  The growth of bacterial flagellar filaments is a self-assembly process where
flagellin molecules are transported through the narrow core of the flagellum
and are added at the distal end. To model this situation, we generalize a
growth process based on the TASEP model by allowing particles to move both
forward and backward on the lattice. The bias in the forward and backward jump
rates determines the lattice tip speed, which we analyze and also compare to
simulations. For positive bias, the system is in a non-equilibrium steady state
and exhibits boundary-induced phase transitions. The tip speed is constant. In
the no-bias case we find that the length of the lattice grows as
$N(t)\propto\sqrt{t}$, whereas for negative drift $N(t)\propto\ln{t}$. The
latter result agrees with experimental data of bacterial flagellar growth.
