Lactate Module

Introduction and Goals

Our idea was to distinguish cancer and normal cells based on their different lactate production rates. We assumed that during the measurement time frame, the lactate production does not reach steady state and we have to measure the relative concentration of lactate rather than the absolute concentration. In other words, our goal was to detect the fold-change in lactate concentration over time. Therefore, we designed a sensor with the topology of a fold-change sensor based on the simple detection system of lactate in E. coli.

Description of the design

The network depicted here is equivalent to an incoherent feed forward loop. Lactate activates production of LacI and GFP.

In the absence of lactate, the regulatory protein LldR binds to the promoter and represses transcription of LacI and represses transcription of GFP. When Lactate is present, lactate binds to LldR and both transcription of LacI and GFP are initiated. When LacI reaches a certain threshold, LacI represses transcription of GFP after a certain delay. Both LacI and LldR should be absent in order to have transcription of GFP. Under certain parameters, the system behaves as a fold-change sensor. In our system, if repression of LacI is less strong it will repress transcription of GFP for low levels of lactate and allow GFP transcription for high levels of lactate.

In order to allow fine tuning of the levels of active LacI inside the cells, we decided to add IPTG to the model.

We characterized various promoters and included them in the simulations for the fold-change sensor.

Comparison of lactate sensor models

The lactate production behavior of the cancer and normal cells determines which type of lactate sensor is appropriate (fold-change sensor or simple lactate detection of E. coli). This is why we first investigated the response of the lactate fold-change sensor to various lactate inputs. The two main parameters concerning lactate are the difference in steady-states between normal and cancer cell and the time point when the steady-state is reached. From these observations we can derive two different situations (represented on the scheme).

1. If both normal and cancer cells have the same lactate concentration level at steady-state, the measurement has to take place before the steady-state is reached and we have to use a fold-change sensor, otherwise, we will obtain the same response for both cancer and normal cells.
2. If both normal and cancer cells have different lactate concentration levels at steady-state, we can use both a fold-change sensor and the nautural lactate detection system.

In the following, we describe the functioning of the fold-change sensor and discuss important parameters defining its behavior.

Obviously, the specifications on the output behavior strongly depends on the second module: the two signals has to be coherent to produce the desired output. To learn more about the AND gate, click here.

Defining parameters

During the design of our system, we evaluated several factors that could greatly influence the response of the system.

1. The half-maximal substrate concentration and the cooperativity of LldR.
2. The delay of LacI induction.
3. The amount of introduced IPTG.

We characterized our lactate sensor and we obtained a range of different K_M values. Depending on the lactate input concentration, the appropriate construct with the correct sensitivity has to be chosen. The cooperativity of LldR is a further important parameter: If the cooperativity of the lactate detection system is higher than LacI, then the lactate sensor amplifies the signal and does not behave like a fold-change sensor. However, in the characterization of our system we obtained a cooperativity equivalent to LacI's Hill coefficient. In this case, our lactate sensor behaves like a fold-change sensor.

We simulated the following system, to gain more information about how the system would behave under different conditions to identify the optimal sensor: a simple lactate detection system or the fold-change sensor.

CASE 1) Same steady state

As mentioned above, if the lactate production reaches steady state before we are able to perform the measurement, the simple lactate detection system would lead to the same output in both cases. This does not fulfill the specifications.

That is why we show here the response of the fold-change sensor when the input from cancer and normal cells reach the same steady state.

Assumptions and modeling

Protein transcription and protein translation are lumped into one reaction. Since this results in a shorter time delay in protein expression than in a real-life situation, we introduced in the following a delay in LacI transcription.

Simulation

Comparison of behavior when a LacI transcription delay is introduced

When a delay is introduced, we have a typical fold-change behavior: the height of the GFP production peak in the case of cancer cells is three times higher than in the case of normal cells. In this case, the simple lactate detection system would not work because the time difference in LacI transcription would not be sufficient to introduce a significant delay.

Possible biological implementation of a delay

If the present design does not introduce a sufficient delay in LacI transcription, one could modify the system by adding an intermediary protein. This protein would be under the control of lactate, and would induce LacI when present in the cell.

IPTG influence

In order to activate the AHL module, the height and also the broadness of the peak are important parameters. Indeed, if the area of the peak is not elevated enough, the system will never trigger, no matter the percentage of colocalization. As you can see, adding some IPTG broadens the peak of GFP.

Effect of LldR promoter sensitivity

Thanks to our nice promoter library, we can tweak the sensitivity to correspond to the range of lactate production considered.

CASE 2) Different steady states

Below, we study the behavior of the simple lactate detection system, compared to the behavior of the fold-change system.

Behavior of the fold-change sensor

We can see that the response of the fold-change sensor is not optimal when the steady states of lactate are separated. Indeed in the case of normal cells, then the concentration of LacI is not high enough to repress the GFP output, compared to the cancer cell case where the concentration of LacI is higher allowing repression of the output. The resulting consequence is a higher steady state GFP concentration in case of normal cell.

Behavior of the simple detection system

As expected the simple lactate detection system reproduces the input. In this case, the GFP outputs are well separated. To know if this response is more appropriate in the context of the full model, visit the combined compartment model.

Comparison between the simple detection system and the fold-change sensor

The simple detection systems shows a different dynamic behavior than the fold-change sensor. Depending on the behavior of the quorum sensing module, we will be able to derive proper conclusions. However, from the simulation results we can suggest that:

1. In case of separated steady states, it is not clear which system would be the more appropriate one, since in both cases the response for high and low lactate production is similar.
2. However, if the steady states of lactate are similar, the fold-change sensor leads to the best output, since for the other system, the GFP responses are almost identical.

Parameter Search and amplification

In our initial model , the difference in non linearity was driving an amplification. Here, we want to see under which parameters the system still performs amplification. But when we included the difference in non linearity we discovered that the Hill coefficient for LldR was inferior to our first hypothesis. We can nicely see on the following graphs that the maximal possible amplification is 15-fold for a weak absolute response of GFP. As already described earlier the objective of this system is to provide fold-change sensing.

Our system should be able to amplify the difference of production between cancer and normal cells. To see what are the conditions on the parameters for the system to amplify the ratio of GFP for cancer and normal cells, we calculated this ratio using the equations for the non-dimensionalized system. For this search we assumed the fold-change production of lactate between cancer and normal cells is 3 .

Equations of the non-dimensionalized system

Range of parameters chosen

Results of the parameter search

In the diagrams below, two parameters are plotted against each other. The left-over parameters in each graph are set to their optimal values. The optimal values were computed using constrained non-linear optimization.

The first figure represents the ratio of GFP output for cancer versus normal cells. The second figure represents the absolute values of GFP concentrations. Indeed, we want to have a ratio of at least 8 fold between the output for cancer and normal cells. But we also want to have high "absolute" values. Indeed, if the percentage of activation is not elevated enough, the quorum sensing module will never be activated. That is why we plotted both conditions.

As we can see on the graphs, the two parameter searches do not coincide. The areas with the best ratio do not coincide with a high output.

Early stage modeling

Overview

In the following we describe our initial model. Thanks to that model, we were able to make decisions concerning the design of our system. We also derived precise functional specifications for our system.

How did we derive the model?

In this system, the mechanism of action of LacI is well known, whereas the action of LldR and lactate is not. Therefore, we derived the model for the mechanism of LldR by analogy to similar metabolic pathway. The paper from [Aguilera 2008], indicates that members of FadR family, including LldR, are highly similar. For example, GntR binds to two operator sites to negatively regulate the transcription of the gntT gene. Total repression of gntT was suggested to be achieved by DNA looping through interaction between the two GntR molecules. From this, we assumed that :

• LldR exists as a dimer in solution.
• 2 molecules of lactate bind to one LldR dimer (L2).
• Lldr dimer bind to the two operator sites when no LldR is present.
• Lactate releases the binding of LldR dimer to the operators.

Chemical species

Name	Description
Lactout	Lactate produced by mammalian cells
Lactin	Lactate inside E. coli cells
L2	Dimer of LldR, regulatory protein of lld operon, acts as a repressor
DLL	Dimer formed between Lactate and LLdr dimer
LacI	Lac operon repressor, DNA-binding protein, acts as a protein
IPTG	Isopropyl β-D-1-thiogalactopyranoside, prevents LacI from repressing the gene of interest
IL	Dimer formed between LacI and IPTG
LldP	Lactate permease, active transporter

Reactions

\begin{align*} \text{Lact}_{\text{out}}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{M,LldP}},v_\mathrm{LldP}} \text{Lact}_{\text{in}}\\ 2 \cdot \text{Lact}_{\text{in}} + \text{L}_{2} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{DLL}}}^{k_{\mathrm{-DLL}}} \text{DLL}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\bot}^{\text{L}_2}} \text{LacI}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\bot}^{\text{L}_2}} \text{GFP}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\bot}^{\text{LacI}}} \text{GFP}\\ \text{IPTG} + \text{LacI} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{IL}}}^{k_{\mathrm{-IL}}} \text{IL}\\ \end{align*}

Equations

Assumptions

1. We used the quasi steady state approximation to model the fast dimerization of lactate to L₂ and of IPTG to LacI.
2. We assumed that the Hill coefficient for Lldr was equal to two, since two lactate molecules bind to one dimer of Lldr. The Hill coefficient for LacI was also set to two.

Equations

\begin{align*} [L_2] &=\frac{L_\mathrm{2tot}}{\frac{[Lact_\mathrm{in}]^2}{K_\mathrm{d,DLL}}+1}\\ \frac{d[LacI]}{dt}&=\frac{a_{\mathrm{LacI}}}{1+(\frac{[L_2]}{K_{\mathrm{R,L}}})^{n_1}}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP}}{1+(\frac{[L_2]}{K_{\mathrm{R,L}}})^{n_1}}*\frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}})^{n_2}}-d_{\mathrm{GFP}}[GFP]\\ \end{align*}

Non dimensionalized equations

For the initial model, we chose to model the input of lactate as a step input. We non-dimensionalized the system in order to simplify the system.

\begin{align*} l_0 &=\frac{\gamma_1}{F_c^2 \cdot \alpha^2 \cdot B+1}\\ \frac{dl_1}{d\tau}&=\frac{a_1}{1+l_0^{n_1}}-l_1\\ \frac{dl_2}{d\tau}&=\frac{b_1}{1+l_0^{n_1}}\cdot \frac{1}{1+(\frac{l1}{\gamma_2 +1})^{n_2}}-b_2l_2\\ \end{align*}

\begin{align*} l_0&= [ \tilde{L_2}]=\frac{[L_2]}{K_{\mathrm{R,L}}}\\ l_1&=[\tilde{LacI}]=\frac{[LacI]}{K_{\mathrm{R,LacI}}}\\ l_2&=[GFP]\\ \tau &=d_{\mathrm{LacI}}\cdot t\\ B&=\frac{Lact_\mathrm{initial}^2}{K_\mathrm{d,DLL}}\\ a_1&=\frac{a_\mathrm{LacI}}{d_\mathrm{LacI}\cdot K_{RLacI}}\\ b_1 &= \frac{a_\mathrm{GFP}}{d_\mathrm{LacI}}\\ b_2&= \frac{d_\mathrm{GFP}}{d_\mathrm{LacI}}\\ K_\mathrm{d,DLL} &= \frac{k_\mathrm{-DLL}}{k_\mathrm{DLL}}\\ \gamma_1 &= \frac{L_\mathrm{2tot}}{K_\mathrm{R,L}}\\ \gamma_2&=\frac{IPTG_\mathrm{tot}}{K_\mathrm{d,IL}}\\ \end{align*}

Initial States

Every time, we set the initial states of our model to be the steady states when only some Lactate in the medium.

Characteristics of the system

Fold-change behaviour

The model displays perfect fold-change behaviour when the steady state of GFP does not depend on the input lactate. In order to do so, we need to supress all the saturation terms and then :

\begin{align*} [LldR]&\propto \frac{1}{[Lact]^2}\\ [LacI]&\propto \frac{1}{(\frac{1}{[Lact]^2})^{n_1}}\\ [GFP]&\propto \frac{1}{(\frac{1}{[Lact]^2})^{n_1}} \cdot \frac{1}{[LacI]^{n_2}}\\ [GFP]&\propto \frac{[Lact]^{2\cdot n_1}}{[Lact]^{2\cdot n_1 \cdot n_2}} \end{align*}

In order to satisfy this condition, we need:

\begin{align*} n_2&=1 \end{align*}

If we apply the two necessary conditions in the MATLAB model, we obtain a perfect fold-change sensor.

The Fold-change represents the fold-change between different production rates between mammalian cells. Hence, Fold Change =1 represents the production of a normal cell and the other curves represent potential production of cancer cells.

Observation: The fold change pulse will probably be too short, and the basal level of GFP is probably too high with this system.

Amplification behavior

If this module would be separated from the quorum sensing module, we would like to obtain a system that amplifies the difference in production rates between cancer and normal cells. Under certain parameters, the system displays the following response:

Here we can observe that for a fold change of 5 for the input, we obtain a 200 fold change at the output. We have amplified the response compared to the input, but also compared to the simple lactate sensor (fold change in the response is about 15). In the next section, we will discuss the influence of the parameters on that ratio.

Parameter search

Using the literature and our own estimations, we estimated a reasonable range of parameters in which we think the set of biological parameters is located.

In this case, we want to amplify the signal difference between cancer and normal cells' production of lactate. That's why our objective function is to maximize the following ratio:

\begin{align*} \frac{\text{GFP}_\mathrm{\text{ss,Cancer}}}{\text{GFP}_\mathrm{\text{ss,Normal}}} \end{align*}

To obtain the following figure, we had first to compute the optimal parameters in the chosen ranges. The set of optimal parameters was obtained thanks to constrained non-linear optimization. We then computed the cost for every pair of parameters on a 2D grid, fixing the other parameters to their optimal values.

What do the variables represent?

• \(\gamma_1\) represents the repression by LldR.
• \(\alpha \cdot \sqrt{B}\) represents the production of lactate by a normal cell.
• \(\frac{a_1}{\gamma_2 +1}\) represents the repression by LacI.\(\gamma_2\) represents the equivalent amount of IPTG. So the more we increase \(\gamma_2\), the more we reduce the amount of active LacI in the cell.
• \(n_1\) is the Hill coefficient of LldR.
• \(n_2\) is the Hill coefficient of LacI.

Observations

From this figure, we can make the following observations:

• If we increase \(\gamma_1\) then we increase the range where our system show high amplification.
• If we increase \(\frac{a_1}{\gamma_2 +1}\) then we increase the range of possible values for \(\gamma_1\) .
• \(n_1\) has a strong influence on the GFP ratio.

Specifications of the system

The specifications for the lactate module are highly connected to the behaviour of the AHL module. If the lactate sensor would be isolated from the AHL module, we would aim for an amplification of the fold change production between cancer and normal cells, as described here. However, the AHL module has the particularity to be leaky. Leakiness is required to initiate the activation of the AHL sensor but might also lead to an self-activation of the AHL sensor. In consequence, the leakiness has to be high enough initiate an initial concentration of AHL, but low enough to not self-activate the AHL sensor.

Summary

The lactate sensor behaves as a fold-change sensor if there is a delay in LacI transcription. If the lactate concentration inside an E. coli cell reaches the same steady state as in a normal cell, the use of a fold-change sensor is appropriate. However, if cancer and normal cells have different steady states in lactate concentration, the advantages of a fold-change sensor is not clear. To properly derive conclusions about the last two points, we need to study the combined model and to describe the behavior of the AHL module when a pulse of LuxR is introduced. Using the fold-change sensor model, we defined important parameters that greatly influence the output of the system.

• The height and the broadness of the peak can be tuned either by adding IPTG to the medium or by increasing the LacI transcription's delay .

More generally, we learned that depending on the non-linearity of LacI and LldR, the incoherent feed forward loop (i.e. the fold-change sensor) can fulfill various functions. If the non-linearity difference between the two proteins is high, then the system can amplify the input. Whereas, if the non-linearities are comparable, the network displays fold-change behavior.

Outlook

The effect of the lactate inputs' behavior on LuxR dynamic expression profiles needs to be studied in more detail for both the fold-change sensor and the simple lactate detection system. To answer this question, we investigated in the combined compartment model.