Take the limit in which the two leaflets of a bilayer are only weakly coupled and…

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch The fluid membrane: bending, flicker and shape

Problem

Take the limit in which the two leaflets of a bilayer are only weakly coupled and slide freely past one another as the membrane bends, so each leaflet, thickness

h/2

and its own area (stretch) modulus

K_A/2

, bends independently about its own mid-plane (at

\pm h/4

from the bilayer's centre). ((a)) Following the strain argument used for a slab of uniform stiffness, show that one leaflet stores bending energy per unit area

u_1 = \int_{-h/4}^{h/4}\tfrac12\left(\dfrac{K_A}{h}\right)c^2z^2\,dz

when bent to curvature

c

, and that adding the two leaflets and comparing with

u=\tfrac12\kappa c^2

gives

\kappa = K_Ah^2/48

. ((b)) A synthetic bilayer has

K_A = 0.22\,\mathrm{N\,m^{-1}}

and hydrocarbon core thickness

h = 3.2\,\mathrm{nm}

. Evaluate this lower bound for

\kappa

in joules and in units of

k_BT

at body temperature,

310\,\mathrm{K}

. ((c)) A patch of this membrane of lateral size

L = 10\,\mathrm{\mu m}

carries a tension

\sigma = 2.0\times10^{-7}\,\mathrm{N\,m^{-1}}

. Starting from the undulation spectrum

\langle|h_{\mathbf q}|^2\rangle = k_BT/[A(\kappa q^4+\sigma q^2)]

, where

h_{\mathbf q}

are the Fourier modes of the membrane's height (not its thickness

h

), and converting the mode sum to an integral,

\sum_{\mathbf q}\to \dfrac{A}{(2\pi)^2}\int d^2q

, with the smallest accessible wavevector set by the patch size,

q_{\min}=2\pi/L

, show that the total mean-square height fluctuation is

\langle h^2\rangle = \frac{k_BT}{4\pi\sigma}\ln\!\left(1+\frac{\sigma}{\kappa q_{\min}^2}\right),

using the identity

\displaystyle\int_{q_{\min}}^{\infty}\frac{dq}{q(\kappa q^2+\sigma)} = \frac{1}{2\sigma}\ln\!\left(1+\frac{\sigma}{\kappa q_{\min}^2}\right)

(provable by partial fractions). ((d)) Evaluate

\langle h^2\rangle

and its square root (the rms roughness) for the values above, and find the crossover wavelength

2\pi/q^* = 2\pi\sqrt{\kappa/\sigma}

. State whether this

10\,\mathrm{\mu m}

patch is mostly in the tension-dominated or the bending-dominated regime.

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