Protein Crystallography
Important points and definitions:
 Crystal:
 a regular repeat of molecules, usually with some sort of internal rotational
symmetry. Protein crystals are usually about 4060% solvent by
weight and are thus fragile and sensitive to drying out.
 Unit cell:
 the smallest repeating unit that can generate the crystal with only translation
operations.
 Lattice:
 the regular spacing (defined by lengths and angles) of the origins of the
individual unit cells.
 Mosaicity:
 angular measure of the degree of longrange order of the unit cells within a
crystal. Lower mosaicity indicates better ordered crystals and hence better diffraction
 Asymmetric unit:
 the smallest unit that can be rotated and translated to generate one
unit cell using only the symmetry operators allowed by the
crystallographic symmetry. The asymmetric unit may be one molecule
or one subunit of a multimeric protein, but it can also be more than
one.
 Symmetry:
 description of how the asymmetric units are arranged in the unit cell by
rotation and translation operators. The symmetry operators are
expressed in coordinate frame of the lattice and thus may not be
orthogonal and are not in cartesian coordinates (Å). The
translations are expressed as fractions of a unit cell axial length.
Application of these operators to molecular coordinates must also be
in the frame of fractional coordinates.
 Noncrystallographic symmetry:
 symmetry operations (rotation and translation) that
relate monomers within the asymmetric unit.
 Scattering
 of Xrays is due to electrons, hence heavier atoms with more electrons scatter
more strongly.
 Diffraction:
 scattering of Xrays (in this case) from a crystal. It depends on the "long
range" order in the crystal. More disorder means poorer diffraction especially at higher
resolution.
 Resolution:
 minimum interplanar spacing of the real lattice for the corresponding
reciprocal lattice point (reflection) that is being measured. It is
directly related to the optical definition in which it is the minimum
distance that two objects can be apart and still be seen as two
separate objects. Resolution is normally quoted in Ångstroms (Å).
 Amplitude:
 magnitude or "intensity" of waves
 Phase:
 position of a wave's maximum relative to an origin
 Bragg's law:
In this figure, the Xrays are entering from the upper left, reflecting
off of the two parallel planes and exiting to the upper right. One can
show that AB + CD = 2*d*sin(theta) . This is usually more generally
stated as:
2d*sin(theta) = n*lambda
where n is an integer, theta is the angle of scattering and lambda is the wavelength
 Reciprocal lattice:
 the discrete set of rays, also known as reflections that result from
diffraction. The reciprocal lattice vectors are perpendicular to the real lattice planes from
which they are derived and the relationships of size of the reciprocal lattice are inversely
related to those of the real lattice. Thus large unit cells result in a very closely space
reciprocal lattice and small unit cells result in a reciprocal lattice with large intervals.
 Fourier transform:
 the mathematical relationship between the electron density and the
diffraction by Xrays:
where F(h) is the reflection at reciprocal lattice point h and f(x) is the scattering function
of the electron density at point x. F(h) is a complex number (a vector) containing the
amplitude and the phase of the reflection. The integration is over the complete unit cell.
Thus every atom contributes to the amplitude and phase of each individual reflection, but
to varying extents. This means that a partial data set still gives information about the
complete structure.
There is also an inverse Fourier transform:
in which the electron density at each point rho(x) is made up of a sum of all of the
reflection amplitudes and phases. Again, note that each point in the electron density
contains contributions from all of the reflections. To get the best electron density at each
location within the unit cell a complete data set is necessary.
 Phase problem:
 Only amplitudes can be measured. Phases must be derived by other
means (multiple isomorphous replacement, molecular replacement and multiple
wavelength anomalous dispersion). The phases used initially are always approximate and
yet the phases are very important to the quality of the electron density, thus good quality
initial phases are critical.
 Multiple isomorphous replacement:
 the primary method for determining the initial
phases for a new structure. These phases are derived from multiple
(two or more) data sets collected on crystals into which heavy atoms
have been soaked.
 Molecular replacement:
 a method for deriving initial phases by superimposing a
known homologous structure onto the diffraction data for the unknown structure.
 Multiple wavelength anomalous dispersion (MAD) phasing:
 a new method for
deriving initial phases by measuring diffraction data at several different wavelengths near
the absorption edge of a heavyatom. The anomalous signal that results from this can
give very accurate phases. This commonly done by replacing methionine with seleno
methionine when producing the protein.
 Rfactor:
 "residualfactor" or agreement factor:
 Free Rfactor:
 an Rfactor calculated on a partial data set that is not used in the
refinement of a structure.
 Refinement:
 improvement of the Rfactor by adjustment of the model to better agree
with the measured data. There are limits as to how close the model must be before a
refinement method can find a better structure. This is the radius of convergence.
 Bfactor:
 "temperaturefactor" or "DebyeWaller factor." A factor that can be applied
to the Xray scattering term for each atom (or for groups of atoms) that describes the
degree to which the electron density is spread out. While the theory is that the Bfactor
indicates the true static or dynamic mobility of an atom, it can also indicate where there
are errors in model building. The Bfactor is given by:
where Ui2 is the mean square displacement of atom i. This produces a weighting factor
on the contribution of atom i to the Fourier transform by:
As U increases, B increases and the contribution of the atom to the scattering is
decreased. If atoms are incorrectly built, their Bfactors will tend to be higher than
correctly built atoms nearby.
References:
"Protein Crystallography" T.L. Blundell & L.N. Johnson, Academic Press Inc. London,
(1976).
"Proteins. Structures and Molecular Properties" T.E. Creighton, W.H. Freeman & Co.,
New York (1984) Chapter 6 pp 204220.
Brändén, C.I. and Jones, T.A. (1990). Between objectivity and subjectivity. Nature 343,
687689.
Kleywegt, G.J. and Brünger, A.T. (1996). Checking your imagination: applications of the
free R value. Structure 4, 897904.
Laskowski R A, MacArthur M W, Moss D S & Thornton J M (1993). PROCHECK:
a program to check the stereochemical quality of protein structures. J.
Appl. Cryst., 26, 283291. See also the
PROCHECK web
site for more information regarding the checks that it performs.
Morris A L, MacArthur M W, Hutchinson E G & Thornton J M (1992).
Stereochemical quality of protein structure coordinates. Proteins, 12,
345364.
See also the teaching links below.
Links to sites for teaching crystallography
Protein crystallography lecture outline page
Protein crystallography lecture home page
