A membrane held under micropipette aspiration has bending modulus at and is put under…

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

Soft Matter at the Thermal Scale, Ch Membranes, fluctuations and soft matter in biology

Problem

A membrane held under micropipette aspiration has bending modulus κb=25kBT\kappa_{\rm b} = 25k_BT at T=310 KT = 310\,\mathrm{K} and is put under tension Σ=5×10−7 N m−1\Sigma = 5\times10^{-7}\,\mathrm{N\,m^{-1}}. The mean-square amplitude of a Fourier mode of wavevector qq on a membrane under tension is ⟨∣hq∣2⟩=kBT/[A(κbq4+Σq2)]\langle|h_q|^2\rangle = k_BT/[A(\kappa_{\rm b}q^4 + \Sigma q^2)], with AA the membrane area. ((a)) Derive an expression for the wavevector q∗q^* at which the bending term κbq4\kappa_{\rm b}q^4 and the tension term Σq2\Sigma q^2 contribute equally to the mode's stiffness, and hence for the crossover length ξΣ=1/q∗\xi_\Sigma = 1/q^* separating bending-dominated fluctuations at short length scales (large qq) from tension-dominated fluctuations at long length scales (small qq). ((b)) Evaluate ξΣ\xi_\Sigma in nanometres for the values given.

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