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Unformatted text preview: 6. X-ray Crystallography and Fourier Series Most of the information that we have on protein structure comes from x-ray crystallography. The basic steps in finding a protein structure using this method are: • a high quality crystal is formed from a sample of protein • the crystal is placed in an x-ray beam and the intensities of the diffraction spots are measured • after finding the phases, an electron density map is computed from the diffraction intensities and phases using Fourier analysis. • The coordinates of the atoms are found from the electron density. • The structure is refined by checking that, for example, the atoms do not get too close to each other. We will briefly discuss some of the mathematics involved in finding the electron density from the diffraction intensities and phases. This requires studying Fourier series for functions periodic on lattices. 6.1. Lattices. The basic structure of a crystal is that of a lattice. A crystal is formed by many copies of the same protein in a pattern is formed which fills a unit cell. The unit cell is a parallelepiped which is used as a tile whose translations fill up space. For mathematical simplicity, we can suppose that there are an infinite number of copies of the same protein. A lattice is easy to describe mathematically. A lattice L in three dimensions is generated by three linearly independent vectors a , b , c ; it is the set of all vectors h a + k b + l c where h,k and l are integers. A unit cell is the set of points x a + y b + z c for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. Similarly we can define a lattice in two dimensions as the set of integer combinations of two linearly independent vectors a and b and the unit cell is the parallelogram formed by the points x a + y b for ≤ x ≤ 1, 0 ≤ y ≤ 1. In the pdb file for a crystal structure, you can find the lengths and angles between the vectors a , b and c which generate the lattice for the crystal. The origin of the coordinate system can be put at any point crystallized protein. If the origin is placed at an atom in the protein, then every lattice point will be on exactly the same atom in a translation of the protein. 6.1.1. Examples of lattices. First consider two dimensions. The vectors a = (1 , 0) and b = (0 , 1) generate the square lattice . The vectors a = (1 , 0) and b = (1 / 2 , √ 3 / 2) generate the hexagonal lattice . 1 2 In three dimension the vectors a = (1 , , 0), b = (0 , 1 , 0), and c = (0 , , 1) generate the cubic lattice . The vectors a = (1 , 1 , 0), b = (1 , , 1), c = (0 , 1 , 1) generate the face centered cubic lattice . The face centered cubic lattice can also be described as the set of points ( x,y,z ) with integer coordinates such that x + y + z is even....
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- Fall '11
- Fourier Series, Periodic function, electron density