Experiments Meet Modeling

Thanks to the excellent cooperation between our wetlab biologists and our modellers, we successfully characterized our lactate promoter library, as well as the effect of the AHL-degrading enzyme AiiA.

Goals

1. investigate the biological mechanism of LldR. That is why during the course of this project, we first made an assumption on its functioning. Then, we evaluated our model against the results. We then established a new model.
2. Characterize the influence of our AiiA construct

Characterization of the LldR promoters and Biophysical Model

To see, how this model was derived in the first place, click here.

A closer look at the mechanism of LldR

After looking at puzzling results, we realized that our first model of the mechanism of action of LldR was not realistic. Searching literature, we finally found a compatible explanation, depicted in Figure 1. In the paper by Aguilera et al. (2008), they suggest that LldR may be required for the transcription machinery. Hence, instead of having only repression by LldR, LldR might play a dual role as a repressor and an activator. It suggests that when lactate is present, it destabilizes the DNA loop and induces a conformational change of LldR.This results in the transcription of the gene of interest (goi). This mechanism is consistent with our results. In the following, we will describe the mathematical equations corresponding to this mechanism.

Reactions

\begin{align} \label{eq:1} \varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LldR}}} \text{LldR}\\ 2 \cdot \text{LldR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}^{K_{\mathrm{d,1}}} \text{LldR}_2\\ 2 \cdot \text{Lact}+\text{LldR}_2 &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}^{K_{\mathrm{d,2}}}2 \cdot \text{LactLldR} \hspace{2em}\\ \text{LldR} &\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{lldR}}} \varnothing\\ \text{LldR}_2 &\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{lldR_2}}} \varnothing\\ \text{LactLldR} &\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LactLldR}}} \varnothing\\ \end{align}

Production of LldR. Dimerization of LldR in solution. Unbinding of the dimer of LldR with lactate. Degradation of all the species.

Mathematical model

According to the previous description, the gene of interest is activated by LactLldR and repressed by LldR dimer. Hence, if gfp is the gene of interest, we get the following ODE:

\begin{align*} \frac{d[GFP]}{dt}&=\frac{a_{GFP}}{1+\left(\frac{[LldR_2]}{K_R}\right)^{n_r}}\cdot \frac{\left(\frac{[LactLldR]}{K_A}\right)^{n_a}}{1+\left(\frac{[LactLldR]}{K_A}\right)^{n_a}}-d_\mathrm{GFP}[GFP] \end{align*}

Assuming mass conservation for LldR, the total amount of LldR is given by:

\begin{align*} \text{LldR}_{tot}&=[LldR]+[LactLldR]+2 \cdot [LldR_2] \end{align*}

Simplification

This equation is unidentifiable and we simplified the system by approximating the transcription of gfp by an Hill activation function:

\begin{align*} \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP} \cdot \left(\frac{[Lact]}{K_\mathrm{A,Lact}}\right)^{n}}{1+\left(\frac{[Lact]}{K_\mathrm{A,Lact}}\right)^{n}}-d_\mathrm{GFP}[GFP]\\ \end{align*}

Characterization of the lactate responsive promoter in LB

We characterized the LldR promoter in different media. While all promoters has been tested in LB medium, only the best synthetic promoter has been tested in different mammalian media.

To the extent of our knowledge, no characterization of the lldPRD operon is available in the literature, nor in the iGEM registry. We characterized our synthetic promoters and the natural promoter in two lactate titration experiments: one with overexpressed LldP, and one with the natural expression of LldP. For a further description of this experiment, click here. However, we encountered several problems. In this setup, the lldPRD operon is not knocked out in our E. coli strains. Thus, we don't know how much LldP is expressed by the E. coli . We only know that the natural promoter is weak. Therefore, we assumed that when LldP is not overexpressed, there is no active transport of lactate by the permease. As the lactate molecule is small, we consider lactate passes through the E. coli only thanks to diffusion mechanisms, and then: \([Lact_{in}]=[Lact_{out}]\).

To see the details about the characterization, click here.

Experiment 1): No overexpressed LldP in LB

Equations

Because, we consider that only diffusion is happening, then we have

\begin{align*} \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP} \cdot \left(\frac{[Lact_{in}]}{K_\mathrm{A,Lact}}\right)^{n_1}}{1+\left(\frac{[Lact_{in}]}{K_\mathrm{A,Lact}}\right)^{n_1}}-d_\mathrm{GFP}[GFP]\\ \end{align*}

Parameter Fitting

We fitted our model using the Least Absolute Residual method, using the fitting toolbox of Matlab. We designed different constructs of the LldR responsive promoters.We were thus able to compare all the promoters thanks to their ON/OFF ratio and K_M values. For a more detailed description of the experiment and the characterization, go to the registry, by clicking on the different links provided below.

Promoter	Promoter Strength	ON/OFF ratio	KM (μM)
K822000	unknown (natural)	10.35	955
K1847008	162	15.26	1075
K1847009	1429	1.56	977.5
K1847007	2547	1.34	697.7

We can observe that our construct K1847008 has the best ON/OFF compared to the other synthetic promoters which are very leaky. Also, during all these experiments, the levels of LldR inside the cell were kept constant. The two binding sites of LldR were also conserved (same distance to the promoter and same sequences). It is therefore nice to see that the K_M values do not vary a lot depending on the construct.

Experiment 2): Overexpressed LldP in LB

We then designed other constructs including the symporter LldP. We expected an increase in lactate import such that the E. coli cells became more sensitive. However, the designed promoters show completely different LldR levels compared to the previous experiments. That is why we can not extract the parameters for LldP symporter using both experiments, because the data sets are not comparable. We will therefore use the same fitting function as before. Below, the promoter levels computed with a RBS calculator and the registry are indicated.

Construct	Expression of LldR(A. U.)	Expression of LldP(A. U.)
LldR	51100
Low LldP- lldR	664	8400
High LldP- lldR	12	23.4

Parameter Fitting

We then fitted our model as explained before,and we obtained the following values for ON/OFF ratio and K_M values.

Observations

It is difficult to make a correct explanation here, since, both the levels of LldP and lldR change. However, in the first construct, we can clearly see that the leakiness is increased for small amounts of LldP/LldR. Consistent with our model, it is probably due to the insufficient levels of LldR. Since LldR is thought to repress the transcription, this could explain the leakiness.

Characterization of the lactate responsive promoter in mammalian medium

Below is displayed the response of one synthetic promoter in < a href= "https://2015.igem.org/Team:ETH_Zurich/Materials">RPMI- serum free medium. The promoter harboring the permease and LldR displays a much higher sensitivity than the one only harboring LldR. In the following, these parameters are going to be used to simulate the fold-change sensor.

Characterization of the effect of AiiA

To characterize this effect, we ran two experiments. In one of the experiments, AiiA was constituvely produced and in the other one, AiiA was not present. Below we compare both responses. The first visible thing is that AiiA drastically shifts the curve until lower sensitivities.

From the parameters, we know AiiA acts with Michaelis Menten Kinetics. The K_M value and turnover number are already known from literature, see parameters. However, for our model, we need to know how much AiiA is produced by the cell in order to characterize the behaviour of the system.

Dose response curves and apparent K _M values

Here we compared the fitted curves of AiiA and the K values for both situations. AiiA drastically shifts the curve towards less sensitivities. We calculated the apparent half substrate maximum concentration for both cases.

Concentration of AiiA inside the E. coli

Below we fitted two ordinary differential equations thanks to MEIGO toolbox. The equations are below.

Equations

Here, AHL is the input. There is no amplification by LuxI.

\begin{align*} \frac{d[AHL]}{dt}&=- \frac{v_{AiiA} \cdot [AHL]}{K_{M,AiiA}+[AHL]} \\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP} \cdot \left(\frac{[AHL]}{K_\mathrm{Lux}}\right)^{n_1}}{1+\left(\frac{[AHL]}{K_\mathrm{Lux}}\right)^{n_1}}-d_\mathrm{GFP}[GFP]\\ \end{align*}

Parameter fitting

Due to the practical identifiability problem, we could not estimate a value for K. We then re-used the value from the literature, in the millimolar range, which is consistent with the rest of the equations.

We find:

\begin{align*} v_{AiiA}= 9.46 \cdot 10^4 nM min^{-1} \\ \end{align*}