Simulation of the lo-ld phase boundary in DSPC/DOPC/cholesterol ternary mixtures using pairwise interactions

Abstract

Recently, a number of ternary phase diagrams of lipid mixtures have been constructed using various experimental techniques with a common goal of understanding the nature of lipid domains. An accurate experimental phase diagram can provide rich thermodynamic information and can also be used to extract molecular interactions using computer simulation. In this study, the liquid-ordered and liquid-disordered (l(o)-l(d)) phase boundary of DSPC/DOPC/Cholesterol ternary mixtures is simulated in a lattice model using pairwise interactions. The block composition distribution (BCD) technique was used to locate accurately the compositions of coexisting phases and thermodynamics tie-lines in the two-phase region, and the Binder ratio method was used to determine the phase boundary in the critical region. In simulations performed along a thermodynamic tie-line, the BCD method correctly samples the compositions as well as the relative amounts of coexisting phases, which is in excellent agreement with the lever rule. A "best-fit" phase boundary was obtained that has a top boundary closely resembling the experimental boundary. However, the width of the simulated two-phase region is significantly wider than the experimental one. The results show that pairwise interactions alone are not sufficient to describe the complexity of molecular interactions in the ternary lipid mixtures; more complex forms of interactions, possibly multibody interaction or domain interfacial energy, should be included in the simulation.

Figure 1

Three types of particle moves are used in simulations. A phospholipid is treated as physically linked dimer chains occupying two adjacent sites, and a cholesterol molecule occupies one site. DSPC acyl chains: red; DOPC acyl chains: yellow; cholesterol: black. (A) Nearest-neighbor dimer pair rotation; (B) long-range dimer-dimer exchange; and (C) nearest-neighbor cholesterol-dimer flipping.

Figure 2

Snapshots of lateral lipid distributions for some DSPC/DOPC/cholesterol mixtures simulated with pairwise interaction energies (ΔE_OS=0.54 kT, ΔE_OC=-1.0 kT, ΔE_SC=-1.3) reveals phase separations and the range of cluster sizes. Black dots, cholesterol; Yellow dots, DOPC acyl chains; Red dots, DSPC acyl chains. Two chains from the same PC were physically linked. All mixtures have the same R ratio (i.e., DSPC/(DSPC+DOPC)) of 0.5, but different cholesterol mole fraction χ_C. (A) and (B) are 2-phase mixtures, (D) and (E) are 1-phase mixtures, and (C) is a mixture at the phase boundary. (F) is an ideal (random) mixing distribution at χ_C = 0.48.

Figure 3

Local composition histograms for the five mixtures in Fig. 2 reveals coexisting phases. A local lipid composition is specified by local R ratio and local χ_C. The vertical axis is the number of the lattice sites having a particular local lipid composition. Some histograms have been shifted in local χ_C axis to facilitate viewing. χ_C is the bulk (overall) cholesterol mole fraction of a simulated mixture.

Figure 4

The upper boundary of phase coexistence of l_o + l_d is shown on the ternary phase diagram of DSPC/DOPC/cholesterol simulated with pairwise interaction energies (ΔE_OS=0.54 kT, ΔE_OC=-1.0 kT, ΔE_SC=-1.3). Open circles and open squares are boundary points obtained by the BCD method. Dashed lines are simulated thermodynamic tie-lines. The filled square is the boundary point obtained from Binder ratio method. The thin solid line is the phase boundary determined using the BCD method and the Binder ratio method together. The thick solid line is the experimental phase boundary. Filled triangles represent overall (bulk) lipid compositions at which simulations with the BCD method were performed. Filled triangles labeled as “a” and “b” indicate the overall lipid compositions of the mixtures “a” and “b” in Fig. 2 and Fig. 3.

Figure 5

The Binder ratio as a function of ΔE_OS calculated using phase mask size of 91×91, 121×121, and 141×141 at the bulk composition of R=0.5 and χ_C=0.40. The crossover occurs at ΔE_OS≅ 0.54 kT, at which the Binder ratio becomes independent of system size.

Figure 6

(a) Pair-correlation functions for the 2-phase ternary lipid mixture with R = 0.5 and χ_C = 0.36. (b) Pair-correlation lengths as a function of χ_C for mixtures with constant R ratio of 0.5. The phase boundary is at χ_C = 0.40.

Figure 7

Simulated left and right phase boundaries using the BCD method with different sizes of phase-mask. Simulation lattice size was fixed at 300×300.

Figure 8

The BCD method is consistent with the Lever Rule. Upper: Local lipid composition histograms for five ternary lipid mixtures along a thermodynamic tie-line. The lipid compositions of the left and the right end-points of the tie-line are (R = 0.022, χ_C = 0.174) and (R = 0.973, χ_C = 0.294), respectively. Triangle inset: Schematic representation of a thermodynamic tie-line (dashed line) and a phase boundary (solid line). Points “a” to “e”, at which the simulations were performed, evenly divide the length of the tie-line. Bottom: The “weight” fractions of the left and right peaks in the local composition. The solid straight lines are the expected weight fractions according to the Lever Rule. X and Y are the distances from a given point on the tie-line to the two end-points.