The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

Abstract: The inertialess fluid-structure interactions of active and passive inextensible filaments and slender- rods are ubiquitous in nature, from the dynamics of semi-flexible polymers and cytoskeletal filaments to cellular mechanics and flagella. The coupling between the geometry of deformation and the phys- ical interaction governing the dynamics of bio-filaments is complex. Governing equations negotiate elastohydrodynamical interactions with non-holonomic constraints arising from the filament inex- tensibility. Such elastohydrodynamic systems are structurally convoluted, prone to numerical erros, thus requiring penalization methods and high-order spatiotemporal propagators. The asymptotic coarse-graining formulation presented here exploits the momentum balance in the asymptotic limit of small rod-like elements which are integrated semi-analytically. This greatly simplifies the elas- tohydrodynamic interactions and overcomes previous numerical instability. The resulting matricial system is straightforward and intuitive to implement, and allows for a fast and efficient computation, over than a hundred times faster than previous schemes. Only basic knowledge of systems of linear equations is required, and implementation achieved with any solver of choice. Generalisations for complex interaction of multiple rods, Brownian polymer dynamics, active filaments and non-local hydrodynamics are also straightforward. We demonstrate these in four examples commonly found in biological systems, including the dynamics of filaments and flagella. Three of these systems are novel in the literature. We additionally provide a Matlab code that can be used as a basis for further generalisations.