The production of recombinant proteins is a problem of significant interest in bioengineering. Because of the existing trade-off between cellular growth and protein production, these two processes are separated in time in most commonly-employed strategies: a growth phase is followed by a production phase. Here, we investigate the potential of an alternative strategy using artificial cell specialization and differentiation systems in which cells either grow ("growers") or produce proteins ("producers") and growers can irreversibly "differentiate" into producers. Inspired by an existing two-population system implemented in yeast, we propose a model of a "yeast synthetic stem cell system" and define an optimal control problem to maximize bioproduction. Analytically, we first establish the well-posedness of the problem. Then, we prove the existence of an optimal control and derive non trivial optimality conditions. We finally use these results to find numerical optimal solutions. We conclude by a discussion of extensions of this work to models that capture the heterogeneity of the cell response to differentiation signals.