Fourier shell correlation explained

In structural biology, as well as in virtually all sciences that produce three-dimensional data, the Fourier shell correlation (FSC) measures the normalised cross-correlation coefficient between two 3-dimensional volumes over corresponding shells in Fourier space (i.e., as a function of spatial frequency^[1]). The FSC is the three-dimensional extension of the two-dimensional Fourier ring correlation (FRC);^[2] also known as: spatial frequency correlation function.^[3]

Calculation

FSC(r) = (\displaystyle\sum _{r _{i} \in r} { F _{1(r} _{i)} ⋅ F _{2(r} ^{\ast} ^{i)})/() }

where

F _{1} is the complex structure Factor for volume 1, F ^{\ast} _{2} is the complex conjugate of the structure Factor for volume 2, and r _{i} is the individual voxel element at radius r .^{[4] [5] [6]} In this form, the FSC takes two three-dimensional data sets and converts them into a one-dimensional array.

Applications

The FSC originated in cryo-electron microscopy and gradually proliferated to other fields. To measure the FSC, two independently determined 3D volumes are required. In cryo-electron microscopy, the two volumes are the result of two three-dimensional reconstructions, each based on half of the available data set. Typically, random halves are used, although some programs may use the even particle images for one half and the odd particles for the other half of the data set. Some publications quote the FSC 0.5 resolution cutoff, which refers to when the correlation coefficient of the Fourier shells is equal to 0.5.^{[7] [8]} However, determining the resolution threshold remains a controversial issue, with some arguing fixed-value thresholds to be based on incorrect statistical assumptions.^{[6] [9]} Many other criteria using the FSC curve exist, including 3-σ criterion, 5-σ criterion, and the 0.143 cutoff. The half-bit criterion indicates at which resolution we have collected enough information to reliably interpret the 3-dimensional volume, and the (modified) 3-sigma criterion indicates where the FSC systematically emerges above the expected random correlations of the background noise.^[6] The FSC 0.143 cutoff was proposed in part to make the resolution measurement comparable to measurements used in X-ray crystallography.^[10] Currently, the 0.143 cutoff is the most commonly used criterion for the resolution of cryo-EM reconstructions better than 10 ångström resolution.^[11]

See also

• Resolution (electron density)

References

• Harauz . G. . van Heel M. . Exact filters for general geometry three dimensional reconstruction . Optik . 73 . 146 - 156 . 1986.
• Book: van Heel , M. . Keegstra, W.. Schutter, W.. van Bruggen E.F.J. . Arthropod hemocyanin studies by image analysis, in: Structure and Function of Invertebrate Respiratory Proteins, EMBO Workshop 1982, E.J. Wood . Life Chemistry Reports . Suppl. 1 . 69 - 73 . 1982 . 9783718601554.
• Saxton . W.O. . W. Baumeister . The correlation averaging of a regularly arranged bacterial cell envelope protein . Journal of Microscopy . 127 . 127 - 138 . 1982 . 10.1111/j.1365-2818.1982.tb00405.x . 2. 7120365 . 27206060 .
• Böttcher . B. . Wynne, S.A.. Crowther, R.A. . Determination of the fold of the core protein of hepatitis B virus by electron microscopy . Nature . 386 . 88 - 91 . 10.1038/386088a0 . 1997 . 9052786 . 6620. 275192 .
• Optimal determination of particle orientation, absolute hand, and contrast loss in single-particle electron cryomicroscopy . Journal of Molecular Biology . 2003 . 333 . 721 - 745 . Rosenthal . P.B. . Henderson, R. . 0022-2836 . 14568533 . 4 . 10.1016/j.jmb.2003.07.013.
• Fourier shell correlation threshold criteria . Journal of Structural Biology . 2005 . van Heel . M. . Schatz, M. . 151 . 250 - 262 . 10.1016/j.jsb.2005.05.009 . 16125414 . 3 .
• Book: Frank , J. . Joachim Frank

. Joachim Frank . Three-Dimensional Electron Microscopy of Macromolecular Assemblies . . New York . 2006 . 0-19-518218-9 .

• Reassessing the Revolution's Resolution . bioRxiv . 2017 . van Heel . M. . Schatz, M. . 10.1101/224402 . free .

External links

• EMstats Trends and distributions of maps in EM Data Bank (EMDB), e.g. resolution trends

Notes and References

1. Harauz & van Heel, 1986
2. van Heel, 1982
3. Saxton & Baumeister, 1982
4. Web site: Image Science's FSC: Program to calculate the Fourier Shell Correlation (FSC) of two 3D volumes . fsc . Image Science . 2009-04-09.
5. Web site: RF 3 - Phase Residual & Fourier shell correlation . SPIDER . Wadsworth Center . 2009-04-09.
6. van Heel & Schatz, 2005
7. Böttcher et al., 1997
8. Frank, 2006, p250-251
9. van Heel & Schatz, 2017
10. Rosenthal & Henderson, 2003
11. Web site: The Electron Microscopy Data Bank. www.ebi.ac.uk. 2019-01-07.