Fit parameters and evaluate the performance of our system

Goal

Once the initial guidelines for the genetic design given in a first step, we need to measure on our obtained biological system whether it behaves as intended. Therefore, we will submit it to carefully designed experiments, in order to fit our model on the obtained data and retrieve crucial parameters. The tricky part here is that in a test tube, our system will behave differently than in a tumor, as the repartition of bacteria is not the same and diffusion of AHL does not take place in the lab experiments. We therefore have to extend our model to take this discrepancy into account.

In vitro model extension

Compared to the real-life situation happening in the tumor, during our experiments bacteria are in a very different state: they are in bulk in the liquid culture and not in an open solid medium like they would be around the tumor. Therefore, the diffusion of AHL in the test tube does not take place, and some of our equations used before should be adapted to describe the behavior of our circuit when bacteria are in the situation of our lab experiments: only AHL production and degradation happen in the test tubes.

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= P_{\text{AHL}} - k_{\text{deg}} [\text{AHL}]\end{aligned}\]

Therefore, once reached the steady state, and thanks to the still valid equation describing AHL production\begin{aligned}\[P_{\text{AHL}} &= d_{\text{cell}} a_{\text{AHL}} [\text{luxI}]\end{aligned} the level of AHL is now:

\[\begin{aligned} \text{[AHL]} &= \frac{d_{\text{cell}} a_{\text{AHL}} [\text{luxI}]} {k_{\text{deg}}}\end{aligned}\]

The rest of the model remain unchanged, as it concerns intracellular phenomena that are assumed to happen the same way in vitro than in vivo.

Fitting of LuxR expression

Strategie

LuxR is the regulatory protein binding AHL and subsequently activating a promoter where LuxR-AHL tetramers can bind:

The amount of LuxR present in the cell is crucial to know how much AHL is necessary to fully activate the promoter. And inversely, knowing the AHL dose response curve of the very simple system constituted of the constitutively luxR gene and a fluorescent reporter under the Lux promoter is enough to deduce the amount of LuxR in the bacteria, and therefore their expression level. This is why we have designed and performed an experiment to precisely measure a dose response curve of such a system to be able to fit our luxR expression parameter

Fitting method

We used the the following process to fit our model to experimental data:

From an initial set of parameters for our model, we simulate the response of the system that we should obtain
We quantify the difference (via a cost function) between the model-based response and the actual experimental data
We change the set of parameters in a way that is expected to reduce the cost (with gradient-based method) the and go back to step 1. and iterate

Thanks to the MEIGO toolbox already used for our parameter search, we do not have to implement de novo

References

^ Vander Heiden, Matthew G., Lewis C. Cantley, and Craig B. Thompson. “Understanding the Warburg Effect: The Metabolic Requirements of Cell Proliferation.” Science (New York, N.Y.) 324.5930 (2009): 1029–1033. PMC. Web. 18 Oct. 2017.