Team:ETH Zurich/Modeling/Lactate Module
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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 timeframe, 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 natural detection system of lactate in E. coli. However, with the used parameters, our system does not behave as a fold change sensor but rather amplifies the difference between the production rate of cancer and normal cells.
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.
Goals
The aim of the model is to:
- understand the behaviour and characteristics of our system.
- discuss the specifications of our system.
- define the parameters and components that we will influence the output of our system.
We characterized the different promoters, and included them in the simulations for the fold-change sensor.
Full module simplified model
Questions
Figure 3. Lactate Concentration over time - input of the system.
The choice of using a fold-change sensor or directly the natural detection system of E. coli mainly depend on the behavior of the input. Do the lactate concentration inside E. coli in case of a cancer cell will reach the same steady state than in case of a normal cell? An other important parameter is the time point of the saturation. From these observations we can derive three different situations (represented on the scheme).
- If the measurement happens after the saturation of lactate, then we will need the fold change sensor otherwise, we will obtain the same response for both cancer and normal cells.
- If the measurement happens when the signal are still separated then, we can use both the fold change sensor and the simple lactate detection system.
In the following, we will describe the functioning of the fold change sensor and describe the important parameters defining its behavior.
Obviously, the specifications on the output behavior, strongly relies on the next sequential module: the two signals should be coherent in order 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.
- The half-maximal substrate concentration and the cooperativity of LldR.
- The delay for LacI induction.
- The amount of IPTG we introduce in our system.
We characterized our lactate responsive constructs, and we obtained a range of different KM values. Depending on the lactate input concentration, the appropriate construct with the correct sensitivity has to be chosen. The cooperativity of LldR is also an important parameter. If the cooperativity of the lactate detection system is higher than LacI, then the response will display amplification. When we characterized our system, we obtained a cooperativity equivalent to LacI's Hill coefficient. In this case, the output displays fold-change behaviour.
We simulated the following system, to gain more information about how the system would behave under different conditions, and to know which construct to use: the simple lactate detection system or the fold-change sensor.
Chemical species, reactions and equations
Chemical species
| Name | Description |
|---|---|
| Lactout | Lactate outside E. coli cells |
| Lactin | Lactate inside E. coli cells |
| 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}}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}_{in}}} \text{LacI}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}_{in}}} \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 for the fold change sensor
Consistent with the simplification describes in the previous section, we derive these equations:
\begin{align*} \frac{d[LacI]}{dt}&=\frac{a_\mathrm{LacI}\cdot (\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}{1+(\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP}\cdot (\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}{1+(\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}} \cdot \frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}})^{n_\mathrm{2}}}-d_{\mathrm{GFP}}[GFP]\\ \end{align*}Equations for the natural lactate detection system
\begin{align*} \frac{d[LacI]}{dt}&=\frac{a_\mathrm{GFP}\cdot (\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}{1+(\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}-d_{\mathrm{GFP}}[GFP]\\ \end{align*}
CASE 1) Same steady state
As already mentioned, if the lactate production reaches steady state quickly then, the simple lactate detection system would lead quickly 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
Here, we model directly the transcription and the translation of one protein with only one differential equation. The time delays that we consider are then shortened compared to real-life situation. That is why in the following a delay in LacI transcription is introduced.
Simulation
Comparison of behavior when a delay is introduced
When a delay is introduced, we have the typical fold change behavior. The height of the peak in the case of cancer cells is three times higher than in the case of normal cells. In this case the simple detection system would not work because the time difference in LacI transcription would not be sufficient to introduce a significant delay.
|
No delay in LacI transcription 
|
Delay in LacI transcription 
|
Figure 4. Comparison of the behavior when introducing a delay in LacI transcription. The rate of production fold change between normal and cancer cells is set to 3.
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 will be 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.
Figure 5. GFP response when adding IPTG to the medium
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.
Figure 6. GFP response when adding IPTG to the medium
Behavior of the natural detection system
In the second case, the response of the natural detection system is more appropriate.
Figure 7. GFP response when adding IPTG to the medium
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 maximum amplification possible is 15 fold for small GFP absolute response. 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
| \begin{align*} \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&=\frac{[Lact]}{K_{\mathrm{A,Lact}}}\\ l_1&=\frac{[LacI]}{K_{\mathrm{R,LacI}}}\\ l_2&=[GFP]\\ \tau &=d_{\mathrm{LacI}}\cdot t\\ 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}}\\ \gamma_2&=\frac{IPTG_\mathrm{tot}}{K_\mathrm{d,IL}}\\ \end{align*} |
Range of parameters chosen
| Name | Description | Minimum Value | Maximum Value | References/Estimation |
|---|---|---|---|---|
| \([Lact]\) | Production of lactate by normal cells | 1 μM | 100 μM | estimated |
| \(K_{\mathrm{A,Lact}}\) | Lumped parameter for the lactate sensor | 50 μM | 2000 μM | Based on the characterization of the promoters. |
| \(a_1\) | \(\frac{a_\mathrm{LacI}}{d_\mathrm{LacI}\cdot K_{RLacI}}\) | 0.05 | 1000 | |
| \( a_\mathrm{LacI}\) | Maximal production rate of LacI | 0.05 μM.min-1 | 1 μM.min-1 | Basu, 2005 |
| \( d_\mathrm{LacI}\) | Degradation rate of LacI | 0.01 min-1 | 0.1 min-1 | Basu, 2005 |
| \( K_\mathrm{R,LacI}\) | Repression coefficient of LacI | 0.1 μM | 10 μM | Basu, 2005 |
| \( \gamma_2\) | \(\frac{IPTG_{tot}}{K_{IL}}\) | 0 | 500 | estimated |
| \( \frac{a_1}{\gamma_2+1}\) | 0.001 | 1000 | estimated | |
| \( n_1\) | Hill coefficient of LldR | 1 | 2.5 | estimated |
| \( n_2\) | Hill coefficient of LacI | 1.5 | 2.5 | estimated |
Results of the parameter search
Below you will see two parameters represented 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 there.
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
- We used the quasi steady state approximation to model the fast dimerization of lactate to L2 and of IPTG to LacI.
- 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 natural 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 parameters of the biological system is located.
| Name | Description | Minimum Value | Maximum Value | References/Estimation |
|---|---|---|---|---|
| \(\text{B}\) | \(\frac{Lac_\mathrm{ini}^2}{K_\mathrm{d,DLL}}\) | 0.000001 | 4 | |
| \(\text{Lac}_{\text{ini}}\) | Initial concentration of lactate in the medium | 0.1 μM | 2 μM | Low concentration of lactate in the medium |
| \(K_\mathrm{d,DLL}\) | Dissociation constant between the dimer of Lldr and Lactate | 10 μM2 | 10000 μM2 | |
| \(\alpha\) | Multiplication factor between the initial concentration of Lactate and Production of normal cells | 1 | 150 | estimated |
| \(F_\mathrm{C}\) | Fold change between Lactate production by cancer and normal cells | 2 | 4 | estimated |
| \(a_1\) | \(\frac{a_\mathrm{LacI}}{d_\mathrm{LacI}\cdot K_{RLacI}}\) | 0.05 | 1000 | |
| \( a_\mathrm{LacI}\) | Maximal production rate of LacI | 0.05 μM.min-1 | 1 μM.min-1 | Basu, 2005 |
| \( d_\mathrm{LacI}\) | Degradation rate of LacI | 0.01 min-1 | 0.1 min-1 | Basu, 2005 |
| \( K_\mathrm{R,LacI}\) | Repression coefficient of LacI | 0.1 μM | 10 μM | Basu, 2005 |
| \( \gamma_1\) | \( \frac{L_\mathrm{2tot}}{K_\mathrm{R,L}}\) | 5 | 10000 | estimated |
| \( L_\mathrm{2tot}\) | Total concentration of LldR dimer | 0.5 μM | 10 μM | estimated from paxdb |
| \( K_\mathrm{R,L}\) | Repression coefficient of LldR | 0.001 μM | 0.1 μM | estimated |
| \( \gamma_2\) | \(\frac{IPTG_{tot}}{K_{IL}}\) | 0 | 500 | estimated |
| \( \frac{a_1}{\gamma_2+1}\) | 0.001 | 1000 | estimated | |
| \( n_1\) | Hill coefficient of LldR | 0.5 | 2.5 | estimated |
| \( n_2\) | Hill coefficient of LacI | 1.5 | 2.5 | estimated |
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 ranges chosen. The set of optimal parameters was obtained thanks to constrained non-linear optimization.We then computed the cost for every pair of parameters on 2D grid, fixing the other parameters to their optimal values.
Figure 13. Parameter Search
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 output of the
Our system specifications
The specifications for the lactate module are highly connected to the quorum sensing module's behavior. If the lactate sensor would be isolated from the rest of the system, we would like to have amplification of the fold change production between cancer and normal cells, as described here. However, the quorum sensing module has the particularity to display high leakiness. The leakiness has to high enough to obtain activation of the system, but low enough to see the effect of colocalization and quorum sensing on the output's response.
Summary
The fold-change lactate sensor displays typical fold-change behavior when a delay in LacI transcription is introduced. If the concentration of lactate inside an E. coli cell reach the same steady state than in the case of a normal cell, the use of a fold-change sensor is particularly appropriate. However, when the lactate steady states are separated, the advantage of a fold-change sensor is not clear. To properly derive conclusions about the last two points, we need to study the < a href="https://2015.igem.org/Team:ETH_Zurich/Modeling/Single-cell_Model">combined model and describe the behavior of the quorum sensing module when a pulse of LuxR is introduced. Thanks to this 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 .
We also determined objectives and questions to be answered by the combined compartment model.
- The effects of the lactate inputs' behavior, which result in different LuxR dynamic expression profiles, need to be studied in more detail for both the fold-change sensor and the natural lactate detection system.
More generally, we learned that depending on the non-linearity of LacI and LldR, the incoherent feed forward loop 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.
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