  Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We
provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist
of $H$. In particular, we introduce the notion of a twisting pair for $H$ such
that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use
twisting pairs to describe twists of Manin's universal quantum groups
associated to quadratic algebras and provide twisting of solutions to the
quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan
construction.
