Model

The Quorum sensing module is mainly involved in receiving signals from the sender cells. The sender cells secrete some signaling molecules (inducers) which diffuse out of their membrane, then diffuse in receiver cells membrane, and bind to the regulator molecules in the receiver cells, thus activating the transcription of certain genes. In order to characterize the quorum sensing module with a transfer function, we consider different initial inputs of external AHL, and see how much output is produced, as it was done in the quorum sensing experiments.

As diffusion through the membrane is very fast^[27], according to Fick's law of diffusion, internal and external concentration of AHL can always be considered as equal. When an initial external AHL concentration is given, AHL diffuses in the cells very quickly (less than 20 seconds)^[27] until internal AHL concentration equals external concentration. Then as soon as some internal AHL is consumed in the cell, it is taken up again without affecting external concentration, because external volume is very high compared to internal volume. Therefore we can consider in this module that external AHL (which is equal to internal AHL) only degrades, with the rate of extracellular decay.

Chemical Species

Reactions

For the Lux system: $$ \begin{align} &\rightarrow LuxR \\ LuxAHL+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\ mRNA_{Bxb1}&\rightarrow Bxb1\\ LuxAHL &\rightarrow \\ LuxR &\rightarrow \\ RLux &\rightarrow\\ DRLux &\rightarrow\\ mRNA_{Bxb1} &\rightarrow\\ Bxb1 &\rightarrow \end{align}$$

For the Las system

\begin{align} &\rightarrow LasR \\ LasAHL+LasR & \leftrightarrow RLas \\ RLas+RLas & \leftrightarrow DRLas\\ DRLas+P_{LasOFF} & \leftrightarrow P_{LasON}\\ P_{LasON}&\rightarrow P_{LasON}+mRNA_{\phi C31}\\ mRNA_{\phi C31}&\rightarrow \phi C31\\ Las-AHL &\rightarrow \\ LasR &\rightarrow \\ RLas &\rightarrow\\ DRLas &\rightarrow\\ mRNA_{\phi C31} &\rightarrow \\ \phi C31 &\rightarrow \\ \end{align}

Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations. $$\begin{align*} \frac{d[LuxAHL]}{dt} &= -d_{LuxAHL}[LuxAHL]\\ \frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[LuxAHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\ \frac{d[RLux]}{dt} &= k_{RLux}[LuxAHL][LuxR] - k_{-RLux}[RLux] - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ \frac{d[DRLux]}{dt} &= k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\ \frac{d[P_{LuxON}]}{dt} &= k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\ \frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + k_{mRNA_{Bxb1}} [P_{LuxON}] - d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\ \frac{d[Bxb1]}{dt} &= k_{Bxb1} [mRNA_{Bxb1}] - d_{Bxb1}[Bxb1]\\ \end{align*}$$

The same holds true for the Las system.

From the original set of reactions, we reduce the rate of production of mRNA_Bxb1 as a Hill function of RLux instead of Mass action kinetics in terms of P_LuxON and P_LuxOFF. For more information please check the characterization section.

Characterization: K_mLux and K_mLas

Data

For the Quorum sensing module we used established experimentally determined parameters for the rate of formation of RLux (reference). Since, in the literature the other parameters were estimated or fitted to their data, we decided to determine the parameters specific to our system. Hence, we used our data for the remaining parameters. Our data was mainly a transfer function of normalized GFP concentration as a function of input LuxAHL concentrations. (link to data)

Assumptions

Assumption A

We assumed that the dimerization of RLux to DRLux is quick. Quasi steady state approximation (QSSA) as follows

$$\frac{d[DRLux]}{dt} = k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \approx 0\\$$

Assumption B

Further, from literature, we found that DRLux is specific to DNA and the dissociation constant is low (k_m = 0.1nM) {Reference}. Therefore, we using QSSA again,

$$\frac{d[P_{LuxON}]}{dt} = k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}] \approx 0\\$$

Solving, we get the rate of production of mRNA_Bxb1 as

$$\frac{d[mRNA_{Bxb1}]}{dt} = L_{P_{Lux}} + \frac{k_{mRNA_{Bxb1}}[RLux]^2}{K_{mLux}^2 + [RLux]^2 }- d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\$$

where

$$K_{mLux} = \sqrt{\frac {k_{-P_{LuxON}}}{k_{P_{LuxON}}}.\frac {k_{-DRLux} + d_{DRLux}}{k_{DRLux}}}$$

is a lumped parameter which we fitted to our data.

Similarly, lumped parameter K_mLas was derived for the las system and fitted to a transfer function of normalized GFP concentration as a function of input Las-AHL.

Parameter fitting

We used MEIGO Toolbox to fit the parameters to the experimental data. We used the concentrations at the end of five hours from each simulation and fit it to the experimental concentrations at the same time.

Lux QS Module fitted to experimental data from riboregulated Lux system.

Using the 'DHC' local-solver (Direct search method) in MEIGO, we found the lumped parameters $$K_{mLux} = 0.0124 \pm 0.00051 nM$$

and

$$K_{mLas} = 0.3818 \pm 0.0082 nM$$

respectively.

Las QS Module fitted to experimental data from riboregulated Las system.

Range of validity of the assumptions

These assumptions hold true for all input LuxAHL and LasAHL concentrations.

Retrieving degradation rates

We are considering quorum sensing experiments with riboregulator, where GFP is produced instead of Bxb1, and this time we look at dynamic curves.

Curve

By adding an additional quasi steady state assumptions on R_Lux, and neglecting degradation of R_Lux compared to its unbinding rate, we can find :

$$\frac{d[GFP]}{dt}=LeakyLux+\frac{k_{mRNAGFP} k_{GFP} \alpha_{LuxR}^2}{d_{LuxR}^2(Km_{Lux}^2+\alpha_{LuxR})} \frac{[AHL]^2}{K_{mAHL}^2 + [AHL]^2}-d_{GFP}[GFP]$$

$$\text{with} K_{mAHL}=\frac{K_{mLux}^2 k_{-RLux}}{k_{RLux}(K_{mLux}^2+\alpha_{LuxR}^2/d_{LuxR})}$$

We have dynamic curves for different initial AHL concentrations.

We can see in the equation above that for initial AHL concentrations much higher than K_mAHL=0.3 nM, GFP is only produced and degraded and thus :

$$\frac{d[GFP]}{dt}=LeakyLux+\frac{k_{mRNAGFP} k_{GFP} \alpha_{LuxR}^2}{d_{LuxR}^2(Km_{Lux}^2+\alpha_{LuxR})}-d_{GFP}[GFP]$$

so by taking $$t_{1/2}=\frac{ln(2)}{d_{GFP}}$$

from experimental curves, we find 

$$d_{GFP} = 4.9 . 10^{-3} min^{-1}$$

For initial AHL concentrations much lower than K_mAHL=0.3 nM, we find $$\frac{d[GFP]}{dt}=LeakyLux+\frac{k_{mRNAGFP} k_{GFP} \alpha_{LuxR}^2}{d_{LuxR}^2(Km_{Lux}^2+\alpha_{LuxR})} \frac{[AHL]^2}{K_{mAHL}^2}-d_{GFP}[GFP]$$

and thus a steady state $$[GFP]=\frac{Constant}{(d_{GFP}-2d_{AHL})}(e^{-2d_{AHL}t}-e^{-d_{GFP}t})$$

This curve has a maximum at 

$$t_{max}=\frac{1}{d_{GFP}-2d_{AHL}}ln\big(\frac{d_{GFP}}{2d_{AHL}}\big)$$

This way we can find from experimental curves (exploiting here only the GFP data points) $$d_{AHL}=4,0.10^{-3} min^{-1}$$

Leakiness

The leakiness of promoters is a major issue in our system. As the signal propagates row-wise, error diffusion could lead to a totally different pattern. The goal is then to master the leakiness. This issue was particularly observed in the case of the Lux promoter during our experiments set. This leakiness is dependent on the LuxR concentration in the cell. Therefore, an assumption would be that Lux(see the Registry for more exhaustive information)

Cross-talk

Alternate Design

As cross-talk is a burning issue in quorum sensing, we thought about a theoretical solution. Edinburgh iGEM team 2014also worked on communication between E. coli. They developed new communication channels via metabolic wiring. By assuming that quorum sensing molecules would not cross-talk with metabolites, we used their idea to develop a model on molecular level. The idea is finally to combine our whole-cell model and their idea on metabolites.

Metabolic wiring is based on the fact that when a resource is at disposal, a cell will produce an enzyme to break it down to smaller pieces. There is often a chain of metabolites. Using the fact that metabolites can diffuse through the membrane, this can allow communication (for more information, please see the Edinburgh iGEM team 2014 wiki).

Chemical Species

The species' names are generic because the implementation with particular metabolites implies biological considerations on cell growth and medium used.

Reactions

$$\begin{align*} &\rightarrow A \\ A + P_{Aoff} &\rightarrow P_{Aon} \\ P_{Aon} &\rightarrow P_{Aon} + Enz \\ A + Enz &\leftrightarrow [A.Enz] \\ [A.Enz] &\rightarrow B \\ A &\rightarrow \\ B &\rightarrow \\ [A.Enz] &\rightarrow \\ Enz &\rightarrow \end{align*}$$

Parameters

Parameters are not well-known.

Deterministic Model

We derived this model doing the following assumptions:

Assumption 1

The induction of the promoter P_A by A is supposed to follow an Hill function.

Assumption 2

The enzyme-based reaction from A to B is supposed to follow a Michaelis-Menten function.

$$\begin{align} \frac{d[A]}{dt} &= \alpha_A - d_{A} [A] \\ \frac{d[Enz]}{dt} &= \alpha_{Enz} \frac{[A]^n}{K_{A}^n + [A]^n} - d_{Enz} [Enz] \\ \frac{d[B]}{dt} &= \alpha_B \frac{[Enz] [A]}{K_d + [A]} - d_{B} [B] \end{align}$$

Application to our project

Populations of bacteria will grow into a medium, which provide them metabolite A. The metabolite B will serve as communicating signal, like LasAHL or LuxAHL. It will be an input for the logic construct. As soon as metabolite B is being made available to the cell, the cell will produce the input for the logic construct.
Moreover, one type of cell will produce B as output. Therefore, the output of the logic gate signal will correspond to the production (or absence of production) of the enzyme Enz, so that the A contained in the medium can be transformed to B. Enz will play the same role as LasI or LuxI in our original model.
We separate the promoter activating the production of the enzyme with the production of the enzyme itself. Moreover, cells sense B and want to produce B. In our modules, we have to remplace equations in the sensing module and in the production module.

Sensing Module

For example, B would induce the production of the integrase, Bxb1.

$$ \begin{align*} \frac{d[B]}{dt} &= \alpha_B - d_{B} [B] \\ \frac{d[Bxb1]}{dt} &= \alpha_{Enz} \frac{[B]^n}{K_{B}^n + [B]^n} - d_{Bxb1} [Bxb1] \end{align*} $$

Production Module

Here, output_logic is the output of the XOR logic gate, factorized in one term for simplicty's sake. For more information on this function, see the XOR gate modeling page.

$$\begin{align*} \frac{d[Enz]}{dt} &= output_{logic} - d_{Enz} [Enz] \\ \frac{d[B]}{dt} &= \alpha_B \frac{[Enz] [A]}{K_d + [A]} - d_{B} [B] \end{align*} $$

Simulations

We implemented this solution in our whole-cell model. As no parameter is known, we assumed their values to be in the range of standard rates. It gave a possible valid result that could work in our system.