# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 3 Results and Discussion  
### Nominal TRY as a Function of the Expression Space and Bioreactor Operation  
The analysis performed with different conditions of bioreactor operation and several combinations spanning the expression space reveals, as expected, that the cell growth rate decreases as the cell burden induced by higher expression levels of \(A\) increases. Figure 2 shows the results for the nominal variation of the TRY space as a function of the expression space for the three modes of operation in the bioreactor. We varied the parameters \(N_{A}\omega_{A}\) and \(K_{C_{0}}^{A}(s_{n})\) of the expression space in a range from low to high expression of the protein \(A\) (see Figure 2C).  
Titer and yield show a decreasing monotonous relationship with growth rate. Thus, as observed in Figure 2A, titer and yield decrease as the cell burden decreases (i.e., weak promoter and RBS strengths are used) and, consequently, the cell growth increases. Indeed, to obtain higher titer and yield, the cell must invest a higher fraction of its mass to synthesize the protein \(A\). As a result, it uses fewer cell resources to build ribosomes, and the cell growth decreases [27].  
On the contrary, the maximum productivity rate depends non-monotonously on the growth rate, achieving a maximum for \(\mu\approx 0.075\,\mathrm{min}^{-1}\).  
Thus, we find the same qualitative trade-offs between titer, yield and rate as a function of growth rate encountered when trying to optimize the metabolic flux towards a product of interest using metabolic flux analysis [42; 43]. There is a trade-off between titer and yield on the one hand, and productivity rate on the other. High titer and yield cannot be attained without eventually decreasing the productivity rate.  
Additionally, as expected, the fedbatch mode achieves higher values of productivity rate and titer of the protein \(A\) for all the combinations in the expression space (see Figure 2A). This simply reflects that in fedbatch mode the total amount of substrate fed to the bioreactor is larger than in batch and continuous modes. Yet, the normalized results show no relevant differences (Figure 2B). That is, the normalized titer rate and yield do not depend on the mode of operation of the bioreactor but only on the the promoter and RBS strengths in the expression space.  
This result provides a principle of space-scale separation for multi-scale models. For a given substrate availability, and assuming homogeneity in the population of cells in the bioreactor, the cell growth for each individual local cell and the mass of heterologous proteins it will express depend on the interactions between the cell host and the genes, being independent of the way the substrate is fed to the population of cells. The bioreactor mode of operation will affect the total amount of substrate fed during the fermentation process and, therefore, the size of the population of cells. Although the geometry and other physical characteristics of the bioreactor are not considered here, they could be important to provide a way to characterize heterogeneous distributions of the limiting substrate within the bioreactor. Individual cells with different substrate concentrations will be subject to different interactions between the host and the synthetic circuit, and then they might synthesize the heterologous proteins at different rates.