We theoretically study the electronic structure of small-angle twisted bilayer graphene with a large potential asymmetry between the top and bottom layers. We show that the emergent topological channels known to appear on the triangular AB-BA domain boundary do not actually form a percolating network, but instead, they provide independent, perfect one-dimensional eigenmodes propagating in three different directions. Using the continuum-model Hamiltonian, we demonstrate that an applied bias causes two well-defined energy windows which contain sparsely distributed one-dimensional eigenmodes. The origin of these energy windows can be understood using a two-band model of the intersecting electron and hole bands of single-layer graphene. We also use the tight-binding model to implement the lattice deformations in twisted bilayer graphene and discuss the effect of lattice relaxation on the one-dimensional eigenmodes.