Connection to Vesicles¶

The hollow sphere model is not just a mathematical curiosity — it is the first reasonable physical description of a lipid vesicle: a closed, spherical membrane that encloses an aqueous interior.

What is a lipid vesicle?¶

A lipid molecule has a hydrophilic (water-loving) headgroup and one or two hydrophobic (water-avoiding) hydrocarbon tails. In water, lipids spontaneously assemble into a bilayer: two leaflets arranged tail-to-tail, with headgroups facing outward into the water on both sides.

When a bilayer closes on itself it forms a vesicle — a hollow sphere whose wall is the bilayer membrane.

The bilayer thickness in a typical phospholipid vesicle is roughly 30–40 Å. The vesicle radius depends on preparation conditions but is commonly 200–1000 Å (20–100 nm) for laboratory samples.

Simulating a vesicle¶

Use the multi_shell_form_factor function with parameters that represent a realistic lipid vesicle:

R_vesicle   = 300.0   # outer radius in Å (30 nm)
t_bilayer   = 35.0    # bilayer thickness in Å
R_interior  = R_vesicle - t_bilayer

rho_water   = 9.47e-6   # Å^-2 — both interior and exterior
rho_lipid   = 7.0e-6    # Å^-2 — hydrocarbon chain region (below water)

P_vesicle = multi_shell_form_factor(
    q,
    radii=[R_interior, R_vesicle],
    slds=[rho_water, rho_lipid],    # core = matrix = water
    rho_matrix=rho_water,
)

fig, ax = plt.subplots(figsize=(8, 5))
plot_form_factor(q, P_vesicle,
                 label=f"Vesicle: R = {R_vesicle} Å, t = {t_bilayer} Å",
                 ax=ax, color="steelblue")
ax.set_title("Scattering from a simple vesicle model")
plt.tight_layout()
plt.show()

The curve looks different from anything in Section 2. Note the pronounced peak at intermediate \(q\) — this is characteristic of hollow sphere scattering and is one of the signatures used to identify vesicles in experimental data.

Reading structural parameters from the curve¶

The scattering curve from a vesicle encodes two geometric parameters: the outer radius \(R\) and the shell thickness \(t\).

Outer radius: The first prominent minimum in the hollow sphere form factor falls at approximately \(q_{\min} \approx 4.5/R\), the same as for a solid sphere of the same outer radius. Features at small \(q\) are governed by the large-scale size of the object.

Shell thickness: At larger \(q\), the oscillation period is related to the shell thickness. For a thin shell, minima are spaced by approximately \(\Delta q \approx \pi / t\). For shells that are not thin compared to \(R\), both length scales contribute and the pattern is more complex — but the general trend holds: thicker shells produce more closely spaced high-\(q\) features.

Compare vesicles of different sizes and thicknesses to build intuition:

configs = [
    (300, 35, "steelblue"),
    (500, 35, "tomato"),
    (300, 60, "seagreen"),
]

fig, ax = plt.subplots(figsize=(9, 5))

for R, t, color in configs:
    R_in = R - t
    P = multi_shell_form_factor(
        q,
        radii=[R_in, R],
        slds=[rho_water, rho_lipid],
        rho_matrix=rho_water,
    )
    plot_form_factor(q, P, label=f"R = {R} Å, t = {t} Å", ax=ax, color=color)

ax.set_title("Vesicle scattering: effect of radius and thickness")
plt.tight_layout()
plt.show()

Observe that increasing \(R\) shifts the main peak to smaller \(q\), while increasing \(t\) (with \(R\) fixed) shifts the finer high-\(q\) oscillations to smaller \(q\).

The limits of this model¶

The simple hollow sphere treats the bilayer as a uniform slab with a single SLD. A real lipid bilayer is not uniform — it has a layered structure with distinct regions:

1. Outer headgroup layer — phosphocholine or similar, higher electron density
2. Hydrocarbon chain region — two monolayers tail-to-tail, lower electron density
3. Inner headgroup layer — mirror of the outer layer

Each of these regions has a different SLD, and together they produce a characteristic electron density profile that is asymmetric across the bilayer. The multi-shell model you have already built can represent this — the bilayer becomes three shells (outer headgroup, chain region, inner headgroup) instead of one.

R_outer       = 300.0  # Å
t_head_outer  = 9.0
t_chains      = 26.0
t_head_inner  = 9.0
t_total       = t_head_outer + t_chains + t_head_inner  # 44 Å

# Radii from inside out
R_inner        = R_outer - t_total          # interior water surface
R_chain_start  = R_inner  + t_head_inner    # start of chain region
R_chain_end    = R_chain_start + t_chains   # end of chain region (= outer head start)
# R_outer = R_chain_end + t_head_outer = 300 Å ✓

rho_water    = 9.47e-6
rho_headgroup = 12.0e-6
rho_chains   =  7.0e-6

P_bilayer = multi_shell_form_factor(
    q,
    radii=[R_inner, R_chain_start, R_chain_end, R_outer],
    slds=[rho_water, rho_headgroup, rho_chains, rho_headgroup],
    rho_matrix=rho_water,
)

# Single-layer reference
P_single = multi_shell_form_factor(
    q,
    radii=[R_outer - t_total, R_outer],
    slds=[rho_water, rho_chains],
    rho_matrix=rho_water,
)

fig, ax = plt.subplots(figsize=(8, 5))
plot_form_factor(q, P_single,  label="Single-layer bilayer",  ax=ax, color="steelblue")
plot_form_factor(q, P_bilayer, label="Three-layer bilayer",   ax=ax, color="tomato")
ax.set_title("Bilayer structure: single-layer vs three-layer model")
plt.tight_layout()
plt.show()

Where to go from here¶

The model you have built — a multi-shell hollow sphere — is a practical starting point for analyzing SAXS data from vesicle systems. Real experimental analysis extends this in several directions:

• Polydispersity in vesicle radius — real samples contain a distribution of sizes, broadening and dampening the oscillations (the same effect you explored in Section 2)
• Bilayer asymmetry — the inner and outer leaflets of a biological membrane can differ in lipid composition, making the electron density profile asymmetric
• Structure factor — at high vesicle concentrations, inter-vesicle interactions produce additional features in the scattering pattern
• Gaussian electron density profiles — fitting the continuous bilayer electron density with Gaussian functions provides more structural detail than discrete shell models

These topics are addressed in the research literature referenced in the context materials for this tutorial.

Final commit¶

Save everything you have built:

$ git add chapter_04_shells.ipynb
$ git commit -m "add multi-shell and hollow sphere form factor models"
$ git push

You now have a complete, version-controlled scattering toolkit: a sphere form factor, polydisperse averaging, a general multi-shell model, and its application to the hollow sphere geometry that describes vesicle scattering.

What's next: Section 5 → — replacing the sharp-interface shell model with continuous Gaussian electron density profiles that more accurately describe a real lipid bilayer.