# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 3 Results and Discussion
### There Exists a Trade-Off in the Relative Variation Indexes  
In the previous sections, we have shown how \(N_{A}\omega_{A}\) and \(K^{A}_{C_{0}}(s_{n})\) determine trade-offs in the measures of titer, productivity rate and yield; in this section we investigatedwhether there is also a trade-off in the relative variation indexes and how \(N_{A}\omega_{A}\) and \(K^{A}_{\mathcal{C}_{0}}(s_{n})\) affect it. Next, we show which conditions are necessary to minimize each of the relative variation indexes. The titer and yield relative variation indexes are zero when the mass fraction between host cells and protein \(A\) is constant with changes in substrate. The titer and yield equations are independent of growth rate; therefore, for their relative variation indexes to be zero, it is sufficient for the cell to invest the same mass fraction as protein \(A\) regardless of substrate changes. The productivity relative variation index is zero when the product of growth rate by the fraction of mass invested as protein \(A\) is constant with changes in substrate. The productivity equation depends on the growth rate; then, for its index to be zero, it is not sufficient to achieve a constant mass fraction invested as protein \(A\) (as was the case with titer and yield). To achieve a productivity relative variation index of zero, it is necessary that the product of mass fraction and growth rate remains constant. Therefore, there exists a trade-off between the titer and yield indexes versus the productivity index since they require different conditions to minimize their values that are incompatible with each other. Figure 3B shows that none of the combinations of the colored dots is able to make all three indices zero at the same time (this also true for the gray dots, although it is not shown in the graph). To take a deeper look into the relative variation index \(RV_{X}\) and their meaning, it is necessary to analyze the variation of indices for changing levels of substrate. Figure 4 shows that a change in the substrate can affect the TRY values by causing them to go down or up with respect to the TRY nominal value (it may even be the case that the TRY has one section where TRY goes up and another where TRY goes down). For example, Figure 4B shows that in the case of titer it is possible to select a value of \(k_{u}\) (solid blue line) that allows us to increase the titer when the substrate decreases, and we can also select a different value of \(k_{u}\) (dotted blue line) that decreases the titer when the substrate decreases. Figure 4: TRY at different nominal substrate concentrations for high and low protein expression levels for fedbatch operation. Orange and yellow (high and medium burden) lines correspond to different \(N_{A}\omega_{A}\) and \(K^{A}_{\mathcal{C}_{0}}(s_{n})\) combinations that achieved the same productivity value of \(1.31\,\mathrm{g/L/h}\) at the nominal substrate \(3.6\,\mathrm{g/L}\), while blue (low burden) lines correspond to combinations that achieved \(4.3\times 10^{-4}\mathrm{g/L/h}\). We chose \(k_{b}=15\) for all the combinations, \(k_{u}=\{6,20,135\}\) for orange and yellow, \(k_{u}=\{6,80,135\}\) for blue, and we used \(N_{A}\omega_{A}\) to achieve the same productivity level. Low levels of \(k_{u}\) are drawn with dotted lines, mid levels with dashed lines and high levels with solid lines. Then, for these combinations, we reduced the nominal substrate up to \(0.1\,\mathrm{g/L}\) to see its effect on titer, productivity rate, yield and growth rate. (**A**) TRY for medium and high burden \(N_{A}\omega_{A}\) and \(K^{A}_{\mathcal{C}_{0}}(s_{n})\) combinations. (**B**) TRY for low burden \(N_{A}\omega_{A}\) and \(K^{A}_{\mathcal{C}_{0}}(s_{n})\) combinations. From this, we can see that both the absolute variation and the direction of change in production are important. For industrial production purposes it is not the same for production to go down than for production to go up with respect to the nominal value.