casein_micelle

** BETA version **
testing in progress...

The model is decribed in
Pedersen et al. (2022) Soft Matter 18: 8613, DOI: 10.1039/d2sm00724j
changes:
- numerical integration for polydipserse spheres to avoid discontinuities at low q, including the option to reduce steps in numerical integrals while refining parameters

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 or fat globules, 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 internal structure - could be cavities or incompressible regions (polydisperse spheres)
* $I_{P}$: protein particles (star of rods with hard-sphere structure factor)
* $I_{C}$: colloidal 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:

$$ I_{total} = I_{aggr} + I_{micelle} + I_{int} + I_{P} + I_{C} + I_{PC} + I_{L} + I_{B}$$

the aggregate contribution is given as a power law:

$$ A_{power} q^{-power} $$

where $A_{power} $ is a scaling constant.

The scattering from the casein micelle overall shape is modelled as a polydipserse sphere with normal size distribution - with the option of a hard-sphere structure factor.

The scattering from the intermediate stuctures were also modelled with a polydisperse sphere form factor.

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

$$ P(q) = (P_{rod}/n_a + (1-1/n_a) A_{rod}^2) A_{xs}^2 $$

$P_{rod}$ 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:

$$ V_{rel,P} = R_{HS,P}^3/R_{HS,av}^3 $$

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

$$ S_P = 1 + fraction * weight * (S_{HS}(q,eta,R_{HS})-1) $$

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

$$ S_{eff,P} = 1 + \beta(S_P-1) $$

where $\beta$ is given in terms of the form factor amplitude, $A(q)$:

$$ \beta = \frac{<A(q)>^2}{<A^2(q)>} $$

where $<..>$ is the orientation average, and $<A^2(q)>$ 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 excess scattering length density - or contrast - times the volume of one protein subparticle):

$$I_P = N_p * x * M_P^2 * P_P * S_{eff,P}$$

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

$$ P_C = int_0^{pi/2} A_{sphere}(q,R_{eff})^2 \sin(\alpha)d\alpha $$

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

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

$$ I_C = N_p * x * M_C^2 * P_C * S_{eff,C} $$

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:

$$ P_{PC} = A_P A_C $$

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

$$I_{PC} = 2 N_p \sqrt{x(1-x)} M_PC S_{eff_PC}$$

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

$$I_{Lorentz} = \frac{A_{Lorentz}}{1 + (\frac{q-q_{Lorentz}}{w_{Lorentz}})^2}$$

where $A_{Lorentz}$ is a scaling constatn.

The model uses the SasView built in constant background to model imperfect background subtraction.

Notes on scaling:

$\cdot$ Overall scaling is effectively fitted through protein concentration ($c_P$) and calsium phosphate concentraiont ($c_C$)
$\cdot$ This is because all the terms $I_{micelle} + I_{int} + I_{P} + I_{C} + I_{PC}$ are scaled with $N_p = n_C + n_P$ where $n_C=c_C/mass_C$ and $n_P=c_P N_A/MW_P$ ($N_A$ is Avogadros number and $MW_P$ is the protein mol weight).
$\cdot$ Therefore, do NOT use SasView built-in scaling if these are fitted.
$\cdot$ Also, be carefull if you fit c_P and $MW_P$ - they are very correlated - and both affect overall scattering along with relative protein scattering.
$\cdot$ The relative scattering of casein micelles compared to that from intermediate structures can be adjusted with $n_{int}$.
$\cdot$ The relative scattering of casein micelles compared to that from protein and calcium phosphate can be adjusted with $n_{sub}$

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