^{1}and Markus Deserno

^{1,a)}

### Abstract

We present a simple and highly adaptable method for simulating coarse-grained lipid membranes without explicit solvent.Lipids are represented by one head bead and two tail beads, with the interaction between tails being of key importance in stabilizing the fluid phase. Two such tail-tail potentials were tested, with the important feature in both cases being a variable range of attraction. We examined phase diagrams of this range versus temperature for both functional forms of the tail-tail attraction and found that a certain threshold attractive width was required to stabilize the fluid phase. Within the fluid-phase region we find that material properties such as area per lipid, orientational order, diffusion constant, interleaflet flip-flop rate, and bilayer stiffness all depend strongly and monotonically on the attractive width. For three particular values of the potential width we investigate the transition between gel and fluid phases via heating or cooling and find that this transition is discontinuous with considerable hysteresis. We also investigated the stretching of a bilayer to eventually form a pore and found excellent agreement with recent analytic theory.

We thank Oded Farago, Friederike Schmid, Hiroshi Noguchi, Gregoria Illya, Kurt Kremer, and Bernward Mann for valuable discussions. We also gratefully acknowledge financial support by the fGerman Science Foundation under Grant De775/1-2.

I. INTRODUCTION

II. BASIC PRINCIPLES OF THE MODEL

III. BILAYER STABILITY AND SELF-ASSEMBLY

IV. PROPERTIES OF THE FLUID PHASE

A. Observables

1. Orientational order parameter

2. Cross-bilayer density profile

3. Bending modulus

4. Diffusion constant

5. Flip-flop rate

B. Constant-temperature cuts

C. Constant profiles: Gel-fluid transition

V. MEMBRANE STRETCHING AND PORE OPENING

VI. CONCLUSIONS

## Figures

Phase diagram resulting from cohesion [Eq. (4)] in the plane of potential width and temperature at zero lateral tension. Each symbol corresponds to one simulation and identifies different bilayer phases: (×) gel, (●) fluid, and (+) unstable. The lines are merely guides to the eye. The inset shows the pair potential between tail lipids (solid line) and the purely repulsive head-head and head-tail interactions (dashed line).

Phase diagram resulting from cohesion [Eq. (4)] in the plane of potential width and temperature at zero lateral tension. Each symbol corresponds to one simulation and identifies different bilayer phases: (×) gel, (●) fluid, and (+) unstable. The lines are merely guides to the eye. The inset shows the pair potential between tail lipids (solid line) and the purely repulsive head-head and head-tail interactions (dashed line).

Phase diagram resulting from cohesion [Eq. (5)] in the plane of potential width and temperature at zero lateral tension. The meaning of all symbols is the same as in Fig. 1.

Phase diagram resulting from cohesion [Eq. (5)] in the plane of potential width and temperature at zero lateral tension. The meaning of all symbols is the same as in Fig. 1.

Self-assembly sequence for the bilayer system with 1000 lipids in a cubic box of side length . Lipid cohesion was set to and temperature to . A random gas of lipids quickly forms small clusters which slowly coarsen and eventually “zip up” to form a box-spanning bilayer sheet. The numbers indicate the MD time.

Self-assembly sequence for the bilayer system with 1000 lipids in a cubic box of side length . Lipid cohesion was set to and temperature to . A random gas of lipids quickly forms small clusters which slowly coarsen and eventually “zip up” to form a box-spanning bilayer sheet. The numbers indicate the MD time.

Profile of the density as a function of vertical distance from the bilayer midplane for a system of 4000 lipids at constant zero tension and with simulation parameters and . The plotted lines are bead densities for head beads (long dashed), first tail beads (short dashed), terminal tail beads (dotted), and the sum of all beads (solid).

Profile of the density as a function of vertical distance from the bilayer midplane for a system of 4000 lipids at constant zero tension and with simulation parameters and . The plotted lines are bead densities for head beads (long dashed), first tail beads (short dashed), terminal tail beads (dotted), and the sum of all beads (solid).

Asymptotic scaling of the power spectrum for the bilayer system with and .

Asymptotic scaling of the power spectrum for the bilayer system with and .

Scaled distribution of squared displacements of lipid molecules for the system with , and (crosses), (open circles), and (filled circles). The lines are fits to Eq. (12). These always yield and values of the two diffusion constants as given in the inset. While is constant, the data are compatible with approaching with a asymptotics (inset, dashed line).

Scaled distribution of squared displacements of lipid molecules for the system with , and (crosses), (open circles), and (filled circles). The lines are fits to Eq. (12). These always yield and values of the two diffusion constants as given in the inset. While is constant, the data are compatible with approaching with a asymptotics (inset, dashed line).

Basic static properties of the fluid bilayer phase as a function of and , where indicates a rescaled attractive potential . is the value of on the liquid-unstable (gas) transition line. Each plot shows four isotherms: (filled squares), (asterisks), (open circles), and (filled circles). The values of for each of these isotherms were 0.815, 1.025, 1.2, and 1.27, respectively. In all cases statistical errors were smaller than the size of the plotted points.

Basic static properties of the fluid bilayer phase as a function of and , where indicates a rescaled attractive potential . is the value of on the liquid-unstable (gas) transition line. Each plot shows four isotherms: (filled squares), (asterisks), (open circles), and (filled circles). The values of for each of these isotherms were 0.815, 1.025, 1.2, and 1.27, respectively. In all cases statistical errors were smaller than the size of the plotted points.

Diffusion constant as a function of rescaled potential width . The symbols and shifts are the same as in Fig. 7.

Diffusion constant as a function of rescaled potential width . The symbols and shifts are the same as in Fig. 7.

Flip-flop rate as a function of rescaled potential width . The symbols and shifts are the same as in Fig. 7.

Flip-flop rate as a function of rescaled potential width . The symbols and shifts are the same as in Fig. 7.

Variation of the area per lipid across the gel-fluid-phase boundary. Each figure shows a cooling-heating hysteresis for a particular value of . From top to bottom the values of used were 1.0, 1.4, and 1.6. The arrows indicate the direction of temperature change. The rate of temperature change was for the top plot and for the bottom two plots. The three vertical lines in the uppermost plot indicate the temperatures where the order parameter of Fig. 11 has been measured.

Variation of the area per lipid across the gel-fluid-phase boundary. Each figure shows a cooling-heating hysteresis for a particular value of . From top to bottom the values of used were 1.0, 1.4, and 1.6. The arrows indicate the direction of temperature change. The rate of temperature change was for the top plot and for the bottom two plots. The three vertical lines in the uppermost plot indicate the temperatures where the order parameter of Fig. 11 has been measured.

Probability density of the height difference between a lipid and its six immediate neighbors. The solid, dashed, and dotted curves correspond to the temperatures indicated by lines 1, 2, and 3 in Fig. 10, respectively.

Probability density of the height difference between a lipid and its six immediate neighbors. The solid, dashed, and dotted curves correspond to the temperatures indicated by lines 1, 2, and 3 in Fig. 10, respectively.

Bilayer tension as a function of (projected) area *A* for a flat membrane sheet with at . The bold solid line is a fit to the model of Farago (Ref. 19) and Tolpekina *et al.* (Ref. 26) [see also Eq. (B5) in Appendix B]; the fine solid and dashed curves indicate the metastable and unstable branches, respectively.

Bilayer tension as a function of (projected) area *A* for a flat membrane sheet with at . The bold solid line is a fit to the model of Farago (Ref. 19) and Tolpekina *et al.* (Ref. 26) [see also Eq. (B5) in Appendix B]; the fine solid and dashed curves indicate the metastable and unstable branches, respectively.

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