Team:ETH Zurich/Modeling/AHL Module

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

AHL Module

Introduction and Goals

Cancer cells, sensitive to sTRAIL-induced apoptosis, flip phosphatidylserine to their outer membrane. Our E. coli can bind phosphatidylserine thanks to Annexin V, which is displayed on their membrane. This module is designed to detect colocalization of E. coli (MicroBeacon) cells on the mammalian cell surface. Since the E. coli densely populate the mammalian cell surface, they can communicate via AHL signaling to produce GFP, whereas the E. coli localized in the bulk will be too far away to communicate in the case of normal cells. However, the dimensions of the chip in which the bacteria and the mammalian cells are located is really small (1 nL). We expect then the signaling molecule to diffuse almost instantaneously in the well. Also, from the results of last years' ETH Zurich team, we know that the LuxRAHL responsive operator is leaky in the presence of LuxR. This results in self-activation of all the bacteria, independently of their density. To address this problem, we studied two different ways to control the activation of the construct. During the first part of the project, we privileged integrating AiiA, an AHL degrading enzyme, into the final design. We thought it would be sufficient to prevent activation in unwanted situations. Our second approach consisted in replacing the LuxI controlling promoter by a riboregulated promoter. As shown by ETHZ 2014 iGEM Team, the resulting promoter is not leaky. In the following we will describe the influence of the two approaches.

Description of the design

Figure 1: AHL sensor genetic design

When AHL concentration is high enough due to high density of cells, the lux promoter is triggered, which produces more AHL via the synthesis of LuxI. When the threshold concentration of activated LuxR (LuxRAHL) is reached, GFP production starts (to see the reactions, click reactions)

Goals

In the following, we present a single-cell model and a compartment model. The purpose of the single cell model is to:

1. check the implemented equations.
2. study the influence of AiiA on the single cell.
3. study the influence of a riboregulator on the LuxI controlling promoter.

The compartment model compares the behavior of the E. coli in two different situations, when E. coli are highly concentrated around the mammalian cell and when they are spread in the bulk. Thanks to this model, we hope to know:

4. which approach, i.e. the riboregulator or AiiA degradation, obtains the desired AND gate.

Jump to Summary.

Chemical species

Name	Description
AHL	Signaling protein, Acyl homoserine lactone (30C6-HSL)
LuxR	Regulator protein, that can bind to AHL to form a complex
LuxRAHL	Complex of LuxR and AHL, activates transcription of LuxI
LuxI	Autoinducer synthase
Aiia	AHL-lactonase, N-Acyl Homoserine Lactone Lactonase

Compartment Model

Overview

We investigated a simplified and a full compartment model. The simplified model serves as a sanity check of the activation time: We tested which of the previously described approaches results in a detectable difference between cancer and normal cells with respect to different cell densities. Here, the E. coli located in the bulk are isolated from the ones in the doughnut (no exchange by diffusion). In the full compartment model, we compared two situation: one of them represents the situation when there is colocalization on the mammalian cell surface, and some E. coli floating in the bulk, and the other situation represents the situation when E. coli are evenly spread in the bulk, because the mammalian cell does not present phosphatidylserine.

We want activation only in one case: when E. coli are colocalized on the mammalian cell surface.

Self-activation of the E. coli for different densities

To see the derivations of the ordinary differential equations or why we chose these approaches, jump to single-cell model.

Before implementing the solution with a more realistic geometry, we wanted to estimate easily which solution (AiiA or riboregulator) would be the best for our system. In order to do that, we implemented a compartment model. The compartment model represents two different situations. In the first one, the cells are dense in the doughnut. In the second one, the cells are less dense because spread in the bulk. We calculated the number of E. coli that would fit on a mammalian cell surface. In all the rest, we assume that LuxR is constitutively produced and AHL is able to diffuse in the bulk. We will compare the activation time in both cases.

The present model has two compartments: One represents the E. coli, the second one represents the external volume around the bacterium.Only AHL can diffuse through the membrane. The model is meant to check for the self-activation in case of different E. coli cell densities.

To evaluate the model, we considered that the E. coli would be located either in a "doughnut" representing the higher density of cells around the mammalian cells and the "bulk". The E. coli can be localized in the bulk for two different reasons. Either because the cancer cell surface is saturated, or because the normal cell does not express phosphatidylserine. So there is no colocalization on the surface.

To represent the two different situations, we calculated the volume per cell in the two different cases. In the case of the denser cells, the volume per cell is much smaller. In this case, we expect an earlier activation time, compared to non-sense cells.

Below you will be able to find the equations for the compartment model and the results of the simulation.

Equations of the compartment model for both cases

To simulate this model in Matlab we introduced a new state: AHL_e representing concentration of AHL in the volume the volume around the E. coli.AHL_i represents the concentration of AHL inside the bacterium.

Assumptions

• LuxR constitutively produced and constant.
• Instant diffusion of AHL in the external volumes.
• Flow of AHL through the E. coli membranes.
• No external degradation of AHL.

Equations with AiiA degradation
\begin{align*} [LuxRAHL_i]&= \frac{[AHL_i]\cdot LuxR_\mathrm{tot}}{K_{\mathrm{d,LuxRAHL_i}}+[AHL_i]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL_i])+\frac{a_{\mathrm{LuxI}}(\frac{[LuxRAHL_i]}{K_{\mathrm{a,LuxRAHL_i}}})^{n_{lux}}}{1+(\frac{[LuxRAHL_i]}{K_{\mathrm{a,LuxRAHL_i}}})^{n_{lux}}}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL_i]}{dt}&=a_{\mathrm{AHL_i}}[LuxI]-d_{\mathrm{AHL_i}}[AHL_i]-\frac{v_\mathrm{Aiia}\cdot [AHL_i]}{K_{\mathrm{M,AiiA}}+[AHL_i]}+Dm \cdot ([AHL_e]-[AHL_i])\\ \frac{d[AHL_e]}{dt}&=\alpha_{db}\cdot Dm \cdot ([AHL_i]-[AHL_e])\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL_i])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL_i]}{K_{\mathrm{a,LuxRAHL_i}}})^{n_{lux}}}{1+(\frac{[LuxRAHL_i]}{K_{\mathrm{a,LuxRAHL_i}}})^{n_{lux}}}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL_i} &= \frac{k_\mathrm{-LuxRAHL_i}}{k_\mathrm{LuxRAHL_i}}\\ \end{align*}
Equations with riboregulator
\begin{align*} [LuxRAHL_i]&= \frac{[AHL_i]\cdot LuxR_\mathrm{tot}}{K_{\mathrm{d,LuxRAHL_i}}+[AHL_i]}\\ \frac{d[LuxI]}{dt}&=L_{Lux,ribo}+\frac{a_{\mathrm{LuxI,ribo}}(\frac{[LuxRAHL_i]}{K_{\mathrm{LuxRAHL_i,ribo}}})^{n_{lux}}}{1+(\frac{[LuxRAHL_i]}{K_{\mathrm{LuxRAHL_i,ribo}}})^{n_{lux}}}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL_i]}{dt}&=a_{\mathrm{AHL_i}}[LuxI]-d_{\mathrm{AHL_i}}[AHL_i]+Dm \cdot ([AHL_e]-[AHL_i])\\ \frac{d[AHL_e]}{dt}&=\alpha_{db}\cdot Dm \cdot ([AHL_i]-[AHL_e])\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL_i])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL_i]}{K_{\mathrm{a,LuxRAHL_i}}})^{n_{lux}}}{1+(\frac{[LuxRAHL_i]}{K_{\mathrm{a,LuxRAHL_i}}})^{n_{lux}}}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL_i} &= \frac{k_\mathrm{-LuxRAHL_i}}{k_\mathrm{LuxRAHL_i}}\\ \end{align*}

Simulation

Here we will compare the results of the simulation, for the three cases:

1. No AiiA degradation, no riboregulator.
2. Only AiiA degradation.
3. System with riboregulator

On the following graph you can observe the response of the system, when the E. coli are dense in the doughnut. When they are distributed in the bulk, we consider two situations. The worst case is when they are concentrated in the bulk called: \(V_{cell,b,worst}\). A better situation is represented when they are less dense in the bulk:\(V_{cell,b,norm}\).

We can observe, that for cases A) and B), there is not a significant difference between the E. coli for the volumes considered. AiiA here acts here as a time-delay for activation rather than amplifying the difference between the response of the E. coli .

At the opposite for case C), with the riboregulator, there is a significant difference between the different densities of E. coli . Therefore the riboregulator seems to be a better solution.

A)	B)
C)

Figure 1: GFP concentration with A) No AiiA degradation and no riboregulator. B) With AiiA degradation, C_AiiA=800 μM. C) With Riboregulator

Full compartment model

Figure 2. Compartment model description. Two situations are depicted: on the left, some E. coli are colocalized on the mammalian cell surface (doughnut) and some E coli are spread in the bulk. On the right, phosphatidylserine is not displayed on the surface, there is no colocalization on the cell's surface, E. coli spread evenly in the bulk

In the following, we want to see under which conditions (volume of the a nano-well plate, AiiA degradation, riboregulator), we can observe a difference in the activation time for the E. coli located next to a cancer cell, versus the E. coli next to a normal cell.

Assumptions

We assume instant diffusion in this model.

The number of E. coli in the bulk (i.e. the volume of the well without the doughnut) is the same for both cancer and normal cells. Since we assume a pre-incubation time of the bacteria and the mammalian cells before being spread in the chip, we assume that the total number of E. coli in the whole nanowell is higher in case of cancer cells than for normal cells, because E. coli accumulate on cancer cell surfaces but not on the normal cell surfaces.

Equations

Based on the previous equations, we just added an additional convection equation between the doughnut compartment and the bulk.

\begin{align*} \text{AHL}_d &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}\text{AHL}_b\\ \end{align*}

To compute the number of molecules that pass through the surface, we scaled the estimated number of molecules of AHL that pass through one E. coli cell per minute, by including the maximal number of E. coli fitting on the mammalian surface and the percentage of the exposed area of one E. coli “doughnut” cell to the bulk. Taking all together leads to the following equation:

\begin{align*} D_{m,db}&= D_m\cdot \text{percentage of area exposed} \cdot N_d \\ \end{align*}

Simulation

Influence of AiiA and Riboregulators

We checked the influence of both methods but for the set of parameters at stake, the system showed no activation for none of the conditions tested. The leakiness is not high enough to activate the construct. We will concentrate then on the natural system with no degradation.

Important parameters

Here we study different parameters that have a big influence on the activation time of the outputs.

1. The volume of one well in the nanowell plate
2. The total amount of LuxR. As LuxR drives the leakiness of the promoter, it is an important parameter.

GFP response for LuxRtotal,max	GFP response for LuxRtotal,max/2
GFP response for LuxRtotal,max/5

Figure 3. GFP response for several nano-well volumes and several LuxR_total concentrations

As we can notice on the graphs, decreasing the total concentration of LuxR_totalinside the cell, increase the difference between the case where there is colocalization and the case where there is no colocalization.

Single Cell Model

The single cell model describes the basic chemical reactions and equations concerning the quorum sensing module. It also shows the basic behavior of the system.

In all the following, we consider that LuxR is constitutively produced and constant. In the complete design, LuxR is however regulated by the lactate amplifier. Also, here, no AHL diffuses out of the cell. It is therefore not a realistic case in the context of our system.

Single cell model with AiiA degradation

Reactions

The Reactions depicted here are based on simple kinetics and Michaelis Menten kinetics. They describe the design provided above.

\begin{align*} \varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LuxR}}} \text{LuxR}\\ \text{AHL} + \text{LuxR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{LuxRAHL}}}^{k_{\mathrm{-LuxRAHL}}} \text{LuxRAHL}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{LuxI},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{LuxI}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{GFP},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{GFP}\\ \text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{AHL}}}\text{AHL}+\text{LuxI}\\ \text{LuxR}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxR}}}\varnothing\\ \text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{AHL}}}\varnothing\\ \text{LuxRAHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxRAHL}}}\varnothing\\ \text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxI}}}\varnothing\\ \text{Aiia}+\text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{M}},v_{\mathrm{Aiia}}}\text{Aiia}\\ \end{align*}

Equations

The equations are based on mass action kinetics and basic rate laws such as Hill equation.

Assumptions

1. As LuxR is constitutively produced in this model, we considered that LuxR was constant. We used then the conservation of mass to derive the equations
2. On the other hand, the binding and unbinding of LuxR to AHL is fast compared to the synthesis of LuxI. We used the quasi steady state approximation (QSSA)
3. In this model we considered that no AHL diffuses out of the cell.

Simplified equations including degradation by AiiA

\begin{align*} [LuxRAHL]&= \frac{[AHL]\cdot LuxR_\mathrm{tot}}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL])+\frac{a_{\mathrm{LuxI}}(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL]}{dt}&=a_{\mathrm{AHL}}[LuxI]-d_{\mathrm{AHL}}[AHL]-\frac{v_\mathrm{Aiia}\cdot [AHL]}{K_{\mathrm{M,AiiA}}+[AHL]}\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \end{align*}

To see the parameters used, in this section of the model, click here.

Simulations

In the figure below, we simulated the different states of the system: LuxI, AHL and GFP with different AiiA concentrations in the E. coli.

Figure 4. GFP response of the AHL module for various total AiiA concentrations

Single cell model with riboregulated LuxR responsive promoter

Adding a riboregulator

Figure 5. AHL sensor genetic design with riboregulator. A riboregulator controls the expression of LuxI protein

We wanted to check the influence of a riboregulator on GFP output. Looking at the characterization from ETHZ 2014, we saw that riboregulators were able to reduce the leakiness by 65 fold. However, as well as reducing the leakiness, riboregulators reduce the final expression levels by 10 fold. As we have two different promoters controlling the expression of LuxI and GFP. We decided to use one riboregulator only for LuxI, to prevent LuxI accumulation in the cell.

Simplified equations

The equations presented below are equivalent to the ones present below, except that the production rate of LuxI is reduced by 10 fold, and the \(K_M\) value of LuxRAHL is shifted due to the addition of a riboregulator.

\begin{align*} [LuxRAHL]&= \frac{[AHL]\cdot LuxR_\mathrm{tot}}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=L_{Lux,ribo}+\frac{a_{\mathrm{LuxI,ribo}}(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL,ribo}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL,ribo}}})^{n_{lux}}}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL]}{dt}&=a_{\mathrm{AHL}}[LuxI]-d_{\mathrm{AHL}}[AHL]-\frac{v_\mathrm{Aiia}\cdot [AHL]}{K_{\mathrm{M,AiiA}}+[AHL]}\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \end{align*}

Simulation

Below we simulated the system with a riboregulator controlling LuxI. We can see that: The activation time is delayed compared to the case with no riboregulator. Also, AiiA has a much stronger influence on GFP concentration. Indeed, here we need only a concentration of 0.5 μM to drastically reduce the GFP levels, whereas, in the previous model, a concentration of 800 μM was needed to reduce significally GFP concentration.

Figure 6. GFP response of the AHL module with riboregulator. C_AiiA=0.5 μM

Summary and expected limitations

From these models, we figured out that the degradation action of AiiA, our first preferred approach, is not sufficient to produce a significant difference on the GFP output. The riboregulator's leaky expression seem to be too tight to initiate GFP transcription.

Our system should respond to the colocalization of the e-coli cells on the mammalian cell surface. The different concentrations of e-coli in the bulk and around the mammalian cells, as well as the concentration of LuxR, will determine the response of the sensor. However, we anticipate the following problems:

1. If the density of cells on the mammalian surface is not high enough, we will not obtain the desired results.
2. In this model, we were not able to simulate the growth of E. coli on the output.

The reaction-diffusion model is meant to answer these questions. Click here for reaction diffusion models

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