DILUTE SOLUTIONS

The experimental conditions are good if the particles have a radius of gyration¹ of the order of 10 to 50 A, or a volume of approximately 8 X 10³ to 2 X 10⁵ A3• The corresponding weight of the particle is from I to 50 X 10-²⁰ g. if the density is between I and 2. The gram-molecular weight of such molecules is between 5,000 and 250,000. This is the

1 The introduction of this po.re.meter, the radius of gyre.tion, is found on p. 24, where it was defined as the root mee.n aqua.re of the distances of a.toms from the center of gravity of the pe.rticle, ee.ch diste.nce being modified by a coefficient eque.l to the atomic nwnber of the e.tom.

167

168 SMALL-ANGLE SCATTERING OF X-RAYS

order of magnitude of the molecular weights of many important biological compounds, as, for example, the proteins.

These remarks explain why the study of proteins offers one of the best applications of this method. As a matter of fact a large number of investigations have already been carried out on proteins, as is shown in the bibliography at the end of the text. We believe that it is in this field that small-angle scattering can give the most valuable and most important results from a general point of view.

The experimental work begins with the preparation of a dilute solution in which the molecule to be studied is found in a pure state (within several per cent).

It is essential that the electronic density of the solvent be as different as possible from that of the molecule, because the smaller this difference, the weaker will be the low-angle scattering.

Frequently aqueous solutions produce a rather intense scattering, so that solutions down to a concentration of only a few per cent can be studied.

The object of the experiment is to determine the curve of the scattered intensity as a function of scattering angle. A photographic determination of the scattering pattern must therefore be followed by a microphoto-metering of the film. It is certain that Geiger-Miiller counter measurements can be made to a higher degree of accuracy than photographic measurements, and this accuracy is very desirable, particularly for solutions of molecules, since the theoretical interpretation of this case is on a rather sound basis.

The measurements of the radius of gyration of the molecules can be made by the method of §4.1.1.1. If the shape of the molecules is not too far from spherical, the description of the experimental curve as an exponential in -kh² is a sufficiently accurate approximation over a large angular region. Here an incident beam of large height can be employed without having to make corrections before interpreting the curves of log /(h2) (seep. 114). This is important from the point of view of rapidity of measurement. As we have already indicated, the radius of gyration does not completely define the form and the size of the molecules, but (and here is a point of view on which we must insist) it is a well-defined geometrical parameter which can be considered as a characteristic of a molecule of any shape.

The experimental techniques, as illustrated, for example, in the counter-equipped apparatus of Beeman (Fig. 25, p. 95), have reached such a degree of simplicity and accuracy that one can envisage a broad program of systematic measurements of radii of gyration of different proteins or other large molecules in biochemical laboratories.

There are many services which these measurements could render. We

169 have already seen how a knowledge of the radius of gyration, together with other data, can give indications of the shape and the real dimensions of the molecule.¹ Figure 53a gives the curves of log J(h2) for a series of proteins, and it can be seen that the straight lines are defined well enough to permit an accurate determination of the radius of gyration. In Fig.

53b the curves of log I vs. log h are plotted for two ellipsoids of the same radius of gyration with axial ratios of 2 and 3. The experimental points for ovalbumin fall just between these two curves, indicating that the

0 log I

Lysocym

1·10-² 2·10-² 3·10-² h²(A-2) (a)

log I 2

OL---'-~~-'-~,-1-::----,:-'::-~

0.02 0.05 0.1 0.2 h(A-1) (b)

Fig. 53. (a) Log I vs. h' for solutions of different proteins; (b) log I vs. log h for ellipsoids of axial ratios 2 and 3 (theoretical). The small circles O show the experimental points obtained for a solution of

ovalbumin. (Ritland, Kaesberg, and Beeman [492].)

ovalbumin molecule has an axial ratio of about 2.5. With the latest improvements in experimental techniques (use of a powerful X-ray tube for studies of dilute solutions in which interparticle interferences are negligible, correction of the curves for the effect of beam height), the error in a determination of the radius of gyration is of the order of 2 to 3 per cent. This parameter could also be used as a means of identification, if no simpler method were possible. Finally, the single aspect of the linearity of the curve of log J(h2 ) is an indication of the uniformity of size of the particles and thus constitutes a means of inspection of the purity of preparations. For example, the curves of log J(h2) for euglobulin and pseudoglobulin extracts from the serum of horse blood are strongly

1 In addition to the other data previously mentioned (p. 130),the sedimentation and diffusion constants can be used toward this end (Ritland, Kaesberg, and Beeman (492]).

170 SMALL-ANGLE SCATTERING OF X-RAYS

convex (Ft<urnet [48]), and other techniques have shown that the globulins are complex mixtures.

Dervichian, Fournet, and Guinier [419] have studied the denaturing of hemoglobin and albumin serums with urea. When the small-angle scattering characteristic of the molecules disappears, the molecules have been broken up into small fragments. If all the molecules are not

100.---,~~-,-~~-,-"T'"--rl~-.--~--.

80 60 40

~ 20

=

~ ~ 10

~ 8

"'

c: 6

~

4

2

5% T.B.S.V.

0.5 x 10-mm. slits

1 60-=---40::'="-~--:2⁷o~~~l0:--'-8~6'---4..l..-~--'2 Angle, radians x 10-3

Fig. 54. Scattered intensity for a l'i per cent solution of tomato bushy stunt virus. The dotted curve is the theoretical curve for spheres of 310-A radius. (Leonard, Anderegg, Shulman, Kaesberg, and Beeman

[478].)

destroyed, the patterns decrease in intensity without changing their appearance. A change of appearance of the patterns occurs when new particles are formed.

Beeman and coworkers have obtained especially accurate results for virus macromolecules [376]. The experimental curve showed several peaks which the authors were able to make coincide with the successive peaks in a scattering pattern of homogeneous spheres. They thus demon-strated that these viruses were spherical and found their diameters (Fig.

54); for example, the diameter of the tomato bushy stunt virus was 310 A. Then, by a comparison of this diameter with other molecular

APPLICATIONS 171 constants, they were able to determine the degree of internal hydration of the molecules. different order of difficulty. The crystallographic method requires a considerable amount of work (and sometimes only for an uncertain result) both in the preparation of a usable crystal and in the interpretation of the patterns, whereas a low-angle scattering investigation of a solution can now be considered a routine operation.