Simulation of Multiscale Hydrophobic Lipid Dynamics via Efficient Integral Equation Methods

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 nonlocal 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 Nystrom 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. A mobility problem formulation supplies the motion for the rigid particles in a viscous fluid. 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.