Line tension and interaction energies of membrane rafts calculated from lipid splay and tilt

Abstract

Membrane domains known as rafts are rich in cholesterol and sphingolipids, and are thought to be thicker than the surrounding membrane. If so, monolayers should elastically deform so as to avoid exposure of hydrophobic surfaces to water at the raft boundary. We calculated the energy of splay and tilt deformations necessary to avoid such hydrophobic exposure. The derived value of energy per unit length, the line tension gamma, depends on the elastic moduli of the raft and the surrounding membrane; it increases quadratically with the initial difference in thickness between the raft and surround; and it is reduced by differences, either positive or negative, in spontaneous curvature between the two. For zero spontaneous curvature, gamma is approximately 1 pN for a monolayer height mismatch of approximately 0.3 nm, in agreement with experimental measurement. Our model reveals conditions that could prevent rafts from forming, and a mechanism that can cause rafts to remain small. Prevention of raft formation is based on our finding that the calculated line tension is negative if the difference in spontaneous curvature for a raft and the surround is sufficiently large: rafts cannot form if gamma < 0 unless molecular interactions (ignored in the model) are strong enough to make the total line tension positive. Control of size is based on our finding that the height profile from raft to surround does not decrease monotonically, but rather exhibits a damped, oscillatory behavior. As an important consequence, the calculated energy of interaction between rafts also oscillates as it decreases with distance of separation, creating energy barriers between closely apposed rafts. The height of the primary barrier is a complex function of the spontaneous curvatures of the raft and the surround. This barrier can kinetically stabilize the rafts against merger. Our physical theory thus quantifies conditions that allow rafts to form, and further, defines the parameters that control raft merger.

FIGURE 1

A schematic representation of the raft boundary. The raft, on the left, is thicker than the surround on the right. (A) A step in monolayer thickness creates a large hydrophobic surface. (B) Monolayer deformation at the raft boundary alleviates any creation of hydrophobic surfaces exposed to water.

FIGURE 2

An illustration of tilt and splay. (A) In the unperturbed flat monolayer, the normal N and director n are parallel. (B) Lipid tilt is illustrated. The tilt vector is parallel to the neutral surface. Tilt does not alter monolayer thickness. (C) Splay is illustrated. In splay, adjacent directors are not parallel to each other and monolayer thickness changes. The direction of the x axis is indicated.

FIGURE 3

Monolayer deformations. (A) Undisturbed monolayer, (B) negative splay, (C) positive splay, and (D) tilt. The sketch illustrates why splay changes the monolayer thickness (B and C) whereas tilt (D) does not. The volume per lipid is not altered by any of the deformations.

FIGURE 4

Dependence of bilayer line tension, 2γ, on equilibrium thickness mismatch δ for J_r = J_s = 0. Curve 1 is a flexible raft for K_r = 10 kT/nm², curve 2 is a firm raft for K_r = 10 kT/nm², and curve 3 is a firm raft for K_r = 40 kT/nm². The dotted curve is drawn for a perfectly firm raft (B_r → ∞ and K_r → ∞). For this and all subsequent figures, h₀ = 20 Å.

FIGURE 5

Dependence of bilayer line tension, 2γ, on equilibrium thickness mismatch δ for nonzero-spontaneous curvature. (A) A firm raft. J_r = 0 for curves 1, 2, and 3; J_s = 0 (curve 1), J_s = −0.1 nm⁻¹ (curve 2), J_s = +0.1 nm⁻¹ (curve 3). J_r = 0.1 nm⁻¹ for curves 4, 5, and 6; J_s = 0.1 nm⁻¹ (curve 4), J_s = 0 (curve 5), J_s = −0.1 nm⁻¹ (curve 6). Elastic moduli for the raft are B_r = 40 kT and K_r = 10 kT/nm², δ = 5 Å; all other parameters are as in Fig. 4. (B) A flexible raft. J_r = 0, J_s = 0 for curve 1; J_s–J_r = ±0.2 nm⁻¹ for curve 2. The elastic moduli are B_s = B_r = 10 kT and K_s = K_r = 10 kT/nm².

FIGURE 6

Phase diagram for stability of the raft as a function of the spontaneous curvature of raft, J_r, and surround, J_s. A raft is stable only in the region γ > 0. (A) Firm raft. B_r = 4B_s = 40 kT, K_r = K_s = 10 kT/nm². (B) Flexible raft. B_r = B_s = 10 kT and K_r = K_s = 10 kT/nm². For both panels A and B, δ = 4 Å.

FIGURE 7

The profile of the neutral surface near the raft boundary. The monolayer shape is plotted as h(x) − (h_r + h_s)/2. The numerical values thus provide, in Å, the deviation of monolayer height from the average thickness of the raft and surround. The dotted line indicates the underformed, step-like raft boundary. For the flexible raft (curve 1), B_r = B_s = 10 kT and K_r = K_s = 10 kT/nm². For the perfectly firm raft (curve 2), B_r → ∞ and K_r → ∞. In both cases, δ = 5 Å.

FIGURE 8

The energy barrier between interacting rafts for zero spontaneous curvature everywhere. The total energy per unit length of two straight parallel raft boundaries as a function of their separation distance, L, is plotted. The line tension of an isolated raft is the energy at large separation, divided by two. The interaction energy at any separation distance is the deviation of that energy from the energy at large separation. The curve number is rank ordered with increasing raft firmness: B_r = B_s = 10 kT and K_r = K_s = 10 kT/nm² (curve 1), B_r = 4B_s = 40 kT and K_r = K_s = 10 kT/nm² (curve 2), and B_r = 4B_s = 40 kT and K_r = 4K_s = 40 kT/nm² (curve 3). δ = 5 Å.

FIGURE 9

The energy barrier between rafts for J_s = 0 and varied J_r. The total energy per unit length of two straight parallel raft boundaries as a function of their separation distance, L, is plotted. (A) Firm rafts (B_r = 4B_s = 40 kT, K_r = K_s = 10 kT/nm²). J_r = 0 (curve 1), J_r = −0.1 nm⁻¹ (curve 2), and J_r = +0.1 nm⁻¹ (curve 3). (B) Flexible rafts (B_r = B_s = 10 kT, K_r = K_s = 10 kT/nm²). J_r = 0 (curve 1), J_r = −0.2 nm⁻¹ (curve 2), and J_r = +0.2 nm⁻¹ (curve 3). For panels A and B, δ = 5 Å.

FIGURE 10

The dependence of energy barriers between rafts on J_s for positive and negative J_r. The total energy per unit length for two straight parallel boundaries of firm rafts is plotted. (A) J_r = −0.1 nm⁻¹ < 0. Curves are drawn for different spontaneous curvature of the strip of surround that separates the rafts. J_s = 0, curve 1; J_s = −0.1 nm⁻¹, curve 2; J_s = +0.1 nm⁻¹, curve 3. B_r = 4B_s = 40 kT, K_r = K_s = 10 kT/nm², δ = 5 Å. (B) J_r = +0.1 nm⁻¹ > 0. J_s = 0, curve 1; J_s = −0.1 nm⁻¹, curve 2; J_s = +0.1 nm⁻¹, curve 3. B_r = 4B_s = 40 kT, K_r = K_s = 10 kT/nm², δ = 5 Å. The differences in energies in Figs. 8, 9, and 10 illustrate the complex dependence of interactions on J_r and J_s.