# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 2 Materials and Methods
### Multi-Scale Model
The resources recruitment strength of a given gene, gene \(A\) in this case, \(J_{A}(\mu,r,s)\) can be understood as a measure of its eagerness to capture cell resources to get expressed. As seen from Table 2, it can be expressed as:  
\[J_{A}(\mu,r,s)=0.62\frac{l_{p}^{A}}{l_{e}}\frac{\omega_{A}}{K_{0}^{A}(s)}+\mu r \tag{9}\]  
where \(l_{p}^{A}/l_{e}\) is the ribosomes density (protein length over ribosomes occupancy length), the product \(\mu r\) of growth rate and number of free ribosomes is the flux of free resources, and \(d_{m}^{A}\) is the degradation rate of the transcript. The transcription rate is \(\omega_{A}\), the RBS strength is \(K_{C_{0}}^{A}(s)\), and the gene copy number is \(N_{A}\). In this work we analyze the roles that gene transcription and translation may play in shaping the trade-offs between titer, yield and productivity at the bioreactor level. Thus, on the one hand we will consider the transcription rate per gene \(\omega_{A}\) times the gene copy number \(N_{A}\). Notice \(\omega_{A}\) is directly related to the promoter strength. On the other hand we consider the RBS strength. The effective translation rate depends on the availability of the substrate. In our model, the RBS strength for a gene expressing the protein \(A\) is defined as:  
\[K_{C_{0}}^{A}(s)=\frac{k_{b}^{A}}{k_{u}^{A}+k_{e}(s)} \tag{10}\]  
where \(k_{b}^{A}\) and \(k_{u}^{A}\) are the association and dissociation rates between the ribosome binding site and the transcript, respectively, and \(k_{e}(s)\) is a Monod-like function of the extracellular substrate \(s\) that models the translation initiation rate as a function of the maximum translation rate per ribosome, the ribosomes density, the availability of substrate that can be catabolized to build aminoacids, and the affinity of the host for the substrate (see Tables 2 and 3 and reference [27]). The fact that this RBS strength definition depends on the available substrate as in Equation (10) has several implications. First, notice that an infinite number of different combinations of \(k_{b}^{A}\) and \(k_{u}^{A}\) can give the same RBS strength. This is not different from the situation when considering the substrate-independent saturated RBS strength obtained for substrate saturation:  
\[K_{C_{0},\text{sat}}^{A}=\frac{k_{b}^{A}}{k_{u}^{A}+\
u_{\text{max}}/l_{e}} \tag{11}\]  
Most important, the sensitivity of the substrate-dependent RBS strength to changes in the substrate concentration in the culture depends on the actual values of \(k_{b}^{A}\) and \(k_{u}^{A}\). Notice that for higher values of the saturated RBS strength, the variation of the substrate-dependent RBS-strength as a function of \(k_{e}(s)\) is larger, with:  
\[\frac{1}{K_{C_{0}}^{A}(s)}\frac{\partial K_{C_{0}}^{A}(s)}{\partial k_{e}(s)}= -\frac{1}{k_{b}^{A}}K_{C_{0}}^{A}(s) \tag{12}\]Therefore, in our analysis we evaluated the values of titer, productivity rate and yield for a nominal value of the substrate concentration in the culture (see Section 2.2) as a function of the expression space determined by the parameters \(N_{A}\omega_{A}\), \(k_{b}^{A}\) and \(k_{u}^{A}\). In addition, we also evaluated the sensitivity of the TRY space to changes in the substrate concentration as a function of the expression space.