Asymmetric exclusion processes, with periodic or open boundaries, have been studied extensively in the mathematics and statistical physics communities, as paradigm models for stochastic particle transport far from equilibrium. Though significant progress was made only recently, the original model was actually introduced decades ago to model protein synthesis.

In this presentation, we describe recent efforts to develop a comprehensive theory for protein synthesis, building on asymmetric exclusion processes with extended objects, modeling robosomes covering multiple codons. We discuss the effects of ribosome size and local hopping rate on density profiles and current-density relationships. The latter translate directly into synthesis rates for the corresponding protein. Turning to more realistic scenarios, we also investigate the competition of multiple mRNAs for a finite supply of ribosomes and aa-tRNAs.