# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 2 Materials and Methods
### Multi-Scale Model  
In order to take into account the different scales involved in the model of a bioprocess, we consider a multi-scale model integrating the different occurring phenomena. From top to bottom (Figure 1A), our model describes the interactions in the three levels considered: the biomass and substrate dynamics in the bioreactor, the dynamics of the host and those of the synthetic gene circuit expressing the proteins of interest. At the top of our multi-scale model we find the macroscopic dynamics of the substrate and biomass in the bioreactor (Figure 1A, Bioreactor model). This bioreactor model takes as inputs the substrate inflow and outflow, together with the specific growth rate of the cell  
Figure 1: Bioprocess multi-scale model. (**A**). Schematic representation of the multi-scale model. The three levels considered including the bioreactor, the host cell and the synthetic circuit models and their interactions. (**B**). Simulation obtained with the multi-scale model. The colors correspond to the interactions in panel (A). An example of a batch operation, where the synthetic circuit parameters are \(\omega_{A}=50\), \(k_{b}^{A}=5\) and \(K_{a}^{A}=117\). Top panel shows the macroscopic scale with the substrate availability (left axis) and biomass concentration (right axis). Middle panel shows the growth rate (left axis) and a measure of the cell burden (right axis). Bottom panel shows the amount of protein A expressed in a cell (left axis) and its resource recruitment strength \(J_{A}\) (right axis). (**C**). TRY relative variation indices for substrate variations. Shaded area \(RV_{X}\) represents the integral from minimum substrate \(s_{min}\) to the nominal substrate \(s_{n}\) of the difference between the nominal index value and the actual value. population in the bioreactor, and it provides the concentration of available substrate and biomass in the bioreactor. These dynamics are defined by the following set of equations:  
\[\dot{V} =F_{in}-F_{out} \tag{1}\] \[\dot{n} =\mu n-\frac{F_{in}}{V}n\] (2) \[\dot{s} =\frac{F_{in}}{V}(s_{feed}-s)-y^{-1}\mu x(n,\mu) \tag{3}\]  
where \(V\) is the volume of culture in the bioreactor in \(L\), \(n\) is the concentration of cells in the bioreactor (cell \(\cdot\) L\({}^{-1}\)), and \(s\) is the amount of available substrate (g \(\cdot\) L\({}^{-1}\)). \(F_{in}\) and \(F_{out}\) represent the input and output fluxes (L \(\cdot\) min\({}^{-1}\)), and \(\mu\) the specific cell growth rate in min\({}^{-1}\). At the intermediate level (Figure 1A) the host model represents the dynamics of the cell and takes as inputs the available substrate from the bioreactor model and a measure of the gene expression resources demanded by the synthetic gene circuit. This host model provides the cell-specific growth rate and a measure of the cell burden, and it is defined as follows:  
\[\mu =\frac{m_{aa}}{m_{p}(\mu)}\
u(s)\phi_{b}(\mu,r,s)\phi_{t}r_{t}(m_ {r}) \tag{4}\] \[r =\frac{\phi_{t}r_{t}(m_{r})}{1+I_{sum}(\mu,r,s)}\] (5) \[\dot{m}_{r} =\left[m_{p}(\mu)\frac{N_{r}I_{r}(\mu,r,s)}{I_{sum}(\mu,r,s)}-m_ {r}\right]\mu\] (6) \[\dot{m}_{nr} =\left[m_{p}(\mu)\frac{N_{nr}I_{nr}(\mu,r,s)}{I_{sum}(\mu,r,s)}-m _{nr}\right]\mu \tag{7}\]  
where the main variables can be found in Table 1, and the terms \(J_{r}(\mu,r,s),J_{nr}(\mu,r,s)\) are the resources recruitment strengths for the host ribosomal and non-ribosomal lumped ensembles of genes, and \(J_{sum}(\mu,r,s)=\sum_{i=r,nr,A}N_{i}I_{i}(\mu,r,s)\) is the total resources recruitment strength of the cell which represents the cell burden. These are the key functional coefficients that allow us to explain the distribution of resources in the host and the relationship between the usage of resources, cell growth and protein productivity. Details of the host-circuit interaction model derivation can be found in our previous work [27]. The parameters used in this work were fitted from experimental data from [28; 29] and correspond to the wild-type _E. coli_ K-12 strain MG1655 [30] and _E. coli_ B/r strain.