Model

The Quorum sensing module is mainly involved in receiving signals from the sender cells. The sender cells secrete some signalling molecules (inducers) which bind to the regulator molecules in the receiver cells, thus activating the transcription of certain genes. The model for this module is presented below.

Chemical Species

Reactions

For the Lux system

$$ \begin{align} &\rightarrow LuxR \\ LuxAHL+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\ mRNA_{Bxb1}&\rightarrow Bxb1\\ AHL &\rightarrow \\ LuxR &\rightarrow \\ RLux &\rightarrow\\ DRLux &\rightarrow\\ mRNA_{Bxb1} &\rightarrow\\ Bxb1 &\rightarrow \end{align}$$

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. $$\begin{align*} \frac{d[LuxAHL]}{dt} &= k_{-RLux}[R_{Lux}]-k_{RLux}[LuxAHL][LuxR]-d_{LuxAHL}[LuxAHL]\\ \frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[LuxAHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\ \frac{d[RLux]}{dt} &= k_{RLux}[LuxAHL][LuxR] - k_{-RLux}[RLux] - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ \frac{d[DRLux]}{dt} &= k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\ \frac{d[P_{LuxON}]}{dt} &= k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\ \frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + k_{mRNA_{Bxb1}} [P_{LuxON}] - d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\ \frac{d[Bxb1]}{dt} &= k_{Bxb1} [mRNA_{Bxb1}] - d_{Bxb1}[Bxb1]\\ \end{align*}$$

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 $$K_{mLux} = 0.0124 nM (error 0.0821)$$

and

$$K_{mLas} = 0.3818 nM (error 0.0791)$$

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.

Leakiness

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 ressource 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.

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

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.

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

Production Module

Here, output_logic is the output of the logic factorized in one term.

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

Simulations

We implemented this solution in our whole-cell model. As no parameter is none, we assumed their values to be in the range of standard rates. It gave a possible valid result.