Patriots aren't laying down for this game. They still have the #1 seed to play for and as it stands right now NE @ Mia is a 1 pm game and Oak @ Den is a 4:25 pm game. There is a possibility the Miami game gets flexed to 4pm, but either way they wont know the outcome until the end of the Miami game. NE will be playing to win.
If I am Gase, I play to win as there is a chance for the 5th seed and I would much rather play at Houston than at Pittsburgh. However, if NE abuses Miami's LBs and DB (which I kinda expect) and they get out to a big lead then I absolutely would start pulling starters with any type of nagging injury out of the game.