In nature, microorganisms are continuously facing nutrient availability changes in the environment, and thus they have evolved to dynamically adapt their physiology to cope with this phenomenon, by dynamically allocating resources to different cellular functions. In order to study their behaviours, the fitness of such microorganisms can be represented as a dynamical growth maximization strategy, which is formulated as an OCP (Optimal Control Problem) in coarse-grained self-replicator systems. This study inspired the use of biotechnological engineering to maximize the production of a metabolite of interest in E. coli by means of both analytical and computational techniques. Motivated by this approach, we incorporate the metabolite production scheme in a CSTR (Continuous Stirred-Tank Reactor) Bioreactor, which can be interpreted as a general case of the preceding models. We then derive two particular cases and study the associated OCP, so as to stress the importance of singular regimes and chattering arcs in optimal solutions. From a biological point of view, our results show that the natural allocation of resources of bacteria has to be modified in order to achieve optimal metabolite production. Finally, we go over the computations of the second order singular arc, and provide a numerical check of the Legendre-Clebsch condition.