This paper focuses on the development of computationally-efficient predictive control algorithms for nonlinear parabolic PDEs with state and control constraints arising in the context of diffusion-reaction processes. Specifically, we consider a diffusion-reaction process described by a nonlinear parabolic PDE and address the problem of stabilization of an unstable steady-state subject to input and state constraints. Galerkin's method is used to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDF, which are subsequently used for controller design. Various MPC formulations are constructed on the basis of the finite dimensional approximations that differ in the way the evolution of the fast eigenmodes is accounted for in the performance objective and state constraints. The impact of these differences on the ability of the predictive controller to enforce state constraints satisfaction in the infinite-dimensional system is analyzed. Finally, the MPC formulations arc applied, through simulation, to the problem of stabilizing an unstable steady-state of a nonlinear model of a diffusion-reaction process subject to state and control constraints.