Fit of the hybrid promoter

What experiment do we need to fit the activation behavior of the hybrid promoters?

The most simple way to check the response of an inducible promoter is to measure its activity (via a reporter gene like gfp) under different inducer concentrations. In our case, we are dealing with a hybrid promoter that should respond to two different inducers. Therefore, decided for an experiment covering the 2D space of the two inputs: lactate and AHL. However, we must not forget that AHL alone cannot induce the promoter, and needs to bind to the LuxR protein for that. This is why we also included the luxR gene, whose expression has been previously characterized, in our strain for this experiment.

First attempt of fit

We first tried to fit the parameters of the hybrid promoters relying on following model we had used until then:

\[\frac{\mathrm{d} [\text{luxI}]}{\mathrm{d} t} = a_{\text{LuxI}} (k_{\text{LuxI}} + (1 - k_{\text{LuxI}}) P_{\text{Lux-Lac}}) - d_{\text{LuxI}} [\text{luxI}]\]

where

\[\begin{aligned} P_{\text{Lux-Lac}} &= \frac{\left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR}}} \right)^{n_{\text{LuxR}}}}{1 + \left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR}}} \right)^{n_{\text{LuxR}}}} \times \frac{\left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}}{1 + \left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}} \\ \end{aligned}\]

However, given the experimental data we wanted to fit, it had become clear that this model had to be adapted. Indeed, it only takes into account a "global leakiness" k_luxI, which means that the promoter is considered to have the same leakiness in regard to LuxR-AHL and lactate. This is not at all what we observed on the activation pattern of the AND-gate

Assumptions

Parameters

We assumed fixed the following parameters of our model, which are considered known well enough from previous characterizations, including our own characterization of luxR expression:

Constant	Description	Value	Reference
K_LuxRAHL	LuxR-AHL quadrimer binding constant	5.10-10 nM-3	[2]
d_luxR	LuxR degradation rate	0.023 min-1	[3]
K_luxR	Half-activation LuxR-AHL concentration of the hybrid promoter	5 nM	iGEM ETH 2013
a_AHL	AHL synthesis rate by LuxI	0.01 min-1	[2]
k_deg	AHL degradation rate	5.10-4 min-1	[4]
d_luxI	LuxI degradation rate	0.017 min-1	[4]

We will let the value of a_luxR (expression level of luxR) vary between the two bounds that our first characterization has issued. We don't give any bound for k_luxI, the leakiness, as we should be able to fit it again independently.

Disclaimer: Strong assumptions about the interpretation of the experimental data

We were not able to find any significant fit on the raw experimental data. To be able to still get information out of them, we had to consider that the maximal activation of our system was reached by the most dense and most fluorescent bacteria (which grew in high-glucose-content cultures), even though a clear plateau cannot be directly observed on our data (see Figure 3 below). This rather liberal interpretation of the data was motivated by the hypothesis that fully activated bacteria were experiencing a high metabolic burden preventing them from reaching the OD they should have reached without this burden.

In practice, what we did to get exploitable data for our fit is the following: we fitted first a linear model on the first data points (unaffected by any burden) of the A600 vs. glucose to get the OD that each culture should have reached without burden. We used this adjusted OD value in the (therefore adjusted) data to fit our model. Here is how the adjusted look like:

FIGURE HERE

From the experimental data of the quorum sensing end-point experiment, we could fit two parameters of our model: the expression level of luxI, and once again the leakiness of our promoter (both in log scale):

NEEDS A BIG HIGHLIGHT THERE

From our statistically significant fit and under the assumptions made, we can gather with a 95% confidence that:

1. The expression level of luxI in our system is comprised between 1.0 and 6.8 µM.min^-1, with its most probable value being 3.2 µM.min^-1
2. The leakiness of the P_Lux promoter in our system is comprised between 2.0 and 6.2 %, with its most probable value being 3.7 %

In addition, we can verify that the leakiness of the P_Lux promoter was successfully fitted two times with excellent consistency on two completely different experiments, supporting the validity of our approach and the assumptions we made.