Johnson suffered the injury in New England's season-opening 31-0 loss in Buffalo and is expected to be out for the "foreseeable future," the team said.
It is the second loss in a week for the Patriots' defense as safety Lawyer Milloy was cut just before camp broke.
A nine-year veteran, Johnson has had an injury-plagued career, including two torn biceps, but had been healthy dating to the 2001 season. He was rotating with Tedy Bruschi and Roman Phifer at inside linebacker in the Pats' 3-4 set.
Couple mistakenly served bag of cash at McDonald's drive-thru
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need