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The holographic idea may also be applied to the problem of structure completion in the X-ray crystallography of proteins [5]. During the process of structure solution, a partial model of the molecule may have been constructed, with the remainder unknown. Treating the diffraction amplitudes from the known parts of the structure as a reference wave, the electron density from the unknown part may be reconstructed, using the intensities of the diffraction pattern, now interpreted as a hologram.

The reconstruction algorithm proposed by Szöke and co-workers [5, 6, 7] relies on holographic notion that, if the known part of the structure is a significant part of the total, the measured Bragg intensities are, to a first approximation, a linear function of the unknown electron densities. Within this approximation the latter may be found by the solution of a set of linear equations by any of a number of standard numerical methods. Incorporation of this solution into the known part of the structure and subsequent iteration allows refinements of this initial estimate.

Sometimes the amount of data from the experiment be insufficient to
uniquely determine the electron density sought. Such a circumstance
calls for a technique capable of distinguishing the most likely of a
set of feasible solutions. Such a problem falls within the domain of *maximum
entropy* methods [8, 9, 10].

Fig. 3 shows the electron density of the
residues 2-7 and 35-40 of the 207 residue sweet-tasting protein
thaumatin as determined from a partial model containing only residues
121-207, and calculated Bragg intensities from a model of the entire
protein, using the classical *difference Fourier* method [11]. The wire mesh surface depicted has
been contoured at a value of 1.5 times the RMS deviation above the mean
electron density. Comparison with the stick figure of the corresponding
residues shows some false gaps in the reconstructed electron density.

The electron distribution of the same deleted residues as calculated by a maximum entropy solution of the same holographic problem is shown in Fig. 4, in which is plotted an electron density isosurface corresponding to the same RMS deviation of the density above the mean. The 3D electron density of the same amino acid residues 2-7 and 35-40 are here seen to be recovered with much better connectivity and greater fidelity than that from the difference Fourier method (Fig. 3).

**Next:** Surface X-Ray
Diffraction **Up:** Reconstructing
the Atomic Architecture **Previous:** Photoelectron Diffraction