Comparing to the Shell Model¶

You now have two ways to compute the scattering form factor of a lipid vesicle: the three-shell model from Section 4 and the Gaussian profile model from this section. This page puts them side by side and examines when they agree, when they diverge, and what that means for interpreting experimental data.

Setting up the comparison¶

For a fair comparison, both models should represent the same nominal bilayer geometry. Map the POPC Gaussian parameters onto equivalent three-shell parameters:

# Gaussian model parameters (POPC)
R       = 300.0
d_HH    = 37.1
sigma_H =   3.0
sigma_C =   5.5
A_H     =  3.0e-6
A_C     =  1.2e-6
rho_water = 9.47e-6

# Equivalent three-shell parameters
# Treat each layer thickness as the FWHM of its Gaussian: FWHM = 2*sqrt(2*ln2)*sigma ~ 2.35*sigma
t_head  = 2.35 * sigma_H          # ~7 Å headgroup thickness
t_chain = d_HH - 2 * t_head       # chain region thickness
R_outer = R
R_chain_end   = R_outer - t_head
R_chain_start = R_chain_end - t_chain
R_inner       = R_chain_start - t_head

rho_head  = rho_water + A_H        # headgroup SLD
rho_chain = rho_water - A_C        # chain SLD

P_shell = multi_shell_form_factor(
    q,
    radii=[R_inner, R_chain_start, R_chain_end, R_outer],
    slds=[rho_water, rho_head, rho_chain, rho_head],
    rho_matrix=rho_water,
)

P_gauss = vesicle_form_factor_gaussian(q, R, d_HH, sigma_H, A_H, sigma_C, A_C,
                                        rho_water)

fig, ax = plt.subplots(figsize=(8, 5))
plot_form_factor(q, P_shell, label="Three-shell model",   ax=ax, color="tomato")
plot_form_factor(q, P_gauss, label="Gaussian profile",    ax=ax, color="steelblue")
ax.set_title("Shell model vs Gaussian profile — POPC vesicle")
plt.tight_layout()
plt.show()

What you should observe¶

The two curves agree closely at small \(q\) and begin to diverge at larger \(q\):

Key Concept: Where the Models Differ

At small to intermediate \(q\) — below roughly \(q \approx 0.2\) Å\(^{-1}\) for a typical lipid bilayer — both models predict essentially the same curve. The scattering at these length scales is dominated by the overall bilayer thickness and vesicle radius, which both models represent the same way.

At larger \(q\), the Gaussian profile predicts systematically lower oscillation amplitude than the shell model. The reason is the \(\exp(-q^2\sigma^2/2)\) Gaussian envelope: each interface contributes a scattering amplitude that decays exponentially with \(q^2\). Broader interfaces (larger \(\sigma\)) decay faster. A sharp interface has \(\sigma = 0\) and no decay at all — it contributes equally at all \(q\), which is why the shell model over-predicts high-\(q\) oscillations compared to a real membrane.

Convergence at narrow interfaces¶

When \(\sigma_H \to 0\) and \(\sigma_C \to 0\), the Gaussian profile approaches a step function and the two models should converge. Verify this:

fig, ax = plt.subplots(figsize=(8, 5))
plot_form_factor(q, P_shell, label="Three-shell model", ax=ax,
                 color="tomato", linestyle="--")

for sigma, color in [(3.0, "steelblue"), (1.0, "seagreen"), (0.3, "darkorchid")]:
    P = vesicle_form_factor_gaussian(q, R, d_HH, sigma, A_H, sigma, A_C, rho_water)
    plot_form_factor(q, P, label=rf"Gaussian $\sigma$ = {sigma} Å", ax=ax, color=color)

ax.set_title("Convergence to shell model as Gaussian widths narrow")
plt.tight_layout()
plt.show()

As \(\sigma\) decreases from 3.0 to 0.3 Å the Gaussian curve approaches the shell model from below, confirming that the shell model is the \(\sigma \to 0\) limit of the Gaussian model.

Try It Yourself

At what value of \(\sigma\) do the Gaussian and shell models become visually indistinguishable across the full \(q\) range? Is this physically achievable in a real bilayer?
Fix sigma_H = 3.0 Å and vary only sigma_C between 1 and 15 Å. At which \(q\) values does the chain width most strongly affect the curve?
Set both A_H and A_C to very small values (near zero contrast). What happens to both curves? Does the Gaussian model degrade more gracefully than the shell model in the low-contrast limit?
Conceptual: A SAXS experiment on POPC vesicles measures data up to \(q_{\max} = 0.35\) Å\(^{-1}\). Looking at your comparison plot, would you be able to distinguish the Gaussian model from the shell model at this \(q\) range? What if \(q_{\max} = 0.6\) Å\(^{-1}\)?

Solution

Part 1: The curves become indistinguishable by eye around \(\sigma \approx 0.5\) Å. This is far narrower than any real fluid-phase lipid bilayer (where \(\sigma_H \sim 3\)–\(5\) Å), so a sharp-interface model always has measurable error relative to the physical truth for fluid-phase membranes.

Part 2: sigma_C affects the intermediate-\(q\) region most strongly — roughly between the first minimum and third minimum of the form factor. The headgroup peaks at higher \(q\) are relatively insensitive to chain width because they arise primarily from the headgroup Gaussians.

Part 3: Both curves approach a flat line \(P(q) = 1\) as contrast vanishes, since the vesicle becomes indistinguishable from the surrounding water. The Gaussian model degrades smoothly; the shell model behaves identically because the normalization absorbs the contrast.

Part 4: Up to \(q_{\max} = 0.35\) Å\(^{-1}\) the two models overlap closely for POPC parameters — you would not be able to reliably distinguish them from data alone. Above \(q \approx 0.4\)–\(0.5\) Å\(^{-1}\) the models diverge measurably, so data to \(q_{\max} = 0.6\) Å\(^{-1}\) would allow the Gaussian widths to be determined from a fit. This is the primary reason that structural SAXS studies of bilayers push to higher \(q\) whenever beam time and signal-to-noise allow.

Git checkpoint

$ git add chapter_04_gaussian_bilayer.ipynb
$ git commit -m "compare Gaussian and shell model form factors for POPC"

What's next: Asymmetric Bilayer — extending the Gaussian model to allow the inner and outer leaflets to have different structural parameters.