Increased local load in branching atrial tissue (muscle fibers and bundle insertions) influences wave propagation during atrial fibrillation (AF). This computer model study reveals two principal phenomena: if the branching is distant from the driving rotor (>19 mm), the load causes local slowing of conduction or wavebreaks. If the driving rotor is close to the branching, the increased load causes first a slow drift of the rotor towards the branching. Finally, the rotor anchors, and a stable, repeatable pattern of activation can be observed. Variation of the bundle geometry from a cylindrical, volumetric structure to a flat strip of a comparable load in a monolayer model changed the local activation sequence in the proximity of the bundle. However, the global behavior and the basic effects are similar in all models. Wavebreaks in branching tissue contribute to the chaotic nature of AF (fibrillatory conduction). The stabilization (anchoring) of driving rotors by branching tissue might contribute to maintain sustained AF.