  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.
