^{1}, Raymond Kapral

^{2}, Alexander S. Mikhailov

^{3}and Hsuan-Yi Chen

^{1,4,5}

### Abstract

A mesoscopic coarse-grain model for computationally efficient simulations of biomembranes is presented. It combines molecular dynamics simulations for the lipids, modeled as elastic chains of beads, with multiparticle collision dynamics for the solvent. Self-assembly of a membrane from a uniform mixture of lipids is observed. Simulations at different temperatures demonstrate that it reproduces the gel and liquid phases of lipid bilayers. Investigations of lipid diffusion in different phases reveals a crossover from subdiffusion to normal diffusion at long times. Macroscopic membrane properties, such as stretching and bending elastic moduli, are determined directly from the mesoscopic simulations. Velocity correlation functions for membrane flows are determined and analyzed.

Financial support from the Humboldt Foundation and the Deutsche Forschungsgemeinschaft (DFG) Training Research Group (GRK 1558) “Nonequilibrium collective dynamics in condensed matter and biological systems” in Germany is gratefully acknowledged. The research of R.K. is supported in part by the Natural Sciences and Engineering Research Council of Canada. The research of M.J.H. and H.Y.C. is supported by the National Science Council of Taiwan (NSCT) under Grant No. NSC 98-2112-M-008-004-MY3 and by the National Center for Theoretical Sciences, Taiwan.

I. INTRODUCTION

II. MESOSCOPIC MODEL FOR LIPID BILAYER DYNAMICS

A. Lipid interactions

B. Lipid-solvent interactions

C. MD-MPC dynamics

D. Simulation details

III. MEMBRANE PROPERTIES AT DIFFERENT TEMPERATURES

IV. SELF-ASSEMBLY OF THE MEMBRANE

V. MACROSCOPIC MEMBRANE PROPERTIES AND VELOCITY CORRELATION FUNCTIONS

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Lipids
- 154.0
- Solvents
- 70.0
- Cell membranes
- 17.0
- Diffusion
- 15.0
- Hydrophobic interactions
- 13.0

## Figures

The lipid chain (left) and its schematic representation as a rod (right). The lipid consists of four beads linked by elastic FENE bonds (solid lines) and straightened by elastic bonds (dashed lines). The first bead (dark blue) is hydrophilic. Three other beads are hydrophobic, the terminal hydrophobic bead is shown as light blue.

The lipid chain (left) and its schematic representation as a rod (right). The lipid consists of four beads linked by elastic FENE bonds (solid lines) and straightened by elastic bonds (dashed lines). The first bead (dark blue) is hydrophilic. Three other beads are hydrophobic, the terminal hydrophobic bead is shown as light blue.

Membrane structures at three different temperatures: (a) k B T/ε = 0.4, (b) k B T/ε = 1.0 and (c) k B T/ε = 2.0. The rod representation is used to display lipids. The solvent particles are not shown.

Membrane structures at three different temperatures: (a) k B T/ε = 0.4, (b) k B T/ε = 1.0 and (c) k B T/ε = 2.0. The rod representation is used to display lipids. The solvent particles are not shown.

Radial distribution functions of lipid head beads in the membrane at three different temperatures: (a) k B T/ε = 0.4 (dashed black line), (b) k B T/ε = 1.0 (thick blue line) and (c) k B T/ε = 2.0 (thin red line).

Radial distribution functions of lipid head beads in the membrane at three different temperatures: (a) k B T/ε = 0.4 (dashed black line), (b) k B T/ε = 1.0 (thick blue line) and (c) k B T/ε = 2.0 (thin red line).

A cut through the simulation box showing the vertical structure of the bilayer and solvent particles at k B T/ε = 1.0.

A cut through the simulation box showing the vertical structure of the bilayer and solvent particles at k B T/ε = 1.0.

Vertical density profiles for hydrophilic head beads (ρ h , thick black line), hydrophobic tail beads (ρ t , dashed blue line) and solvent particles (ρ s , thin red line) at three different temperatures: (a) k B T/ε = 0.4, (b) k B T/ε = 1.0 and (c) k B T/ε = 2.0. The scale is different for the solvent density profile.

Vertical density profiles for hydrophilic head beads (ρ h , thick black line), hydrophobic tail beads (ρ t , dashed blue line) and solvent particles (ρ s , thin red line) at three different temperatures: (a) k B T/ε = 0.4, (b) k B T/ε = 1.0 and (c) k B T/ε = 2.0. The scale is different for the solvent density profile.

Diffusion of lipids in the membrane. Log-log plots of the mean square displacements (MSD) of the center of mass of a lipid are shown as functions of time for (a) k B T/ε = 0.4, (b) k B T/ε = 1.0 and (c) k B T/ε = 2.0. The dashed and solid straight lines are linear fits for the subdiffusive and normal diffusive regimes, respectively.

Diffusion of lipids in the membrane. Log-log plots of the mean square displacements (MSD) of the center of mass of a lipid are shown as functions of time for (a) k B T/ε = 0.4, (b) k B T/ε = 1.0 and (c) k B T/ε = 2.0. The dashed and solid straight lines are linear fits for the subdiffusive and normal diffusive regimes, respectively.

Self-assembly of the lipid bilayer. Six configurations at time moments (a) 0 δt, (b) 6000 δt, (c) 18 000 δt, (d) 160 000 δt, (e) 246 400 δt and (f) 300 000 δt are shown. The initial state (a) corresponds to the uniform mixture of lipids and the solvent (enhanced online). [URL: http://dx.doi.org/10.1063/1.4736414.1]doi: 10.1063/1.4736414.1.

Self-assembly of the lipid bilayer. Six configurations at time moments (a) 0 δt, (b) 6000 δt, (c) 18 000 δt, (d) 160 000 δt, (e) 246 400 δt and (f) 300 000 δt are shown. The initial state (a) corresponds to the uniform mixture of lipids and the solvent (enhanced online). [URL: http://dx.doi.org/10.1063/1.4736414.1]doi: 10.1063/1.4736414.1.

Dependence of the surface tension γ on the membrane area A. The first five data points were used to determine the membrane stretching modulus.

Dependence of the surface tension γ on the membrane area A. The first five data points were used to determine the membrane stretching modulus.

Power spectrum S(q) of membrane height fluctuations. The solid line is the best fit of the simulation data, using the theoretical dependence (Eq. (12) ).

Power spectrum S(q) of membrane height fluctuations. The solid line is the best fit of the simulation data, using the theoretical dependence (Eq. (12) ).

Time dependence of the longitudinal (a) and transverse (b) velocity correlation functions for three different separations: x = 0 (solid lines), x = 2.5 (dashed lines), and x = 5 (dotted lines).

Time dependence of the longitudinal (a) and transverse (b) velocity correlation functions for three different separations: x = 0 (solid lines), x = 2.5 (dashed lines), and x = 5 (dotted lines).

Longitudinal (circles) and transverse (squares) correlation functions G L (x) and G T (y). The solid and dashed lines show the respective logarithm approximations given by Eq. (15) .

Longitudinal (circles) and transverse (squares) correlation functions G L (x) and G T (y). The solid and dashed lines show the respective logarithm approximations given by Eq. (15) .

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