  Brane tilings are infinite, bipartite, periodic, planar graphs that are dual
to quivers. In this paper, we examine the del Pezzo 2 (dP$_2$) quiver and its
brane tiling, which arise from the physics literature, in terms of toric
mutations on its corresponding cluster. Specifically, we give explicit formulas
for all cluster variables generated by toric mutation sequences. Moreover, for
each such variable, we associate a subgraph of the dP$_2$ brane tiling to it
such that its weight matches the variable.
