# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 1 Introduction  
Optimal gene expression is central to the development of both bacterial expression systems for production of heterologous proteins and microbial cell factories to produce metabolites of industrial interest. Both applications seek to obtain high levels of products of interest by means of metabolic engineering. The goal is to reach industry-level overproduction demands in an optimal way, as measured using the key performance indices titer, productivity (rate) and yield (henceforth TRY). In practice, some trade-offs in the TRY space must be reached and be adaptive to the growth and environmental dynamic changes that occur in a bioreactor set-up [1, 2]. Indeed, biomass growth and product yields cannot be simultaneously maximized. For a given substrate uptake rate, a higher growth yield will lead to a higher growth rate at the expense of the product yield. In silico constraint-based metabolic genome-scale models have proved very valuable in engineering biosynthetic metabolic pathways in which the combined catalytic activity of a collection of enzymes is coordinated so as to produce the desired metabolites. These models provide predictions on maximum theoretical yields, optimal flux distribution to maximize flux towards some metabolite reaction bottlenecks and the required gene expression leading to increases in fluxes towards the final products [3, 4, 5]. It is possible to deal with the trade-off between yield and productivity using dynamic flux balance analysis(dFBA), which incorporates both process dynamics and the constraint-based metabolic network model and relies on dynamic optimization methods [6; 7; 8]. In practice, the metabolic costs of producing proteins, enzymes, and other cell macromolecules during gene over-expression often lead to shifts away from the optimal predictions. Yet, the focus of these constraint-based models on metabolism excludes proteins and other macromolecules, the major biomass constituents, from the model. Models that integrate metabolism, biomass composition and the cost for the cell to produce a certain protein or enzyme can potentially yield better predictions than those focused on metabolism in isolation [9]. Thus, for instance, dynamic enzyme-cost FBA [10; 11] includes dynamic changes in substrate concentration, a detailed description of biomass composition, and takes into account how much it costs to the cell to produce the desired enzyme. As an alternative to metabolism-centered models, in the recent years there has been a increasing interest in the development of models of gene expression accounting for cellular resources competition [12; 13]. These range from very coarse-grain ones [14; 15; 16; 17] to semi-mechanistic ones with varied degrees of granularity [18; 19; 20]. This class of models may consider the interplay between substrates uptake, cell growth rate, gene expression and interaction with the cell host caused by competition for cellular resources. Therefore, they have the potential to deal with the related issues of dynamic gene regulation, cell resource allocation and the varying industry-scale bioreactor environment. As an intermediate modeling strategy, whole-cell models connecting gene expression, metabolism and growth, such as the coarse-grained self-replicator models, take into account the dynamic feedback from gene expression and growth to metabolism [21; 22]. So far, the main interest in the literature has been in determining the required optimal gene expression levels that fulfill the design specifications using any of the modeling approaches described above. The dynamic regulation of the the specified expression levels has also been addressed both for heterologous protein expression systems [23] and for metabolic pathways [24]. Indeed, major improvements in yield, titer and productivity of engineered metabolic pathways can be accomplished by dynamic balancing of pathway gene expression [25; 26], where the application of dynamic feedback and feedforward regulation of gene expression addresses the robustness pitfalls of static regulation.