An MRI scan after the game confirmed the player damaged his medial collateral ligament before leaving the pitch in the first half and is expected to be out for several weeks.
Scotland had already lost number eight Josh Strauss and captain Greig Laidlaw before Saturday's game, making their final two matches away to England and at home to Italy even tougher.
Wing Tommy Seymour suffered bruised ribs while scoring a try on Saturday but has been passed fit to continue.
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(Reporting by Mitch Phillips, editing by: Toby Davis)
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