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

Lactate Module

Overview

Figure 1: Lactate sensor design

Our idea was to distinguish cancer and normal cells based on their different lactate production rates. We assumed that in our 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

Figure 1: Topology of the lactate sensor

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.
set the specifications of our system.
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.
characterize the different constructs.
define the parameters and components that we will influence the output of our system by integrating the previous points.

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

Figure 1: Mechanism LldR

After looking at puzzling results, we realized that our first model of the mechanism of action of LldR was not realistic. In the literature, we found a compatible explanation, depicted here. 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+(\frac{[LldR_2]}{K_R})^{n_r}}\cdot \frac{(\frac{[LactLldR]}{K_A})^{n_a}}{1+(\frac{[LactLldR]}{K_A})^{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 (\frac{[Lact]}{K_\mathrm{A,Lact}})^{n}}{1+(\frac{[Lact]}{K_\mathrm{A,Lact}})^{n}}-d_\mathrm{GFP}[GFP]\\ \end{align*}

Characterization of the lactate responsive promoter in LB

Figure 1: Experiments for the characterization of the LldR promoters

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, only diffusion is happening and thus: \([Lact_{in}]=[Lact_{out}]\).

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 (\frac{[Lact_{in}]}{K_\mathrm{A,Lact}})^{n_1}}{1+(\frac{[Lact_{in}]}{K_\mathrm{A,Lact}})^{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	K_M (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.

**Low LldP-LldR**
Promoter	Promoter Strength	ON/OFF ratio	K_M (μM)
K822000	unknown (natural)	1.16	720.2
K1847008	162	1.42	337.7
K1847009	1429	0.96	459.8
K1847007	2547	1.29	1337

**High LldP-LldR**
Promoter	Promoter Strength	ON/OFF ratio	K_M (μM)
K822000	unknown (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 < 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.

Parameter	value
K_M	173 μM
ON/OFF	7
n	1.7

Characterization of the low engineered promoter

Full module simplified model

Questions

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 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
Lact_out	Lactate produced by mammalian cells
Lact_in	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 full module

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*}

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 is 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

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.

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.

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.

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.

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.

Figure 1: Parameter search representing the ratio of GFP output for cancer versus normal cells

Figure 1: Parameter Search representing the absolute values of GFP concentrations against the different parameters

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?

Figure 1: Assumption on the mechanism of LldR

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
Lact_out	Lactate produced by mammalian cells
Lact_in	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.

Figure 1: Behaviour of the non-dimensionalized system with the previouly stated conditions

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.

Our system specifications

We want a system that amplifies the difference in production rates between cancer and normal cells. One example of such behavior is the following:

Figure 1: Specifications of our system

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

According to our specifications, 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.

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

Team:ETH Zurich/Modeling/Lactate Module

Lactate Module

Overview

Description of the design

Goals

Characterization of the LldR promoters and Biophysical Model

A closer look at the mechanism of LldR

Reactions

Mathematical model

Simplification

Characterization of the lactate responsive promoter in LB

Experiment 1): No overexpressed LldP in LB

Experiment 2): Overexpressed LldP in LB

Observations

Characterization of the lactate responsive promoter in mammalian medium

Full module simplified model

Questions

Defining parameters

Chemical species, reactions and equations

Chemical species

Reactions

Equations for full module

CASE 1) Same steady state

Assumptions and modeling

Simulation

CASE 2) Different steady states

Parameter Search and amplification

Equations of the non-dimensionalized system

Range of parameters chosen

Results of the parameter search

Early stage modeling

Overview

How did we derive the model?

Chemical species

Reactions

Equations

Assumptions

Equations

Non dimensionalized equations

Initial States

Characteristics of the system

Fold change behaviour

Our system specifications

Parameter search

What do the variables represent?

Observations

Summary

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