Does the first amendment protect federal (and state) whistle blowers?

A case to be heard by the Suprmem Court this week pits a Los Angeles deputy prosecutor against "the system." The details are here. [N]o law effectively protects federal workers who report malfeasance as part of their job duties. And coverage of state workers is patchy. As a result, those workers we depend on for our safety have often faced a terrible conundrum: either remain quiet and allow fraud and wrongdoing to occur, or speak out and risk retaliation. It seems to me that support for whistle-blowers is needed. But is whistle-blowing protected by the first amendment in such a way that the government cannot punish the whistle-blower by, for example, demotion? I'm not realy sure what the issue is here. I guess we'll here more as the case is presented.