtained from the data. Finally, it is necessary to have a description of the 1D rotationally averaged power spectrum of the complex *P*_{f} . One possibility is to carry out X-ray solution scattering experiments (Gabashvili *et al.*, 2000; Saad *et al.*, 2001) that yield a 1D power spectrum of the complex in solution. However, these experiments require large amounts of purified sample and the accuracy of the results in terms of the overall fall-off of the power spectrum can be disputed. For the purpose of cryo-EM, a simple approximation of the protein power spectrum by analytical functions is satisfactory.

### References

Fernandez, J.-J., Sanjurjo, J. R. & Carazo, J. M. (1997).*A spectral estimation approach to contrast transfer function detection in electron microscopy*.

*Ultramicroscopy*,

**68**, 267–295.

Gabashvili, I. S., Agrawal, R. K., Spahn, C. M., Grassucci, R. A., Svergun, D. I., Frank, J. & Penczek, P. (2000).

*Solution structure of the E. coli 70S ribosome at 11.5 Å resolution*.

*Cell*,

**100**, 537–549.

Huang, Z., Baldwin, P. R., Mullapudi, S. R. & Penczek, P. A. (2003).

*Automated determination of parameters describing power spectra of micrograph images in electron microscopy*.

*J. Struct. Biol.*

**144**, 79–94.

Mallick, S. P., Carragher, B., Potter, C. S. & Kriegman, D. J. (2005).

*ACE: automated CTF estimation*.

*Ultramicroscopy*,

**104**, 8–29.

Mindell, J. A. & Grigorieff, N. (2003).

*Accurate determination of local defocus and specimen tilt in electron microscopy*.

*J. Struct. Biol.*

**142**, 334–347.

Saad, A., Ludtke, S. J., Jakana, J., Rixon, F. J., Tsuruta, H. & Chiu, W. (2001).

*Fourier amplitude decay of electron cryomicroscopic images of single particles and effects on structure determination*.

*J. Struct. Biol.*

**133**, 32–42.

Sander, B., Golas, M. M. & Stark, H. (2003).

*Automatic CTF correction for single particles based upon multivariate statistical analysis of individual power spectra*.

*J. Struct. Biol.*

**142**, 392–401.

Welch, P. D. (1967).

*The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short modified periodograms*.

*IEEE Trans. Audio Electroacoust.*

**AU-15**, 70–73.

Zhu, J., Penczek, P. A., Schröder, R. & Frank, J. (1997).

*Three-dimensional reconstruction with contrast transfer function correction from energy-filtered cryoelectron micrographs: procedure and application to the 70S Escherichia coli ribosome*.

*J. Struct. Biol.*

**118**, 197–219.