Fourier shell correlation threshold criteria

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Abstract

The resolution value claimed for an electron microscopical three-dimensional reconstruction indicates the overall quality of the experiment. The Fourier shell correlation (FSC) criterion has now become the standard quality measure. However, what has continued to be controversial is the issue of the FSC threshold level at which one defines the reproducible resolution. Here, we discuss the theoretical behaviour of the FSC in conjunction with the various factors which influence it: the number of “voxels” in a given Fourier shell, the symmetry of the structure, and the size of the structure within the reconstruction volume. Both the theoretical considerations and our model experiments show that fixed-valued FSC threshold (like “0.5”) may never be used in a reproducible criterion. Fixed threshold values are—as we show here—simply the result of incorrect assumptions in the basic statistics. Two families of FSC threshold curves are discussed: the σ-factor curves and the new family of bit-based information threshold curves. Whereas σ-factor curves indicate the resolution level at which one has collected information significantly above the noise level, the information curves indicate the resolution level at which enough information has been collected for interpretation.

Introduction

Three-dimensional electron microscopy of vitrified individual biological macromolecules (“single-particle cryo-EM”) has become a successful branch of structural biology as exemplified by the exponentially growing number of “cryo-EM” papers that have been published in recent years. The excellent specimen preservation associated with this specimen preparation technique (Adrian et al., 1984) is superior to that obtained with the earlier dry negative-stain preparations. Significant further progress with the cryo-EM specimen handling and data collection has been achieved since the spectacular first cryo-EM images of viruses were obtained. Equally impressive developments of data processing methods and of the computing hardware have made that the resolution levels achievable by single-particle cryo-EM are now better than one nanometer and are commencing to approach “atomic” resolution levels (see review Van Heel et al., 2000). With “atomic” resolution we here mean a resolution level at which one can interpret the resulting maps in terms of an atomic model. With the rapidly increasing importance of the technique, the long-standing discussion on what resolution actually means has again come to the foreground. The purpose of this paper was to discuss the theoretical and practical aspects of resolution assessment in single-particle cryo-EM. We discuss the implicit assumptions on which fixed-valued FSC threshold criteria have been based. In cryo-EM resolution tests, one typically splits the full data set in two half data sets, and then calculates two independent three dimensional (3D) reconstructions. Subsequently, one compares the consistency of the two independent 3D reconstructions as function of spatial frequency.

The three-dimensional Fourier shell correlation (FSC) was introduced by Harauz and van Heel (1986). It measures the normalised cross-correlation coefficient between two 3D volumes over corresponding shells (ri) in Fourier space, i.e., as a function of spatial frequency (r):FSC12(ri)=rriF1(r)·F2(r)rriF12(r)·rriF22(r).In this formula, F (r) is the complex “structure factor” at position r in Fourier space, and the “*” denotes complex conjugation (we drop its explicit mentioning henceforth). The summations are over all Fourier-space voxels “r” that are contained in the shell “ri.” The FSC is the straightforward three-dimensional (3D) generalisation of the earlier two-dimensional (2D) Fourier ring correlation (FRC) function. The 2D FRC had been introduced independently by van Heel et al. (1982) and Saxton and Baumeister (1982) (in the latter work under the name: “spatial frequency correlation function”). The FRC was, in turn, a reaction to the Differential Phase Residual proposed as a 2D resolution criterion for single particle averaging, (“DPR,” Frank et al., 1981). Phase residuals already had a history both in X-ray crystallography and in electron microscopy (see DeRosier and Moore, 1970, Unwin and Klug, 1974). The particular definition of the DPR phase residual introduced by Frank et al. (1981) however, included sums of Fourier amplitudes, making the result dependent of the relative scaling of the densities in the two images, and thus making it unsuitable as a reproducible resolution criterion (Orlova et al., 1997, Van Heel, 1987). Although corrected phase residuals have been defined to remove this flaw (Van Heel, 1987), one of the main criticisms against the use of phase residuals in general remained the necessity to use a fixed threshold value (Orlova et al., 1997, Van Heel, 1987). In recent years, the DPR has been abandoned in favour of the FSC.

Various authors have, however, again started to use fixed-value resolution thresholds, but now in conjunction with FSC curves. Most fashionable is currently a “0.5” value as the resolution-defining threshold (Beckmann et al., 1997, Böttcher et al., 1997). More recently the value “0.143” has been proposed as more realistic value (Rosenthal et al., 2003). A fixed threshold cannot account for, for example, the varying number of voxels in a Fourier “shell” and exactly this is one of the strong points of the “2σ”-threshold criterion, originally introduced by Saxton and Baumeister (1982) in connection with their definition of the “spatial frequency correlation function.” The 2σ-threshold criterion was proposed as an approximate significance threshold value based on the expected theoretical pure-noise behaviour of the Fourier-space cross-correlation coefficient (Saxton and Baumeister, 1982). Note that when the original 3D FSC curve was introduced (Harauz and van Heel, 1986), it was argued that a curve was a richer representation of the quality of the data than a single valued resolution statement and no explicit resolution threshold criterion was introduced. FSC curves, were nevertheless already shortly after their introduction always accompanied by “σ-factor” curves. The currently most widely used threshold curve in connection with 3D FSC data is the σ-factor curve (Orlova et al., 1997):σ(ri)=σfactorn(ri)/2·nasym.In this formula, n (ri) is the number of voxels contained in Fourier shell of radius ri; the extra factor of “2” is required since the FSC summations include all Hermitian pairs in Fourier space; nasym is the number of asymmetric units within the given pointgroup symmetry (“1” for an asymmetrical object, up to “60” for objects with icosahedral symmetry) (Orlova et al., 1997). The most frequently used value for the “σ-factor” threshold is “3,” meaning that one chooses three standard deviations above the expected random noise fluctuations as a significance threshold (known as “3σ-curve”).

Section snippets

FSC theoretical behaviour

A good understanding of the expected statistical behaviour of the FSC itself is a prerequisite for appreciating the requirements for resolution-defining FSC threshold criteria.

Model experiments

We here conduct a number of straightforward computer model experiments to study the behaviour of the FSC and its threshold criteria. By plotting a large number of FSC plots onto the same frame, one yields a visual impression of the typical extent of the FSC curves as function of spatial frequency (Fig. 2). First, whole series of random-noise 3D densities or “phantoms” have been generated and correlated with each other, leading to hundreds of FSC curves. The behaviour of all FSC model

The σ-factor threshold curves

The FSC curves between the pure random noise phantoms confirm the expected statistical behaviour of random-noise correlations with respect to the family of σ-factor curves. These experiments have been conducted with asymmetrical (“C1”) and with symmetrised random-noise phantoms (“D4,” etc.; Fig. 2). The experiments were also conducted with different degrees of filling of the 3D volume (Fig. 3). In all cases, the expected theoretical behaviour was confirmed by the computer model experiments.

Let

Take home lessons

In this paper, we have studied the theoretical behaviour of FSC threshold criteria: the σ-factor criterion, and the new family of information-based threshold criteria. The meaning of these families of threshold curves is different and complementary. In detail:

(1) Assuming the data have been collected with a sufficiently high sampling rate (see below), the FSC should oscillate around the value zero close to the highest possible frequency, the Nyquist frequency. The 3σ criterion indicates where

Conclusions

There is no scientific justification for the use of fixed FSC threshold criteria (such as the “0.5” criterion) as is emphasised by our theoretical considerations and our model experiments. All fixed-valued threshold criteria are implicitly or explicitly the result of flawed assumptions on the statistical behaviour of the noise and the signal components of the data. Fixed FSC-threshold values can be overly “conservative” or excessively “liberal,” depending on various secondary factors unrelated

Acknowledgments

We thank Drs. Ardan Patwardhan and Richard Henderson for discussions and for their comments on the manuscript, and Ralf Schmidt for software maintenance. This research was financed in part by various grants from the BBSRC and the European Union (especially FP6 NOE). The work contained in this paper was first presented in public during an EU-IIMS Software Developers meeting in Hinxton in November 2002.

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