Introduction

Bacterial conjugation is the transfer of genetic material (plasmids) between bacterial cells by direct cell-to-cell contact or by a bridge-like connection between two cells. It is also the way by which we planned to grant C.C. the ability of color print. As the significance of this process is self-evident, we established a model to explore the mechanism of concentration change of bacteria during conjugation, answering the question that how many E. Coli & C.C. one should put into the system initially to ensure more C.C. yield. This model serves as a useful guidance of this experiment.

First, according to the types of bacterium (E.Coli/C.C.) and their content (with/without plasmids), bacteria involved in this scenario are divided into 4 categories as shown in Fig. 1. Each category has a few numbers to represent its property of growth: $r$ stands for the rate of birth, while $d$ stands for the rate of death.

To be specific: $r_1$ and $r_2$ represent the rate of birth of E. Coli and C.C., respectively. Accordingly, $d\_1$ and $d_2$ respectively represent the rate of death of them. $d_3$ stands for the rate of death causing by lack of plasmid.

Then, we made several assumptions.

Assumptions:

1. All kinds of bacteria involved are uniformly distributed in the system where conjugation happens. Using one number, the concentration n1, n2, n3, to describe a specific type of bacterium, respectively (n4=0 in the case of conjugation). That means, the concentration varying among different locations is ignored, which, significantly simplifies the math in the model without hampering the major part of the biological process to be evaluated.
   To be specific, eliminating the variable of location automatically make the form of possible governing equations going from PDE to ODE: a huge simplification math-wise and an acceptable compromise bio-wise. It's natural to interpret n1, n2, n3 as mean values of bacteria concentrations.

2. The concentration of each type of bacterium is governed by logistic function when an individual type peacefully grows, free from interference of others. r, rate of birth, d, rate of death, and K, environmental capacity are introduced as required by logistic function
   $$ \frac{dN}{dt}=[r(1-\frac{N}{k})-d]N $$ and its integral form.
   $$ N(t)=\frac{\frac{r-d}{r}K}{1+(\frac{(r-d)K}{rN\_0}-1)e^{-(r-d)t}}
   $$ This assumption generally settles down the shape of the equations in the model and establishes a bridge connecting the experimental observation and the theoretical prediction. They will be detailed after.

3. The different types of bacterium interfere with each other in two ways: competition & conjugation.
   Competition is originated from the contradiction between limited nutrition in the medium and endless reproduction desire of bacteria. Regarding n in $(1-\frac{n}{k})$ of logistic function as the total concentration of all types of bacterium by convention, empowers the equations to describe this phenomenon. Conjugation is the way by which engineered plasmids transfer from E.Coli to C.C.. By appropriately adding a term proportional to the concentration of E. Coli with plasmids and C.C. without plasmids, enables us to evaluate the influence on the growth of bacteria caused by conjugation.

With all the assumptions above, we established our model in principal, which is able to describe almost everything which can affect the amount of bacteria: birth, death, competition and conjugation. We write down our formalism with ODE.
$$\frac{dn_1}{dt}=[r_1(1-\frac{n_1+n_2+n_3}{K}-d_1)]n_1\ \frac{dn_2}{dt}=[r_2(1-\frac{n_1+n_2+n_3}{K}-d_2-d_3)]n_2-kn_1n_2\ \frac{dn_3}{dt}=[r_3(1-\frac{n_1+n_2+n_3}{K}-d_2)]n_3+kn_1n_2$$

However, it's obviously not enough to just stay here. We must acquire all the rational values of the parameters mentioned above to possibly make any meaningful prediction with this model, which is almost the trickiest part of most biological models.

Extracting parameters from experiment

We did an experiment to explore the growth pattern of E. Coli and C.C. individually. Tracking the OD values of those bacteria enables us to know their relative concentration during growth. Without interference, the growth curves are generally "S"-shaped except for a few points deviating from the trend, as shown in Fig. 2.
After removing 3 of those points, we fit the data points with logistic function, as shown in Fig. 3.
$$ y=\frac{a}{1+be^{-kx}}
$$

The function fits well with the data points, indicating the assumptions are, in a sense, rational. Furthermore, comparing the fitting function with the theoretical function, we got 4 constraint equations which incorporates 6 undetermined parameters.
Apparently, more experiments are needed to settle down each of the 6 parameters, but conceiving a method to experimentally quantify the the amount of plasmids in cells goes beyond the best of our knowledge. However, we are still able to move forward by intuitively set $K=1.5$, $k=0.05$ and $d\_3=0.02$. By doing this, all the parameters in the model are determined, as shown in Fig. 4.

To this point, the whole model is formally established, can't waiting to make predictions.

Try it out

After we set the initial condition to be
n1=0.03, n2=0.07;
the model make its first attempt of prediction. (Fig. 5)

From Fig. 5, we can safely say that the model's debut is really not bad. It got reasonable results. As time went by, $n_2$ and $n_3$ rised at first, but went down and faded away at the end, cannot competing with the soar of $n_3$ because of the preference of medium. After 24h， the ratio of $n_3$ (C.C. with plasmids) went over 75%, a pretty good consequence.

However, happy ending is always fragile. Consequence is hypersensitive to the subtle change of initial concentration of different types of bacterium. For example, when n1(0)=0.8, n2(0)=0.2, situation becomes entirely different as shown in Fig. 8.

Now comes the question:
What's the best choice of initial condition which ensure the prevail of $n3$ in a relatively short time?

Phase-space Analysis

Trying to answer the question above, we did phase-space analysis. In our case, phase space is such a 3D rectangular coordinate space in which each points (x,y,z) represents a phase (n1,n2,n3). From this perspective, the time evolution of the whole system can be represented by a trajectory in phase space.

Let's take the debut results mentioned above as a concrete example.
Fig 9. shows its trajectory in phase space, where it begins at n1-n2 bottom plane, goes all the way up, and finally ends at n3 axes. It means that initially n3 equals 0, but prevails soon after, and finally dominates the whole system and eliminates other components.

We can do more of this. Fig. 10 is a collection of trajectories, from which, one can easily tell that despite where a trajectory started, it always ended to a single point.

To further evaluate this phenomenon, we singled out the only stable steady state in phase space shown in Fig. 11, which was unsurprisingly consistent with the common ending point in Fig. 10.

Now we have known a lot about this system： any arbitrary initial condition in the region of interest will lead to that single end, where n3 is the only kind to survive and thrive.

However, this is still not enough. Practically, we don't satisfy only with n3 to survive and thrive. We hope it thrives in a short time (24h). So we evaluated the ratio of $n_3$ at t=24h when varying initial condition and plotted the colorful map in Fig. 12.

The map shows n1-n2 plane, in which each point represents an initial condition. The color stands for the ratio of $n_3$ at t=24h. The redder, the higher. The dark ribbon marks the line of 95%~96%. With this map, every time experimenters need to do a conjugation experiment, they can conveniently know how many E. Coli & C.C. they, should add into the system by looking at the map and doing simply calculation.

Qualitatively speaking, the map illustrates that experimenters should put way more C.C. into the system than E. Coli (>2:1) initially, as the map is diagonally unsymmetrical. Besides, the initial concentrations should be relatively high if one expects to complete this part of the experiment in a short time. Practically, the point at the turning point of the black ribbon represents an ideal initial condition, for it has high potential of n3 yield, but requires relatively low initial concentration.

Conclusion

Founded by reasonable and practical assumptions, upheld by concrete experimental data, this model simulated the concentration change in the process of conjugation, taking birth, death, competition and conjugation all into consideration. In this light, we deepened our understanding of conjugation from a new quantitative perspective and went boldly beyond to make practical predictions.

• This modeling work is chiefly done by Fangming Xie, assisted by Hongda Jiang and Zui Tao.
• The experimental data is offered by Juntao Yu.
• This article is written by Fangming Xie.