Team:ETH Zurich/Modeling/Experiments Model

"What I cannot create I do not understand."
- Richard Feynmann

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, after finding that literature on the system is incomplete. During the course of this project, we first made a model based on an assumption on its functioning. We evaluated it against the results, and found that the model could not explain the data. We then studied the results and established an improved model.
2. Characterize the influence of our AHL-degrading enzyme AiiA construct.

Understanding the biological mechanism of LldR

Lab rat: Here are your parameters:

Experiment 1): Characterization of lldR sytem 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 KM 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)
K822000unknown (natural) 10.35 955
K1847008 162 15.26 1075
K1847009 1429 1.56 977.5
K1847007 25471.34 697.7

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 KM values do not vary a lot depending on the construct.

Nerd: Therefore we have to update our model, let's include a activation term?

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 importer in LB

Figure 2: Experiments for the characterization of the LldR promoters

To the extent of our knowledge, no characterization of the lldPRD operon is available in 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: $$[Lact_{in}]=[Lact_{out}]$$. When LldP is overexpressed, we consider that the lactate concentration is higher inside the E. coli than in the extracellular space.

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
High LldP- lldR 664 8400
Low 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 KM values.

Low LldP-LldR

Promoter Promoter Strength ON/OFF ratio KM (μM)
K822000unknown (natural) 1.16 720.2
K1847008 1621.42 337.7
K1847009 1429 0.96 459.8
K1847007 2547 1.29 1337

High LldP-LldR

Promoter Promoter Strength ON/OFF ratio KM (μM)
K822000unknown (natural) 8.04 1930
K1847008 162 23.96 1751
K1847009 1429 24.34 2361
K1847007 2547 3.85 1977

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 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.

Parameter value
KM 173 μM
ON/OFF 7
n 1.7

Figure 3: Characterization of the low engineered promoter

Geometry of the chip and design of the microfluidic chip

Lab rat: I have an idea for that!

Single-cell analysis is a pre-requisite for our cancer detection system. Circulating tumor cells are really scarce in the blood compared to healthy cells. Implementing a microfluidic chip was then the ideal solution because it enables high-throughput single-cell analysis. However, having small reaction volumes was very challenging. Indeed, the natural quorum sensing system is meant to work on large length scales, as proven by the last year’s ETHZ IGEM team. The signalling molecule AHL diffuses then almost instantaneously in the chip. Biologically, we thought first that including AiiA degradation in the design or tightening the leakiness of the PLux promoter would solve the problem. It turned out that implementing a fold-change sensor solved the problem, since a pulse of LuxR will limit over-activation of the system.

Characterization of the effect of AiiA

Nerd: Hm, we need more date to estimate that...

To characterize the effect of AiiA on our system, we ran two experiments. In one of them, 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. As AiiA degrades AHL, this phenomenon is expected. In addition, since AiiA is on a medium copy plasmid and under the control of a strong constitutive promoter, the large effect seen in our experiments is expected.

Nerd: Ok, that should help us decide.

From the parameters, we know AiiA acts with Michaelis Menten Kinetics. The KM 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.

 KM 9.983 nM
 KM 106 nM

Figure 4: Experimental and fitted curves for the plasmid: A) Without AiiA, B) With expressed AiiA.

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*}

Can AiiA be used in our system?

As already mentioned, AiiA shifts the active curve until lower sensitivities. This means that higher concentrations of AHL will be required to activate the construct. In our case, the leakiness is necessary to initiate the signal. The amount of AiiA present here, will prevent any form of activation. We will therefore, not use AiiA in our final design.