Measuring Gaussian Rigidity Using Curved Substrates

The Gaussian (saddle splay) rigidity of fluid membranes controls their equilibrium topology but is notoriously difficult to measure. In lipid mixtures, typical of living cells, linear interfaces separate liquid ordered (LO) from liquid disordered (LD) bilayer phases at subcritical temperatures. Here, we consider such membranes supported by curved substrates that thereby control the membrane curvatures. We show how spectral analysis of the fluctuations of the LO-LD interface provides a novel way of measuring the difference in Gaussian rigidity between the two phases. We provide a number of conditions for such interface fluctuations to be both experimentally measurable and sufficiently sensitive to the value of the Gaussian rigidity, while remaining in the perturbative regime of our analysis.

Figure 1

A surface with the geometry of a catenoid, closed (arbitrarily) with a spherical cap, serving as a substrate for a bound membrane. A domain of LD (red, labeled $D_{\mathrm{LD}}$) is shown surrounded by LO (cyan). (a) The equilibrium LO-LD interface $\mathrm{\Gamma }$ is a symmetric circle. (b) Thermal fluctuations displace the LD domain (the $m=1$ mode shown), the mean squared magnitude of which provides a probe of Gaussian rigidity. (c) The arclength variable $t$, measured from the minimal catenoid radius $r_0$ along a line of constant $\varphi$, parametrizes a family of circles of radius $r(t)$ on the surface. (d) The interface displacement projected from above ${\epsilon }_{\perp }$, as might be observed experimentally.

Figure 2

Values of the nondimensionalized coupling ${\eta }_{\overline{k}}=-\mathrm{\Delta }\overline{k}/(\sigma r_0)$ that can be probed with our method as a function of the rescaled domain size $\widehat{t}=t/r_0$, provided all three feasibility constraints are satisfied. In this figure, we take relatively stringent limits on the constraint parameters, as explained in the main text. The blue, red, and black line correspond, respectively, to constraints I, II, and III. The yellow region is the intersection of these, and thus shows the values of ${\eta }_{\overline{k}}$ that could be measured for a suitable domain size.

Figure 3

The yellow region indicates the space of experimental systems that could allow for accurate measurement of the Gaussian rigidity contrast between two membrane phases by measuring the $m=1$ displacement mode sketched in Fig. 1, under the assumption $L_{\overline{k}}=400\mathrm{nm}$ and $L_{\sigma }=1\mathrm{nm}$. The catenoidal surface feature(s) have a neck radius $r_0$ and the equilibrium location of the interface is at (projected) distance $r$ from the symmetry axis. The solid lines correspond to criteria I with $L_{\mathrm{res}}=100\mathrm{nm}$ (blue), II with ${\eta }_{\mathrm{pert}}=1/40$ (red), and III with $\nu =0.5$ (black). Dashed lines show variations of these values of, respectively, $\pm 10\%$, $\pm 20\%$, and $\pm 50\%$. We disregard $r<r_0$ as the interface would lie on the spherical cap.