Chapter 5 – Detection and Imaging Tools that Use Nonoptical Waves  177

can determine the distribution of interlayer spacing. The Laue method uses instead a poly­

chromatic x-​ray source, which produces a range of different diffraction peaks as a function

of wavelength that can be used to determine the interlay spacing distribution provided the

sample consists of just a single crystal (the combination of a polycrystalline sample with a

polychromatic x-​ray source generates a diffraction pattern, which is difficult to interpret in

terms of the underlying distribution of interlayer spacings). The most useful approach is the

single-​crystal monochromatic radiation method, which generates the most easily interpreted

diffraction pattern concerning interlayer spacings.

The intensity of the diffraction pattern can be modeled as the Fourier transform of a

function called the “Patterson function,” which characterizes the spatial distribution of elec­

tron density in the crystal. The pattern of all the scattered rays appears as periodic spots of

varying intensity and may be recorded behind the crystal using a CCD. Typically, the crystal

will be rotated on a stable mount so that diffraction patterns can be collated from all pos­

sible orientations. However, growing a crystal from a given type of biomolecule with min­

imal imperfections can be technically nontrivial (see Chapter 7). To maximize the effective

signal-​to-​noise ratio of the scattered intensity from a crystal, there is a benefit of growing one

large crystal as opposed to multiple smaller ones, and this larger scale is also a benefit due to

radiation damage destroying many smaller crystals. In many examples of biomolecules, it is

simply not possible to grow stable crystals.

The intensity and spacing of the spots in the diffraction patterns is the 2D projection of

the Fourier transform of spatial coordinates of the scattering atoms. The coordinates can be

reconstructed using intensive computational analysis, hence, to solve the molecular struc­

ture, with a typical resolution being quoted as a few angstroms (which equals 10⁻¹⁰ m, useful

since it is of a comparable length scale to covalent bonds). However, an essential additional

requirement in this analysis is information concerning the phase of scattered rays. For con­

ventional x-​ray crystallography, which uses either incoherent x-​ray tube or synchrotron radi­

ation, the intensity and position of the maxima in the diffraction pattern alone do not provide

this, since there is no x-​ray “lens” as such to form a direct image that can be done using visible

light wavelengths, for example.

Crystallographers refer to this as the phase problem, and this phase information is often

then obtained indirectly using a variety of additional methods such as doping the crystals

with heavy metals at specific sites, which have known phase relationships. Phase information

is normally generated by using iterative computational methods, the most common being

the hybrid input–​output algorithm (HIO algorithm). Here, a Fourier transformation and an

inverse Fourier transformation are iteratively applied to shift between real space and recip­

rocal space under specific boundary conditions in each. This approach is also coupled to

oversampling by sampling the diffraction intensities in each dimension of reciprocal space

at an interval of at least twice as fine as the Bragg peak frequency (the highest spatial fre­

quency detected for a diffraction peak in reciprocal space). For the real space part of struc­

tural refinement, molecular dynamics and structural modeling/​validation are also widely

used (see Chapter 8).

X-​ray crystallography has been at the heart of the development of modern biophysics.

For example, the first biomolecule structure solved was that of cholesterol as early as 1937

by Dorothy Hodgkin, and the first protein structures solved were myoglobin in 1958 (John

Kendrew and others) followed soon after by hemoglobin in 1959 (Max Perutz and others).

There are important weaknesses to the method, which should be noted, however. A key disad­

vantage of the technique, as with all techniques of diffraction, is that it requires an often arti­

ficially tightly packed spatial ordering of molecules, which is intrinsically nonphysiological.

In addition, the approach is reliant upon being able to manufacture highly pure crystals,

which are often relatively large (typically a few tenths of a millimeters long, containing ~10¹⁵

molecules), which thus limits the real molecular heterogeneity that can be examined since

the diffraction information obtained relates to mean ensemble interference properties from

a given single crystal. In some cases, smaller crystals approaching a few microns of length

scale can be generated.

Also, the crystal-​manufacturing process is technically nontrivial, and many important

biomolecules, which are integrated into cell membranes, are difficult, if not impossible, to