## 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.

• Updated Feb. 20, 2003