*E. coli*cells.

Therefore the glucose model uses ordinary differential equations (ODE) to model both the glucose and the

*E. coli*concentration, assuming both are independend of each other. This is plausible because the medium used in the experiments contained carbon sources other than glucose. The glucose consumption rate per

*E. coli*is assumed to be independent of the glucose concentration.

# Theory - Glucose

Modeling the glucose concentration in a**PACE**experiments lagoon is based on a set of steady state equations while the model for PREDCEL has to take the development over time into account. The glucose concentration in the turbidostat \(c_{G_{T} }\) is increased with the incoming medium with a flow rate of \(\Phi\) and a glucose concentration of \(c_{G_{M} }\). It is decreased by with the medium that leaves the turbidostat with the same flow rate, but a glucose concentration of \(c_{G_{T} }\). Additionally

*E. coli*with a concentration of \(c_{E}\) take up glucose at a rate of \(q\).

*E. coli*titer, glucose concentration and flow rate into the lagoon are constant, a second steady state equilibrium can be assumed:

*E. coli*(see ), hence the

*E. coli*titer is assumed to be the same as in the turbidostat. This simplifies the combination of both equations and results in the full equation

When a

**PREDCEL**experiment is carried out the steady state assumption does not apply as the glucose in a flask is consumed over time, resulting in end concentrations that differ from the initial ones.

The glucose concentration in a flask \(c_{G_{F} }\) only changes by consumption since no glucose is added during PREDCEL. The consumption of glucose until a given timepoint equals to the integral over the

*E. coli*concentration \(c_{E}\) multiplicated with the glucose consumption factor \(q\). That is the amount of glucose that is consumed by a given amount of

*E. coli*during a given duration.

*E. coli*concentrations the logistic growth model should be more precise since it takes into account that there is an upper limit for the

*E. coli*concentration.

**Exponential growth**is defined as

**Logistic growth**is defined as

**logistic growth**is assumed, the term for \(c_{E}(t)\) is substituted by expression above. Here \(c_{c}\) is the capacity, the maximum concentration of

*E. coli*under the present conditions.

**Further calculations**for simplification of entering data:

The glucose concentration in grams of dryweight is needed in order to work with the literature value of \(q\). It can be calculated from the optical density at a wavelength of \(\lambda=600 \: nm\) as

*Milo et al.*

*Neubauer et al.*

**
Table 1: Variables and Parameters used for the calculation of the glucose and E. coli concentrations. **
List of all paramters and variables used in the analytical solution of this model and in the interactive webtool that is provided.

Symbol | Value and Unit | Explanation |
---|---|---|

\(c_{G_{T} }\) | [g/ml] or [mmol/ml] | Glucose concentration in Turbidostat |

\(c_{G_{M} }\) | [g/ml] or [mmol/ml] | Glucose concentration in medium |

\(c_{G_{L} }\) | [g/ml] or [mmol/ml] | Glucose concentration in lagoon |

\(t\) | [min] | Time |

\(\Phi_{T}\) | [ml/min] | Flow rate through Turbidostat |

\(\Phi_{L}\) | [ml/min] | Flow rate through Lagoon |

\(c_{E}\) | [cfu/ml] or OD600 | E. coli concentration |

\(q\) | \([g_{glucose} \: g_{DW}^{-1} h^{-1}]\) | Glucose consumption by E. coli |

\(t_{E}\) | [min] | E. coli generation time |

# Practice - Glucose

Many values can be taken from literature but since phage infection has an effect on the performance of*E. coli*, the maximum capacity for the

*E. coli*concentration in our setup had to be determined with our conditions. For one accessory plasmid the optical density was determined with and without phage infection over more then ten hours. Both cultures reached the maximum density after 510 min with an OD600 of 6.09 for the phage free culture and 5.133 for the infected culture. In the context of PREDCEL these values may serve as an estimation of the maximum

*E. coli*capacity.

# Theory - Arabinose

Arabinose functions as an inducer for the mutagenesis plasmids and it is assumed to not be degraded by*E. coli*in this model. Thus in a PREDCEL experiment arabinose concentration \(c_{A}\) is constant over time, because neither the total volume changes nor the amount of arabinose in the flask. $$ \frac{\partial c_{A_{L} } }{\partial t} = 0 $$ and the concentration is the starting concentration: $$ c_{A_{L} }(t) = c_{A_{L} }(t_{0}) $$ If the arabinose concentration is modeled for a lagoon supplied by a turbidostat with a separate supply of arabinose solution, the system converges to a steady state, when turbidostat volume \(V_{T}\), arabinose influx \(\Phi_{S}\) and flow rate of turbidostat \(\Phi_{T}\) are constant. In that case $$ \frac{\partial c_{A} }{\partial t} = 0 $$ is true. The change in lagoon arabinose concentration \(c_{A_{L} }\) is described by $$ \frac{\partial c_{A_{L} }(t)}{\partial t} = \Phi_{S} \cdot c_{A_{S} } - \Phi_{L} \cdot c_{A_{L}(t)} $$ \(\Phi_{S}\) and \(\Phi_{L}\) are measured relative to the lagoon volume. The arabinose concentration in the lagoon \(c_{A_{L} }\) can then be calculated using the concentration of the arabinose solution with which the lagoon is supplied \(c_{A_{S} }\). $$ c_{A_{L} } = \frac{\Phi_{S} }{\Phi_{L} } \cdot c_{A_{S} } $$ However, in some cases it may be relevant to estimate when a given percentage of the equilibrium concentration is reached. To make statements about that, the differential equation is solved to $$ c_{A_{L} }(t) = \frac{\Phi_{S} }{\Phi_{L} } \cdot c_{A_{S} } - \big(\frac{\Phi_{S} }{\Phi_{L} } \cdot c_{A_{S} } - c_{A_{L} }(t_{0})\big) \cdot e^{-\Phi_{L} t} $$

**
Table 2: Variables and Parameters used for the calculation of the glucose and E. coli concentrations **
List of all paramters and variables used in the numeric solution of this model. Also have a look at the interactive webtool based on this model

Symbol | Value and Unit | Explanation |
---|---|---|

\(c_{A_{L} }\) | [g/ml] | Arabinose concentration in Lagoon |

\(c_{A_{L}(t_{0}) }\) | [g/ml] | Initial Arabinose concentration in Lagoon |

\(\Phi_{S}\) | [lagoon volumes/h] | Flow rate of inducer supply relative to lagoon volume |

\(\Phi_{L} \) | [lagoon volumes/h] | Flow rate of lagoon |

# Practice - Arabinose

In all performed PACE experiments the inducer flow rate was \(\Phi_{S} = 2 ml/h \) and the lagoon volume was \(V_{L} = 100 ml\), resulting in a normalised \(\Phi_{S} = 0.02\).In the first experiments, \(c_{A_{S} }\) was set to 50 g/l, corresponding to 333 mmol/l. The conditions result in \(c_{A_{L} } = 1 g/l\), or 6 mmol/l.

Later the inducer concentration was doubled to \(c_{A_{S} } = 100 g/l\), resulting in a lagoon concentration of \(c_{A_{L} } = 2 g/l\) or 12 mmol/l.

To circumvent low arabinose concentrations in the beginning, the experiment can be started with starting concentrations above the equilibrium concentration. Since no negative effects of high arabinose concentrations on the outcome of PACE

### References