Motivated by recent observations in biological cells, we study Turing patterns in bounded regions where the nonlinear chemical reactions occur on the boundary and where reagent transport occurs in the bulk. Within a generic model, we formulate the stability problem and discuss the conditions for the occurrence of a Turing instability. By choosing other model parameters to be unequal, we find that Turing patterns exist even in the case of equal diffusion constants. Finally, a recently introduced computation technique is utilized to follow the nascent pattern into the highly nonlinear regime.