Cellular biology abound with filaments interacting through fluids, from intracellular microtubules, to rotating flagella and beating cilia. While previous work has demonstrated the complexity of capturing nonlocal hydrodynamic interactions between moving filaments, the problem remains difficult theoretically. We show here that when filaments are closer to each other than their relevant length scale, the integration of hydrodynamic interactions can be approximately carried out analytically. This leads to a set of simplified local equations, illustrated on a simple model of two interacting filaments, which can be used to tackle theoretically a range of problems in biology and physics.While one tends to think of biological cells as stubby, their environment is in fact rich with filamentous structures. Inside cells, polymeric filaments of microtubules, actin, and intermediate filaments fill the eukaryotic cytoplasm [1] and provide it with its mechanical structure [2]. Outside cells, the motion of flagella and cilia allows cells to generate propulsive forces [3][4][5] and induces flows critical to human health [6,7].In all cases, these biological filaments are immersed in a viscous fluid in which they move at low Reynolds number, be it due to their polymerisation, to fluctuations and thermal forces, or to the action of molecular motors [8]. At low Reynolds number, the flows induced locally by the motion of filaments relative to a background fluid have a slow spatial decay as ∼ 1/r [9,10]. In situations where filaments are close to each other, we thus expect nonlocal hydrodynamic interactions to be important [11].Integrating long-ranged hydrodynamic interactions between filaments has long been recognised as a challenging problem, and one where the theoretical approach has consisted of either full numerical simulations or very simplified analysis. A variety of computational methods have been developed to tackle it including slender-body theory [12][13][14], boundary elements to implement boundary integral formulations [15], the immersed boundary method [16,17], regularised flow singularities [18] and particlebased methods [19,20].While these computational approaches allow to address complex geometries and dynamics, the difficulty of integrating long-range hydrodynamic interactions has prevented analytical approaches from providing insight beyond simplified setups. The two most common approaches in biophysics consist in replacing the dynamics in three dimensions by a two-dimensional problem for which the analysis may be easier to carry out [21,22], or by focusing on far-field hydrodynamic interactions and ignoring the geometrical details of near-field hydrodynamics, a popular approach to study synchronisation of flagella and cilia [23][24][25][26][27][28][29] Fig. 6. In each event, the head became the tail (at least once). For an angle change of 0°, the cell exhibited successive reversals; this was rare. For an angle change of 180°, the cell backed up without changing the orientation of its long axis (the head became th...
