casein_micelle

Definition
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This model calculates the scattering from casein micelles

The total scattering intensity $I(q)$ has eight contribtuions:

* $I_{aggr}$: Large aggregates, substantially larger than the casein micelles (power law)
* $I_{micelle}$: The overall casein micelle (polydisperse spheres with hard-sphere structure factor )
* $I_{int}$: Intermediate sized submicelles (polydisperse spheres)
* $I_{P}$: protein (star of rods with hard-sphere structure factor)
* $I_{C}$: calcium phosphate (oblate ellipsoids with hard-sphere structure factor)
* $I_{PC}$: protein-calcium phosphate cross terms
* $I_{L}$: protein stacking (Lorentzian peak)
* $I_{B}$: imperfect background subtraction correction (constant)

and the total scattering is the sum of these terms:

.. math:: I_{total} = I_{aggr} + I_{micelle} + I_{int} + I_{P} + I_{C} + I_{PC} + I_{L} + I_{B}

the aggregate contribution is given as a power law:

.. math:: scale_power q^{-power}

the scattering from the casein micelle overall shape is modelled as a polydipserse sphere with Schulz size distribution

.. math:: P_{sphere}

the scattering from the submicelles were modelled with the same polydisperse sphere form factor

the scattering from the protein was modelled with the form factor of a star of rods with $n_a$ arms:

.. math:: P_P(q) = (P_rod/n_a + (1-1/n_a) A_rod^2) A_{xs}^2

$P_{rod}$ is the form factor of an infinitely thin rod, $P_{rod} = 2Si(qL)/qL - sinc(qL/2)$, where $sinc(x) = sin(x)/x$ and $Si(x)$ is the sine integral. The arm cross section was taken into account by $A_{xs} = 2J_1(qR_{xs})/(qR_{xs})$, where $J_1$ is the first order Bessel function of the first kind.

where L is the arm length, n_a is the number of arms and...

the proteins interacts with a hard-sphere structure factor, given as usual. The structure factor was weighted by the the fraction of subparticles being proteins, x. The fraction of calcium phosphate is consequently (1-x). The structure factor was also and weighted with the relative volume:

.. math:: V_{rel,P} = R_{HS,P}^3/R_{HS,av}^3

where R_{HS,av} = ((1-x)*R_{HS,C}**3 + x*R_{HS,P}**3)**(1/3). The final weighted structure factor is then:

.. math:: S_P = 1 + fraction * weight * (S_{HS}(q,eta,R_HS)-1)

Due to aniosotropy, the structure factor was modified by the decoupling approximation:

.. math:: S_{eff,P} = 1 + beta * (S_P-1)

where beta is given in terms of the form factor amplitude:

.. math:: beta = <A>^2/<A^2>

where <..> is the orientation average, and <A^2> is the form factor.

The scattering from the protein is weighted with the total number of proteins, in terms of the fraction of proteins (x), and the total number density of subparticles (N_p), as well as the protein scattering mass (M_P, which is the contrast or excess scattering length density times the volume of one protein subparticle):

.. math:: I_P = N_p * x * M_P**2 * P_P * S_{eff,P}

The scattering from the calcium phosphate subparticles is similar, but with the form factor described as oblate ellipsoids with axes $R,R,\vareps R$ and averaged all orientations ($\alpha$):

.. math:: P_C = int_0^(pi/2) psi_{sphere}(q,R_{eff})^2 sin(\alpha)d\alpha

where R_{eff} is the effective radius, $R_{eff} = R \sqrt(\sin^2(\alpha) + \vareps \cos^2(\alpha))$, and psi_{sphere}(x) is the form factor amplitude of a sphere, $psi_{sphere}(x) = 3(\sin(x) - x \cos(x))/x^3$.

The fraction of calcium phosphates is (1-x), so the scattering from calcium phosphate is:

.. math:: I_C = N_p * x * M_C**2 * P_C * S_{eff,C}

with the effective and weighted structure factor provided as described for the protein.

The calcium phosphate and proteins are similar is size, so we must include their cross term. Their effective form factor is provided as a product of the form factor amplitudes:

.. math:: P_{PC} = A_P A_C

likewise for the scattering mass: M_{PC} = M_P M_C. The effective fraction is sqrt(x*(1-x)). Using the approximations R_{HS,PC} \approx R_{HS_P} and eta_{HS,PC} \approx eta_{HS_P} and beta_{PC} = 1, the effective and weighted structure factor can be calculated as described for the protein. The scattering is

.. math:: I_{PC} = 2 N_p sqrt(x(1-x)) M_PC S_{eff_PC}

Finally, the protein form repeated stacking, modelled as a Lorentzian peak at q_{Lorentz} with width w_{Lorentz}:

.. math:: I_{Lorentz} = scale_{Lorentz} = scale_{Lorentz}/[1 + (q-q_{Lorentz})^2/w_{Lorentz}^2]

Use the SasView built in constant background to model imperfect background subtraction. Do NOT use SasView built-in scaling.

References
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Jan Skov Pedersen, Thea Lykkegaard Møller, Norbert Raak, and Milena Corredig. Soft Matter (2022) 18: 8613–8625. DOI: 10.1039/d2sm00724j

Authorship and Verification
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* **Author:** Andreas Haahr Larsen **Date:** 10 December 2025