Soft Condensed Matter

Effective area-elasticity and tension of micro-manipulated membranes

Fluid membranes in aqueous solutions often consist of a fixed number of highly insoluble lipid or surfactant molecules. Since stretching a flat membrane involves very high energies, while macroscopically bending it involves energies of order $k_{\rm B}T$, membranes are commonly modeled as fluctuating two-dimensional sheets with a prescribed microscopic area $\bar A$ and a curvature elasticity. At the macroscopical scale, however, membranes appear completely different: their optically visible area $A$ is unconstrained, and it fluctuates about some value depending on the temperature and on the external constraints. Part of the total area $\bar A$ is stored in short scale fluctuations that are optically unresolved. For such a critically fluctuating system, a coarse-grained effective Hamiltonian ${\cal H}_{\rm eff}$, integrating all sub-optical details, is clearly more adequate than the microscopic Hamiltonian. We have evaluated this effective macroscopic Hamiltonian ${\cal H}_{\rm eff}$ and investigated the associated area-elasticity and tension. We start by considering a quasi-planar membrane with a fixed microscopic area $\bar A$, which is attached to a frame of area $L^2$. We choose the simplest microscopic curvature Hamiltonian: the lowest-order, quadratic approximation of the Canham-Helfrich Hamiltonian [1,2]:

$${\cal H}_c[h_m]=\int\!d^2x\,\frac{\kappa}{2}\left(\nabla^2 h_m\right)^2.$$ (1)

Here $h_m({\bf x})$ is the height of the membrane above a reference plane (Monge gauge), as resolved microscopically. We then determine the coarse-grained Hamiltonian ${\cal H}_{\rm eff}[h]$, where $h({\bf x})$ is the height of the membrane as resolved optically. This Hamiltonian is such that $\exp(-\beta{\cal H}_{\rm eff}[h])$ gives the probability for the occurrence of any optically visible membrane shape $h$, whatever its fluctuating microscopic detail. Technically, this is a one-step renormalization of the fixed area constraint. We have found  [3] that ${\cal H}_{\rm eff}$ involves a non-linear area-elasticity energy ${\cal H}_s(A)$ for the coarse-grained, optically visible, area $A$. This is the effective potential which is probed by pulling a membrane in an optically resolved micro-manipulation. Depending on the microscopic excess area $\alpha_m\!=\!(\bar A-L^2)/L^2$ and on the constraints exerted on the membrane, we find three distinct regimes: a floppy regime, an entropic-tense regime, and a stretched-tense regime. We have provided explicit formulae for the effective tension $\sigma=d{\cal H}_s(A)/dA$ in these three regimes.

Figure 1: Coarse-grained area-elasticity ${\cal H}_s(A)$ as a function of the apparent area $A$. Since $A>L^2$, the hatched region is physically unaccessible. Depending on the microscopic excess area, the membrane is initially floppy ($a_1$) or tense ($b_1$). Further stretching can be externally induced ($a_2,b_2$). The floppy membrane ($a_1$) can, e.g., be led into a state of tension ($a_2$) similar to that of the unperturbed tense one ($b_1$).

Our results can be applied to giant vesicles. This allows us to perform a first test of our theory: re-analyzing the micro-pipette experiments of Evans and Rawicz [4,5], we find an excellent fit for the cross-over between the entropic-tense and the stretched-tense regimes.

Figure 2: Fit of the data obtained in a recent micropipette experiment by Evans et al. [5].

1

P. Canham, J. Theor. Biol. 26, 61 (1970).

2

W. Helfrich, Z. Naturforsch. 28C, 693 (1973).

3

J.-B. Fournier, A. Ajdari and L. Peliti, Phys. Rev. Lett. 86, 4970(2001).

4

E. Evans and W. Rawicz, Phys. Rev. Lett. 64, 2094 (1990).

5

W. Rawicz et al., Biophys. J. 79, 328 (2000).

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