  The aim of this paper is to contribute to the understanding of the pattern
formation phenomenon in reaction-diffusion equations coupled with ordinary
differential equations. Such systems of equations arise, for example, from
modeling of interactions between cellular processes such as cell growth,
differentiation or transformation and diffusing signaling factors. We focus on
stability analysis of solutions of a prototype model consisting of a single
reaction-diffusion equation coupled to an ordinary differential equation. We
show that such systems are very different from classical reaction-diffusion
models. They exhibit diffusion-driven instability (Turing instability) under a
condition of autocatalysis of non-diffusing component. However, the same
mechanism which destabilizes constant solutions of such models, destabilizes
also all continuous spatially heterogeneous stationary solutions, and
consequently, there exist no stable Turing patterns in such
reaction-diffusion-ODE systems. We provide a rigorous result on the nonlinear
instability, which involves the analysis of a continuous spectrum of a linear
operator induced by the lack of diffusion in the destabilizing equation. These
results are extended to discontinuous patterns for a class of nonlinearities.
