# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 2 Materials and Methods
### Multi-Scale Model
The internal variables in the model and the values of the parameter can be found in Table 2 and Table 3 respectively. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Name** & **Description** & **Units** \\ \hline \(r\) & Free mature ribosomes in the cell. & \(\text{mloc}\cdot\text{cell}^{-1}\) \\ \(m_{r}\) & Total mass of ribosomal proteins in the cell. & \(\text{fg}\cdot\text{cell}^{-1}\) \\ \(m_{nr}\) & Total mass of non ribosomal proteins in the cell. & \(\text{fg}\cdot\text{cell}^{-1}\) \\ \(m_{A}\) & Total mass of protein \(A\) in the cell. & \(\text{fg}\cdot\text{cell}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: States and main variables of the host model. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Name** & **Description** & **Equation** & **Units** \\ \hline \(v(s)\) & Effective translation rate per ribosome. & \(v(s)=v_{max}s/(s+K_{s})\) & aa \(\cdot\) min\({}^{-1}\) \\ \(k_{s}(s)\) & Translation initiation rate. & \(k_{s}(s)=v(s)/l_{s}\) & min\({}^{-1}\) \\ \(l_{sum}(\mu,r,s)\) & Total sum of all the J in the cell. & \(J_{sum}(\mu,r,s)=\sum_{i=r,\mu,r,s}N_{i}l_{i}(\mu,r,s)\) & adim \\ \(K_{s}^{\prime}(s)\) & Effective RRS affinity of ribosomal mRNA. & \(K_{s}^{\prime}(s)=k_{s}^{\prime}(s+k_{s}(s))\) & cell \(\cdot\) molec\({}^{-1}\) \\ \(E_{m}\) & Ribosomes density related term for ribosomal mRNA. & \(E_{m}^{\prime}=0.62f_{s}/l_{s}\) & adim \\ \(J(\mu,r,s)\) & Average J value of one ribosomal _E. coli._ gene. & \(J_{s}(\mu,r,s)=E_{max}^{\prime}/(l_{m}^{\prime}/K_{s}^{\prime}(s)+\mu r)\) & adim \\ \(r_{s}(m_{r})\) & Number of mature and immature ribosomes. & \(r_{s}(m_{r})=m_{r}/r_{m}\) & molec\(\cdot\) cell\({}^{-1}\) \\ \(\Phi_{t}(\mu,r,s)\) & Fraction of translating ribosomes of \(\phi_{t}(\mu,r,s)\) & \(\phi_{t}(\mu,r,s)\) & \(\phi_{t}(\mu,r,s)\) & adim \\ \(K_{s}^{\prime}(s)\) & Effective RRS affinity of non-ribosomal mRNA. & \(K_{s}^{\prime\prime}=k_{s}^{\prime\prime}(s+k_{s}(s))\) & cell \(\cdot\) molec\({}^{-1}\) \\ \(E_{m}^{\prime\prime}\) & Ribosomes density related term for non-ribosomal mRNA. & \(E_{m}^{\prime\prime}=0.62f_{s}^{\prime\prime}/l_{s}\) & adim \\ \(J_{m}(\mu,r,s)\) & Average J value of one non-ribosomal _E. coli._ gene. & \(J_{m}(\mu,r,s)=\sum_{i=r,\mu,r,s}N_{i}l_{i}(\mu,r,s)\) & adim \\ \(K_{s}^{\prime}(s)\) & Effective RRS affinity of protein A mRNA. & \(K_{s}^{\prime}(s)=k_{s}^{\prime}(s+k_{s}(s))\) & cell \(\cdot\) molec\({}^{-1}\) \\ \(E_{m}^{\prime\prime}\) & Ribosomes density related term for protein A mRNA. & \(E_{m}^{\prime\prime}=0.62f_{s}/l_{s}\) & adim \\ \(I_{A}(\mu,r,s)\) & Average J value of one protein A gene. & \(J_{A}(\mu,r,s)=E_{m}^{\prime\prime}a_{A}/(d_{m}^{\prime\prime}/K_{s}^{\prime} (s)+\mu r)\) & adim \\ \(m_{p}(\mu)\) & Total protein mass of the cell. & \(m_{p}(\mu)=c_{1}\mu^{2}+c_{2}\mu+c_{3}\) & fg \(\cdot\) cell\({}^{-1}\) \\ \(\pi(\mu,r)\) & Concentration of biomass in the bioreactor. & \(x(\mu,r)=m_{p}(\mu)\) & g/L \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Internal variables of the model. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Name** & **Description** & **Value** & **Units** & **Reference** \\ \hline \(K_{s}\) & Half activation threshold of growth rate. & 0.1802 & g \(\cdot\) L\({}^{-1}\) & [31] \\ \(w_{max}\) & Maximum effective translation rate per ribosome. & 1260 & aa \(\cdot\) min\({}^{-1}\) & [32] \\ \(m_{max}\) & Average aminoacid mass. & 182.6 \(\times 10^{-9}\) & fg \(\cdot\) aa\({}^{-1}\) & [33] \\ \(l_{e}\) & Ribosome occupancy length. & 25 & aa & estimated [34, 35, 36, 37] \\ \(d_{p}^{\dagger}\) & Fraction of mature available ribosomes relative to the total. & 0.7976 & adim & [17, 29] \({}^{*}\) \\ \(l_{p}^{\dagger}\) & Mean length of ribosomal proteins. & 195 & aa & calculated from [28] \\ \(d_{m}^{\prime}\) & Mean degradation rate of ribosomal mRNA. & 0.16 & min\({}^{-1}\) & calculated from [28] \\ \(k_{m}^{\prime}\) & Dissociation rate RRS-ribosome for ribosomal mRNA. & 135 & min\({}^{-1}\) & [27] \({}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the model.