# Model II

## Competition model

Raising different adapted

The growth behavior of one

*E. coli*cultures together in one flask, one can observe that they compete for the limited space and nutrients. To determine how much better one*E. coli*culture must be adapted in comparison to its rivals and how long our system takes to produce a winner*E. coli*culture, we decided to model the competitive behavior.The growth behavior of one

*E. coli*culture can be modeled with a logistic growth function (Bacaër 2011):
were

For different adapted

*x*stands for the population [OD],*r*for the reproduction rate [h^{-1}] and*C*for the maximal capacity [OD].For different adapted

*E. coli*cultures equation (15) must be modified to:
where

The growth behavior of one adapted

*x*stands for the different adapted_{1},...,x_{n}*E. colis*and*r*for the different reproduction rates of the different adapted_{1},...,r_{n}*E. colis*.The growth behavior of one adapted

*E. coli*culture depends on its specific reproduction (*r*) rate. But one adapted_{1},..,r_{n}*E. coli*culture can only reach the maximal capacity*C*. Thus, the change in growth also depends on the amount of free capacity ((*C*-(*x*))/_{1}+...+x_{n}*C*).## Experimental validation

A theoretically implemented model must always be validated with experimentally measured data. To improve model II we did a row of cultivation experiments. One part of the cultivation experiments was used for fitting the growth parameter

*r*, and separate experiments were used for validating the model.### Cultivation experiments

For the determination of the experimental growth rate and for the production of a standard growth curve of the JS200 strain with an active error prone polymerase I we cultivated JS200 for around 18 hours at 37°C. We also wanted to investigate how a JS200 cell grows and, thus, beta-lactamase reacts, when it encounters different ampicillin concentrations. Therefore, the cultures were provided with a plasmid that carries a beta-lactamase sequence. Additionally, we let the cultures grow with 1, 3, 5 and 10 g ampicillin per liter, respectively (figure 1 (A)). As a control we also cultivated JS200 with an active wild-type polymerase I (figure 1 (B)). All experiments were done with three biological replicates. The measurements were taken every hour. The results can be seen here:

These measurements were used to calculate which reproduction rates can be reached. The maximal reproduction rate of the JS200 strain with the error prone polymerase is 0.67 h

A close look at our experimentally measured error prone growth curves (figure 1 (A)) led to the following interesting observation: Although the curves showed different growth behavior and diverging lag phases their rates seemed to undergo a shift, once the ampicillin was out of the system. Without ampicillin pressure the growth rates seem to be similar (figure 1 (A), exponential phase). To account for varying growth behavior at the beginning and an almost uniform behavior after a certain time, we had to work with a dynamical

As we could not find information about the behavior of different adapted

^{-1}. It were determined through fitting (figure 2) with the logistical function with the dynamical*r*.A close look at our experimentally measured error prone growth curves (figure 1 (A)) led to the following interesting observation: Although the curves showed different growth behavior and diverging lag phases their rates seemed to undergo a shift, once the ampicillin was out of the system. Without ampicillin pressure the growth rates seem to be similar (figure 1 (A), exponential phase). To account for varying growth behavior at the beginning and an almost uniform behavior after a certain time, we had to work with a dynamical

*r*. After fitting the growth curves of our theoretical model to the experimentally measured ones, we observed a change of the growth rate r at an OD of about 0.38.As we could not find information about the behavior of different adapted

*E. colis*together in one flask, we performed another experiment for further improvement of our mathematical model. Therefore, we cultivated two different adapted JS200 cultures, culture A and culture B, together in one flask. To be able to compare the two cultures we transformed a GFP labeled plasmid in culture A and an RFP labeled plasmid in culture B. Culture B had a little disadvantage with respect to growth in comparison to culture A. As a control we also cultivated the cultures alone. The experiments were performed with 1:1 A to B and 1:9 A to B proportion of the two different cultures to be able to study the behavior in detail.
As can be seen in figure 2 (A) the culture mixtures do not reach the same high plateau as the separately cultivated culture A. Still they reach a higher plateu as the separately cultivated culture B.
This means that the reproduction rate is influenced by different cultures.
Also in figure 2 (B) it can be seen that the curve of the culture mixtures lies in between culture A and B, but the curve resembling the culture mixtures with A 9:1 B is nearly the same as the curve representing culture A.
Thus, it can be seen that in the culture mixture culture A grows normally, but can not reach the same final OD.
It can also be seen that in the 1:1 culture mixture the GFP concentration is not that much lower compared to the concentration in the separately cultivated culture A.
This means that culture B does not influence the growing of culture A.
Figure 2 (C) shows that in the culture mixtures also culture B grows, but not as fast. So it never reaches the same OD amount as the single culture B.

In this experiment we observed that the differently growing

While comparing all growth curves, we made an additional observation. There is always a little fluctuation in the growth behavior. This leads to the assumption that even when one strain is cultivated more than once its growth behavior will vary from time to time. So we incorporated a fluctuating

In this experiment we observed that the differently growing

*E. coli*cultures compete with respect to space and nutrients. That is why we decided to use a modified logistical growth equation for the modelling of the competition.While comparing all growth curves, we made an additional observation. There is always a little fluctuation in the growth behavior. This leads to the assumption that even when one strain is cultivated more than once its growth behavior will vary from time to time. So we incorporated a fluctuating

*r*, following a normal distribution.### References

- Bacaër, Nicolas (2011): A Short History of Mathematical Population Dynamics. London: Springer-Verlag London Limited. Online accessible at http://site.ebrary.com/lib/alltitles/docDetail.action?docID=10445161.