Having browsed exactly three webpages on the topic, I would say there will be minimal Zn in the Dulbecco Minimal Essential Medium (I had to look that up too). Your cells may grow a little with whatever trace Zn and other elements they can scavenge, but you might want to check what is present in your DMEM stock before dosing them with any extra trace elements.

My suggestion is you may want to grow at least two cultures, one with minimal Zn (so the cells are viable) and one supplemented with Zn (but not so much that it becomes toxic), then you can see whether there is a functional difference in protein activity between the baseline and supplemented proteins. If you then extract the protein from the baseline cells and monitor activity with/without supplied Zn in vitro, and compare that to the supplemented proteins, then you can say it was the zinc. If there is no effect, then perhaps you can't conclude that.

Thank you Matthew Wheal for the suggestion