In this work, we develop a detailed, stochastic, dynamical model for the tryptophan operon of E. coli, and estimate all of the model parameters from reported experimental data. We further employ the model to study the system performance, considering the amount of biochemical noise in the trp level, the system rise time after a nutritional shift, and the amount of repressor molecules necessary to maintain an adequate level of repression, as indicators of the system performance regime. We demonstrate that the level of cooperativity between repressor molecules bound to the first two operators in the trp promoter affects all of the above enlisted performance characteristics. Moreover, the cooperativity level found in the wild-type bacterial strain optimizes a cost-benefit function involving low biochemical noise in the tryptophan level, short rise time after a nutritional shift, and low number of regulatory molecules.