Dynamic implicit-solvent coarse-grained models of lipid bilayer membranes: Fluctuating hydrodynamics thermostat

We introduce a thermostat based on fluctuating hydrodynamics for dynamic simulations of implicit-solvent coarse-grained models of lipid bilayer membranes. We show our fluctuating hydrodynamics approach captures interesting correlations in the dynamics of lipid bilayer membranes that are missing in simulations performed using standard Langevin dynamics. Our momentum conserving thermostat accounts for solvent-mediated momentum transfer by coupling coarse-grained degrees of freedom to stochastic continuum fields that account for both the solvent hydrodynamics and thermal fluctuations. We present both a general framework and specific methods to couple the particle and continuum degrees of freedom in a manner consistent with statistical mechanics and amenable to efficient computational simulation. For self-assembled vesicles, we study the diffusivity of lipids and their spatial correlations. We find the hydrodynamic coupling yields within the bilayer interesting correlations between diffusing lipids that manifest as a vortex-like structure similar to those observed in explicit-solvent simulations. We expect the introduced fluctuating hydrodynamics methods to provide a way to extend implicit-solvent models for use in a wide variety of dynamic studies.

Figure 1

Implicit-solvent coarse-grained lipid model: (a) Individual lipids are modeled by three coarse-grained units that interact to mimic amphiphilic molecules [2]. Starting from a concentrated solution of model lipids, structures are self-assembled. (b) The self-assembly proceeds initially through the formation of aggregates that resemble micelles which subsequently merge into larger disklike structures. (c) These structures merge further, and after reaching a critical size they either orient to span the entire periodic domain as a planar bilayer or wrap spontaneously to form a vesicle. (d) Solvent degrees of freedom are represented by a stochastic velocity field to capture both hydrodynamic momentum transfer and thermal fluctuations. (e) The fluctuating hydrodynamic fields of the solvent are represented on an Eulerian lattice and coupled bidirectionally to the Lagrangian degrees of freedom of the lipids through two interrelated force terms.

Figure 2

The diffusivity and correlations of lipids within the vesicle bilayer. While Langevin dynamics with Stokes drag and the SELM fluctuating hydrodynamics exhibit similar lipid diffusivities, the Langevin dynamics with Stokes drag exhibits almost no lateral correlations between lipid motions. Retaining such spatial correlations, even for small lipid diffusivities, is an important consequence of the momentum conservation of the SELM fluctuating hydrodynamics thermostat.

Figure 3

Lipid flow correlations $\mathbf{w}=\Psi {\mathbf{e}}_1$ under Langevin dynamics with small drag ${\rm Y}=0.06$ $m_0/\tau$, Langevin with Stokes drag ${\rm Y}=7210$ $m_0/\tau$, and SELM fluctuating hydrodynamics with ${\rm Y}=7210$ $m_0/\tau$. The Langevin dynamics with small drag and the SELM fluctuating hydrodynamics exhibit vortex-like structures similar to those in explicit-solvent lipid models [19]. In the color map, red indicates flow to the right and purple indicates flow to the left.