A matrix notation coupled to macroscopic principles is introduced as a means to develop first- principles models in an efficient and structured way within PAT applications. The notation was evaluated for developing an integrated biological, chemical (pH modeling) and physical (gas-liquid exchange) model for describing antibiotic production with Streptomyces coelicolor in batch fermentations. The model provided statistically adequate fits to all the monitored macroscopic biological, chemical and physical data of the process, except the phosphate uptake dynamics. This phosphate discrepancy is hypothesized to result from the internal storage of phosphate as polyphosphate prior to the exponential growth phase. The antibiotic production was associated with the stationary phase and its kinetics was adequately described using a modified Luedeking-Piret equation. Further, the maintenance was best described by employing a combination of Pirt and Herbert models, a result that was supported by a model-based hypothesis testing. Overall the process knowledge currently incorporated in the model is believed to be useful both for process optimization purposes and for further testing of hypotheses aiming at improving the mechanistic understanding of antibiotic production with S. coelicolor. Last but not least, the matrix notation is believed to be a promising supporting tool for efficient development and communication of complex dynamic models within a PAT framework.