X-ray crystallography utilizes the coherent scattering of short-wavelength X-rays (~0.5–1.5 Å) by the periodic electron density of a crystalline macromolecular lattice to calculate atomic coordinates.
1. Bragg’s Law of Diffraction
Constructive interference of scattered X-ray waves occurs only when the path length difference between reflections from parallel crystal lattice planes is an integer multiple of the X-ray wavelength:
$$nlambda = 2d sintheta$$
Where $n$ is an integer (order of reflection), $lambda$ is X-ray wavelength, $d$ is interplanar lattice spacing, and $theta$ is the angle of incidence.
2. Solving the Phase Problem
X-ray detectors record the intensities ($I(h,k,l) propto |F(h,k,l)|^2$) of diffraction spots, but all phase information ($alpha(h,k,l)$) is lost. Phases are required to compute electron density via inverse Fourier transformation ($rho(x,y,z) = frac{1}{V}sum F(hkl) e^{-2pi i(hx+ky+lz)}$). Phases are solved via:
- Molecular Replacement (MR): Uses a homologous known structure as an initial search model via Patterson rotation and translation functions.
- Single/Multiple Isomorphous Replacement (SIR/MIR): Soaking crystals with heavy metal salts (Hg, Pt, Au, U) whose intense scattering shifts diffraction intensities.
- Single/Multi-Wavelength Anomalous Dispersion (SAD/MAD): Incorporating selenomethionine (Se-Met) into recombinant proteins and tuning synchrotron radiation near the selenium absorption edge to exploit anomalous resonant scattering.