Bending fluctuations in semiflexible, inextensible, slender filaments in Stokes flow: towards a spectral discretization

TL;DR

This paper introduces a spectral discretization method for simulating the bending fluctuations of semiflexible, inextensible filaments in Stokes flow, improving computational efficiency and accuracy over traditional bead-link models.

Contribution

The authors develop a novel spectral discretization approach using Chebyshev grids and a new temporal integrator for efficient simulation of semiflexible filaments with hydrodynamics.

Findings

Chebyshev grid discretization closely matches blob-link equilibrium distribution.

The new temporal integrator accurately samples the equilibrium distribution.

Fewer than 12 Chebyshev nodes suffice for realistic dynamics.

Abstract

Semiflexible slender filaments are ubiquitous in nature and cell biology, including in the cytoskeleton, where reorganization of actin filaments allows the cell to move and divide. Most methods for simulating semiflexible inextensible fibers/polymers are based on discrete (bead-link or blob-link) models, which become prohibitively expensive in the slender limit when hydrodynamics is accounted for. In this paper, we develop a novel coarse-grained approach for simulating fluctuating slender filaments with hydrodynamic interactions. Our approach is tailored to relatively stiff fibers whose persistence length is comparable to or larger than their length, and is based on three major contributions. First, we discretize the filament centerline using a coarse non-uniform Chebyshev grid, on which we formulate a discrete constrained Gibbs-Boltzmann equilibrium distribution and overdamped Langevin equation. Second, we define the hydrodynamic mobility at each point on the filament as an integral of the Rotne-Prager-Yamakawa kernel along the centerline, and apply a spectrally-accurate quadrature to accurately resolve the hydrodynamics. Third, we propose a novel midpoint temporal integrator which can correctly capture the Ito drift terms that arise in the overdamped Langevin equation. We verify that the equilibrium distribution for the Chebyshev grid is a good approximation of the blob-link one, and that our temporal integrator samples the equilibrium distribution for sufficiently small time steps. We also study the dynamics of relaxation of an initially straight filament, and find that as few as 12 Chebyshev nodes provides a good approximation to the dynamics while allowing a time step size two orders of magnitude larger than a resolved blob-link simulation. We conclude by studying how bending fluctuations aid the process of bundling in cross-linked networks of semiflexible fibers.