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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)={\frac {\displaystyle \sum _{r_{i}\in r}{F_{1}(r_{i})\cdot F_{2}(r_{i})^{\ast }}}{\displaystyle {\sqrt[{2}]{\sum _{r_{i}\in r}{\left|F_{1}(r_{i})\right|^{2}}\cdot \sum _{r_{i}\in r}{\left|F_{2}(r_{i})\right|^{2}}}}}}

where $F_{1}$ is the complex structure Factor for volume 1, $F_{2}^{\ast }$ 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]

Notes

^ Harauz & van Heel, 1986
^ van Heel, 1982
^ Saxton & Baumeister, 1982
^ "Image Science's FSC: Program to calculate the Fourier Shell Correlation (FSC) of two 3D volumes". fsc. Image Science. Retrieved 2009-04-09.
^ "RF 3 - Phase Residual & Fourier shell correlation". SPIDER. Wadsworth Center. Retrieved 2009-04-09.
^ ^a ^b ^c van Heel & Schatz, 2005
^ Böttcher et al., 1997
^ Frank, 2006, p250-251
^ van Heel & Schatz, 2017
^ Rosenthal & Henderson, 2003
^ "The Electron Microscopy Data Bank". www.ebi.ac.uk. Retrieved 2019-01-07.

References

Harauz, G.; van Heel M. (1986). "Exact filters for general geometry three dimensional reconstruction". Optik. 73: 146–156.
van Heel, M.; Keegstra, W.; Schutter, W.; van Bruggen E.F.J. (1982). Arthropod hemocyanin studies by image analysis, in: Structure and Function of Invertebrate Respiratory Proteins, EMBO Workshop 1982, E.J. Wood. Vol. Suppl. 1. pp. 69–73. ISBN 9783718601554. {{cite book}}: |journal= ignored (help)
Saxton, W.O.; W. Baumeister (1982). "The correlation averaging of a regularly arranged bacterial cell envelope protein". Journal of Microscopy. 127 (2): 127–138. doi:10.1111/j.1365-2818.1982.tb00405.x. PMID 7120365. S2CID 27206060.
Böttcher, B.; Wynne, S.A.; Crowther, R.A. (1997). "Determination of the fold of the core protein of hepatitis B virus by electron microscopy". Nature. 386 (6620): 88–91. doi:10.1038/386088a0. PMID 9052786. S2CID 275192.
Rosenthal, P.B.; Henderson, R. (2003). "Optimal determination of particle orientation, absolute hand, and contrast loss in single-particle electron cryomicroscopy". Journal of Molecular Biology. 333 (4): 721–745. doi:10.1016/j.jmb.2003.07.013. ISSN 0022-2836. PMID 14568533.
van Heel, M.; Schatz, M. (2005). "Fourier shell correlation threshold criteria". Journal of Structural Biology. 151 (3): 250–262. doi:10.1016/j.jsb.2005.05.009. PMID 16125414.
Frank, J. (2006). Three-Dimensional Electron Microscopy of Macromolecular Assemblies. New York: Oxford University Press. ISBN 0-19-518218-9.
van Heel, M.; Schatz, M. (2017). "Reassessing the Revolution's Resolution" (PDF). bioRxiv. doi:10.1101/224402.