# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 3 Results and Discussion
### Mapping between Gene Expression and TRY Spaces  
In this section, we investigated different combinations of promoter and RBS strength and their corresponding regions in TRY space. Figure 3 shows the results of TRY space for fedbatch mode. The colored dots represent different combinations of \(N_{A}\omega_{A}\) and \(K^{A}_{\mathbb{C}_{0}}(s_{n})\) and TRY space, and the gray dots show the entire expression space. Figure 2: TRY of protein \(A\) across the gene expression space for batch, fedbatch and continuous operation obtained by simulation with the multi-scale model. All plots share the growth rate as the y-axis. Each dot corresponds to a different combination of transcription \(N_{A}\omega_{A}\) and RBS \(K^{A}_{\mathbb{C}_{0}}(s_{n})\) strengths. The dot color corresponds to the following: blue, batch; red, fedbatch; yellow, continuous. (**A**) Absolute titer, productivity rate and yield. (**B**) Normalized titer, productivity rate and yield. (**C**) Shows the combinations of \(N_{A}\omega_{A}\) and \(K^{A}_{\mathbb{C}_{0}}(s_{n})\) used and their associated induced growth rate (note that we plot \(K^{A}_{\mathbb{C}_{0}}(s_{n})\) at nominal substrate \(s_{n}=3.6\cdot\mathrm{L}^{-1}\), since \(K^{A}_{\mathbb{C}_{0}}(s)\) is not constant because \(s\) changes in simulation time). The wild-type growth rate is shown with a black square (right). At low protein expression levels, the promoter strength mainly determines the TRY values, while the RBS strength weakly influences its value. Figure 3A shows that the low protein expression region (\(\mu\in[0.015,0.025]\) min\({}^{-1}\)) corresponds to low values of \(N_{A}\omega_{A}\) (blue dots), and the medium to high values (yellow and red dots) are outside this region. However, varying the value of \(K_{\mathbb{C}_{0}}^{A}(s_{n})\) (represented in the size of the dots) hardly changes the value of the TRY space, and they remain within the region of low protein expression levels. This distinct behavior between varying \(N_{A}\omega_{A}\) and \(K_{\mathbb{C}_{0}}^{A}(s_{n})\) is because in (9) the value of \(J_{A}(\mu,r,s)\) is proportional to the value of \(N_{A}\omega_{A}\). Whereas the value of \(K_{\mathbb{C}_{0}}^{A}(s)\) is in the denominator of \(\omega_{A}/(d_{m}^{A}/K_{\mathbb{C}_{0}}^{A}(s)+\mu r)\), so the value of \(\mu\) limits the effect of \(K_{\mathbb{C}_{0}}^{A}(s_{n})\) on  
Figure 3: TRY values and relative variation indices of protein \(A\) for all the gene expression space in fed-batch operation obtained by simulation with the multi-scale model. All the plots share the growth rate as the y-axis. Each one dot corresponds to a different combination of transcription rate \(N_{A}\omega_{A}\) and effective RBS \(K_{\mathbb{C}_{0}}^{A}(s_{n})\) (note that we plot \(K_{\mathbb{C}_{0}}^{A}(s_{n})\) at nominal substrate \(s_{n}=3.6\cdot\mathrm{L}^{-1}\), since \(K_{\mathbb{C}_{0}}^{A}(s)\) is not constant because \(s\) changes in simulation time). The colors blue, yellow and red represent low, medium and high \(N_{A}\omega_{A}\) respectively, while the sizes of the markers are small, medium and big and show low, medium and high \(K_{\mathbb{C}_{0}}^{A}(s_{n})\). The gray color represent the rest of \(N_{A}\omega_{A}\) and \(K_{\mathbb{C}_{0}}^{A}(s_{n})\) combinations. (**A**) TRY measurements: titer, productivity and yield. (**B**) TRY relative variation indices for substrate variations. (**C**) Transcription rate \(N_{A}\omega_{A}\) and the effective RBS strength \(K_{\mathbb{C}_{0}}^{A}(s_{n})\). increasing the value of \(J_{A}(\mu,r,s)\). Moreover, in the case of low protein expression is when the value of \(\mu r\) is maximized with respect to the case of high protein expression, as there is less competition for recruiting ribosomes. At high protein expression, both the promoter and RBS strength determine the TRY value. Figure 3A shows that the high protein expression region (\(\mu\in[0,0.015]\) min\({}^{-1}\)) corresponds to medium to high values of \(N_{A}\omega_{A}\) (yellow and red dots), as we discussed in the previous paragraph.