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 Incoming links: Membrane-mediated interaction potential

Membrane-mediated interaction potential Created 2026-09-28 Updated 2026-11-05

A membrane-mediated interaction potential is the separation-dependent energy produced when deformation fields around two membrane-bound objects overlap. Its sign and range depend on membrane tension, bending stiffness, inclusion geometry, and which side of the membrane each object occupies.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 355 / 1 / b / Solution 2026-09-28

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Reflection symmetry makes the two outer free tails identical. Their combined energy is

E_{o} = \frac{k _{c} δ _{o}^{2}}{ξ R ^{2}} .

(1)

Between the cylinders the free profile is even. Up to an irrelevant additive height it has the form

h_{i} (x) = C cosh (x / ξ), ∣ x ∣ \leq L - δ_{i} .

(2)

Slope matching at

x = L - δ_{i}

gives

C = \frac{ξ δ _{i}}{R s i n h [( L - δ _{i} ) / ξ ]} .

(3)

Direct integration, or the boundary expression from part (a), yields

E_{i} = \frac{k _{c} δ _{i}^{2}}{ξ R ^{2}} coth \frac{L - δ _{i}}{ξ} .

(4)

At the retained order

δ_{i} / ξ ≪ 1

, replace the argument by

L / ξ

. The four contact halves contribute bending plus adhesion energy

E_{c} = \frac{k _{c}}{R ^{2}} (1 - 2 U) (δ_{o} + δ_{i}) .

(5)

Thus

E_{2} = \frac{k _{c}}{ξ R ^{2}} [δ_{o}^{2} + δ_{i}^{2} coth (L / ξ) - 2 ξ (U - \frac{1}{2}) (δ_{o} + δ_{i})] .

(6)

Independent minimization gives

δ_{o} = ξ (U - \frac{1}{2}), δ_{i} = ξ (U - \frac{1}{2}) tanh (L / ξ),

(7)

and hence

E_{2} (L) = - \frac{k _{c} ξ}{R ^{2}} (U - \frac{1}{2})^{2} [1 + tanh (L / ξ)] .

(8)

At infinite separation the energy is twice the one-cylinder minimum. The membrane-mediated interaction potential is therefore

V (L) = E_{2} (L) - E_{2} (\infty) = \frac{k _{c} ξ}{R ^{2}} (U - \frac{1}{2})^{2} [1 - tanh (L / ξ)] .

(9)

It is positive and decreases monotonically to zero, so the two cylinders repel. The physical cause is the overlap of their exponentially relaxing membrane deformations.

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