Team:Hong Kong HKUST/Model/TimeControl
Team Hong Kong - HKUST
Single Cell ODE Model for the Time Control Module
1 Overview
The second module in our construct is the Time Control module. The purpose of this module is to produce a time delay in responding to the AHL signal inside the cell and to control the translation of two repressors downstream the sensing promoter in order to turn on the expression of Cre recombinase, which is necessary for the recombination reactions in the Recombination module. In transducing the signal through the Time Control module instead of directly passing it to the Recombination module, we seek to allow the Sensing module sufficient time to amplify the AHL signal with its positive feedback loop which would allow for more effective quorum sensing when the AHL diffuses into surrounding cells. Here, we will model the delay produced by the module.
In this module, the most important species considered are AHL, cI, PhlF, and Cre. To recap, AHL and LuxR are integral to the function of the module, both being required to induce the promoter pLuxR. Specifically, a (LuxR_AHL)2 molecule induces pLuxR by binding to a region in front of it on the plasmid. Because we are simulating only the Time Control module in this simulation, we will assume AHL has a constant concentration, being constitutively produced by the Sensing module. Thus, we can treat pLuxR as operating at the full transcription rate.
The promoter pLuxR is involved in the transcription of mRNA, which in turn is needed for the translation of cI.
The promoter pcI is the next promoter in the stream and it is repressed by cI2 and is needed for the translation of PhlF molecules. On pcI, there are two binding sites for cI2 with cooperativity. This means that a cI2 already bound to one site will increase the rate at which cI2 binds to the other site. It will only produce PhlF when both sites are not bound to cI2.
The promoter pPhlF is another promoter needed repressed by PhlF for the translation of Cre. It will only produce Cre when it is not bound and repressed by PhlF.
Under no induction, only the mRNA transcript for PhlF production is produced. Upon receiving (LuxR_AHL)2 inducer, the mRNA transcript is translated to produce cI molecules. Under induction, however, cI will be translated from the mRNA transcript and will inhibit pcI in its dimer form cI2 . This will lead to decreased production of PhlF and this in turn will allow pPhlF to produce Cre.
2 System of Reactions
We can describe the Sensing module as elaborated above with this set of reactions:
Note that in this simulation, since pLuxR is treated as being induced at its maximum level, we simply use its induced transcription rate as its transcription rate.
Reactions which we treat as a regular first order reaction are represented with a one-way arrow ( → ), while those that we apply the quasi-steady-state approximation to are represented with a two-way arrow ( ←→ ).
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.
We apply this approximation to the following reactions:
Furthermore, we also make the approximation that the promoters are much fewer than the number of repressors, so we make the approximation that reaction of a repressor binding to a promoter does not reduce the repressor’s concentration. We can make this approximation because the number of promoters is equal to the plasmid copy number while the number of repressor is usually much higher for any significant concentration.
3.2 Non-Cooperativity Approximation
Because modelling the effects of cooperativity of the two binding sites on pcI would add significant complexity to the model, we decided to ignore it and simply treat the binding reactions as independent from each other. That is, the binding of cI to a site of pcI has the same rate regardless of whether the other site has a cI bound.
Thus, under this simplification, the percentage of unbound pcI simply becomes the product of the unbound percentages of both types.
4 Simulations
To simulate the system, we implemented Euler’s method for approximating ordinary differential equations and applied the approximations as stated.
4.1 Control without Time Control
Fig. 1 Control setting where Cre is directly induced
As can be seen, from a high AHL concentration produced by the Sensing module, Cre saturates quite quickly and already reaches 80% of its maximal level at around 50 minutes.
Because we want to allow the Sensing module to reach sufficiently high levels of AHL before activating the recombinase production, it is then necessary to add a Time Control module. As can be seen in the Sensing module simulation, it takes about 200 minutes to reach 80% of maximal AHL concentration levels.
4.2 Adding Time Control
Above is the simulation result of the Time Control module and a control where Cre is directly induced.
On the left, we have a simulation of the module without any initial AHL. As can be seen from the graph, the concentration of cI2 and Cre do not increase from 0 as desired, while the concentration of PhlF remains high, continuously repressing Cre. This shows that if the Sensing module does not produce AHL, then there will be no production of Cre as desired
On the right, the initial AHL is at the maximal level, and cI2 is constitutively produced. The concentration of cI2 increases quickly which then represses PhlF and causes its concentration to fall. When PhlF has a sufficiently low concentration, Cre begins to be produced. Compared to the control without our Time Control module, the Cre takes about 150 minutes longer to reach 80% of maximum Cre levels. This should allow the Sensing module more time to produce AHL and aid in efficient quorum sensing activation of surrounding cells.
5 Parameters
Due to the limited availability of constants in the literature, we estimated multiple constants based on the rates of similar proteins or reactions.
| Parameter | Value | Justification | Description |
|---|---|---|---|
| [CI]0 | 0 M | Controlled Variable | Initial concentration of cI |
| [CI2]0 | 0 M | Controlled Variable | Initial concentration of cI2 |
| [PhlF]0 | 1.33E-04 M | Estimated constitutively produced concentration of PhlF | Initial concentration of PhlF |
| [Cre]0 | 0 M | Controlled Variable | Initial concentration of Cre |
| [pLuxR] | 1.00E-06 M | Estimated | Concentration of pLuxR |
| [pcI] | 1.00E-06 M | Estimated | Concentration of pcI |
| [pPhlF] | 1.00E-06 M | Estimated | Concentration of pPhlF |
| kd,cI2-A | 2.00E-07 M | Estimated | pcI-cI2 Dissociation constant (Site 1) |
| kd,cI2-B | 2.00E-07 M | Estimated | pcI-cI2 Dissociation constant (Site 2) |
| k-cI2 | 3 /M/min | [3] | cI2 dedimerization rate |
| k+cI2 | 3.00E+09 /min | [3] | cI2 dimerization rate |
| kd,PhlF | 2.00E-07 M | Estimated | pPhlF-PhlF Dissociation constant |
| αcI | 6.94 /min | Estimated | Synthesis constant of cI |
| αCre | 2 /min | Estimated | Synthesis constant of Cre |
| αpPhlF | 5.574 /min | [3] | Synthesis constant of PhlF |
| dcI | 0.042 /min | [3] | Degradation constant of cI |
| dPhlF | 0.042 /min | Estimated | Degradation constant of PhlF |
| dCre | 0.042 /min | Estimated | Degradation constant of Cre |
