Section 5: Gaussian Bilayer Profiles¶
The shell models in Section 4 represent the bilayer as a stack of uniform slabs with perfectly sharp boundaries. That is a useful first approximation, but it does not reflect how a real membrane is structured. Lipid molecules fluctuate thermally, water penetrates into the headgroup region, and the interfaces between structural regions are gradients rather than steps.
This section replaces the sharp-interface model with a Gaussian electron density profile — a continuous function that describes the bilayer structure more physically. We will build the profile for a POPC bilayer, compute its scattering form factor numerically, compare it to the three-shell model, and then extend it to handle asymmetric bilayers where the inner and outer leaflets differ.
By the end of this section you will have:
- A function that builds a continuous radial electron density profile for a symmetric POPC bilayer
- A numerical form factor computed from that profile via a spherical Fourier transform
- A quantitative understanding of where the Gaussian and shell models agree and where they diverge
- An asymmetric bilayer model with independent parameters for each leaflet
All code belongs in chapter_04_gaussian_bilayer.ipynb. Copy plot_form_factor and
multi_shell_form_factor from your earlier notebooks into the first cell so everything
is self-contained.