Section 19.4.3.1. Variation of solvent density

The principle of contrast variation was studied in early work by Bragg & Perutz (1952), who observed that the magnitudes of low-order reflections in X-ray studies of protein crystals were reduced as the salt concentration in the solvent was raised. Following their concept, the effective scattering density of a dissolved particle is where is the average solvent scattering density. If we take a mean scattering density for the particle, a mean contrast, ρ, is defined: If the solute and solvent have equal densities, they are `contrast matched', and the scattering from the particle vanishes at zero angle. The particle will nonetheless scatter radiation at larger angles as a consequence of internal density fluctuations, which can be useful in obtaining structural information. In the case of neutron scattering, the solvent density is most often varied by mixing D₂O and H₂O to obtain different percentages of D₂O. This strategy gives a range of solvent densities that includes the densities of most biological molecules. However, biological molecules contain hydrogen atoms that exchange with solvent, so deuteration of the environment alters their scattering density to some extent (see below). In general, all hydrogen atoms not bonded to carbon are potentially exchangeable, but not all of these will actually exchange in a typical experiment.

To describe the variation of the radius of gyration with contrast, Stuhrmann derived the useful relationship (Stuhrmann, 1976; Stuhrmann et al., 1976) which separates the contributions of the internal structure of the particle to its radius of gyration from the contributions of the shape . Scattering from the internal structure is independent of contrast; scattering from the shape is contrast dependent. The shape function is defined as having a value of one inside the particle and zero outside. The total scattering density is then The contrast-independent terms in the Stuhrmann equation are where is the radius of gyration of the shape function and is its volume. The sign and magnitude of α give information on the radial density distribution of scattering density in the particle: if the outer region is higher in density than the inner region, α is positive (as, for example, in lipoproteins); if the inner region is denser, α is negative. The β coefficient represents the displacement of the centre of mass as a function of the contrast and is always positive; in real cases, β is often negligible. The Stuhrmann equation leads to a useful way to represent graphically the radius of gyration data obtained from a series of contrasts: the observed is plotted versus . If β is negligible, the plot is a straight line of slope α, intercepting the axis at . Thus, is obtained by extrapolation to a point where , and so is often termed the radius of gyration at infinite contrast. This quantity is a representation of the shape of the particle as if it had uniform internal scattering density. In a particle with two discrete regions of density, the radius of gyration for each region can be obtained from such a graph by evaluating where ρ is equal to the density of one region, so that of the non-contrast-matched region is determined. Such measurements can also be made by adjusting the solvent to match the scattering of one region to reveal the scattering of the other.

A parameter that is often useful is the contrast-match point for the particle, which reflects its overall composition including exchange. where , the match point, is the solvent scattering length density at which the contrast is zero and n is the number of exchanged hydrogens multiplied by d, the fractional deuteration of the water at the match point. Typically, the match point is obtained by measuring small-angle scattering at a series of D₂O:H₂O ratios, plotting each using a Guinier plot to obtain a value for I(0) by extrapolation, and then plotting , where C is the particle concentration. It is often convenient to represent as per cent D₂O. The plot should be a straight line, passing through zero at the contrast-match point. As noted above, the vast majority of biological molecules have contrast-match points at densities between those of H₂O and D₂O. If the particles are compositionally heterogeneous, the observed plot will be a weighted sum of the curves for each of the compositions present and will deviate from a straight line at low contrasts. Thus, the contrast-matching experiment can provide information on both composition and homogeneity.

While contrast variation is most often based on variation of the deuteration level in water, it is also possible to create variation by adding molecules to the solvent. As an example, a study of hydration layers was conducted by adding solute molecules, such as glycerol, at high concentration; the solute molecules alter the solvent scattering length density but do not penetrate the hydration layer (Lehmann & Zaccai, 1984).