Our goals were 1) To test if our genetic design works as planned; 2) To get preliminary ideas on the main parameters of the two-compartment reactor to be built for the lab experiments. In vivo tests would be clearly too costly and time consuming for this purpose.

Principle we designed a simple model in which QS bacteria have two states. In the i) ground state, glucose intake and signal production is at a low level. If the signal concentration reaches a threshold, the cells enter into an ii) active state characterized by higher glucose intake, higher signal production and the production of glucose from cellobiose. (Figure 1).

The cells are in a perfectly mixed, closed environment. At the beginning of the simulation, a given number of ground state cells are placed into the medium, and the simulation proceeds in discrete time steps. At every step, the cells take up nutrients and carry out the signal and/or glucose production depending on their state of activation. When the cells accumulate a certain amount of energy ("glucose equivalents"), they divide. This program is repeated at every time step. As a result, the cells produce a growth curve quite similar to that seen in liquid cultures.

Figure 1. The two states of the bacterial model.

This is a very simple model, in which the cells do not move and the signal is not diffusing within the compartment, since the medium is perfectly mixed. At each time step we can record the number of cells, the concentration of the solutes, etc. This is a so-called agent-based model since each cell-agent executes its own program that depends on its state of activation and stored energy.

Two-compartment reactor. In this setup, two different kinds of cells are put into two equal, perfectly mixed compartments that are separated by a large pore-size semi-permeable membrane. The cells themselves can’t pass through the membrane, but the dissolved materials can freely move between the compartments. As compared to the previous one-compartment model, the only difference is that the solutes need to equilibrate between the two compartments at every time step. If we assume perfect equilibration between the compartments, the concentration of a solute would be the average of those measured in the two compartments at each time step. Here we employ a simple trick, we introduce a virtual diffusion coefficient that regulates the exchange between the two compartments.
The diffusion coefficient D was defined in such a manner that its values be between zero and 1.0. If it is zero, the two compartments do not communicate. It if is 1.0, the compartments are fully equilibrated, i.e. the equilibrated concentration will be the average of the concentrations within the two compartments at every time-step.

The model was implemented using an existing Matlab code written by S. Netotea and A. Kerenyi for modeling the swarming of quorum sensing bacteria on agar plates (Netotea et al, 2009, Venturi et al, 2010) [1][2], kindly provided to us by the authors. This code had to be slightly simplified, as described in Appendix 1.

We carried out simulation runs at a large number of parameter settings, changing the number of starting population, the ratio of the populations, the starting concentrations of the various solutes, etc. We evaluated the results in a qualitative way, i.e. growth vs. no growth, slower or faster growth. As a comparison, we used a wt model that responds to its own signal.

Figure 2a. Growth of wild type cells that produce a signal and respond to it by producing glucose from cellobiose.

Figure 2b. Carbon source transformation - from cellobiose to glucose - and consumption.

Then we extended the experiments to the species designed to depend on each other. The main finding were as follows:
-   Mutually dependent bacterial cells that respond only to the signal of the other species, can grow within the same compartment, same as the wild type. The ratio of the species is 50:50 %. This is a stable equilibrium that can be reached even if one of the populations starts with a single cell.

Figure 3. When the two bacterial populations are in the same compartment (c = 1) their growth is synchronized.

-   When put into separate but communicating compartments, mutually dependent cell populations will grow, but the speed of growth will depend on the intensity of material exchange between the compartments. If there is no exchange, mutually dependent cells will not grow.

Figure 4a. Growth of the two bacterial populations in separated compartments. c = mixing coefficient.

Figure 4b. In this last condition (Figure 3) we can observe the usual pattern in the carbon source transformation and consumption. The concentration of the signals molecules (OC8 and OC 12 AHL) follow the cells growth pattern for both species.

-   If we put a ternary community into the two compartments, in such a way that lactamase producing partner is in one compartment, and the mutually dependent bacteria are in the other compartment, the system will start only if we jump start it by adding glucose and signal.

Figure 5a. Cell growth of the ternary system without and with the "jump start".

Figure 5b. The bacterial signaling also switch-on the production of beta-lactamase by HeLa cells that allows the bacterial growth.

We used a highly simplified qualitative model in order to have preliminary insights into the logics of our system, in hope of obtaining indications regarding the physical setup of the reaction chamber to be used in the lab experiments. The model shows that the system is in princple "viable", howeer the growth can be limited by the material exchange between the compartments. Also, the system may need to be jump-started by adding a certain amount of glucose and signal. Naturally, these predictions are qualitative and have to be checked by experiment.

References

[1] Netotea S, Bertani I, Steindler L, Kerényi A, Venturi V, Pongor S. (2009) .A simple model for the early events of quorum sensing in Pseudomonas aeruginosa: modeling bacterial swarming as the movement of an "activation zone". Biology Direct, 4:6 . PDF

[2] Venturi V., Bertani I., Kerényi Á., Netotea S. and Pongor S. (2010) Co-Swarming and Local Collapse: Quorum Sensing Conveys Resilience to Bacterial Communities by Localizing Cheater Mutants in Pseudomonas aeruginosa, PlosOne, 5,4, PDF

Appendix 1.

The QS modeling program of S. Netotea and A. Kerenyi (Netotea et al, 2009, Venturi et al, 2010) is designed to model the swarming of QS bacteria. In this modell i) bacteria are modeled as individuals freely moving on a 2D plane, i.e. in an open environment; ii) bacteria have an individual program that activates them depending on the threshold concentration of a) signals, b) public goods, according to the known rules of QS; and iii) nutrients and solutes freely diffuse on a 2D plane which is discretized into square zones (Netotea et al, 2009). This model was instrumental in showing that QS regulation is sufficient for a model population to show density-dependent activation, tracking of external signals, co-swarming of species and community collapse.

Our system is simplified, since due to the perfect mixing within a reactor compartment, a) cells do not move by themselves; b) solutes do not diffuse (concentration is uniform throughout the compartment). So, in order to model growth within a closed compartment, the diffusion and the movement part of the original program had to be simply switched off.

In order to simulate growth in a two-compartment system, one can model the growth of bacteria in such a way, that solutes accumulated separately, i.e. the same program is executed for the two separate components. If we now want to model the passage of solutes between the compartments, we can use a diffusion-like concentration equilibration at each time step. This setup corresponds to two compartments separated by a semi-permeable membrane. The concentration of a solute can be calculated as follows:

c1(t), equilibrated = c1(t) – 0.5 * D* [(c1(t) - c2(t)]

where c1 and c2 are the concentrations in compartment 1 and 2, respectively, and D is the virtual diffusion coefficient, with a value between zero and 1.0. It is easy to see, that at D=0, the concentration remains the same, at m=1, the concentration will be the average of c1 and c2.

Based on the above design, the Matlab code was modified by A. Kerenyi and put to our disposal for modeling as a *.exe file that runs under Windows. The results are cell counts, signal and food concentrations as a function of the time steps These results were visualized by Excel.