On Wednesday, Browns coach Mike Pettine said the injury is a sprain and that Tate will miss "some time."
Tate was injured during a 25-yard run in Cleveland's Week 1 loss to the Pittsburgh Steelers and did not return. He had six carries for 41 yards before the injury. Tate tweeted on Monday the injury is "nothing that serious."
Backup running back Terrance West, who the Browns selected in the third round of the 2014 draft, had 100 yards on 16 carries in relief of Rice. Undrafted rookie Isaiah Crowell added 32 yards and two touchdowns on five carries against the Steelers.
Cleveland signed Tate this offseason after he spent his first four years with the Houston Texans. The running back rushed for 771 yards and four touchdowns for the Texans last season.
The Browns host the New Orleans Saints on Sunday in Week 2.
- Paul Palladino