  Josephson junctions made of closely-spaced conventional superconductors on
the surface of 3D topological insulators have been proposed to host Andreev
bound states (ABSs) which can include Majorana fermions. Here, we present an
extensive study of the supercurrent carried by low energy ABSs in
Nb/Bi$_2$Se$_3$/Nb Josephson junctions in various SQUIDs as we modulate the
carrier density in the Bi$_2$Se$_3$ barriers through electrostatic top gates.
As previously reported, we find a precipitous drop in the Josephson current at
a critical value of the voltage applied to the top gate. This drop has been
attributed to a transition where the topologically trivial 2DEG at the surface
is nearly depleted, causing a shift in the spatial location and change in
nature of the helical surface states. We present measurements that support this
picture by revealing qualitative changes in the temperature and magnetic field
dependence of the critical current across this transition. In particular, we
observe pronounced fluctuations in the critical current near total depletion of
the 2DEG that demonstrate the dynamical nature of the supercurrent transport
through topological low energy ABSs.
