Slender fibers are ubiquitous in biology, physics, and engineering, with
    prominent examples including bacterial flagella and cytoskeletal fibers. In
    this setting, slender body theories (SBTs), which give the resistance on the
    fiber asymptotically in its slenderness $\epsilon$, are useful tools for both
    analysis and computations. However, a difficulty arises when accounting for
    twist and cross-sectional rotation: because the angular velocity of a filament
    can vary depending on the order of magnitude of the applied torque, asymptotic
    theories must give accurate results for rotational dynamics over a range of
    angular velocities. In this paper, we first survey the challenges in applying
    existing SBTs, which are based on either singularity or full boundary integral
    representations, to rotating filaments, showing in particular that they fail to
    consistently treat rotation-translation coupling in curved filaments. We then
    provide an alternative approach which approximates the three-dimensional
    dynamics via a one-dimensional line integral of Rotne-Prager-Yamakawa
    regularized singularities. While unable to accurately resolve the flow field
    near the filament, this approach gives a grand mobility with symmetric
    rotation-translation and translation-rotation coupling, making it applicable to
    a broad range of angular velocities. To restore fidelity to the
    three-dimensional filament geometry, we use our regularized singularity model
    to inform a simple empirical equation which relates the mean force and torque
    along the filament centerline to the translational and rotational velocity of
    the cross section. The single unknown coefficient in the model is estimated
    numerically from three-dimensional boundary integral calculations on a
    rotating, curved filament.

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