Asymmetric Bilayer¶

The symmetric Gaussian model assumes both leaflets of the bilayer are structurally identical — same headgroup width, same amplitude. For many purposes this is a reasonable first approximation, but real vesicle membranes break this symmetry in ways that are physically meaningful and measurable with SAXS.

Why bilayers are not symmetric¶

Two sources of asymmetry are relevant for vesicle scattering:

Geometric asymmetry from curvature. In a spherical vesicle the outer leaflet occupies more surface area than the inner leaflet. For a vesicle of radius \(R\) with bilayer thickness \(d_{HH}\), the ratio of outer to inner leaflet area is \((R / (R - d_{HH}))^2\). For a 300 Å radius POPC vesicle this is approximately \(1.28\) — the outer leaflet has 28% more area per lipid than the inner leaflet. This difference in packing density leads to different chain order, slightly different headgroup exposure, and a measurably different electron density profile on each side.

Structural disorder asymmetry. Brzustowicz & Brunger (2005) analyzed SAXS data from unilamellar SOPS vesicles and found that the inner headgroup layer is consistently rougher — has a larger Gaussian width — than the outer layer. They attributed this to the inner leaflet relieving curvature strain by adopting a more conformationally disordered state. For POPC this translates to:

Outer headgroup: \(\sigma_H^{\text{out}} \approx 3.0\) Å
Inner headgroup: \(\sigma_H^{\text{in}} \approx 4.5\) Å

The asymmetric model¶

The symmetric model forced both headgroup Gaussians to share the same amplitude \(A_H\) and width \(\sigma_H\). The asymmetric model gives each its own parameters:

\[\rho(r) = \rho_w + A_H^{\text{out}} \exp\!\left(-\frac{(r - R)^2}{2(\sigma_H^{\text{out}})^2}\right) + A_H^{\text{in}} \exp\!\left(-\frac{(r - (R - d_{HH}))^2}{2(\sigma_H^{\text{in}})^2}\right) - A_C \exp\!\left(-\frac{(r - r_C)^2}{2\sigma_C^2}\right)\]

The chain region remains symmetric about the midplane \(r_C = R - d_{HH}/2\).

Implementation¶

def asymmetric_bilayer_electron_density(r, R, d_HH,
                                         sigma_H_out, A_H_out,
                                         sigma_H_in,  A_H_in,
                                         sigma_C, A_C, rho_water):
    """
    Electron density profile for an asymmetric lipid bilayer vesicle.

    Allows independent Gaussian parameters for the outer and inner
    headgroup layers.

    Parameters
    ----------
    r : array-like
        Radial coordinate in Å.
    R : float
        Outer vesicle radius in Å.
    d_HH : float
        Head-to-head distance in Å.
    sigma_H_out : float
        Outer headgroup Gaussian width in Å.
    A_H_out : float
        Outer headgroup amplitude (positive, in Å^-2).
    sigma_H_in : float
        Inner headgroup Gaussian width in Å.
    A_H_in : float
        Inner headgroup amplitude (positive, in Å^-2).
    sigma_C : float
        Chain Gaussian width in Å.
    A_C : float
        Chain amplitude (positive, subtracted, in Å^-2).
    rho_water : float
        Background SLD in Å^-2.

    Returns
    -------
    numpy.ndarray
        Electron density rho(r) in Å^-2.
    """
    r_H_out = R
    r_H_in  = R - d_HH
    r_C     = R - d_HH / 2

    return (rho_water
            + A_H_out * np.exp(-0.5 * ((r - r_H_out) / sigma_H_out)**2)
            + A_H_in  * np.exp(-0.5 * ((r - r_H_in)  / sigma_H_in)**2)
            - A_C     * np.exp(-0.5 * ((r - r_C)      / sigma_C)**2))


def asymmetric_vesicle_form_factor(q, R, d_HH,
                                    sigma_H_out, A_H_out,
                                    sigma_H_in,  A_H_in,
                                    sigma_C, A_C, rho_water, n_points=2000):
    """
    Form factor P(q) for a vesicle with an asymmetric Gaussian bilayer profile.

    Parameters
    ----------
    q : array-like
        Scattering vector in Å^-1.
    R, d_HH, sigma_H_out, A_H_out, sigma_H_in, A_H_in, sigma_C, A_C, rho_water :
        See asymmetric_bilayer_electron_density.
    n_points : int
        Number of points in the radial integration grid.

    Returns
    -------
    numpy.ndarray
        P(q), normalized to 1 at q -> 0.
    """
    sigma_max = max(sigma_H_out, sigma_H_in)
    r = np.linspace(0.1, R + 5 * sigma_max, n_points)

    rho = asymmetric_bilayer_electron_density(r, R, d_HH,
                                               sigma_H_out, A_H_out,
                                               sigma_H_in,  A_H_in,
                                               sigma_C, A_C, rho_water)
    delta_rho = rho - rho_water

    F = np.array([
        np.trapz(delta_rho * r * np.sin(qi * r), r) * 4 * np.pi / qi
        for qi in q
    ])
    F0 = 4 * np.pi * np.trapz(delta_rho * r**2, r)

    return (F / F0)**2

Visualizing the asymmetric profile¶

Compare the symmetric and asymmetric profiles side by side:

r_plot = np.linspace(R - d_HH - 4*max(sigma_H, 4.5), R + 4*max(sigma_H, 4.5), 1000)

rho_sym = bilayer_electron_density(r_plot, R, d_HH, sigma_H, A_H, sigma_C, A_C,
                                    rho_water)
rho_asym = asymmetric_bilayer_electron_density(r_plot, R, d_HH,
                                                sigma_H_out=3.0, A_H_out=A_H,
                                                sigma_H_in=4.5,  A_H_in=A_H,
                                                sigma_C=sigma_C, A_C=A_C,
                                                rho_water=rho_water)

fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(r_plot, rho_sym  * 1e6, color="steelblue", label="Symmetric (σ = 3.0 Å both)")
ax.plot(r_plot, rho_asym * 1e6, color="tomato",    linestyle="--",
        label=r"Asymmetric ($\sigma_{\rm out}$ = 3.0, $\sigma_{\rm in}$ = 4.5 Å)")
ax.axhline(rho_water * 1e6, color="gray", linestyle=":", linewidth=0.8)
ax.set_xlabel(r"$r$ (Å)")
ax.set_ylabel(r"$\rho(r)$ ($10^{-6}$ Å$^{-2}$)")
ax.legend(fontsize=9)
ax.set_title("Symmetric vs asymmetric POPC bilayer profile")
plt.tight_layout()
plt.show()

The outer headgroup peak is unchanged. The inner headgroup peak is broader and shorter — the same area under the curve (same total electron count) but spread over a wider range.

Effect on the scattering curve¶

P_sym  = vesicle_form_factor_gaussian(q, R, d_HH, sigma_H, A_H, sigma_C, A_C,
                                       rho_water)
P_asym = asymmetric_vesicle_form_factor(q, R, d_HH,
                                         sigma_H_out=3.0, A_H_out=A_H,
                                         sigma_H_in=4.5,  A_H_in=A_H,
                                         sigma_C=sigma_C, A_C=A_C,
                                         rho_water=rho_water)

fig, ax = plt.subplots(figsize=(8, 5))
plot_form_factor(q, P_sym,  label="Symmetric",   ax=ax, color="steelblue")
plot_form_factor(q, P_asym, label="Asymmetric",  ax=ax, color="tomato")
ax.set_title("Effect of inner leaflet broadening on P(q)")
plt.tight_layout()
plt.show()

The asymmetric curve is nearly identical to the symmetric one at small \(q\) and diverges only at large \(q\) — reflecting the fact that the asymmetry is encoded in the fine structure of the bilayer profile, which requires high spatial-frequency information to resolve.

Try It Yourself

Increase sigma_H_in progressively from 3.0 to 8.0 Å in steps of 1 Å, keeping sigma_H_out = 3.0 Å fixed. At what sigma_H_in does a difference between the symmetric and asymmetric scattering curves become visible on your log-log plot?
Now also reduce A_H_in to 80% of A_H_out while keeping sigma_H_in = 4.5 Å. How does the combination of broader and lower-amplitude inner headgroup change the profile? Does this make the scattering difference larger or smaller?
Conceptual: Given the difference between the symmetric and asymmetric curves, what \(q_{\max}\) would your experiment need to reach to distinguish the two models? What signal-to-noise ratio would you need at those \(q\) values?

Solution

Part 1: A visible difference typically appears when sigma_H_in reaches approximately 5–6 Å. Below that, the broadening of a single peak contributes only a small perturbation to the total amplitude, and the difference is buried in the log scale. At 7–8 Å the inner headgroup begins to approach the chain region in width and the asymmetry becomes clearly visible above \(q \approx 0.4\) Å\(^{-1}\).

Part 2:

rho_asym2 = asymmetric_bilayer_electron_density(r_plot, R, d_HH,
                                                 sigma_H_out=3.0,
                                                 A_H_out=A_H,
                                                 sigma_H_in=4.5,
                                                 A_H_in=0.8*A_H,
                                                 sigma_C=sigma_C,
                                                 A_C=A_C,
                                                 rho_water=rho_water)

Reducing \(A_H^{\text{in}}\) lowers the inner headgroup peak further below the outer one, making the profile visibly asymmetric even by eye. However, the effect on the scattering curve is still small at accessible \(q\) values — SAXS is relatively insensitive to single-leaflet amplitude differences compared to width differences.

Part 3: The asymmetric and symmetric curves diverge meaningfully above \(q \approx 0.4\)–\(0.5\) Å\(^{-1}\). In practice, data to \(q \approx 0.6\) Å\(^{-1}\) with good signal-to-noise (\(I/\sigma > 5\) in the oscillations) would be needed to reliably fit leaflet asymmetry parameters. This is achievable at synchrotron SAXS beamlines but challenging with laboratory instruments.

Git checkpoint

$ git add chapter_04_gaussian_bilayer.ipynb
$ git commit -m "add asymmetric Gaussian bilayer model for POPC"

You have now built a complete progression of bilayer scattering models — from a uniform hollow sphere, through a symmetric Gaussian profile, to an asymmetric bilayer that reflects real membrane physics. The next steps in a real analysis would be fitting these models to experimental data, accounting for polydispersity in vesicle size, and including the structure factor for concentrated samples.