Team:Hong Kong HKUST/Model/Sensing
Team Hong Kong - HKUST
Single Cell ODE Model for the Sensing Module
1 Overview
The first module in our construct is the Sensing module. The purpose of this module is to respond to the concentration of AHL in the cell and amplify the amount of AHL as a signal for further modules of the construct by using a positive feedback loop. In this module, the most important species considered are AHL, LuxR, mRNA, and GFP. The parameter in the simulation is the extracellular AHL concentration which diffuses into an initially AHL free cell.
AHL and LuxR are integral to the function of the function of the module, both being required to induce the pLuxR. Specifically, a (LuxR_AHL)2 molecule induces pLuxR by binding to a region in front of it on the plasmid.
pLuxR is involved in the transcription of mRNA, which in turn in needed for the translation of LuxI and GFP. While ideally, the promoter would only function when induced by (LuxR_AHL)2, in practice it has leakiness and still promotes transcription even in the absence of any AHL, which is problematic due to our usage of a positive feedback loop. The effects of this leakiness and how we chose to combat it shall be described momentarily.
The mRNA transcript is translated to produce LuxI and GFP. LuxI is a synthase for AHL, and completes the positive feedback loop. GFP is used in this module as a reporter for the activity of the system as well as a proxy for the concentration of mRNA.
In our later models, we will also model the effect os adding antisense RNA (asRNA). The binding of asRNA to mRNA to form double stranded RNA (dsRNA) can prevent it from being used to translate any protein. It also increases the degradation rate of mRNA when it is bound in dsRNA due to the presence of the Hfq binding site on the asRNA sequence which is suspected to recruit RNase to degrade the targeted RNA chain.
Fig. 1 Positive Feedback loop of the Sensing Module
2 System of Reactions
We can describe the Sensing module as elaborated above with this set of reactions:
Note that in our construct LuxR is constitutively produceed, and thus, treating its production and degradation as in equilibrium, we assume they are conserved.
Note that in the second equation, which accounts for the diffusion through the membrane, if the intracellular AHL concentration is lower than the extracellular concentration, the rate would become negative and this would represent AHL diffusing into the cell. If the rate is positive, it would represent AHL diffusing out of the cell.
3 Approximations
3.1 Quasi-Steady-State Approximation
To simplify our model, we apply the quasi-steady-state approximation (QSSA). Because the rate of binding and dissociation are much faster than the rate of transcription and translation, we can simplify the set of reactions by treating some reactions as being in steady-state equilibrium.
This approximation is applied to the reactions of
3.2 Solution Volume Approximation
Because the volume of a cell is generally much smaller than the volume of the solution it is in, we apply the approximation that AHL diffusing into or out of the cell has no effect on the extracellular AHL concentration.
This approximation is applied to the diffusion reaction of AHL into and out of the cell.
4 Simulations
To simulate the system, we implemented Euler’s method for approximating ordinary differential equations and applied the approximations as stated.
4.1 Simulation without asRNA
First, we simulate the model without asRNA to show how leakiness can potentially cause problems due to the positive feedback loop.
Fig. 2 Positive feedback loop simulation from [AHL]extracellular = 0 M
The graph shows a simulation where the initial concentration of AHL is 0. The amount of AHL in the cell steadily increases even though no AHL was added, which means that the leakiness of the promoter is creating more AHL, which then induces the promoter in a positive feedback loop. Since the purpose of the Sensing module is to respond to an AHL signal before amplifying it, this is not an acceptable behavior of the system.
4.2 Simulation with asRNA
In the next simulation, we add asRNA. First, we will show simulation results for two levels of concentration, with one high enough to activate the positive feedback loop, and the other being too low.
Note that to show the relative concentration differences between the simulation with and without asRNA, we used the same values for [mRNA]const , [GFP]const , and [AHL]const , setting them as the constitutively produced concentrations in the simulation without asRNA.
The figure on the left shows a simulation from an initial AHL concentration of 0, while the right uses an initial concentration of 10−5 M. As can be seen, the concentration remains low for the first simulation, and rises to an eventual plateau for the second as expected for the module.
The production of mRNA in the left graph can be attributed to the leakiness of the promoter. Meanwhile, the slightly higher value in the middle of the simulation for [AHL] in the right graph can be attributed to the lower net diffusion due to external AHL concentration. As the extracellular AHL gradually degrades, the intracellular concentration also gradually decreases, but always remains at least the constitutively produced level with asRNA.
The following figure shows the AHL concentration for each initial concentration after 200 minutes. For the simulation with asRNA, as can be seen from the graph, the eventual concentrations will have a sharp increase as AHL level increases to more than around 10−8 M. This behavior suggests that the module has concentrations where it is on or off which is desirable in a sensor to respond to changes in AHL . Meanwhile, without asRNA, the response concentration of AHL is at a high value for most initial concentrations, which would hamper its function as a sensor. Both concentrations increase when extracellular [AHL] increases above 10−7, which can be attributed to the higher net diffusion of AHL into the cell independent of the activation of the positive feedback loop itself. However, it can also be seen that the addition of asRNA reduces the response of the system at high concentrations of AHL and so its main advantage is in the sharp boundary between on and off concentrations.
Fig. 5 Sensitivity comparison of construct with and without asRNA at various concentrations
5 Parameters
| Parameter | Value | Justification | Description |
|---|---|---|---|
| [AHL]0 | 1.00E-20 M | Controlled Variable | Initial intracellular AHL concentration |
| [LuxI]0 | 0 M | Controlled Variable | Initial concentration of LuxI |
| [mRNA]0 | 0 M | Controlled Variable | Initial concentration of mRNA |
| [GFP]0 | 0 M | Controlled Variable | Initial concentration of GFP |
| [asRNA]0 | 1.00E-07 M | Estimated | Initial concentration of antisense RNA |
| [pLuxR] | 1.00E-06 M | Estimated | Concentration of pLuxR (equal to concentration of plasmid) |
| [LuxR] | 2.5E-08 M | ETH, 2015 |
Constitutively produced concentration of LuxR |
| kd,LuxR_AHL | 1.00E-07 M | [1] | Dissociation constant of AHL_LuxR |
| kd,(LuxR_AHL)2 | 2.00E-08 M | [1] | Dissociation constant of (AHL_LuxR)2 |
| kd,pLuxR-(LuxR_AHL)2 | 2.00E-07 M | [1] | Dissociation constant of pLuxR-(AHL_LuxR)2 |
| kLeaky | 0.01 | [1] | Ratio between unactivated and activated rate of transcription of pLuxR |
| KpLuxR | 2.5 /min | [1] | Transcription constant of luxR |
| αLuxI | 6.94 /min | [1] | Translation constant of LuxI |
| αAHL | 0.04 /min | [1] | Synthesis constant of AHL via LuxI |
| αGFP | 6.94 /min | [1] | Translation constant of GFP |
| αLuxI | 0.01 /min | [1] | Degradation constant of LuxI |
| d
|
0.001 /min | [1] | Degradation constant of AHL |
| dmRNA | 0.347 /min | [1] | Degradation constant of mRNA |
| dGFP | 0.01 /min | [1] | Degradation constant of GFP |
| dasRNA | 0.44 /min | [2] | Degradation constant of asRNA |
| KLux pL | 4.40E-08 /min | Estimated | Transcription constant for antisense RNA |
| kd,dsRNA | 6.60E-20 M | Obtained in collaboration with CUHK | Dissociation constant between asRNA and mRNA |
| DAHL | 0.167 /min | [1] | Diffusion constant of AHL through membrane |
