We present a biophysical model for the propulsion of the cellulose synthase complex, the motile transmembrane protein complex responsible for the biosynthesis of cellulose microfibrils, the dominant architectural component of the cell walls of higher plants. Our model identifies the polymerization and the crystallization of the cellulose chains as the combined driving forces and elucidates the role of polymer flexibility and membrane elasticity as force transducers. The model is elaborated using both stochastic simulations and a simplified analytical treatment. On the basis of the model and approximate values for the relevant physical constants, we estimate the speed of the cellulose synthase complex to be in the range vp {approx} 10-9-10-8 m/s, consistent with the recently reported experimental value of 5.8 x 10-9 m/s.