Lamellar Slab Partition Constant

Note: model title and parameter table are inserted automatically

This model (Tan et al. J. Appl. Cryst. 2022) provides the scattering intensity, $I(q)$, for a lyotropic lamellar phase, randomly distributed in solution in the presence of a co-solvent which partitions into the lamellar phase. The lamellar phase structure is considered here as three layers; a core layer centered on the lamellar phase midpoint and excluding the primary solvent, and two symmetric outer layers (solvent-exposed
and solvent-containing). The co-solvent is permitted to partition into both layers of the lamellar phase.

This model describes the lamellar phase structure and co-solvent partitioning. The model is intended to be used with input parameters including; the molecular volume and bound coherent scattering length of the molecules comprising the lamellar structure, along with those of the primary solvent and co-solvent. The user is also required to enter the molar concentration of the constituent molecules.

The variable parameters are intended to be the APL, the average area per amphiphile (lipid) molecule at the core/outer layer interface, the n_W, the number of solvent (water) molecules residing in the outer layer of the bilayer, Kp, the partition constant, and P_s, the co-solvent localization constant which defines the fraction of the co-solvent located in the core layer.

This model is adapted from the lamellar_APL_Nw model in Tan et al. (2021) and earlier models (Nagle and Wiener, 1988) and the fitting function lamellar_hg (Nallet 1993) and (Berghausen 2001).

Note that the model can be applied with other combinations of input parameters/assumptions; and is ideally applied as a simultaneous fit to datasets with multiple independent measurements; such as neutron contrast variation strategies.

Definition
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The scattering intensity $I(q)$ is

$$P(q) = \frac{4}{q^2}
\left\lbrace
\Delta \rho_H
\left[\sin[q(\delta_H + \delta_T)\ - \sin(q\delta_T)\right]
+ \Delta\rho_T\sin(q\delta_T)
\right\rbrace^2$$

where $\delta_T$ is $length_{tail}$, $\delta_H$ is $length_{head}$,
$\Delta\rho_H$ is the head contrast ($sld_{head} - sld_{solvent}$),
and $\Delta\rho_T$ is tail contrast ($sld_{tail} - sld_{solvent}$),
$length_{tail}$ equals $(V_c+P_sV_sN_s)/APL$,
$length_{head}$ equals $(V_h+N_wV_w+(1-P_s)V_sN_s)/APL$,
$sld_{head}$ equals $(B_h+N_wB_w+(1-P_s)B_sN_s)/(V_h+N_wV_w+(1-P_s)V_sN_s)$,
$sld_{tails}$ equals $(B_c+P_sB_sN_s)/(V_c+P_sV_sN_s)$. Where $V_h$, $V_c$, $V_w$, $V_s$ are the molecular volume of lipid headgroup, lipid tails, solvent (water), and co-sovlent. $N_w$, $N_s$ are number of solvent and co-solvent molecules partitioning in lipid on a per lipid basis.

The total thickness of the lamellar sheet is $\delta_H + \delta_T + \delta_T + \delta_H$.
Note that in a non aqueous solvent the chemical "head" group may be the
"Tail region" and vice-versa.

The 2D scattering intensity is calculated in the same way as 1D, where
the $q$ vector is defined as

$q = \sqrt{q_x^2 + q_y^2}$

References
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Tan, L., Smith, M. D., Scott, H. L., Yahya, A., Elkins, J. G., Katsaras, J., ... & Nickels, J. D. (2022). Modeling the partitioning of amphiphilic molecules and co-solvents in biomembranes. Journal of Applied Crystallography, 55(6).
Tan, L., Elkins, J. G., Davison, B. H., Kelley, E. G., & Nickels, J. (2021). Implementation of a self-consistent slab model of bilayer structure in the SasView suite. Journal of Applied Crystallography, 54(1), 363-370.
Nagle, J., & Wiener, M. (1988). Structure of fully hydrated bilayer dispersions. Biochimica et Biophysica Acta (BBA)-Biomembranes, 942(1), 1-10
F Nallet, R Laversanne, and D Roux, *J. Phys. II France*, 3, (1993) 487-502
J Berghausen, J Zipfel, P Lindner, W Richtering, *J. Phys. Chem. B*, 105, (2001) 11081-11088

Authorship and Verification
----------------------------

Luoxi Tan, Micholas Dean Smith, Haden L. Scott, Ahmad Yahya, James G. Elkins, John Katsaras, Hugh M O'Neill, Sai Venkatesh Pingali, Jeremy C. Smith, Brian H Davison, Jonathan D. Nickels