## The Mathematical Model

In a coculture where the organisms are dependent on each other for survival, a mathematical model can be useful to predict the stability of the system under different conditions. To that end, we worked on developing a model to aid us in our work of creating a stable coculture system. The mathematical model described here is based on a paper by Kerner et al. [1].

### A Substrate-Constrained System

Our model relies on the assumption that the system is substrate-constrained. For the production organism, this is quite realistic, as the only substrate available will be the acetate produced by *Synechocystis*. In the case of our cyanobacterium, the limiting substrate in the model is the amino acid produced by the production organism (arginine for *E. coli* and *B. subtilis*, and glutamine for *S. cerevisiae* and *Y. lipolytica*), for which *Synechocystis* is auxotrophic. While this amino acid is required by the cyanobacterium, it is not guaranteed to always be rate limiting. For instance, the amount of light could be the limiting factor under certain conditions, especially as the OD increases with the growth of the cells, restricting the penetration of light through the medium. Another conceivable limitation is the CO_{2} used by *Synechocystis* during photosynthesis, but that should be relatively simple to maintain abundant levels of. For this model we have assumed that we will be able to maintain conditions where the amino acid will be the growth-limiting factor.

### Mathematical Relations

For a substrate-constrained system governed by Monod kinetics, growth can be described with the following equations [1]. Just like in the article, we have not taken cell death into account in this first simple system.

The first two rows of equations describe regular substrate-constrained Monod-governed growth, where µ is the growth rate [1/h], n is the cell density [g DW/L], k is the maximum growth rate [1/h], K is the substrate affinity [mol/L], and c is the concentration of cross-fed metabolite [mol/L]. In the last equation, we look at the change in the extracellular concentration of the respective limiting nutrients. Here, α represents the export rate of a nutrient [mol/(g DW*h)] in the organism that produces it. β is the required amount of the same nutrient for growth, for the organism that is limited by its availability [mol/g DW].

## MATLAB Simulation

We used MATLAB to solve the differential equations above for two organisms simultaneously, with a simple ordinary differential equation (ODE) solver. While the coding itself did not present any problems, finding the correct values for all parameters turned out to be trickier. For that reason, we have only looked at parameters for *Synechocystis* and *Y. lipolytica*, the latter being the production organism to have shown the most promise in the lab.

### Parameter Estimation

Making a parameter estimation for the maximum growth rate was not very difficult for either organism, as we had data from our growth trials. The remaining parameters proved more difficult, since there are of course no papers published on the effects of changes we have made ourselves during the project. The ideal solution would be to perform experiments ourselves to measure export rates and growth requirements of the essential nutrients, as well as substrate affinities. Lacking enough time in the lab for this, however, we have elected to make some rough estimations of what these parameters might be, in order to be able to show the model at work. See table 1 for a list of all parameter values.

**Table 1.**Parameter estimations for

*Synechocystis*and

*Y. lipolytica*

Parameter | Value | Source |
---|---|---|

k_{syn} | 0.012 h^{-1} | Growth trial |

K_{syn} | 5*10^{-3} M | [5] |

α_{syn} | 4.2*10^{-6} mol/(g DW*h) | [3] |

β_{syn} | 1.3*10^{-4} mol/(g DW) | |

k_{yar} | 0.15 h^{-1} | Growth trial |

K_{yar} | 5*10^{-3} M | [5] |

α_{yar} | 9*10^{-4} mol/(g DW*h) | [4] |

β_{syn} | 0.12 mol/(g DW) | [2] |

For substrate requirement, we found the biomass yield on acetate for *Y. lipolytica* [2] and inverted it. Glutamine requirement for *Synechocystis* proved harder to find a source for, so we did some back-of-the-envelope calculations based on the approximate percentage of cell dry weight that is proteins, and the percentage of amino acid residues in proteins that is glutamine. Naturally, this provides only a rough estimate.

As we got our *Synechocystis* strain from Josefine Anfelt in Paul Hudson’s group at KTH, information on its acetate export rate was readily available in one of their articles [3]. The closest thing we could find on arginine export rates in *Y. lipolytica* was the same information for *C. glutamicum* [4]. Obviously, this is not optimal.

Finally, substrate affinity also proved difficult to find information on. Since no measurements seem to be available for either of the organisms and their limiting substrates, we decided to use the Michaelis-Menten constants for uptake of the relevant substrates as approximations. For substrate affinity to be equal to a Michaelis-Menten constant, we have to assume a very rapid equilibrium. This is most likely not quite true, but it will suffice for this approximation. We could only find the desired constant for *Synechocystis* [5], so we used that value for *Y. lipolytica* as well. Naturally, this might not be a good approximation, but as we need measurements of our own to get the model to be predictive anyway, this just serves as a first step to see the model in action.

### Collaboration

At this stage we collaborated with the Manchester iGEM team in order to get help with parameter estimations. They helped us with feedback, as well as with a way to make more accurate estimations based on values from other organisms using a weighting system. Unfortunately, we ended up not having time to incorporate this last part in the project, but we’d like to direct a big “thank you” their way for all their help!

## Model Results

With this basic model we can study the ratio between *Synechocystis* and *Y. lipolytica* that stabilises over time, as can be seen in figure 1. In our case, the final ratio between the organisms is 350:1, even though they started at 1:1. Since we have not had a chance to measure the necessary parameters ourselves, this value should not be trusted. However, it does show how a reaching a stable, albeit different ratio could look.

Even if the ratio our model returned is not to be taken too seriously, it is not unrealistic that the cyanobacterium should outnumber the production organism, given the premises of the model. The production organism needs quite a lot of acetate to grow, and its only source is through the production of *Synechocystis*. The amount of acetate exported by the cyanobacterium is not very high. Therefore, after a predictive model is reached, the problem of getting *Synechocystis* to produce enough acetate for a production organism in co-culture will almost certainly remain a difficult challenge, and one that has to be surmounted when going forward with this project.

## References

- [1] Kerner A, Park J, Williams A, Lin XN. A programmable Escherichia coli consortium via tunable symbiosis. PloS one. 2012;7(3):e34032.
- [2] Daran-Lapujade P, Jansen MLA, Daran J, van Gulik W, de Winde JH, Pronk JT. Role of transcriptional regulation in controlling fluxes in central carbon metabolism of Saccharomyces cerevisiae. A chemostat culture study. The Journal of biological chemistry. 2004;279(10):9125-38.
- [3] Anfelt J, Kaczmarzyk D, Shabestary K, Renberg B, Rockberg J, Nielsen JB, et al. Genetic and nutrient modulation of acetyl-CoA levels in Synechocystis for n-butanol production. . 2015.
- [4] Lee SY, Kim HU. Systems strategies for developing industrial microbial strains. Nature biotechnology. 2015;33(10):1061.
- [5] Labarre J, Thuriaux P, Chauvat F. Genetic analysis of amino acid transport in the facultatively heterotrophic cyanobacterium Synechocystis sp. strain 6803. Journal of Bacteriology. 1987;169(10):4668-73.