  We provide a concrete realization of the cluster algebras associated with
Q-systems as amalgamations of cluster structures on double Bruhat cells in
simple algebraic groups. For nonsimply-laced groups, this provides a
cluster-algebraic formulation of Q-systems of twisted type. It also yields a
uniform proof of the discrete integrability of these Q-systems by identifying
them with the dynamics of factorization mappings on quotients of double Bruhat
cells. On the double Bruhat cell itself, we find these dynamics are closely
related to those of the Fomin-Zelevinsky twist map. This leads to an explicit
formula expressing twisted cluster variables as Laurent monomials in the
untwisted cluster variables obtained from the corresponding mutation sequence.
This holds for Coxeter double Bruhat cells in any symmetrizable Kac-Moody
group, and we show that in affine type the analogous factorization mapping is
also integrable.
