The origins of both coordinate systems are the same point. Coordinates in the Brookhaven Protein Data Bank (PDB) are reported in Cartesian coordinates, and the files contain a 3 × 3 matrix for converting the Cartesian coordinates to fractional coordinates. Usually Cartesian space is used for atomic models, especially since this makes it easier to compare bond angles and lengths.Ĭartesian coordinates are expressed as three mutually perpendicular axes, x, y, and z, with angstroms as the units. It is obvious in fractional space that (0.5, 0, 0) and (1.5, 0, 0) are the same point.
Certain operations such as comparing symmetry relations are more easily done in fractional space. In any space group, 1.0 or –1.0 can be added to any fractional coordinate without changing the meaning of the coordinate because of the infinitely repeating nature of crystal lattices. The point (0.5, 0.5, 0.5) is in the center of the cell regardless of the cell dimensions and shape. Thus the origin of the unit cell is (0, 0, 0), and the point one cell edge away in b is (0, 1.0, 0).
Fractional coordinates are parallel to the crystallographic axes and thus are not necessarily at right angles to each other. Fractional coordinates are expressed as fractions of the unit cell in each of the three directions a, b, c separated by the angles α,β, γ. Two main coordinate systems are used in crystallography: fractional coordinates and Cartesian coordinates. McREE, in Practical Protein Crystallography (Second Edition), 1999 Coordinate Systems
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