  We consider the evolution of linear perturbations in models with a nonminimal
coupling between dark matter and scalar field dark energy. Growth of matter
inhomogeneities in two examples of such models proposed in the literature are
investigated in detail. Both of these models are based on a low-energy limit of
effective string theory action, and have been previously shown to naturally
lead to late acceleration of the universe. However, we find that these models
can be ruled out by taking properly into account the impact of the scalar field
coupling on the formation of structure in the dark matter density. In
particular, when the transition to acceleration in these models begins, the
interaction with dark energy enchances the small scale clustering in dark
matter much too strongly. We discuss the the role of an effective small scale
sound speed in such models with a coupled dark sector.
