Protein Chemistry. Structure, Properties, Research Methods - Shendryk A.N. 2022

Methods for Experimental Investigation of Protein Structure
X-ray Structural Analysis
The Phase Problem in X-ray Analysis

Before moving on to the phase problem in X-ray crystallography (XRD), let us introduce METABOLISM/2.html">THE CONCEPT OF the crystal lattice unit Cell. The unit cell is the fundamental structural building block of a crystal lattice that captures all of its geometric properties. It is defined by six scalar parameters (edge lengths and the angles between them) or three unit vectors along the X, Y, and Z axes (which are not necessarily orthogonal). Consequently, the entire infinite three-dimensional crystal lattice is formed by the spatial repetition of this unit cell (see Fig.). In the case of protein molecules, the unit cell is represented as the volume of space occupied by one or more protein molecules (see Fig. 4.20 for Myoglobin).

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Fig. 4.20. Packing of two myoglobin molecules within the unit cell of the crystal lattice. The black plates represent the heme groups.

The ultimate goal of a crystallographic study is to determine the electron density distribution within the three-dimensional unit cell of the crystal, ρ(x,y,z). Typically, the electron density within The Cell is considered to be continuously distributed, reaching its maxima at the centers of gravity of the atoms located inside the cell. This density is a periodically repeating function when moving from one unit cell to another. In three-dimensional space, it is expressed via a Fourier transform, which takes the form of a triple Fourier series rather than an integral:

where F(hkl) are the amplitudes and φ(hkl) are the initial Phases of the rays.

While the amplitudes are easily determined experimentally from the intensities of the diffraction maxima (intensity is proportional to the square of the light wave amplitude), the initial phases remain unknown. The approaches used to determine initial phases differ between Protein Crystallography and small-molecule crystallography.

At various stages of The Development of XRD Methods for small molecules, the initial phase problem was addressed as follows. The First stage, prior to 1935, was the period of the "trial and error" method, where a model of atomic arrangement within the crystal unit cell was essentially fitted to the experimental diffraction pattern without any direct attempt to solve the phase problem. Such models were constructed based on specific assumptions about the three-dimensional crystal Structure derived from various indirect physicochemical data.

The Second Stage began in 1935, when Patterson proposed the first direct method of structure analysis based on experimental data regarding the distribution of diffraction spots. This method made it possible to unambiguously determine the coordinates of heavy atoms incorporated into the crystal, while the rest of the crystal structure was deduced in relation to these heavy atoms.

In 1952, three authors—Zachariasen, Cochran, and Sayre—independently and simultaneously developed and proposed a fundamentally new statistical approach to solving the phase problem, which did not require the presence of heavy atoms in the crystal.

However, none of these methods are applicable to the analysis of such complex molecular assemblies as Proteins.