Nanoscale mechanical probing of supported lipid bilayers with atomic force microscopy

We present theory and experiments for the force-distance curve $F\left(z_0\right)$ of an atomic force microscope (AFM) tip (radius $R$) indenting a supported fluid bilayer (thickness $2d$). For realistic conditions the force is dominated by the area compressibility modulus ${\kappa }_A$ of the bilayer and, to an excellent approximation, given by $F=\pi {\kappa }_{A}R{z}_0^2/{\left(2d-z_0\right)}^2$. The experimental AFM force curves from coexisting liquid ordered and liquid disordered domains in three-component lipid bilayers are well described by our model, which provides ${\kappa }_A$ in agreement with literature values. The liquid ordered phase has a yieldlike response that we model as due to the breaking of hydrogen bonds.

Figure 1

(Color online) (a) Schematic geometry of an AFM tip of radius $R$ indenting a fluid bilayer of thickness $2d$ by an amount $z_0$. The leaflet dividing surface at $h_b\left(r\right)$ is shown as a solid line. (b) Cylindrical volume elements of depth $z\left(r\right)$ at the distance $r$ from the center of the tip. (c) The area $2\pi rds$ of such an element in contact with the lipid.

Figure 2

Scaled force $F/\left(\pi {\kappa }_{A}R\right)$ as a function of tip depth $z_0/d$ (solid line) for tip radius $R/d=3$, according to Eq. (5). The dashed line shows only the leading quadratic term in $z_0/d$ [Eq. (6)]. Inset: behavior at small $z_0/d$. On this scale the approximation of Eq. (7) is indistinguishable from the numerical solution of Eq. (5).

Figure 3

(Color online) Phase separated lipid bilayer with liquid ordered and liquid disordered domains. (a) Tapping mode AFM image showing the height profile of the bilayer. (b) One-dimensional section along the dashed line in (a).

Figure 4

(Color online) Force-distance curve for a DOPC rich bilayer in the liquid disordered phase. The data points are from AFM experiments and the line is a fit for the theoretical prediction with ${\kappa }_A=0.12\text{N}/\text{m}$. Inset: SEM image of the tip to measure the tip radius. The dashed lines are along the pyramidal face edges. The circle drawn at the end of the tip has a radius of 10 nm.

Figure 5

(Color online) Force-distance curve from AFM (symbols) in the SM rich liquid ordered phase superposed with two separate theoretical fits [lines, using Eq. (5)] involving two different ${\kappa }_A$’s at small and large tip penetrations $z_0$. Inset: closeup of the crossover region.

Figure 6

(Color online) Force-distance curve from AFM (symbols) in the SM rich liquid ordered phase superposed with a microscopically motivated fit that accounts for a separate energetic contribution from hydrogen bond breaking [solid line, based on Eq. (9)]. Also shown are the separate contributions from the van der Waals interaction (dotted line), from the hydrogen bonds (dotted-dashed line), and the area compressibility term (dashed line).