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# 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 40-60% 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 long-range 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 (Ĺ). 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.

Non-crystallographic symmetry:
symmetry operations (rotation and translation) that relate monomers within the asymmetric unit.

Scattering
of X-rays is due to electrons, hence heavier atoms with more electrons scatter more strongly.

Diffraction:
scattering of X-rays (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 Ĺngstroms (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 X-rays 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 X-rays: 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 heavy-atom. 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.

R-factor:
"residual-factor" or agreement factor: Free R-factor:
an R-factor calculated on a partial data set that is not used in the refinement of a structure.

Refinement:
improvement of the R-factor 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.

B-factor:
"temperature-factor" or "Debye-Waller factor." A factor that can be applied to the X-ray 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 B-factor indicates the true static or dynamic mobility of an atom, it can also indicate where there are errors in model building. The B-factor 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 B-factors 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 204-220.

Brändén, C.I. and Jones, T.A. (1990). Between objectivity and subjectivity. Nature 343, 687-689.

Kleywegt, G.J. and Brünger, A.T. (1996). Checking your imagination: applications of the free R value. Structure 4, 897-904.

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, 283-291. 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, 345-364.