Babinet’s principle and corrections for the scattering contribution of the bulk
solvent in protein crystallography. (Richard Kingston, Feb 26 2008).
Useful review material can be found in (Badger, 1997; Tronrud, 1997; Kostrewa, 1997).
The bulk solvent present in protein crystals scatters X-rays.
Protein crystals generally contain between 30 and 70% solvent (Matthews, 1968; Kantardjieff and
Rupp, 2003), most of which is disordered and fills the channels and interstices between protein
molecules in the crystal. If our model of X-ray scattering from the crystal neglects this “bulk
solvent”, and we calculate structure factor amplitudes from the protein alone, there will be large
discrepancies between the observed and calculated amplitudes at low resolution. This was
recognized even before the first protein structures had been solved. By manipulating the electron
density of the solvent, Lawrence Bragg and Max Perutz were able to observe systematic changes in
the intensity of the low order diffraction data, and infer the approximate dimensions of
Haemoglobin (Bragg and Perutz, 1952).
From these early experimental observations we can deduce that neglecting scattering from the
bulk solvent (i.e placing the protein molecules “in a vacuum”) will lead to calculated structure
factor amplitudes that are generally much too large.
How do we model scattering from the bulk solvent?
First let’s partition the total electron density in the cell into two parts protein and the other due to the bulk solvent
on part due to the
tTOTAL ] r g = tPROTEIN ] r g + tSOLVENT ] r g
Or equivalently, in “reciprocal space”, we can write
FTOTAL ]h g = FPROTEIN ]h g + FSOLVENT ]h g
(This works because the Fourier transform is a linear operator)
We can calculate the structure factors from the protein FPROTEIN(h) readily enough. But how do we
calculate FSOLVENT(h), the contribution from the bulk solvent? One way to proceed is to use a real
space modeling method, as introduced by Simon Phillips (Phillips, 1980).
In this procedure
a) A grid is created, covering the unit cell.
b) The boundary between protein and solvent is defined.
c) All grid points in the solvent region are assigned a value of 1, while all those in protein region
are assigned a value of 0. The resulting binary function is termed the solvent mask
If we Fourier transform the solvent mask we get FMASK(h) - the structure factors of the mask.
From these, FSOLVENT(h), the structure factors of the bulk solvent, can be readily calculated...
2
FSOLVENT ]h g = KSOL exp a -BSOL sin i
m2
k FMASK ]h g
where the parameter Ksol is the mean electron density of the solvent. The resolution dependent
multiplier (involving the parameter Bsol) is introduced in order to blur the sharp boundary between
the protein and solvent regions. The two parameters Ksol and Bsol can be adjusted during
refinement of the model. There are additional complexities in the definition of the protein
boundary, and hence the solvent mask. Details of the procedure now in common use are given in
Jiang and Brünger (Jiang and Brünger, 1994)
There’s also a second way to proceed - which relies on a basic result from optics ... Babinet’s
principle
Babinet’s principle
Babinet's principle, when applied to the diffraction patterns of complementary masks (masks in
which the opaque parts of one correspond to the clear parts of the other and vice-versa), states
that such masks will give identical diffraction patterns except for a small region in the centre. The
diffracted waves from each mask have the same amplitude but opposite phase( = phase difference
of 180°).
Experimental realization of Babinet’s principle: A 1 dimensional example, using optical diffraction
(Adapted from (Tavassoly et al., 2009))
A simple scaling method incorporating Babinet’s principle
If we consider what a protein crystal looks like at very low resolution, there’s little structural
variation in either protein or solvent region. If a scale factor Ksol is applied to account for the
difference in mean density between protein and solvent, and an artificial temperature factor Bsol, is
used to eliminate structural features within the protein region, we can employ Babinet’s principle
and write
2
FSOLVENT ]h g =-KSOL FPROTEIN ]h g exp a -BSOL sin i
m2
k
Now since
FTOTAL ]h g = FPROTEIN ]h g + FSOLVENT ]h g
2
FTOTAL ]h g = FPROTEIN ]h g - KSOL FPROTEIN ]h g exp a -BSOL sin i
2
FTOTAL ]h g = b 1 - KSOL exp a -BSOL sin i
m2
m2
k
kl FPROTEIN ]hg
Hence the problem of modeling the bulk solvent has been reduced to an exponential scaling
problem - a neat trick (the original literature reference is (Moews and Kretsinger, 1975)). Once
again, the values of the parameters Ksol and Bsol need to be determined during refinement of the
model. It’s useful to sketch things out out on an Argand diagram to understand what’s going on.
Advantages and Disadvantages of “Babinet” scaling
The advantage of the bulk solvent scattering correction based on Babinet’s principle is that it’s
simple, quick to calculate, and involves a very limited number of additional parameters (2 !!).
However, calculation speed is no longer a limiting factor in crystallographic refinement. The real
space mask model is more physically realistic, seems in general to slightly outperform Babinet
scaling (Kostrewa, 1997), and is probably to be preferred in most circumstances. However, it’s still
useful to understand Babinet’s principle since it gives useful physical insight into the way the bulk
solvent contributes to the X-ray diffraction pattern.
References
Badger, J. (1997). Modeling and refinement of water molecules and disordered solvent. Method
Enzymol 277, 344-352.
Bragg, W. L., and Perutz, M. F. (1952). The External Form of the Haemoglobin Molecule. I. Acta
Crystallogr 5, 277-283.
Jiang, J. S., and Brünger, A. T. (1994). Protein hydration observed by X-ray diffraction. Solvation
properties of penicillopepsin and neuraminidase crystal structures. J. Mol. Biol. 243, 100-115.
Kantardjieff, K. A., and Rupp, B. (2003). Matthews coefficient probabilities: Improved estimates for
unit cell contents of proteins, DNA, and protein-nucleic acid complex crystals. Protein Sci. 12,
1865-1871.
Kostrewa, D. (1997). Bulk Solvent Correction: Practical Application and Effects in Reciprocal and
Real Space. CCP4 Newsletter on Protein Crystallography 34, 9-22.
Matthews, B. W. (1968). Solvent content of protein crystals. J. Mol. Biol. 33, 491-497.
Moews, P. C., and Kretsinger, R. H. (1975). REFINEMENT OF STRUCTURE OF CARP MUSCLE
CALCIUM-BINDING PARVALBUMIN BY MODEL-BUILDING AND DIFFERENCE FOURIERANALYSIS. J. Mol. Biol. 91, 201-225.
Phillips, S. E.V. (1980). STRUCTURE AND REFINEMENT OF OXYMYOGLOBIN AT 1.6-A
RESOLUTION. J. Mol. Biol. 142, 531-554.
Tavassoly, M., Amiri, M., Darudi, A., Aalipour, R., Saber, A., and Moradi, A. (2009). Optical
diffractometry. J. Opt. Soc Amer. A. 26, 540-547.
Tronrud, D. E. (1997). TNT refinement package. Method Enzymol 277, 306-319.