  Turing's mechanism is often invoked to explain periodic patterns in nature,
although direct experimental support is scarce. Turing patterns form in
reaction-diffusion systems when the activating species diffuse much slower than
the inhibiting species, and the involved reactions are highly non-linear. Such
reactions can originate from co-operativity, whose physical interactions should
also affect diffusion. We here take direct interactions into account and show
that they strongly affect Turing patterns. We find that weak repulsion between
the activator and inhibitor can substantially lower the required differential
diffusivity and reaction non-linearity. In contrast, strong interactions can
induce phase separation, but the resulting length scale is still typically
governed by the fundamental reaction-diffusion length scale. Taken together,
our theory connects traditional Turing patterns with chemically active phase
separation, thus describing a wider range of systems. Moreover, we demonstrate
that even weak interactions affect patterns substantially, so they should be
incorporated when modeling realistic systems.
