Direct versus indirect hydrodynamic interactions during bundle formation of bacterial flagella

Most motile bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. These semi-rigid filaments repeatedly join ('bundle') and separate ('unbundle'), resulting in a two-gait random walk-like motion of the cell. In this process, hydrodynamic interactions between the filaments are known to play an important role and can be categorised into two distinct types: direct interactions mediated through flows that are generated through the actuation of the filaments themselves, and indirect interactions mediated through the motion of the cell body (i.e. flows induced in the swimming frame that result from propulsion). To understand the relative importance of these two types of interactions, we study a minimal singularity model of flagellar bundling. Using hydrodynamic images, we solve for the flow analytically and compute both direct and indirect interactions exactly as a function of the length of the flagellar filaments and their angular separation. We show (i) that the generation of thrust by flagella alone is sufficient to drive the system towards a bundled state through both types of interaction; (ii) that indirect advection dominates for long filaments and at wide separation, i.e. primarily during the early stages of the bundling process; and (iii) that, in contrast, direct interactions dominate when flagellar filaments are in each other's wake, which we characterise mathematically. We further introduce a numerical elastohydrodynamic model that allows us to compute the dynamics of the helical axes of each flagellar filament while analysing direct and indirect interactions separately. With this we show (iv) that the shift in balance between direct and indirect interactions is non-monotonic during the bundling process, with a peak in direct dominance, and that different sections of the flagella are affected by these changes to different extents.