Line tension of multicomponent bilayer membranes

The line tension or edge energy of bilayer membranes self-assembled from binary amphiphilic molecules is studied using self-consistent-field theory (SCFT). Specifically, solutions of the SCFT equations corresponding to an infinite membrane with a circular pore, or an open membrane, are obtained for a coarse-grained model in which the amphiphilic species and hydrophilic solvents are represented by $AB\mathrm{and}ED$ diblock copolymers and $C$ homopolymers, respectively. The edge energy of the membrane is extracted from the free energy of the open membranes. Results for membranes composed of mixtures of symmetric and cone- or inverse cone-shaped amphiphilic molecules with neutral and/or repulsive interactions are obtained and analyzed. It is observed that an increase in the concentration of the cone-shaped species leads to a decrease of the line tension. In contrast, adding inverse cone-shaped copolymers results in an increase of the line tension. Furthermore, the density profile of the copolymers reveals that the line tension is regulated by the distribution of the amphiphiles at the bilayer edge.

Figure 1

Schematics showing the different molecular architectures studied.

Figure 2

Schematics of the model system investigated. A small pore formed in a giant unilamellar vesicle.

Figure 3

2D density profiles for the (a) bilayer and (b) pore configuration. The profile illustrates the overall density of the diblock copolymers $AB+ED$ in the lighter regions and solvent $C$ in the dark regions. These plots correspond to the cross-sectional view of the bilayer and the pore, respectively.

Figure 4

The free energy as a function of pore radius $R$ for blends with ${\chi }_{AB}={\chi }_{ED}={\chi }_{AD}={\chi }_{BE}={\chi }_{BC}={\chi }_{CD}=30$, $f_A=f_B$, ${\kappa }_C=1$, and various $ED$-molecular architectures. The molecular fraction of the $D$ species is defined as $f_D=N_D/N_{AB}$. The pore radius $R$ is measured in units of $R_g$ and is normalized with respect to the bilayer thickness $d$, given as $d\sim 4.3R_g$.

Figure 5

The line tension $\sigma$ as a function of order parameter $\psi$ for blends with ${\chi }_{AB}={\chi }_{ED}={\chi }_{AD}={\chi }_{BE}={\chi }_{BC}={\chi }_{CD}=30$, $f_A=f_B$, ${\kappa }_C=1$, and $f_D=$ 0.25, 0.3, and 0.4.

Figure 6

Relative concentrations of the $AB$, $ED$, and $C$ molecules ${\tilde{\varphi }}_{\gamma }$ ($\gamma =AB,ED$, and $C$) as a function of position for blends with $f_D=0.25$. Inset: 2D density profiles of the (a) $AB$ and (b) $ED$ blends for ${\chi }_{AB}={\chi }_{ED}={\chi }_{AD}={\chi }_{BE}={\chi }_{BC}={\chi }_{CD}=30$, $f_A=f_B$, ${\kappa }_C=1$, $f_D=$ 0.25, and $\psi =$ 0.18.

Figure 7

Line tension $\sigma$ as a function of order parameter $\psi$ for blends with ${\chi }_{AB}={\chi }_{ED}={\chi }_{AD}={\chi }_{BE}={\chi }_{BC}={\chi }_{CD}=30$, $f_A=f_B$, ${\kappa }_C=1$, and $f_E=$ 0.25, 0.35, and 0.45.

Figure 8

Relative concentrations of the $AB$, $ED$, and $C$ molecules ${\tilde{\varphi }}_{\gamma }$ ($\gamma =AB,ED$, and $C$) as a function of position. Inset: 2D density profiles of the (a) $AB$ and (b) $ED$ blends for ${\chi }_{AB}={\chi }_{ED}={\chi }_{AD}={\chi }_{BE}={\chi }_{BC}={\chi }_{CD}=30$, $f_A=f_B$, ${\kappa }_C=1$, $f_E=$ 0.25, and $\psi =$ 0.3.

Figure 9

The line tension $\sigma$ as a function of the order parameter $\psi$ for blends with ${\chi }_{AB}={\chi }_{ED}={\chi }_{AD}={\chi }_{BE}={\chi }_{BC}={\chi }_{CD}=30$, $f_A=f_B=f_E=f_D$, and ${\kappa }_C={\kappa }_{ED}=1$ for ${\chi }_{AE}=$ 0, 2, and 4.

Figure 10

Schematic diagram showing the effect of repulsive head group interaction on the overall geometrical shape of the lipids.