Simulation of Multiscale Hydrophobic Lipid Dynamics via Efficient Integral Equation Methods

Abstract

In this paper a mathematical model for long-range, hydrophobic attraction between amphiphilic particles is developed to quantify the macroscopic assembly and mechanics of a lipid bilayer membrane in solvents. The non-local interactions between amphiphilic particles are obtained from the first domain variation of a hydrophobicity functional, giving rise to forces and torques (between particles) that dictate the motion of both particles and the surrounding solvent. The functional minimizer (that accounts for hydrophobicity at molecular-aqueous interfaces) is a solution to a boundary value problem of the screened Laplace equation. We reformulate the boundary value problem as a second-kind integral equation (SKIE), discretize the SKIE using a Nyström discretization and 'Quadrature by Expansion' (QBX) and solve the resulting linear system iteratively using GMRES. We evaluate the required layer potentials using the 'GIGAQBX' fast algorithm, a variant of the Fast Multipole Method (FMM), yielding the required particle interactions with asymptotically optimal cost. Solving a mobility problem in Stokes flow is incorporated to obtain corresponding rigid body motion. The simulated fluid-particle systems exhibit a variety of multiscale behaviors over both time and length: Over short time scales, the numerical results show self-assembly for model lipid particles. For large system simulations, the particles form realistic configurations like micelles and bilayers. Over long time scales, the bilayer shapes emerging from the simulation appear to minimize a form of bending energy.

Fast and accurate solvers for simulating Janus particle suspensions in Stokes flow

“…In such a situation, the particles are known to form a tetrahedral configuration. Figure 4 shows the initial and final configurations of the spheres as well as cross sectional plots of the resulting potential, agreeing with two dimensional results in [7,43]. The particles seek to minimize contact between the fluid and the hydrophobic tails by shielding them in the center of the configuration.…”

“…We observe the formation of other stable structures, such as bilayer sheets, when the initial configuration is sufficiently close to the final configuration. In two dimensional experiments, bilayers are observed to form spontaneously [7]. We do not notice such spontaneity across our experiments in three dimensions.…”

“…We employ the model developed by Fu et. al [7], which formulates hydrophobic interaction potentials in integral form. The hydrophobic interaction potential φ is defined as the smooth minimizer of the hydrophobic energy functional for a given particle configuration and Dirichlet boundary value f :…”

“…We then explore the dynamics of larger systems of amphiphilic particles. In [7] the authors studied the long-term configurations resulting from two dimensional amphiphilic interactions. We recreate their experiment here in three dimensions.…”

“…In part owing to these challenges, DNS studies of the hydrodynamics of Janus particle suspensions are rather scarce; fluid dynamics of a moderate number of particles have been studied in two [7,8,9,10] and three spatial dimensions [11], while large-scale simulation studies have focused on statistical molecular simulation methods such as Molecular Dynamics, Lattice Boltzmann and Monte Carlo methods [12,13,14]. In this work, we seek to bridge this gap and propose an efficient, fast algorithmic framework for DNS of dense Janus suspensions in three dimensions, building on recent advances in the state-of-the-art rigid body Stokes solvers.…”