Soft membranes

Vesicle Shapes and Bending

How can the same membrane area enclose a long vesicle or a flattened one? A membrane bends into the lowest-energy shape allowed by its area and enclosed volume. Removing water gives the same membrane more room to fold. A preferred local curvature can then favor a different shape, even when the amount of membrane stays fixed.

Deflate a soft vesicle and compare elongated and flattened shapes with equal area and volume. Color shows local mean curvature; bending energy decides which restricted shape branch is favored.

Reduced volume $0.800$

Bending energy / rigidity $35.3$

Other branch energy difference $2.05$

Polar / equatorial radius $2.87$

Deflate the same membrane

Reduced volume

v

0.8

Preferred curvature

c_0R_A

0

Extra polar curvature

\Delta c_0R_A

0

Drag the gold membrane handle to deflate the vesicle. Drag the background to turn it.

The long branch has lower bending energy in this shape family.

Shape family, units and limits

Lengths use the radius of a sphere with the same membrane area. Both candidate shapes enclose the selected volume. Color runs from cyan at lower mean curvature to amber at higher mean curvature. Lipid dots are schematic.

r(\theta)=s\exp[aP_2(\cos\theta)+bP_4(\cos\theta)]

The long and flat branches retain polar radii respectively above and below the equatorial radius, and minimize energy over the retained fourth-order mode. These restricted minima need not be stationary shapes against other deformations. Topology and membrane area are fixed.

c_0(\theta)R_A=c_{\rm global}+c_{\rm polar}\cos^8\theta

The polar pattern is imposed as a preferred-curvature field. A composition model would also need mixing energy, domain boundaries and transport. Uniform bending rigidity multiplies all energies and cannot be inferred from these shapes alone.

Model assumptions

An axisymmetric, two-mode radial family minimizes Helfrich bending energy at fixed area and volume. The polar curvature pattern is prescribed. Topology and area stay fixed; neck fission, composition dynamics, protein transport and the complete vesicle phase diagram are outside this model.

Deterministic variational model

Where the membrane bends

Cyan curve: long branchAmber curve: flat branch

Area, volume and numerical checks

A/(4\pi R_A^2)=1,\quad v=0.800

The saved record contains both shapes, model parameters, quadrature resolution and pointwise curvatures. Area and volume are calculated from the same surface used in the scene; this is simulated geometry, with no experimental image.

Start with the essentials

Focus question: How can the same membrane area enclose a long vesicle or a flattened one?
One-sentence intuition: Fixed area and volume leave several possible shapes; preferred curvature changes their relative bending cost.

Core mathematical model

Reduced volume measures available excess area

R_A=\sqrt{A/(4\pi)},\qquad v=\frac{V}{4\pi R_A^3/3}\leq1

The reference sphere has the same membrane area. A fully inflated sphere has reduced volume one; losing internal water lowers it.

Helfrich bending energy

E=\frac{\kappa}{2}\int(2H-c_0)^2\,dA,\qquad 2H=k_s+k_\phi

The two principal curvatures are positive for an outward-oriented sphere. The Gaussian-curvature term is constant for fixed spherical topology and uniform Gaussian rigidity, so it does not change the branch comparison.

The spherical benchmark

c_0=0:\quad E_{\rm sphere}=8\pi\kappa,\qquad\int K\,dA=4\pi

A sphere of any radius has the same pure bending energy. The curvature integral checks the closed topology independently of the selected deformation.

Common difficulties

Recovering stiffness from a still image

Typical misconceptionThe shape identifies the membrane bending rigidity.

Better mental modelIn this pure bending problem, uniform rigidity multiplies every candidate energy by the same factor. Shape alone cannot determine it; force or thermal fluctuations provide an independent scale.

What a restricted minimum means

Typical misconceptionThe lower of two energies proves that the displayed surface is the true equilibrium vesicle.

Better mental modelThe calculation searches a two-mode axisymmetric family. Additional modes, asymmetric buds or invaginations may lower the energy further. The record states the family and numerical resolution.

Run the experiment

01
Compare two allowed shapes
Start with A long vesicle, then press Show the other shape. Read the energy difference and compare the curvature curves.
What to observe: Both shapes satisfy the same area and volume. They distribute curvature differently and therefore pay different bending energies.
02
Change the volume at fixed area
Drag the gold handle down to deflate the membrane, then choose The fully inflated sphere.
What to observe: Deflation opens a choice of elongated and flattened shapes. At maximum volume, the only closed shape with the fixed area is the sphere.
03
Give the polar regions a preference
Choose Curved polar patches, then reduce Extra polar curvature to zero without moving the other sliders.
What to observe: The favored branch changes from flat to long, with the same area, volume and global preferred curvature. The imposed polar pattern changes the relative energy of the two candidates.

Vesicle Shapes and Bending

Where the membrane bends

How a closed membrane chooses its shape

Core mathematical model

Reduced volume measures available excess area

Helfrich bending energy

The spherical benchmark

Common difficulties

Recovering stiffness from a still image

What a restricted minimum means

Run the experiment

Compare two allowed shapes

Change the volume at fixed area

Give the polar regions a preference

Related experiments and articles

From curvature to active membrane measurements