Bending stiffness depends on curvature of ternary lipid mixture tubular membranes

Abstract

Lipid and protein sorting and trafficking in intracellular pathways maintain cellular function and contribute to organelle homeostasis. Biophysical aspects of membrane shape coupled to sorting have recently received increasing attention. Here we determine membrane tube bending stiffness through measurements of tube radii, and demonstrate that the stiffness of ternary lipid mixtures depends on membrane curvature for a large range of lipid compositions. This observation indicates amplification by curvature of cooperative lipid demixing. We show that curvature-induced demixing increases upon approaching the critical region of a ternary lipid mixture, with qualitative differences along two roughly orthogonal compositional trajectories. Adapting a thermodynamic theory earlier developed by M. Kozlov, we derive an expression that shows the renormalized bending stiffness of an amphiphile mixture membrane tube in contact with a flat reservoir to be a quadratic function of curvature. In this analytical model, the degree of sorting is determined by the ratio of two thermodynamic derivatives. These derivatives are individually interpreted as a driving force and a resistance to curvature sorting. We experimentally show this ratio to vary with composition, and compare the model to sorting by spontaneous curvature. Our results are likely to be relevant to the molecular sorting of membrane components in vivo.

Figure 1

Experimental phase diagram indicating compositions (depicted using stars) of ternary lipid mixture DOPC/Chol/DPPC: I, 70:20:10; II, 55:28:17; III, 40:37:23; IV, 40:40:20; V, 40:46:14; and VI, 40:50:10. Circles mark compositions of vesicles imaged to determine the upper phase boundary and the critical point of phase separation (critical consolute point). Population fractions of vesicles with phase separation are indicated in the left legend. The grayscale line represents the upper phase boundary binodal line at ∼22°C. The binodal line also displays the area fractions of the disordered phase according to the legend on the right. Compositions III and IV are in the neighborhood of the critical point where area fraction is 40–60% (indicated by the thick bar on binodal and right legend). Our compositions I–VI are chosen in the homogeneous phase region outside the miscibility gap and approach the neighborhood of the critical point in two directions: along lines parallel (I, II, and III) and lines perpendicular (VI, V, IV, and III) to the upper phase boundary.

Figure 2

Membrane bending stiffness from projection length (L_p)/tether length (L_t) measurements. (a) Demonstration of L_p versus L_t plots at different membrane tensions. (Square, σ = 1.2 × 10⁻⁴ N/m; triangle, σ = 5.2 × 10⁻⁵ N/m; circle, σ = 3.4 × 10⁻⁵ N/m; and solid and open symbols represent the elongation and relaxation steps of the tether, respectively.) This example is chosen from three tether radius measurements of composition III. From the slopes of these plots, we obtained tether radii (see Eq. 2). (Square, R_t = 20.6 nm; triangle, R_t = 34.0 nm; and circle, R_t = 39.0 nm.) (b) Relation of tether radius and membrane tension of a tether consisting of binary mixture DOPC/Chol = 2:1. Solid line indicates the fit of data points using a single bending stiffness (κ = 7.4 × 10⁻²⁰ J). (c) Bending stiffness versus tether radius plot. Bending stiffness is calculated from panel b and is observed to be roughly constant, under changing curvature. (d) Relation of tether radius and membrane tension of a tether from ternary mixture III. Solid and open circles represent decreasing and increasing R_t, respectively. Solid line and shaded line are plots at the maximum (κ_max = 1.5 × 10⁻¹⁹ J) and the minimum (κ_min = 8.5 × 10⁻²⁰ J) bending stiffness values among these data points. It is thus observed that the bending stiffness decreases with curvature for this mixture. Error bars were determined as explained in the Supporting Material.

Figure 3

Bending stiffness versus (a) tether radius, (b) tether curvature, and (c) squared curvature plots from the same experimental data shown in Fig. 2d. The figures clearly indicate the bending stiffness to decrease with increasing curvature. (d) Bending stiffness and squared curvature C² relation from combination of eight whole-range data sets of composition III. The value n refers to the number of different vesicles examined. Data points in panels c and d are separated into two ranges by dashed lines at C² = 0.0012 nm⁻². The bending stiffnesses obtained at C² values <0.0012 nm⁻² were chosen for fitting our thermodynamic model to bending stiffness-curvature relations κ_eff = κ₀ − ΩC², for all compositions examined.

Figure 4

(a) Linear fits of bending stiffness and squared curvature obtained from multiple data sets in the low curvature range (C² < 0.0012 nm⁻²) for each composition. Compositions are indicated at the upper-left corner of each graph, and the numbers of tethers included for each composition are shown in the upper-right corner of each plot. Data sets of composition III are the same as those in Fig. 3d but display the low curvature region instead of the whole range. (b) Slopes from linear fits for different compositions in panel a. Dashed lines are linear fits depicting the increasing trends of slopes Ω for compositional trajectories parallel (I, II, and III) and perpendicular (VI, V, IV, and III) to phase boundary approaching the neighborhood of a critical demixing point. The graph indicates that the slope Ω, which determines the sorting efficiency (see Eqs. 7 and 8), increases upon approaching the critical region. (c) Bending stiffness at zero curvature (intercepts of plots in panel a) for different compositions. Error bars in panels b and c are the uncertainty of linear fits in panel a.