X-ray Crystallography PDF

Several methods have been developed to determine the structure of molecules and crystals, but the most powerful of them is X-ray crystallography, which is based on the diffraction and reflection of X-rays on crystals. X-rays primarily interact with the electrons of atoms/molecules, but they do not l...

Several methods have been developed to determine the structure of molecules and crystals, but the most powerful of them is X-ray crystallography, which is based on the diffraction and reflection of X-rays on crystals. X-rays primarily interact with the electrons of atoms/molecules, but they do not lose their energy, they are scattered elastically. Recall what we have learned about the interference during the first lecture. At the lattice points of a crystal, there may be atoms, ions, or even molecules that can be considered as scattering centers, since by illuminating the crystal with an X-ray beam, the X- ray photons will be diffracted in all directions. These photons can extinguish (destructive interference, dark spots) or amplify each other (constructive interference, bright intensity maxima). In reality, constructive interference can only be observed in a few preferred directions, which form bright spots in the diffraction image. The phenomenon of X-ray diffraction can only be detected if the distance between the scattering centers is in the order of the wavelength of the X-ray radiation. The crystal structure can be determined from these interference patterns. In order to obtain an X-ray diffraction image, a crystal must be grown from the molecule to be examined, which is usually a rather complicated task. The diffraction pattern was previously recorded on X-ray films, but today a CCD camera sensitized in the X- ray range is used to capture the diffraction pattern rapidly. 28 The figure shows the diffraction of X-rays projected onto a one-dimensional crystal. The scattering centers are located along the z-axis, at a distance of c from each other, to which the X-rays fall perpendicularly, parallel to the x-axis. X-rays are scattered from every scattering center in all directions, however, in most directions this is not detected due to destructive interference. The radiation will be detectable only in those directions in which the condition of constructive interference is fulfilled, i.e. the path difference (Δs) between interfering waves is an integer multiple (l) of the wavelength (λ): s  c  cos   l   , where c is the lattice constant, γ is the angle of diffraction and l=1,2,3,…. In the figure, this path difference is indicated by the thick black lines, which show the distance with which the bottom ray travels more than the top ray before they interfere with each other. The figure shows an example of constructive interference for 2 different angles of diffraction, therefore there are 2 thick black lines. The photon with larger diffraction angle (γ1) has a shorter path difference, which is equal to the wavelength (λ), while the smaller diffraction angle (γ2) corresponds to a longer path difference (longer thick black line), which is twice the wavelength (2λ). 29 The figure shows the diffraction of X-rays on a one-dimensional crystal when the incident X- ray is not perpendicular to the z-axis, i.e. it is inclined. In this case, the path length difference (Δs) consists of two components (Δs1, Δs2) shown in the figure: s  s1  s2  c  (cos  0  cos  )  l   If we want to apply this equation in three dimensions, we get the 4 equations shown on the next slide. 30 Of the 4 equations shown on the slide, the first 3 are the so-called Laue equations, and together the 4 equations describe in three dimensions the directions in which constructive interference is obtained when X-ray photons are scattered at the lattice points. The 4 equations are overdetermined, containing three unknowns (a, b, c), so it has a solution only in certain cases. These special cases can be produced in two ways, but both methods have in common that many combinations of angles are present, and there will be a combination among them where the system of the 4 equations has a solution: 1. Rotating crystal method: during rotation, the crystal will eventually reach such a position in which the conditions of the Laue equations are satisfied. 2. Powdered crystal method: in a powdered sample there are always crystal particles for which the Laue equations are satisfied. 31 According to Bragg, X-rays can not only be scattered from the scattering centers of crystals (Laue’s interpretation), but can also be reflected from the crystal planes, and these reflected rays can also amplify each other. This case can be seen in the figure demonstrated with a two- dimensional crystal. A path difference (Δs, indicated by a red line in the figure) is created between the rays reflected from the parallel planes. The reflected rays produce observable radiation (bright intensity maximum in the diffraction image) if the condition of constructive interference is fulfilled, i.e. the path difference between interfering rays (Δs, red line in the figure) is an integer multiple (l) of the wavelength (λ): s  l   The Bragg equation describes the angles of incidence (α) at which constructive interference occurs with X-rays of a given wavelength: s  2d  cos   l   , where d is the distance between the parallel crystal planes and l=1,2,3,…. If the crystal is irradiated with X-rays of known wavelength (λ) and the direction of the maximum intensities (α) is measured, the distance (d) between the crystal planes can be determined using the Bragg equation. 32 If groups of atoms or molecules are present in lattice points instead of single atoms, scattered rays from adjacent atoms within the molecule can also interact (yellow and blue atoms in the image). A lattice plane drawn across a unit cell is decomposed into several nearby lattice planes corresponding to the lattice planes of each atom that makes up the molecule (blue and red lines in the image, one for the blue atom and the other for the yellow atom). Since the intensity of the reflected rays is influenced by the distance between these lattice planes (the red and blue lines), the internal structure of the molecules can be determined from the X-ray diffraction image. 33 X-ray crystallography can be used to determine the exact spatial arrangement of thousands of atoms in proteins, so it can also be used to explore three-dimensional structures. The figure shows the three-dimensional structure of a potassium ion channel. 34 In order to determine the three-dimensional structure of proteins by X-ray crystallography, the protein must first be synthesized or isolated, purified, and finally crystallized. Knowing their three-dimensional structure, proteins can be used to design drugs more accurately. The slide shows the three-dimensional structure of the epidermal growth factor receptor (EGFR), to which a drug called gefitinib is bound. Gefitinib is able to specifically inhibit EGFR tyrosine kinase activity by binding to the ATP-binding site of the receptor. Because EGF receptor overexpression, observed in many tumors, activates signaling pathways and enhances cell division, such a drug can be used for treating tumors. 35 36