Stuhrmann plot

Non-uniform scattering densities and contrast oariation 2.7.1. The Stuhrmann plot... 

Stuhrmann plots of versus Ap gives Rq, the radius of gyration at infinite contrast (Fig. 5). The tangent to the curve at dp = 0 gives a and this reflects the... 

Fig. 5. Schematic Stuhnnann plots for two-component systems, (a) If the two components are concentric, a linear relationship is found between Rq and dp. The sl< a is positive as shown if the outermost component has a higher scattering density than the iimermost component. It is negative if the lower scattering density is on the outside. Data collected in buffers with H20 contents below the matchpoint lie in the positive dp range, (b) If the centres of the two components are separated within the macromolecule and their matchpoints are sufficiently different, the curvature of the plot represents the separation between the centres of the two components. The value of a can be positive, zero or negative, while is zero or positive, (c) Stuhrmann plots for a two-component system M + P and M + D, in which component P of lower matchpoint than M has been isomorphously replaced by deuterated D with a higher matchpoint than M. The value of Rq for each of M and P/D are obtained by intrapolation as shown.

This expression is identical with the Parallel Axes Theorem above, even though the two components have been distinguished by contrast variation and not by chemical separation into separate entities [48]. This identity is readily shown by equating R and Rqb in the Parallel Axes Theorem with the Rq values in the Stuhrmann plot at Ap values that correspond to the matchout of components B and A respectively, i.e. 

In the Stuhrmann equation, the centre of gravity of scattering density fluctuations is coincident with that of the shape of the protein to a good approximation, thus the term in can be set as zero. The values of a from Stuhrmann plots... 

Fig. 19. Dependence of 7(0) on the volume fraction of HjO and H20 present in neutron scattering experiments. The vertical scales in (a), (b) and (c) are not to the same scale, (a) The I(O) values for myoglobin in 10 different contrasts are a linear function of the solvent scattering density [44]. This is the usual result of neutron contrast variation and indicates the monodispersity of the sample. The match-point corresponds to the so-called dry unhydrated volume of the protein. Typical Stuhrmann plots for proteins and glycoproteins are exempUfied in Fig. 21. (b) The values tor ferritin in 10 different... 

Fig. 24. Stuhrmann plots of the high-density lipoproteins LpA2 ( ) and LpA3 (O) using sucrose contrast variation in X-ray scattering [76]. Note that R%- p has been plotted as a function of Ap, whereupon the slope gives Rc and the intercept on the ordinate axis gives a.

Fig. 27. Stuhrmann plot of the values for the Gene S Protein from phage fd in complexation with protonated DNA (O) and deuterated DNA ( ). The increased n ative slope of the data after deuteration shows that DNA lies at the centre of the cross-section of the rod-like helical complex [368). The experiment with protonated DNA is an important control of this result.

Core particles and nucleosomes are globular particles that have been usefully examined by several groups by neutron and X-ray scattering [382-401]. Stuhrmann plots of the core particle 200,000 45% DNA) show large positive slopes, where Rq is 3.9-4.1 nm and a is 36-51 X 10 (Table 15) [382,387,390,395]. Nucleosomes likewise exhibit a Rq oi 4.0 nm and an a of 45 X 10 [385,393]. Further analysis shows that the DNA component of core particles has an Rq of 4.9 nm, while the protein component has an Rq of 3.3 nm. This shows that the DNA has a more elongated shape in the core particle. In summary, it was shown that the DNA is external to a protein core. Other work shows that 1.8 turns of superhelical DNA are wound around the octamer of core histones. Several workers have examined individual neutron scattering curves in H20 buffers or X-ray curves in H2O buffers... 

Fig. 28. The three basic scattering functions from the shape Iy(Q), the cross-term /vf(0 nd the fluctuations /p(2) fro contrast variation studies on the chromatin core particle [387]. I (Q) corresponds to the scattering from the shape as observed at infinite contrast where there is no influence from the internal structure Pp(r) from the DNA and protein components. Its Guinier region gives the Rq of the Stuhrmann plot. /p(2) is the internal structure function and should correspond to the scattering curve measured at the matchpoint of the core particle in 48% H20. The cross-term /vp(2) is the correlation of the shape and internal structures. The calculated curves from 3 models are shown in...

Other anisometric viruses have rod-like helical or cylindrical structures, such as tobacco mosaic virus [495,496,509,533] or alfalfa mosaic virus [551,561,562]. Thus cross-sectional parameters can be determined using / xs Q) Q q->o addition to Rq d I 0) data [537,550]. Stuhrmann plots of the / xs data lead to information on the cross-sectional distribution of protein and RNA. Shell models for the cross-section can likewise be made by analogy with the isometric viruses [550,561,562]. The radial scattering density of the cross-section can be calculated by applying the Hankel transformation to the scattering curve [509]. 

Summary of X-ray data tor proteins analysed by Guinier Rq and 1(0) plots and Stuhrmann Rq and a plots. These are arranged in order of increasing electron density. Where a matchpoint solution composition is not given, the solute used to vary the solvent contrast is indicated in parentheses... 

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