Rheology of sliding leaflets in coarse-grained DSPC lipid bilayers

Amphiphilic lipid bilayers modify the friction properties of the surfaces on top of which they are deposited. In particular, the measured sliding friction coefficient can be significantly reduced compared with the native surface. We investigate in this work the friction properties of a numerical coarse-grained model of DSPC (1,2-distearoyl-sn-glycero-3-phosphocholine) lipid bilayer subject to longitudinal shear. The interleaflet friction coefficient is obtained from out-of-equilibrium pulling or from relaxation simulations. In particular, we gain access to the transient viscoelastic response of a sheared bilayer. The bilayer mechanical response is found to depend significantly on the membrane physical state, with evidence in favor of a linear response regime in the fluid but not in the gel region. The linear response validity domain is established, and the timescales appearing in the membrane response discussed.

Figure 1

Snapshot of a configuration of a coarse-grained bilayer containing 256 DSPC lipids per leaflet, with 2560 water beads molecules on both sides, in the high temperature fluid state at 340 K. Water molecules are represented by a uniform blue volume, hydrophobic tails by gray thin segments and lipid headgroup beads by colored spheres (green: glycerol, pink and magenta: phosphocholine).

Figure 2

Snapshot of a configuration in the low temperature gel state at 280 K. Same system and color representation as in Fig. 1. Compared to the fluid case, the bilayer is less extended in the $xy$ direction and thicker. No appreciable lipid chain tilt angle is visible.

Figure 3

Geometric parametrization of the system used in the present study, with $L=L_b+L_w$.

Figure 4

When opposing forces are exerted on each leaflet (a) the resulting stationary state sees the two leaflets sliding at constant relative velocity, surrounded by a uniform solvent velocity gradient profile, as emphasized in panel (b) where the simulation box boundary has been purposely shifted to sit exactly at the midplane of the bilayer. Panels (a) and (b) are subsequently referred as a Couette situation. When a uniform force is exerted on both leaflets and an opposing force on the solvent beads (c), a symmetric parabolic velocity profile builds up in the solvent, assuming sticking boundary conditions at the interface with the bilayer (d). Panels (c) and (d) are subsequently referred to as Poiseuille situation. In all cases the total momentum of the system is constant and vanishes.

Figure 5

Constant force pulling experiments in the fluid state: (a) single leaflet COM displacement ${\mathcal{X}}^{\left(\alpha \right)}\left(t\right)$, starting from the simulation box center (ca. 6.6 nm) and averaged displacement $\langle \mathcal{X}\rangle \left(t\right)=1/50{\sum }_{\alpha =1}^{50}{\mathcal{X}}^{\left(\alpha \right)}\left(t\right)$ vs time. A bootstrap procedure (b) estimates the dispersion ${\sigma }_b\left({\mathcal{X}}_u\left(t\right)\right)$ caused by the finiteness of the sample $\left\{\alpha \right\}$. Vertical bars represent the confidence interval of 10 selected points from the second half of the trajectory ($5000<t<10000$ ps), taken as twice the estimated bootstrap standard deviation. The vertical bars are used to provide a confidence interval for the drift velocity (slope of the averaged displacement curve).

Figure 6

Normalized averaged displacements (upper leaflet) ${\langle \mathcal{X}\rangle }_u\left(t\right)/F_u$ for a set of increasing pulling forces $1,...,3000\mathrm{kJ}{\mathrm{mol}}^{-1}{\mathrm{nm}}^{-1}$ (equivalently stresses $\tau \simeq 1,...,300$ bars). The displacement for $F_u=3000$ lies clearly beyond the linear regime and the force $F_u=10$ competes with thermal agitation. For clarity, only a representative subset of normalized displacement curves is shown in the figure.

Figure 7

Average drift velocities $V_u$ vs applied forces $F_u$ (lower horizontal scale bar) or stresses (higher horizontal scale bar) in the constant pull force (CPF) regime. A shear-thinning deviation is seen at $\tau \ge 50$ bars. Inset: focus on the linear regime region.

Figure 8

Normalized averaged displacements $\langle {\mathcal{X}}_u\rangle \left(t\right)/V_0$ for a set of increasing initial velocities $V_0=0.01$ and $0.08,...,0.5\mathrm{nm}{\mathrm{ps}}^{-1}$ in the force kick relaxation (FKR) regime. A fast initial increase in the first 7 ps is observed, followed by a slow relaxation in the opposite direction. As discussed in Sec. 4c, we interpret this peak as related to a fast inertial propagation of stress into the bilayer (acoustic propagation).

Figure 9

Normalized averaged velocities $\langle {\mathcal{V}}_u\rangle \left(t\right)=\langle d{\mathcal{X}}_u/dt\rangle \left(t\right)/V_0$. The velocity starts at an initial value of 1, decreases fast to 0 (coinciding with the sharp peak in the displacement curve) reaches a negative minimum and finally slowly regresses to 0 from below, coinciding with the slowly decreasing approach of the displacement plateau value.

Figure 10

Normalized averaged displacements $\langle {\mathcal{X}}_u\rangle \left(t\right)/V_0$ in the gel state for a set of increasing initial velocities 0.01 and $0.03,...,0.1\mathrm{nm}{\mathrm{ps}}^{-1}$. The plateau value is clearly increasing with the initial applied velocity, and the normalized displacements do not appear to collapse onto a master curve, pointing to an absence of linear response.

Figure 11

Normalized average displacements $\mathrm{\Delta }{X}_u/V_0$ for a set of increasing impulsions and sample size 150, with confidence intervals.

Figure 12

Normalized averaged velocity $\langle d{\mathcal{X}}_u/dt\rangle \left(t\right)/V_0$. Starting from 1, the normalized velocity quickly crosses 0 to reach a minimum, and then relaxes slowly to 0. Curves do not collapse to a master curve. The shape of the normalized relaxation curve is quite different from the fluid case state (Fig. 9).

Figure 13

Velocity $V_u$-applied shear stress ${\varphi }_u$ and inner stress $\tau$ characteristics in the gel and fluid states, from CPF simulations, using logarithmic representation. Dashed lines: linear behavior $V_u\sim \tau$ in the fluid state (triangles) and power-law behavior in the gel state (circles). The fluid regime is linear over the range of stresses considered here, while the gel regime seems consistent with a power-law relation of exponent $V_u\sim {\tau }^{1.50}$ or ${\varphi }_u^{1.44}$.

Figure 14

Martini CG representation of a DSPC molecule. Bead 1: choline, bead 2: phosphate, beads 3–4: glycerol, beads: 5–9 and 10–14 hydrophobic chains. A vector linking the first and last carbons of each chain is used for defining the average lipid tilt angle $\theta$ relative to the membrane normal.

Figure 15

Evolution of the average normalized tilt angle $\langle \theta \rangle$ during a constant pull force experiment for a set of forces ranging from $F=250\mathrm{kJ}{\mathrm{mol}}^{-1}{\mathrm{nm}}^{-1}$ to $3000\mathrm{kJ}{\mathrm{mol}}^{-1}{\mathrm{nm}}^{-1}$.

Figure 16

Average tilt angle curves $\langle \theta \rangle$ during a force kick experiment ($V_0=0.09\mathrm{nm}{\mathrm{ps}}^{-1}$ in the gel and fluid states, respectively).

Figure 17

Apparent friction coefficient $b+\eta /L_w$ from FKR simulations as a function of the initial induced velocity $V_0$ in the fluid (red squares) and gel (blue circles) states. Each point corresponds to 1000 repeated independent simulations. For each point, a vertical error bar $2{\sigma }_b$ is inferred from the bootstrap variance ${\sigma }_b^2$ of 10 synthetic averaged displacement curves. The resulting confidence interval decreases with velocity in all cases. Confidence intervals are about 10% relative value for $V_0=0.08,0.09$ and $1\mathrm{nm}{\mathrm{ps}}^{-1}$, and about 20% relative value for $V_0=0.05,0.06$, and $0.07\mathrm{nm}{\mathrm{ps}}^{-1}$ in the fluid state.

Figure 18

Projected Brownian $xy$ trajectory of the upper leaflet center of mass, observed during 50 ns in the fluid phase (space units in nm).

Figure 19

Mean square displacement curve of the upper leaflet center of mass at 340 K (continuous black lines) and a linear adjustment with $2D=6.8\times {10}^{-5}{\mathrm{nm}}^2{\mathrm{ps}}^{-1}$ (red open circles).

Figure 20

Schematic representation of an ordered bead packing (a), homogeneous elastic shear deformation (b), and plastic deformation (c). The plastic deformation reduces the angle between the crystal axis (blue arrow) and the local normal direction (full black arrow) thus decreasing the shear elastic energy of (c) with respect to (b) at the expense of a crystalline plane shift. The dashed arrow in (c) indicates the global bilayer normal direction.

Figure 21

Evolution of the horizontal and vertical box sizes during the NPT simulation, used for determining the average box size, in the fluid phase.

Figure 22

Evolution of the horizontal and vertical box sizes during the NPT simulation, used for determining the average box size, in the gel phase.