Averaged implicit hydrodynamic model of semiflexible filaments

We introduce a method to incorporate hydrodynamic interaction in a model of semiflexible filament dynamics. Hydrodynamic screening and other hydrodynamic interaction effects lead to nonuniform drag along even a rigid filament, and cause bending fluctuations in semiflexible filaments, in addition to the nonuniform Brownian forces. We develop our hydrodynamics model from a string-of-beads idealization of filaments, and capture hydrodynamic interaction by Stokes superposition of the solvent flow around beads. However, instead of the commonly used first-order Stokes superposition, we do an equivalent of infinite-order superposition by solving for the true relative velocity or hydrodynamic velocity of the beads implicitly. We also avoid the computational cost of the string-of-beads idealization by assuming a single normal, parallel and angular hydrodynamic velocity over sections of beads, excluding the beads at the filament ends. We do not include the end beads in the averaging and solve for them separately instead, in order to better resolve the drag profiles along the filament. A large part of the hydrodynamic drag is typically concentrated at the filament ends. The averaged implicit hydrodynamics methods can be easily incorporated into a string-of-rods idealization of semiflexible filaments that was developed earlier by the authors. The earlier model was used to solve the Brownian dynamics of semiflexible filaments, but without hydrodynamic interactions incorporated. We validate our current model at each stage of development, and reproduce experimental observations on the mean-squared displacement of fluctuating actin filaments. We also show how hydrodynamic interaction confines a fluctuating actin filament between two stationary lateral filaments. Finally, preliminary examinations suggest that a large part of the observed velocity in the interior segments of a fluctuating filament can be attributed to induced solvent flow or hydrodynamic screening.