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

Direct vs indirect hydrodynamic interactions during bundle formation of bacterial flagella