The production of recombinant proteins in a host using synthetic constructs such as plasmids comes at the cost of detrimental effects such as reduced growth, energetic inefficiencies, and other stress responses, collectively known as metabolic burden. Increasing the number of copies of the foreign gene increases the metabolic load but increases the expression of the foreign protein. Thus, there is a trade-off between biomass and product yield in response to changes in heterologous gene copy number. This work proposes a computational method, rETFL (recombinant Expression and Thermodynamic Flux), for analyzing and predicting the responses of recombinant organisms to the introduction of synthetic constructs. rETFL is an extension to the ETFL formulations designed to reconstruct models of metabolism and expression (ME-models). We have illustrated the capabilities of the method in four studies to (i) capture the growth reduction in plasmid-containing E. coli and recombinant protein production; (ii) explore the trade-off between biomass and product yield as plasmid copy number is varied; (iii) predict the emergence of overflow metabolism in recombinant E. coli in agreement with experimental data; and (iv) investigate the individual pathways and enzymes affected by the presence of the plasmid. We anticipate that rETFL will serve as a comprehensive platform for integrating available omics data for recombinant organisms and making context-specific predictions that can help optimize recombinant expression systems for biopharmaceutical production and gene therapy.