Other than the suppliers for various components, the work breakdown has not been decided. Because this is a rather high-cost research mission, it is not likely that private funds will be available. Therefore, we expect that one or more government space agencies (NASA, ESA, NASDA, etc.) will fund the project for its duration.
The builders of Asterius will most likely be the space agencies that fund the project; however, it could also be a spacecraft building company that is contracted. Some of the components of Asterius might be contracted to various businesses; for example, an instrument package could be built to the correct specifications and then integrated into the bus by the main building team. At the lowest level, the components and raw materials are all supplied by numerous companies, only a few of which have already been selected.
The operation of Asterius would most likely be handled by whoever built it. Controlling and monitoring the spacecraft, and other engineering tasks would almost certainly be performed by the builder. Scientific operation and analysis might also be done by the builder, or it could be handled by a separate organization, such as Penn State University.