A cord for supporting a biofilm has a plurality of yarns. At least one of the yarns comprises a plurality of hollow fiber gas transfer membranes. At least one of the yarns extends along the length of the cord generally in the shape of a spiral. Optionally, one or more of the yarns may comprise one or more reinforcing filaments. In some examples, a reinforcing yarn is wrapped around a core. A module may be made by potting a plurality of the cords in at least one header. A reactor may be made and operated by placing the module in a tank fed with water to be treated and supplying a gas to the module. In use, a biofilm covers the cords to form a membrane biofilm assembly.