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Fitting of Our Own Hybrid Promoters

Goal

Fit the behavior of our self-designed hybrid promoters to be able to chose the best one.

Fitting 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.

Need for a better hybrid promoter activation model

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 hybrid promoter: the leakiness towards lactate was far more pronounced than the leakiness towards LuxR-AHL.

This is why we changed our model of the hybrid promoter activation to take into account two separated leakinesses and fit better to reality:

\[\frac{\mathrm{d} [\text{luxI}]}{\mathrm{d} t} = a_{\text{LuxI}} \times P_{\text{Lux-Lac}} - d_{\text{LuxI}} [\text{luxI}]\]

where

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

Fit of the hybrid promoters responses

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:

Symbol	Description	Value	Reference
K_LuxR-AHL	LuxR-AHL quadrimer binding constant	5x10^-10 nM^-3	[2]
d_LuxR	LuxR degradation rate	0.023 min^-1	[3]

We will let the value of a_LuxR (expression level of LuxR) vary between the two bounds that our first characterization has issued. Contrary to previous fits, we don't fix K_luxR, the binding constant (or half-activation concentration) of LuxR-AHL to the hybrid promoter. Indeed, with the addition of other sequences around the operon to add the lactate sensitivity to the promoter, we cannot guaranty that the binding of the LuxR-AHL complex to the Lux operon will happen with the same affinity.

From the experimental data of the hybrid promoter response, we could fit four parameters of our model: both leakinesses of our hybrid promoters in regard to LuxR-AHL and lactate, and both half-activation concentrations of LuxR-AHL and lactate (expect for one hybrid promoter for the latter):

LuxR fit — Figure 1. **Parameter space of the fit of the activation of our hybrid promoter A**. Parameter space fitting the experimental data. Each point represent a parameter vector that significantly fit the experimental data. The blue points fit the data the best (least sum of square) while the yellow ones represent parameters combinations that barely fit the data (but still significant according to the chi2 test of goodness of fit). Fitted parameters are annotated in red.

From our statistically significant fit and under the assumptions made, we were able to determine the following parameters for out two hybrid promoters:

Constant	Description	Best fit for hybrid promoter A	Best fit for hybrid promoter B
k_Lux	Leakiness of the hybrid promoter in regard to LuxR-AHL	6.6 %	1.5 %
k_Lac	Leakiness of the hybrid promoter in regard to lactate	24 %	74%
K_LuxR	Half-activation LuxR-AHL concentration of the hybrid promoter	440 nM	400 nM
K_Lac	Half-activation LuxR-AHL concentration of the hybrid promoter	5.5 mM	3.1 mM (NB: Not significantly determined but still taken into account for further simulations)

These fitted values, along with others previously obtained were used in the final simulation of the response of our system to check the performance of our system including both hybrid promoter versions, and chose the one performing the best in the context of our whole system.