Membrane Elasticity

Lipid Bilayer Mechanics

Biological membranes are fluid lipid bilayers roughly 4–5 nm thick. Despite being only two molecules thick, they resist bending and area changes, controlling cell shape and vesicle formation.

Helfrich Bending Energy

The bending energy per unit area of a membrane is described by the Helfrich Hamiltonian:

$\frac{E_{\text{bend}}}{A} = \frac{\kappa}{2}(c_1 + c_2 - c_0)^2$

where c₁ and c₂ are the two principal curvatures, c₀ is the spontaneous curvature (zero for symmetric bilayers), and κ is the bending modulus.

For a sphere of radius R with c₁ = c₂ = 1/R and c₀ = 0, integrating over the surface area A = 4πR²:

$E_{\text{sphere}} = \frac{\kappa}{2}\left(\frac{2}{R} - c_0\right)^2 \cdot 4\pi R^2$

Remarkably, for c₀ = 0 this simplifies to E = 8πκ, independent of vesicle size!

Typical values: κ ≈ 10–25 k_BT (≈ 4–10 × 10⁻²⁰ J at 37°C).

Area Stretching

The bilayer strongly resists changes in area. The area stretch energy is:

$E_{\text{stretch}} = \frac{K_A}{2} \left(\frac{\Delta A}{A_0}\right)^2 A_0$

where K_A ≈ 0.24 N/m is the area stretch modulus. The membrane tension is:

$\sigma = K_A \cdot \frac{\Delta A}{A_0}$

Membranes lyse (rupture) when stretched by only ~3–5% (σ_lytic ≈ 5–10 mN/m).

Thermal Fluctuations

Thermal energy k_BT sets the scale for bending fluctuations. The dimensionless bending modulus:

$\tilde{\kappa} = \frac{\kappa}{k_B T}$

quantifies how "stiff" a membrane is relative to thermal energy.

Your Task

Implement four functions:

1. bending_energy_sphere_J(kappa_J, R_m, c0_m=0) — Total bending energy of a spherical vesicle in Joules
2. area_stretch_energy_J(K_A_N_m, A0_m2, dA_m2) — Area stretching energy in Joules
3. lytic_tension_N_m(K_A_N_m=0.24, strain=0.05) — Membrane tension at 5% strain in N/m
4. thermal_bending_modulus_kBT(kappa_J, T_K=310) — Bending modulus in units of k_BT