## 4 Lamellar Structures
The data in Figure 9c were used to obtain the electron density profile \(\rho(x)\) across the thickness of the membrane using Equation (14). \[\rho(x)=\int_{q_{min}}^{q_{max}}{(\pm)q\sqrt{I(q)}\text{cos}(qx)dq} \tag{14}\]  
where the \(\sqrt{I(q)}\) is the scattering amplitude of the vesicle, which is a randomly oriented bilayer, and hence is multiplied by \(h\). The electron density profiles shown in the inset show that the MBP does not significantly affect the electron density in the interior of the layer (away from the polar groups). However, the lipid bilayer's hydrophilic polar groups interact with MBP, increasing the electron density in that region, as indicated by large increases in intensity near \(h\) = 0 in the diffraction scan (Figure 9d inset). This, and the increases in the width of the bilayer determined by the position (\(q\)) of the peak in the scattering pattern, support the idea that the protein alters the structure of the lipid membrane [33]. [PERSON] presented a detailed analysis of the lamellar diffraction patterns by focusing on the diffuse scattering that is present in the diffraction patterns using thermal disorder and a paracrystalline model of lamellar structures [112]. In this analysis, the structure factor, \(S(q)\), for perfect and evenly spaced layers in the lamellar structure, is written as:  
\[S(q)=N+2\sum_{k=1}^{N-1}{(N-k)\text{cos}\left(kqd\right)} \tag{15}\]  
Figure 9: **(a)** Structure of myelin; from [PERSON] et al. (2023) [110]. **(b)** SAXS diffraction pattern of a dried sciatic frog nerve with \(d\)-spacings of the reflections; [PERSON] and [PERSON] (1982) [34]. **(c)** SAXS intensity profile for phosphatidylglycerol (PG) [profile a] and PG-myelin basic protein (MBP) complex [profile b] (MBP); the electron density profile derived from these intensity distribution is shown in the inset; from [PERSON] et al. (1984) [33]. **(d)** SAXS pattern of a basic protein-phosphatidylglycerol; from [PERSON] et al. (1984) [33]. **(e)** SAXS data from a multilamellar vesicular dispersion with a Caille structure factor and a lamellar bilayer form factor. Electron density profile is shown in inset. The solid line represents the profile across the bilayer. The dashed layer made with Fourier transformations; reprinted from [PERSON] et al. (2000) [111] with permission from American Physical Society. where \(N\) is the total number of layers inside the scattering domain, \(d\) is the layer spacing, \(q\) is the scattering vector, and \(k\) represents each layer. \(S(q)\) essentially accounts for how the X-rays scattered by one layer interfere with those from other layers. This equation needs to be modified to take into account the various types of defects and imperfections. This thermal disorder model uses the Debye-Waller factors to account for random positional fluctuations. Another main model is the paracrystalline model, in which small fluctuations of layer spacing are considered. In the third model, the Caille model, the fluctuations are quantified in terms of the flexibility of the membranes:  
\[S(q)=N+2\sum_{k=1}^{N-1}{(N-k)\text{cos}\left(kqd\right)}\times\text{exp}\ [- \left(\frac{qd}{2\pi}\right)^{2}\eta\gamma](\pi k)^{-(\frac{qd}{2\pi})^{2}\eta} \tag{16}\]  
where \(\gamma\) is Euler's constant and \(\eta\) is the Caille parameter. \(\eta\) depends on the lamellae's bulk compression modulus, B, and the bending rigidity, K. This model considers fluctuations from elastic membrane deformations. Figure 9e shows an example of the experimental data fitted with this model. \[\eta=\frac{\pi k_{B}T}{2d^{2}{(BK)}^{1/2}} \tag{17}\]  
The paracrystalline and the Caille models were used to model the bilayer form factor between the lipid and peptide. To fit the model, four parameters were used: \(z_{\text{H}}\), \(\sigma_{\text{H}}\) (center and width of the headgroup Gaussian, respectively), \(\sigma_{\text{c}}\), (width of the hydrocarbon chain Gaussian), and \(\rho_{\text{R}}\) (ratio of the methyl-terminus electron density amplitude to the headgroup amplitude) [111; 112].