# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 2 Materials and Methods
### Multi-Scale Model
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Name** & **Description** & **Value** & **Units** & **Reference** \\ \hline \(K_{p}^{d}\) & Association rate RRS-ribosome for ribosomal mRNA. & 8.853 & cell \(\cdot\) min\({}^{-1}\cdot\) molec\({}^{-1}\) & [27] \({}^{*}\) \\ \(N_{r}\) & Number of proteins that make up a ribosome. & 55 & adim & [27] \\ \(\omega_{r}\) & Average transcription rate for ribosomal proteins. & 4.8658 & molec \(\cdot\) min\({}^{-1}\cdot\) cell\({}^{-1}\) & [27] \({}^{*}\) \\ \(r_{n}\) & Weight of a ribosome. & 0.0045 & fg & [29] \\ \(lr^{p}\) & Mean length of non-ribosomal proteins. & 333 & aa & calculated from [28] \\ \(d_{n}^{m}\) & Mean degradation rate of non-ribosomal mRNA. & 0.2 & min\({}^{-1}\) & calculated from [28] \\ \(k_{n}^{m}\) & Dissociation rate RRS-ribosome for non-ribosomal mRNA. & 6.1297 & min\({}^{-1}\) & [27] \({}^{*}\) \\ \(k_{n}^{m}\) & Association rate RRS-ribosome for non-ribosomal mRNA. & 14.9971 & cell \(\cdot\) min\({}^{-1}\cdot\) molec\({}^{-1}\) & [27] \({}^{*}\) \\ \(N_{nr}\) & Number of non ribosomal proteins expressed at one time. & 1735 & adim & [27] \\ \(\omega_{nr}\) & Average transcription rate for non ribosomal proteins. & 0.03 & molec \(\cdot\) min\({}^{-1}\cdot\) cell\({}^{-1}\) & [27] \({}^{*}\) \\ \(f_{n}^{d}\) & Length of protein A. & 195 & aa & \({}^{*}\) \\ \(d_{n}^{d}\) & Mean degradation rate of protein A mRNA. & 0.16 & min\({}^{-1}\) & \({}^{**}\) \\ \(k_{n}^{d}\) & Dissociation rate RRS-ribosome for protein A mRNA. & (6 135) \(\dagger\) & min\({}^{-1}\cdot\) molec\({}^{-1}\) & [38, 39] \\ \(k_{n}^{d}\) & Association rate RRS-ribosome for protein A mRNA. & (3 15) \(\dagger\) & cell \(\cdot\) min\({}^{-1}\cdot\) molec\({}^{-1}\) & [38, 40] \\ \(N_{A}\) & Number of copies of gene \(A\). & (1 70) \(\dagger\) & adim & \({}^{**}\) \\ \(v_{A}\) & Average transcription rate for protein \(A\). & (0 5) \(\dagger\) & molec \(\cdot\) min\({}^{-1}\cdot\) cell\({}^{-1}\) & [27] \({}^{*}\) \\ \(c_{1}\) & First coefficient of mass equation. & 239,089 & fg \(\cdot\) cell\({}^{-1}\cdot\) min\({}^{2}\) & [27] \({}^{*}\) \\ \(c_{2}\) & Second coefficient of mass equation. & 7432 & fg \(\cdot\) cell\({}^{-1}\cdot\) min\({}^{2}\) & [27] \({}^{*}\) \\ \(c_{3}\) & Third coefficient of mass equation. & 37.06 & fg \(\cdot\) cell\({}^{-1}\) & [27] \({}^{*}\) \\ \(y\) & Biomass yield on glucose. & 0.45 & g\({}^{-1}\)Substrate & [41] \\ \(\xi_{/rad}\) & Fresh media substrate concentration. & 3.6 & g \(\cdot\) L\({}^{-1}\) & [31] \\ \hline \end{tabular}  
* These parameters were re-optimized following the methods described in [27] to better fit the wild-type at low growth rate, since that range is the most relevant for this work. ** Without loss of generality in the results, we choose \(l_{n}^{d_{n}}\) and \(d_{n}^{d_{n}}\) to be equal to the ribosomal parameters, and the range of \(N_{A}\) and \(\omega_{A}\) to be in the order of ribosomal and non-ribosomal parameters.
* These parameters define the gene expression space and their value varies in simulations between the minimum value (first value in parentheses) and the maximum value (last value in parentheses).  
\end{table} 
Table 3: _Cont._Finally, at the bottom level, a model of the synthetic gene circuits considers the interaction dynamics with the host and estimates production of the mass fraction of the proteins of interest (8). In this work, as an example, we consider the expression of a generic theoretical gene \(A\). More complex synthetic circuits or metabolic pathways with their corresponding enzymes can be easily incorporated into the model by adding more equations such as (8). \[m_{A}=\left[m_{p}(\mu)\frac{N_{A}J_{A}(\mu,r,s)}{I_{sum}(\mu,r,s)}-m_{A}\right]\mu \tag{8}\]  
where \(m_{A}\) is the total mass of protein \(A\) in the cell, \(N_{A}\) is the number of copies of gene \(A\), and \(J_{A}(\mu,r,s)\) is the resources recruitment strength of the gene \(A\).