## B 2 a Analysis of Scattering Data

The caseins occur in cow's milk as large colloidal complexes of protein and salts, commonly called casein micelles. Removal of calcium results in the dissociation of the micelle into smaller protein complexes called submicelles. The submicelles consist of four proteins, called al s-, a2s-, fi-, and K-casein, in the ratios of 4:1:4:1. These proteins have average monomer molecular weights of 23,300 and are considered to have few specific secondary structural features. The isolated fractions exhibit varying degrees of self-association thought to be driven by hydrophobic interactions.

It has been hypothesized that, upon the addition of calcium, these hy-drophobically stabilized, self-associated casein submicelles further self-

associate via calcium-protein salt bridges. The resulting casein micelle has an average radius of 650 A, as estimated by electron microscopy [6], SAXS has been done on bovine casein to ascertain whether submicellar structures can be detected in the absence of calcium. The structure of the colloidal micelle in the presence of calcium was also investigated [4], The results of this study will be examined in detail to illustrate the analysis of SAXS data. Scattering data for the buffer samples in this work [4] were fitted by the model in Table 9.1 with p = 2, and the fitted curves were subtracted as backgrounds from all protein scattering curves.

SAXS curves for submicellar and micellar casein are shown in Figure 9.3A. The different scattering magnitudes of these two samples are caused by the difference in protein concentration. Close inspection indicates that the shapes of these two curves are qualitatively different.

This difference for the two data sets is quite striking when the SAXS results are plotted in the traditional linearized (Guinier) form, as shown in Figure 9.3B. The submicelle plot is characterized by two linear regions, while the micelle data has three linear regions. However, an objective quantitative analysis of these data by the linear plot method is difficult for several reasons.

In the linearization of the SAXS models, there are difficulties in addition to changes in the error distribution of the dependent variable discussed in Section 2.A.3. When fitting a succession of straight lines to a slightly curving Guinier plot, as in Figure 9.3B, the determination of the location of a break in the curve is a matter for which objective critieria are rarely used. How do we decide where one straight line ends and the next one begins? In usual practice, a best guess at the answer to this question is used. Furthermore, the Guinier plot is an exponential approximation to a series expansion. It begins to deviate appreciably from a straight line at scattering angles beyond the

Figure 9.3 SAXS of submicellar casein (no Ca2+) at 19.38 mg/mL (O) and micellar casein at 16.4 mg/mL (A): (A) absolute intensity; (B) Guinier plots. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

Figure 9.3 SAXS of submicellar casein (no Ca2+) at 19.38 mg/mL (O) and micellar casein at 16.4 mg/mL (A): (A) absolute intensity; (B) Guinier plots. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

Guinier region, in the present case above s = 0.0025 A. In other words, the exact model for the natural log of scattering intensity vs. s2 is not a straight line!

As an alternative to Guinier plots, the use of nonlinear regression analysis onto the model in Table 9.1 provides objective and sensitive criteria for goodness of fit in the form of residual or deviation plots and the extra sum of squares F test (Section 3.C.1). This analysis of SAXS data affords a measure of statistical significance absent in a linearized plot, unless properly weighted linear regression and methods to find statistically best intersection points for multiple-line plots are employed. What is actually optimized in a linear Guinier plot is not the fit to the excess scattering intensity but to its logarithm. This tends to deemphasize the measurements at the smaller angles for which the precision is greatest. Also, the theoretical restriction to keep within the Guinier region is not a serious limitation with nonlinear fits. Here, the data at the higher angles make a much smaller contribution. Although the traditional linear plots may be useful for their heuristic and qualitative value, the analysis of SAXS data benefits greatly from nonlinear regression.

Data for submicellar casein gave best fits to the model in Table 9.1 with two Gaussian components (p = 2), as illustrated in Figure 9.4A. Also shown on the graph are the contributions of each of the two components as computed from the regression parameters. Goodness of fit is supported by the small standard deviation and the small standard error in each parameter (Table 9.3) and also by the random appearance of the residual or deviation plot (Figure 9.4B).

The standard deviation is well below the standard error of the measurement of ±0.004. Note that, contrary to most examples in this book, the deviations have not been divided by SD for the residual plots in this section.

Parameter |
Casein submicelle |
Casein micelle |

0.0326 ± 0.004 |
0.0283 ± 0.003 | |

Rij |
37.98 ± 0.004 |
34.7 ± 1.5 |

jnM |
0.439 ± 0.007 |
0.307 ± 0.011 |

Ra,2 |
81.7 ± 0.9 |
89.96 ± 0.01 |

JnM |
0.287 ± 0.046 | |

R<2,3 |
204 ± 15 | |

SD6 |
0.0022 |
0.0021 |

6 Standard deviation of the regression analysis.

6 Standard deviation of the regression analysis.

H CO

CO UJ

0.003

0.006

0.009

0.012

Figure 9.4 Results of nonlinear regression analysis of SAXS for submicelles of casein (19.4 mgmL"1): (A) experimental scattering data ( + ); line T is best fit to model in Table 9.1 with p = 2; lines B and A are the individual Gaussian components computed from the regression parameters; (B) random residual plots from nonlinear regression onto p = 2 model of submi-celle data in Figure 9.4A (O) and from regression onto p - 3 model for micelle data in Figure 9.6 (A). (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

A different situation occurs when the submicellar data are fitted by a single component model (p = 1). In this case, the deviation plot (Figure 9.5) has a definite nonrandom pattern, indicating that an incorrect model was used for the fit. Moreover, the relative values of the deviations are about an order of magnitude larger than for the p = 2 model, reflecting the larger standard deviation of regression of the p = 1 fit. It is clear then, that even though the submicelle solution is known to contain a single particle, two Gaussian components are required to fit the data adequately. This result is similar to the finding from analysis of the scattering data computed from the crystal structure of ribonuclease.

The micellar casein data gave poor fits to the models in Table 9.1 with

0.003

0.006

0.009

0.012

Figure 9.5 Nonrandom residual plot for submicelle data in Figure 9.4A after regression onto the single component model (p = 1) in Table 9.1. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

p = 1 or 2, but was fit successfully with the p = 3 model (Figure 9.6). Deviation plots were random for p = 3 and nonrandom for p = 1 or 2. The standard deviation of regression for p = 3 was well below the standard error of the measurements of ±0.004. The deviation plot is in Figure 9.4B.

Figure 9.6 Results of nonlinear regression analysis of SAXS for micelles of casein (16.4 mg mL_1): experimental scattering data (+); line T is best fit to model in Table 9.1 with p = 3; lines A, B, and C are the three individual Gaussian components computed from the regression parameters. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

Figure 9.6 Results of nonlinear regression analysis of SAXS for micelles of casein (16.4 mg mL_1): experimental scattering data (+); line T is best fit to model in Table 9.1 with p = 3; lines A, B, and C are the three individual Gaussian components computed from the regression parameters. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

All submicellar casein data examined were fit best by the p = 2 model (Table 9.1), while all micellar casein data were best fit by the p = 3 model. The values of the parameters were used to extrapolate the experimental data to zero scattering angle. The composite curves were then desmeared using the computer program developed by Lake [5], This provides the deconvoluted intensity at zero angle, ¿„(0).

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