## Fourier amplitude correction using low-angle X-ray scattering data

Enhance the Fourier amplitudes of a reconstructed cryo-EM volume
such that they more closely resemble those of experimental
low-angle X-ray scattering data.

This approach is described in Gabashvili et al., *Cell* 100:537-549, March 3, 2000.

EM : 1D power spectrum of EM volume.

X-ray : Low-angle of X-ray scattering data, out to 8 Å.

FSC : Fourier shell correlation of the
EM volume. The Fourier amplitudes need only be corrected out to
the spatial frequency corresponding to the 0.5 FSC level, here 1/14 Å.

The EM Fourier amplitudes should be increased up to the level of the X-ray amplitudes.

To do this, create an enhancement curve from the EM and
X-ray data. The enhancement curve is the factor by which the
EM power spectrum curve must be multiplied so that it will approximate the X-ray curve.

Next, apply the enhancement curve as a filter to the EM volume.
The above data were plotted with a gnuplot script.
The EM data were first multiplied to be in the same numerical range
as the X-ray data (right axis), by running the procedure
b06.pww.

### X-ray data

Experimental X-ray curve, smoothed by Dmitri Svergun:

scattering8.tst exp. curve out to 8 Å

scattering11.tst exp. curve out to 11 Å

These data are in the form of SPIDER document files with 4 columns:

Frequency (1/Å) | intensity multiplied by 60 |
(unknown) | frequency in units of 2.93 Å |

## Amplitude correction

### Step 1 : Create an enhancement curve

Compute an enhancement curve, by comparing a 1D rotationally averaged
power spectrum of the input volume to the X-ray spectrum. This curve
represents the correction that must be applied to the volume's power
spectrum in order to bring it up to the X-ray curve.

enhance.bat
Inputs: | volume to be enhanced, |

| X-ray scattering data |

Outputs: | 1D power spectrum from volume, |

| enhancement curve |

pwsc.bat procedure called by enhance.bat

Given in input spatial frequency (Å),
returns the corresponding intensity from the X-ray data.

### Step 2 : Fit a polynomial to the enhancement curve

The **fit** command in gnuplot can be used to fit a curve to
the enhancement function.

Input: the enhancement curve created by enhance.bat
Output: Fitting parameters A,B (and maybe C)
gnuplot> K(x)=A*x*x+B
gnuplot> fit K(x) 'fen.dat' using 1:3 via A,B
gnuplot> plot 'fen.dat' using 1:3 with lines, K(x)
A = 623.42 +/- 20.02 (3.212%)
B = 13769.2 +/- 1.153e+04 (83.72%)

Or, you can try a 3 parameter expression for a better fit:
gnuplot> K(x)=C*x*x*x+A*x*x+B
gnuplot> fit K(x) 'fen.dat' using 1:3 via A,B,C

Gnuplot script to save parameters to a file,
and plot the curves in a gnuplot window.

Gnuplot script to send output to a postscript file.

### Step 3 : Apply the enhancement parameters to the volume

Output: Filtered volume with corrected higher frequencies.
applyabc.bat using the parameters obtained above,
apply the enhancement/correction to the volume, then filter
the volume at the resolution level.

applyfen.bat Alternatively,
apply the enhancement curve directly to the volume.

Older Procedure files

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Updated Feb. 20, 2003