The deconvoluted intensity at zero angle divided by the corresponding protein concentration yields a value proportional to the molecular weight of the particle (mapp, Table 9.2). This quantity, essentially in(0)/ce, showed no concentration dependence under any of the conditions studied. Hence, extreme particle size polydispersity is unlikely. The particles are likely to have a single, narrow size distribution.

Other models accounting for the double-component (p = 2) character of the submicellar scattering data might be based on particle asymmetry (e.g., rods), or on a spherically symmetrical but inhomogeneous particle having regions of differing electron density. The former would not be in agreement with hydrodynamic, light scattering, and electron microscopic evidence, which indicates that submicellar casein exists as spherical particles [6].

Because the particles result from a hydrophobically driven self-association of monomer units, it has been considered most likely that they contain a hydrophobic inner core surrounded by a hydrophilic outer layer [6], This arrangement would be the most stable on thermodynamic grounds, Such a structure would have different packing densities in regions where predominantly hydrophobic and hydrophilic amino acid side chains reside. This would give rise to two roughly concentric regions of different electron density [4] (Figure 9.7).

Figure 9.7 Schematic representation of casein submicelles within a micellar structure. Cross-hatched area represents core regions of higher electron density and higher concentration of hydrophobic amino acid side chains. The particle on the lower left shows a few representative casein monomer chains. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

Figure 9.7 Schematic representation of casein submicelles within a micellar structure. Cross-hatched area represents core regions of higher electron density and higher concentration of hydrophobic amino acid side chains. The particle on the lower left shows a few representative casein monomer chains. (Reprinted from [4] with permission, copyright © 1988 Academic Press.)

Thus, the submicellar data were analyzed further by means of a model in which the particle has two regions of different electron density within the same scattering center. In this model, the scattered amplitudes rather than the intensities of the two regions must be added because of interference effects of the scattered radiation.

Luzzati et al. [1] developed equations for calculating the molecular and structural parameters of a particle having two regions of different electron density with the same electronic center of mass. These are applicable to smeared SAXS data containing two linear Guinier regions. The model is essentially identical in form to that in Table 9.1 with p = 2. They identified the parameters ;„,i(0) and Ral with the inner electron density region, and U,2(0) and Ra2 with the outer electron density region. They expressed the deconvoluted zero angle scattering intensity ¿„(0), by

[i„(0)]2 = 2VnV3 \j„MRa,i + 7«.2(0)^,2]- (9.5)

The average radius of gyration, R2, of the inhomogeneous particle (i.e., the whole submicelle) is obtained from

In the next models, the subscripts C and L (for compact and loose) [1] are used to designate the higher and lower electron density regions, respectively, and the subscript 2 designates the entire particle composed of these two regions. The respective masses are

where / is the fraction of electrons in the higher electron density region. The fraction / is easily evaluated from the relationship

[i(0)]2 = deconvoluted scattering intensity at zero angle for the whole submicellar particle, [/(0)]c = deconvoluted scattering intensity at zero angle for the higher electron density region.

In Table 9.3, RaA can be identified with the radius of gyration of the denser region, Rc. Thus,

The radius of gyration of the low electron density region, RL, can now be found from the expression

R2 is available from eq. (9.6), the fraction/can be obtained from eq. (9.9), and Rc = R„, 1 obtained from the regression analysis (Table 9.3).

Parameter |
Casein submicelle |
Casein micelle |

M |
882,000 ± 28,000 | |

k2 |
0.308 ± 0.005 (3.: | |

M2 |
285,000 ± 14,600 |
276,000 ± 18,000 |

k |
0.212 ± 0.028 |
0.216 ± 0.003 |

Mc |
60,000 ± 5,650 |
56,400 ± 3,700 |

Ml |
225,000 ± 18,500 |
220,000 ± 18,700 |

A p(e-/Â3) |
0.0081 ± 0.0004 | |

Aft (e-/Â3) |
0.0099 ± 0.0004 |
0.0073 ± 0.0005 |

A Pc(e-/À3) |
0.0148 ± 0.0014 |
0.0128 ± 0.0007 |

A Pz.(e-/À3) |
0.0091 ± 0.0003 |
0.0065 ± 0.0003 |

H (êwater^êprotein) |
7.92 ± 0.42 | |

H2 (gwater^êprotein) |
6.31 ± 0.30 |
8.98 ± 0.44 |

He (gwater/gprotein) |
3.97 ± 0.48 |
4.70 ± 0.31 |

Hl (gwater/gprotein) |
6.90 ± 0.64 |
9.95 ± 0.58 |

From these equations and Table 9.2, the molecular and structural parameters for casein under submicellar conditions were evaluated. No change in any molecular or structural parameter was observed with variation of the protein concentration. The averages of these results are presented in Table 9.4 for the molecular parameters and Table 9.5 for the structural parameters, subscripted as previously. These two tables illustrate the vast amount of quantitative information and the excellent precision available from analysis of SAXS on macromolecular systems with the aid of nonlinear regression.

A similar analysis was done on the SAXS results for the casein micelle solutions. The first two components of the model in Table 9.1 represented by the first four parameters in Table 9.3 reflect the contribution of the submicellar structure to the SAXS results. The third component, which has

Table 9.5 Structural Parameters from SAXS Data

Parameter Casein submicelle Casein micelle

V (À3) 12.72 ± 0.25 x 106 V2 (Â3) 3.33 ± 0.26 x 106 4.44 ± 0.16 x 106

Vc (À3) 0.467 ± 0.002 x 106 0.529 ± 0.003 x 106 VL (À3) 2.86 ± 0.40 X 106 3.91 ± 0.03 x 106

the highest radius of gyration (Ra3 in Table 9.3), reflects the total number of submicellar particles within the cross-sectional scattering profile. Here, at zero angle, the intensity of the larger Gaussian contribution can be simply added to the intensity of submicellar contribution. A new parameter, f2, the ratio of the mass of the submicelles to the total observed mass ascribable to a cross section, is defined as f= (912^

The inverse of f2 is called the packing number; that is, the number of submicellar particles within the observed "cross-sectional mass."

The scattering data for casein micelles were analyzed using the preceding equations for all protein concentrations. Again, no variation of SAXS-derived parameters with protein concentration was observed. The average values with corresponding errors are presented in Tables 9.4 and 9.5. In these tables, subscript 2 denotes corresponding parameters for the submicellar particles when incorporated in an observed scattering volume of the micelle, and the unsubscripted parameters represent total cross-sectional features of the colloidal micelle particle.

The SAXS results on casein were further interpreted by using physical models for the submicelle and micelle [4]. Tertiary and quaternary structural differences between two genetic variants of bovine casein have also been studied by SAXS with nonlinear regression of the data [7] onto the models in Table 9.1. The preceding discussion illustrates that this approach yields a quite complete physical picture of the macromolecular species present in the samples.

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