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Sandbox Physics
Membrane mechanics

L3 · Chapter 7

How composition and curvature reshape each other

Sandbox Physics ·

A mobile molecule that bends a membrane can seek out favorable curvature. Its accumulation changes the curvature the membrane prefers. Shape and composition therefore influence one another. Prescribing a curvature-rich region studies one direction of this interaction; phase separation also requires composition to move or equilibrate.

Make the molecular distribution an unknown

Let dimensionless ϕ\phi be the deviation of a local molecular fraction from its mean. Molecules may prefer a particular curvature and neighbors of their own kind. The first preference changes spontaneous curvature; the second enters mixing free energy. A gradient cost penalizes rapid composition changes and contributes to the cost of interfaces.

A prescribed polar patch specifies where molecules are. Phase separation asks why they gather there. Conserved molecule number also restricts composition dynamics. Jülicher and Lipowsky studied domain boundaries competing with bending to induce buds. Below we define a separate continuous small-deformation teaching model to analyze the onset of instability.

Source: F. Jülicher and R. Lipowsky · W. Helfrich

Write a minimal coupling on a nearly flat patch

Describe height h(x)h(\mathbf x) over planar coordinates x\mathbf x with small slopes. For an upward normal and sphere-positive curvature, 2H≃−∇2h2H\simeq-\nabla^2h. Define signed coupling cc by c0=−cϕc_0=-c\phi. The bending term can then be written as the squared difference between the height Laplacian and composition.

Coefficient aa is the local quadratic mixing stiffness; a negative value favors demixing locally. Positive bb suppresses fine composition ripples and positive quartic coefficient gg limits their amplitude. Rigidity has energy units; σ,a,g\sigma,a,g have energy per area and bb has energy units.

F=∫ ⁣d2x[κ2(∇2h−cϕ)2+σ2∣∇h∣2+a2ϕ2+b2∣∇ϕ∣2+g4ϕ4]F=\int\!\mathrm d^2x\left[\frac{\kappa}{2}(\nabla^2h-c\phi)^2+\frac{\sigma}{2}|\nabla h|^2+\frac a2\phi^2+\frac b2|\nabla\phi|^2+\frac g4\phi^4\right]
This is an explicitly authored continuous small-slope model. Linear onset uses its quadratic terms. Large buds and final domains need nonlinear surface geometry.

Relax height before evaluating composition

Take height and composition cosine waves at the same wave number qq. A symmetric matrix organizes their quadratic energy. Height stiffness comes from bending and tension; the cross term measures how height adjustment can lower the cost of a composition wave.

Hold the composition amplitude fixed, minimize with respect to height and substitute back. The resulting composition stiffness is S(q)S(q); if it is negative, a small fluctuation at that wave number lowers energy. At fixed mean composition, the zero mode is fixed; use the long-wavelength limit to compare large-scale tendencies.

Hhh=κq4+σq2,Hhϕ=κcq2,Hϕϕ=a+bq2+κc2H_{hh}=\kappa q^4+\sigma q^2,\quad H_{h\phi}=\kappa cq^2,\quad H_{\phi\phi}=a+bq^2+\kappa c^2
Both amplitudes use the same area normalization. The cross-term sign follows the curvature and coupling conventions above.
S(q)=Hϕϕ−Hhϕ2Hhh=a+bq2+κc2σκq2+σ(q>0)S(q)=H_{\phi\phi}-\frac{H_{h\phi}^2}{H_{hh}}=a+bq^2+\frac{\kappa c^2\sigma}{\kappa q^2+\sigma}\quad(q>0)
The subtracted term is energy released by height relaxation. A membrane clamped flat cannot release it.

Worked example with stable long waves and an unstable band

Choose length and energy units, then dimensionless teaching parameters κ=20\kappa=20, σ=0.2\sigma=0.2, c=0.4c=0.4, a=−1a=-1 and b=0.1b=0.1. The long-wave limit is positive: S(0)=a+κc2=2.20S(0)=a+\kappa c^2=2.20. At very high wave number the positive gradient term grows. A minimum can lie between them.

Differentiate with respect to q2q^2 to obtain the expression below. It gives q∗≃0.745q_*\simeq0.745 and S(q∗)≃−0.888S(q_*)\simeq-0.888, at wavelength 2π/q∗≃8.432\pi/q_*\simeq8.43 in the chosen length unit. Allowing the surface to bend makes this band of composition fluctuations energetically favorable.

Directly minimize the two-variable quadratic energy over height to check the eliminated formula. This tests linear onset. Final domain sizes, shapes and growth rates also depend on nonlinear terms, conserved dynamics and fluid motion. The fastest-growing wave number also depends on kinetic coefficients and conservation.

q∗2=∣c∣σb−σκq_*^2=|c|\sqrt{\frac{\sigma}{b}}-\frac{\sigma}{\kappa}
A positive right-hand side gives an interior minimum; otherwise the minimum shifts to the long-wave boundary. The route’s composition benchmark preserves these values and their generator.

From fixed patches to mobile molecules

The shape experiment prescribes c0RA=cglobal+cpolarcos⁡8θc_0R_A=c_{\rm global}+c_{\rm polar}\cos^8\theta at two poles. It compares shapes under fixed heterogeneity without solving migration of ϕ\phi. A phase-separation extension must evolve or minimize both fields, conserve molecule number and check convergence with mesh and interface width.

A real protein may also have orientation, occupied area and binding kinetics. When these determine the question, add them as model variables. A curvature parameter alone represents only the specified curvature preference.

Try the reasoning

Explore a fixed curvature-rich polar region

  1. Predict how stronger polar curvature preference changes the energies of the two branches.
  2. Vary only polar curvature and compare contours and energies at fixed area and volume.
  3. Identify the part of the coupling this experiment represents, then propose an additional observable for mobile composition, such as the temporal correlation of concentration with local curvature.

Exercises and explanations

  1. Reverse cc. Does effective linear stiffness change? Does the local height relation change?

    Read the explanation

    Effective stiffness contains the squared coupling and stays fixed. The optimal height changes sign relative to composition. Equal stability does not mean bending toward the same side.

  2. Why is the minimum-stiffness wavelength not automatically the final domain size?

    Read the explanation

    Quadratic analysis tests whether small perturbations lower energy. At larger amplitudes, quartic terms, interfaces, conserved dynamics and surface geometry matter. Final scales require solving and validating those processes.

  3. If height is clamped flat, are the teaching parameters still linearly unstable?

    Read the explanation

    The remaining composition stiffness is a+bq2+κc2=2.2+0.1q2a+bq^2+\kappa c^2=2.2+0.1q^2, always positive. Height relaxation is essential to the finite-wave-number instability in this example.

Sources and calculated scope

  1. W. Helfrich, Elastic Properties of Lipid Bilayers: Theory and Possible Experiments, Z. Naturforsch. C 28, 693–703 (1973)

    Original curvature-elasticity paper, pages 693–703. We use its local quadratic curvature energy with an explicitly stated sphere-positive curvature convention. The sphere, tube and small-slope examples are derived here; no original figure or measured material parameter is copied.

  2. F. Jülicher and R. Lipowsky, Shape transformations of vesicles with intramembrane domains, Physical Review E 53, 2670–2683 (1996)

    Introduction and domain energy: competition between bending and boundary line energy. Our continuous composition–height Hessian and finite-wave-number numerical example are derived teaching extensions, not a reproduction of the published domain-shape diagrams.

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