Mimicking effects of cholesterol in lipid bilayer membranes by self-assembled amphiphilic block copolymers

Abstract

The effect of cholesterol on biological membranes is important in biochemistry. In this study, a polymer system is used to simulate the consequences of varying cholesterol content in membranes. The system consists of an AB-diblock copolymer, a hydrophilic homopolymer hA, and a hydrophobic rigid homopolymer C, corresponding to phospholipid, water, and cholesterol, respectively. The effect of the C-polymer content on the membrane is studied within the framework of a self-consistent field model. The results show that the liquid-crystal behavior of B and C has a great influence on the chemical potential of cholesterol in bilayer membranes. The effects of the interaction strength between components, characterized by the Flory-Huggins parameters and the Maier-Saupe parameter, were studied. Some consequences of adding a coil headgroup to the C-rod are presented. Results of our model are compared to experimental findings for cholesterol-containing lipid bilayer membranes.

Fig. 14

The spatial distributions of the individual rod segments. (a) ${\tilde{n}}_C=0.3$. (b) ${\tilde{n}}_C=1$. (c) ${\tilde{n}}_C=1.4$. Parameters ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and $\eta N=30$ are fixed.

Fig. 15

Schematic of polymer size. The A-coil has a diameter of $2R_g^A$, and the B- and C-rod have lengths of $L_B$ and $L_C$, respectively.

Fig. 16

Schematic of a case of the free energy curve of the two-phase coexistence for a one-dimensional self-consistent field simulation. (a) Phase 1. (b) Phase 2. (c) Free energy curves of Phase 1 $({\widetilde{𝓕}}^{(1)})$ and Phase 2 $({\widetilde{𝓕}}^{(2)})$ in the interval $\left({\tilde{n}}_C^{(1),l},{\tilde{n}}_C^{(1),r}\right)$ and $\left({\tilde{n}}_C^{(2),l},{\tilde{n}}_C^{(2),r}\right)$, respectively. The blue line is the free energy over the interval $\left({\tilde{n}}_C^{(1)},{\tilde{n}}_C^{(2)}\right)$ for two-phase coexistence. The tangent point for this line are $\left({\tilde{n}}_C^{(1)},{\widetilde{𝓕}}^{(1)}\left({\tilde{n}}_C^{(1)}\right)\right)$ and $\left({\tilde{n}}_C^{(2)},{\widetilde{𝓕}}^{(2)}\left({\tilde{n}}_C^{(2)}\right)\right)$. The chemical potential of C-rods is the same for the two-phase region, i.e., ${𝓕}^{{(1)}^{\prime }}\left({\tilde{n}}_C^{(1)}\right)={\widetilde{𝓕}}^{{(2)}^{\prime }}\left({\tilde{n}}_C^{(2)}\right)$.

Fig. 1

Schematic of the polymers. (a) hA homopolymers with a polymerization degree of $N_{hA}$. (b) AB diblock copolymers with a polymerization degree of $N=N_A+N_B$. (c) C homopolymers with a polymerization degree of $N_C$. The arrows indicate the directions that we solve the propagators.

Fig. 2

Effect of truncating the length of hA homopolymer region using the flexible system. (a) The effect of truncation on the density distributions for diblock copolymers. (b) Free energy $\widetilde{𝓕}$ as a function of the number of hA coils, ${\tilde{n}}_C=01$ and 0.5. Parameters are fixed at ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and ${\tilde{n}}_C=0.5$.

Fig. 3

Schematics of the candidate phases of the bilayer self-assembled from rod-coil diblock copolymers. (a) $A_s$-phase. (b) $A_c$-phase. (c) $C_s$-phase. In the $A_s$ and $C_s$ phases, the rods assume an end-to-end arrangement. For the $A_c$ phases, in contrast, the rods are interdigitated.

Fig. 4

Density distributions of the bilayer. (a) ${\tilde{n}}_C=0.3$. (b) ${\tilde{n}}_C=1$. (c) ${\tilde{n}}_C=1.4$. ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and $\eta N=30$ are fixed.

Fig. 5

(a)The density distributions for the two states of the bilayer for ${\tilde{n}}_C=0.5$ ( $l^{*}=9.3$ and 11.5 for the metastable and stable state, respectively). (b) The free energy $\widetilde{𝓕}$ as a function of the computation domain $l$ for ${\tilde{n}}_C=0.3$, 0.5, 1, and 1.4 as indicated. ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and $\eta N=30$ are fixed.

Fig. 6

The area $S$ (left y-axis) and thickness $\Omega$ (right y-axis) of the bilayer as functions of ${\tilde{n}}_C$. The default values of parameters are used: ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and $\eta N=30$.

Fig. 7

(a) The free energy $\widetilde{𝓕}$ and the optimal computational domain $l^{*}$ as functions of ${\tilde{n}}_C$. (b) The chemical potential $\text{Δ}\mu$ of C-rods as a function of their number. The default values of parameters are used: ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and $\eta N=30$.

Fig. 8

The chemical potential $\text{Δ}\mu$ as a function of the number of C homopolymers ${\tilde{n}}_C$. (a) Semiflexible bilayers with ${\chi }_{AB}N={\chi }_{AC}N=20$, 25, 30, and 35. Parameter $\eta N=30$ is fixed. (b) Semiflexible bilayers with $\eta N=0$, 25, 30, and 40 and flexible bilayers (black line, Coil C). Parameters ${\chi }_{AB}N={\chi }_{AC}N=25$ and ${\chi }_{BC}N=0$ are fixed. (c) Flexible bilayer with ${\chi }_{BC}=-10$, −5, 0, 5, and 10. Parameters ${\chi }_{AB}N={\chi }_{AC}N=25$ are fixed.

Fig. 9

The chemical potential of C-polymers with Flory-Huggins parameters ${\chi }_{BC}N=0$, 0.5, and 1.5. The maximum chemical potential is indicated for each value of ${\chi }_{BC}N$. The inset zooms into the plateau region for ${\chi }_{BC}N=0.5$. ${\chi }_{AB}N={\chi }_{AC}N=25$ are fixed for flexible bilayers.

Fig. 10

(a) The free energy $\widetilde{𝓕}$ of the $A_s$-phase and $C_s$-phase as functions of ${\tilde{n}}_C$ with the same parameters ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, and $\eta N=30$. (b) The chemical potential of the $A_s$-phase coexists with the $C_s$-phase as the functions of ${\tilde{n}}_C$. (c) The average tilt angle $\theta$ of B- and C-rods as functions of ${\tilde{n}}_C$.

Fig. 11

(a) The free energy $\widetilde{𝓕}$ of the $A_c$ and $A_s$ phase as functions of ${\tilde{n}}_C$ each phase having the same parameters, ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, $\eta N=30$, and ${\beta }_B={\beta }_C=3.5$. (b) The chemical potential $\mu$ of C-rods as a function of the concentration of C-rods ${\phi }_C$.

Fig. 12

The free energies $\widetilde{𝓕}$ as functions of ${\tilde{n}}_{DC}$ for $f_D=0.1$ and 0.2 in the Rod-Coil/Rod-Coil/Coil system. The default values of parameters are used: ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, $\eta N=30$, and ${\beta }_B={\beta }_C=2$.

Fig. 13

The density distributions of the bilayer formed by the Rod-Coil/Rod-Coil/Coil system at ${\tilde{n}}_{DC}=1$ (a) and 1.4 (b). Parameters are fixed at ${\chi }_{AB}N={\chi }_{AC}N=25$, ${\chi }_{BC}N=0$, $\eta N=30$, ${\beta }_B={\beta }_C=2$, and $f_D=0.1$.