# Gene Expression Space Shapes the Bioprocess Trade-Offs among Tier, Yield and Productivity
## 2 Materials and Methods
### Bioreactor Model--Modes of Operation  
The extracellular macroscopic inflow \(F_{in}\) and outflow \(F_{out}\) in model (1)-(3) depend on the mode of operation considered for the bioreactor: batch, fedbatch or continuous.  
We assumed a well-mixed, homogeneous culture and the values of parameters defined in Table 3. The initial conditions for the concentration of cells in the bioreactor, the substrate concentration and the culture initial volume were set to industrially plausible values for a small size bioreactor with volume up to \(10\,\mathrm{L}\): \(n(0)=5\times 10^{10}\,\mathrm{cell}\cdot\mathrm{L}^{-1}\), \(s(0)=3.6\,\mathrm{g}\cdot\mathrm{L}^{-1}\) and \(V(0)=1\,\mathrm{L}\), respectively. To calculate the initial conditions for the host cell parameters \(\mu\), \(r\), \(m_{r}\), \(m_{nr}\) and \(m_{A}\), we ran a simulation for constant substrate \(s=s(0)\), and we obtained the steady-state values for all the variables in the system. These were used as initial conditions for the host cell parameters.  
For the batch mode of operation in the bioreactor, we set \(F_{in}=F_{out}=0\), and we ran the simulations until the substrate concentration in the bioreactor decreased below \(2\%\) of its initial value.  
For the fedbatch and continuous modes, substrate feeding policies were applied such that the concentration of substrate in the bioreactor was kept constant to the nominal value \(s_{n}=s(0)=3.6\,\mathrm{g}\cdot\mathrm{L}^{-1}\). For the fedbatch mode we used \(F_{out}=0\) and the substrate feeding law  
\[F_{in}(\mu,n,V,s)=y\mu x(\mu,n)\frac{V}{s_{feed}-s} \tag{13}\]  
The feeding law (13) makes the substrate concentration in the bioreactor to remain constant and equal to the initial one [31]. For the substrate feeding concentration we used \(s_{feed}=180.156\,\mathrm{g}\cdot\mathrm{L}^{-1}\), and the bioreactor was fed until the culture volume reached \(10\,\mathrm{L}\). At this point the feeding inflow was set to zero (\(F_{in}=0\)), and the simulation was continued in batch mode until the substrate concentration in the bioreactor decreased below \(2\%\) of its initial value.  
Finally, for continuous mode, we used the feeding law \(F_{out}=F_{in}=VD(\mu,n)\) with:  
\[D(\mu,n)=\mu\frac{x(n,\mu)}{x_{ref}} \tag{14}\]  
where \(D(\mu,n)\) is the dilution and \(x_{ref}\) a reference value for the concentration of biomass in the bioreactor. We used \(x_{ref}=1\,\mathrm{g}\cdot\mathrm{L}^{-1}\) to get a production comparable with batch mode. When the reference is achieved, the concentration of glucose is kept at \(s=1.4\,\mathrm{g}\cdot\mathrm{L}^{-1}\). However, the biomass starts in the simulations approximately at \(0.01\,\mathrm{g}\cdot\mathrm{L}^{-1}\), and it takes almost all the simulation time to get to the reference. Therefore, for most of the simulation time the substrate was close to \(s_{n}=3.6\,\mathrm{g}\cdot\mathrm{L}^{-1}\) as in fedbatch case and in the exponential phase of the batch one.  
The simulation ran for a time interval equivalent to the turnover time, so that \(10\,\mathrm{L}\) of culture was introduced and removed form the bioreactor.  
Notice the conditions in the three modes of operation of the bioreactor were chosen so that the metabolic state of the cells in all three cases were equivalent so as to achieve a fair comparison.  
To keep track of the volume of media and product \(A\) added and removed from the bioreactor we extended the model with the expressions:  
\[\dot{V}_{feed} =F_{in} \tag{15}\] \[\dot{V}_{out} =F_{out}\] (16) \[\dot{S}_{out} =F_{out}\,s\] (17) \[\dot{M}_{A_{out}} =F_{out}\,n\,m_{A}, \tag{18}\]  
where \(V_{feed}\) is the total volume of media fed to the bioreactor, \(V_{out}\) the total volume of media removed from the bioreactor, \(S_{out}\) the total mass of substrate removed from the bioreactor and \(M_{A_{out}}\) the total mass of protein \(A\) removed from the bioreactor. We assumed that the substrate removed from the bioreactor due to \(F_{out}\) is recovered and that cells stop growing once the substrate is removed.