Difference between revisions of "Team:Wageningen UR/Model/QStest"

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<p>As the assumption that AHL diffuses instantaneous breaks down when the system becomes larger, or when the number of activator cells becomes lower, we developed a spatial model that incorporates both diffusion of AHL, but also of the mixing of the split fluorescent proteins.
 
<p>As the assumption that AHL diffuses instantaneous breaks down when the system becomes larger, or when the number of activator cells becomes lower, we developed a spatial model that incorporates both diffusion of AHL, but also of the mixing of the split fluorescent proteins.
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To more realistically model the biological system, we used the spatial model with the following features:</p>
 
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                                <li>Diffusion of the signaling moledule, AHL</li>
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                                <li>Cell movement</li>
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                                <li>Asymmetric cell division</li>
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Im the ODE model the cell density had remained constant throughout the simulation. Now, in the spatial model cell density will increases exponentially.</p>
  
 
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This spatial model is comprised out of a 2d grid where each square is the size of a single bacterial cell. Squares are either external medium, or completely occupied by a single cell. Diffusion occurs through the face that each square has with it's top, left, right, and bottom neighbor.
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This spatial model is comprised out of a 2d grid where each square is the size of a single bacterial cell. Squares are either external medium, or completely occupied by a single cell. The grid system gives every cell a definite position while at the same time discritizing space. The diffusion equation is approximated using the finite volume approximation, which relies on this discritization of space. Diffusion occurs through the face that each square has with its top, left, right, and bottom neighbor.
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In the finite volume method, all volume elements contain a certain concentration of AHL. This concentration is approximates the average concentration pf AHL in that volume element in the analytical solution. The finite volume method conserves the amount of AHL in the system; the flux that leaves a certain square automatically flows into an adjacent square.
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Diffusion rates from a cell to a medium square or from one medium square to another medium square have different diffusion coefficients, where the diffusion coefficient of AHL inside the external medium is much larger. When two cells are in adjacent squares, diffusion is still possible, but at a much slower rate.
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Cell movement in this model is modeled by a random movement where cells have a certain probability to move to the four adjacent squares as well as the four diagonal squares. The model therefore does not realistically model chemotaxis or other forms of bacterial movement.
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When a cell tries to move into a square that is already occupied, movement is prevented and the cell simply stays in their current square. When a cell moves into a unoccupied square, the AHL concentration of the original square and the destination square get switched around, as squares occupied by cells have no external medium AHL values.
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Cells don't change in size in this model. A bacteria is always the size of exactly one square. All cells do have their own growth rate. Once the cell has become big enough to divide, the simulation simply creates a new cell in an adjacent square and distributes all cellular species among both daughter cells. Every species is independently and asymmetrically distributed by drawing a number from a normal distribution (µ=0.5, σ=0.2).Once the cells are divided, the new cell is awarded a growth rate independent of that of the parent cell and both cells have their growth rate clock reset to zero.
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Cells that lyse still release their AHL and fluorescent protein contents the same way as in the ODE model, except in the spatial method the square the lysing cell occupies becomes a external medium square with the same concentration of extracellular AHL as had been the cellular AHL concentration of that cell upon lysis.
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Revision as of 22:39, 25 October 2017

Quorum sensing and cell lysis

We developed a mathematical model of the quorum sensing and cell lysis system. The goals of the mathematical model are the following. To characterize the theoretical properties of the system. To identify which kinetic parameters dictate the behavior of the system. To determine which parameter, that can be changed in the lab, can be varied to obtain a system with the correct behavior. To achieve this goal, the Simple model was first developed. The Simple model was used to sample the global parameter space. The second model that we used is the Spatial Model. This model was used to test hypothesis and to guide wet-lab experiments.
With the modeling of the quorum sensing and cell lysis system we had the following goals.

  • Characterize system behavior
  • Sample global parameter space
  • Identify parameters with high sensitivity
  • Find lab-tunable parameters
  • Test hypothesis

Summary

Our mathematical model shows that the QS system can spontaneously activate, function properly, or be insensitive to AHL. Changing the parameters of LuxR degradation and AHL cellular degradation allow a system to transition from one behavior to the other. In the diffusion model of our QS and cell lysis system, we can change a spontaneously activating system into a system with proper signaling behavior by either increasing LuxR degradation, or by adding the enzyme aiiA to the genetic circuit.

For the system to function properly, it needs the following behaviors.


Bistability: Cells must be able to switch from an OFF-state to an ON-state. This requires that the system of ODE equations has more than one equilibrium state solutions. The bistability of switches based on the lux operon have been explored in the literature. A motif with two feedback loops has been shown to have bistability[1,2]. However, it means unclear if the native lux box has the ability to switch from a clear OFF-state to an ON-state[3].
Cell signaling: Cells in the ON-state should be able to signal cells in the OFF-state to also transition to the ON-state. This is a core property of the quorum sensing system.[4]
Delayed cell lysis: Cells in the ON-state should lyse to release their split fluorescent protein. But they should do so only after signaling cells still in the OFF-state.

ODE model

The main goal of the ODE model is to allow for a sampling of the parameter space of the kinetic rate constants of the model. To effectively do this, the biological picture has to be translated to mathematical equations. Furthermore, this model is more manageable and more straightforward to interpret when the number of kinetic parameters are low. To achieve this, the following assumptions have been made.

System

The system consists out of three compartments; the external medium, antigen-activated bacterial cells, and signal-receiving bacterial cells. The equations of both cells are identical, except that the activator cells receive an external signal intended to put the cells in the ON-state halfway during the simulation. This simulates the addition of antigen-containing blood to the bacteria and detecting of this low concentration of antigen by a minority of the cells. The total size of the system is 1 000 000 volume units. The ratio of external volume to cells is 1 cell volume unit in 1000 total volume units. This approximates a cell density of 1.0 OD600. Therefore, the total cell volume is 1000 volume units. As the volume unit is defined as the volume of a single cell, there are a total 1000 cells in this system. Out of these cells, 900 cells are receiver cells, 100 are activator cells. The two cellular compartments in the model actually represent only a single cell. When cellular compartments exchange species with the external medium compartment, the total amount of exchange is calculated by using the total for each cell type. This means that all receiver cells are identical copies of each other. The same is true for the and activator cells.

Figure 1: The ODE model has three different compartments, the external medium, activator cells, and receiver cells. Diffusion is assumed to be instantaneous inside each compartment. Diffusion across the cell membrane is computed using a linear approximation, as indicated by the 'flux' arrows. ODE equations describe the rate of change of all species inside the three compartments.

Each cell has concentration values for the following species: AHL, LuxI, LuxR, [AHL-LuxR] complex, split fluorescent protein and lysis protein. In the external medium, there are concentration values for AHL, C-terminal split Fluorescent protein, N-terminal split fluorescent protein, and the fused and active fluorescent protein. All compartments are assumed to be well-mixed, having a homogeneous concentration. All cells are identical in size and their size is constant throughout the simulation. There is assumed to be no cell growth and therefore no dilution term. There is no spatial component to any reaction in this model.

Kinetics

The levels of mRNA of all genes expressed is assumed to be in a quasi steady state. The basis for this assumption is that translation is much slower than transcription. Therefore, there are no concentration values for any mRNA species in the model. Similarly, transcription factors binding to DNA is also assumed to quasi steady state. The assumption here is that binding and unbinding of a transcription factor to a promoter occurs at timescales much faster than translation. The third assumption is that the dimerization of [AHL-LuxR] is also in a quasi steady state. While there has been some debate on whether [AHL-LuxR] complex formation or [AHL-LuxR] dimer formation occurs faster. We decided to go with the assumption that dimer formation is faster and in quasi steady state. Therefore, there is no concentration value for the [AHL-LuxR] dimer in the model. To summarize, transcription factor-DNA complexes, mRNA, and [AHL-LuxR] dimer concentrations are implicitly modeled. All promoters are assumed to have a single transcription binding site and the transcription factor is assumed to form a dimer only, giving a Hill coefficient of 2.

The lysis mechanism is not implicitly modeled. Lysis is assumed to occur instantaneously the moment the lysis protein value exceeds a value of 1.0. Before lysis protein reaches this concentration, the lysis protein is inert. Diffusion of AHL across the cell membrane is modeled by a linear diffusion equation depending on the concentration difference inside and outside of the cell, and a diffusion constant for diffusion of AHL across the cell membrane. This diffusion constant is assumed to be the same whether AHL diffuses into out ouf of the cell. As stated earlier, the compartments are considered to be well-mixed, which equates to assuming that both the diffusion of AHL inside each compartment is instantaneous. Similarly, the split fluorescent proteins released upon cell lysis is assumed to be instantaneous.

Except for the fluorescent proteins in the external medium, all species have degradation rate. The split and fused fluorescent proteins are considered to be sufficiently stable so that degradation outside the cell plays no role on simulation length timescales. There are no values of LuxR, LuxI and [AHL-luxR] inside the external medium. Upon lysis of the cell, only AHl and the split fluorescent proteins are released into the external medium. AHL bond to LuxR and LuxI inside lysing cells simply disappear from the model.
Some of these assumptions may result in a model that fails to incorporate important dynamics. To accurately sample the 26-dimensional parameter space, the computational time of a single simulation can not be longer than several seconds. The spatial model, described in the second part of this page, alleviates some of the assumptions that are made here.

Given this biological picture and incorporating the above assumptions, the model can be described by a system of ordinary differential equations (ODEs), which describe the change in time of each biological component part of the system.


Ordinary differential equations

${\mathrm{d}AHL_{cell} \over \mathrm{d}t} = \alpha_{1}LuxI + D_{AHL}(AHL_{ext}-AHL_{cell})+k_{-1}RA-k_1 LuxR \cdot AHL -\beta_{1}AHL_{cell}$

${\mathrm{d}AHL_{ext} \over \mathrm{d}t} = V_{ratio} \cdot D_{AHL} (AHL_{ext}-AHL_{cell}) - \beta_{2}AHL_{cell}$

${\mathrm{d}RA \over \mathrm{d}t} = k_{-1}LuxR \cdot AHL - \beta_{3}RA^{2} - \beta_{4}RA$

${\mathrm{d}LuxI \over \mathrm{d}t} = \alpha_2 + \alpha_3 {{LuxR \cdot RA^2} \over {km_1 + RA^2}} + \alpha_4 + S \cdot \alpha_5 - \beta_5 LuxI$


Positive regulation of LuxR:
${\mathrm{d}LuxR \over \mathrm{d}t} = \alpha_6 + \alpha_7 {{LuxR \cdot RA^2} \over {km_1 + RA^2}} + \alpha_8 + S \cdot \alpha_9 - \beta_6 LuxR$


Negative regulation of LuxR:
${\mathrm{d}LuxR \over \mathrm{d}t} = \alpha_6 + \alpha_7 {{LuxR \cdot km_1} \over {km_1 + RA^2}} + \alpha_8 + S \cdot \alpha_9 - \beta_6 LuxR$

${\mathrm{d}Lysis \over \mathrm{d}t} = \alpha_{10} + \alpha_{11} {{LuxR \cdot RA^2} \over {km_2 + RA^2}} - \beta_7 Lysis$

${\mathrm{d}SplitFP_{cell} \over \mathrm{d}t} = \alpha_{12} - \beta_8 SplitFP$

${\mathrm{d}SplitFP_{N} \over \mathrm{d}t} = -k_3 SplitFP_{N} \cdot dSplitFP_{C}$

${\mathrm{d}SplitFP_{C} \over \mathrm{d}t} = -k_3 SplitFP_{N} \cdot dSplitFP_{C}$

${\mathrm{d}ActiveFP \over \mathrm{d}t} = k_3 SplitFP_{N} \cdot dSplitFP_{C}$


Species

$AHL_{cell}$ Signaling molecule AHL inside the cell

$AHL_{ext}$ Signaling molecule AHL inside the external medium

$LuxR$ Free inactive LuxR protein

$RA$ The LuxR-AHL complex

$RA_2$ The dimer of the LuxR-AHL complex and active transcription factor

$LuxI$ The AHL-producing enzyme LuxI

$LuxR$ The AHL-producing enzyme LuxI

$Lysis$ The lysis protein responsible for the lytic mechanism

$SplitFP_{cell}$ The split fluorescent protein in the cell

$SplitFP_{N}$ The N-terminal part of the split fluorescent protein in the external medium

$SplitFP_{C}$ The C-terminal part of the split fluorescent protein in the external medium

$ActiveFP$ The active full form of the split fluorescent protein in the external medium


Parameters

$\alpha_1$ AHL production rate of LuxI

$\alpha_2$ Leaky production rate of LuxI from pLuxA

$\alpha_3$ Maximum production rate of LuxI from pLuxA

$\alpha_4$ Leaky production rate of LuxI from pLuxB

$\alpha_5$ Maximum production rate of LuxI from pLuxB

$\alpha_6$ Leaky production rate of LuxR from pLuxB

$\alpha_7$ Maximum production rate of LuxR from pLuxA

$\alpha_8$ Leaky production rate of LuxR from pLuxB

$\alpha_9$ Maximum production rate of LuxR from pLuxB

$\alpha_{10}$ Leaky production rate of Lysis from pLuxB

$\alpha_{11}$ Maximum production rate of Lysis from pLuxB

$\alpha_{12}$ Constitutive production of $SplitFP_{cell}$

$k_1$ formation rate of [LuxR-AHL]

$k_{-1}$ dissociation rate of [LuxR-AHL]

$k_{-2}$ dissociation rate of RA

$k_{3}$ formation & maturation rate of the full fluorescent protein

$km_{1}$ Michaelis-Menten constant of pLuxA for $RA$

$km_{2}$ Michaelis-Menten constant of pLuxB for $RA$

$D_{AHL}$ diffusion rate of AHL across the cellular membrane

$\beta_1$ degradation rate of $AHL_{cell}$

$\beta_2$ degradation rate of $AHL_{ext}$

$\beta_3$ degradation rate of the $RA$ dimer

$\beta_4$ degradation rate of $RA$ complex

$\beta_5$ degradation rate of $LuxI$

$\beta_6$ degradation rate of $LuxR$

$\beta_7$ degradation rate of $Lysis$

$\beta_8$ degradation rate of $SplitFP_{cell}$

The system of ODE equations was solved using the scipy.integrate.odeint module available in Python. To sample the global parameter space, 1 000 000 parameter sets were generated using the lhs() function from pyDOE. These parameter sets were simulated for a total of 240 time units, where between t=120 and t=180, the external signal S was set from 0 to 1. A value for 1.0 was chosen to represent enough lysis protein for cell lysis to occur. Every 0.01 timesteps, the value of lysis protein in each cell was checked. In the case that a cell has a lysis protein value above 1.0, the cell would be removed from the simulation and the AHL and split fluorescent protein contents would be released into the external medium. All released fluorescent protein was always 50-50 of both C- and N-terminal versions. This was done to avoid having to introduce random numbers.

Scoring function

To properly score how well the parameter set resulted in the desired behavior, the following scoring scheme was used. The area under the curve of the fluorescence timeseries was approximated using the numpy.np.trapz() function. As an ideal parameter set generates no fluorescence before t=120 and very quickly generates a lot of fluorescence after t=120, the area under the curve before t=120 should be minimal, while after t=120, the area should be as large as possible. The area under the curve represents both how much, as well as how quickly, fluorescence is being generated. Secondly, in absence of an external input, so in case S = 0 during the entire simulation, there should be no fluorescence generated. Fluorescence in absence of an antigen would amount to a false positive. Therefore, each parameter set is simulated twice. Once with no external signal, and once with an external signal between t=120 and T=180. The final score of a parameter set is calculated by combining both simulation runs. As the objective function has to be minimalized. This leads to the following equations for the scoring function.

$Score_{simulation} = \displaystyle\int_{t=0}^{t=120} F \mathrm{d}t + {1 \over {\displaystyle\int_{t=120}^{t=240} F \mathrm{d}t + 0.01}}$

$Score_{set} = {900 \over { |Score_{noAntigen} - Score_{Antigen}|+ 0.9} } + 0.1 \cdot Score_{Antigen}$

Activator and receiver cells

In the Mantis system, antigen concentrations will be low and therefore only a small number of cells that come into contant with antigens. For the ODE model, we assumed that 10% of all cells would come into direct contant with the antigen, allowing them to detect the antigen. These cells are the activator cells. In the model, detection of antigen is simulated by switching on LuxI and LuxR production in 10% of the cells. The receiver cells are identical to the activator cells in all respects, except for this activation of LuxR and LuxI production.

Results

Playing around with the model, the following types of system behaviors can be categorized:

No lysis: None of the cells in the system lyse. Even the activator cells, that are directly induced by the external signal, are unable to produce enough lysis protein within the duration of the simulation.
Activator lysis only: Activator cells are able to lyse, release their fluorescent protein and generate some fluoresence. But the majority of the cell population remain unaffected. There is no cell-to-cell signaling through quorum sensing.
Spontaneous lysis: All cell lyse within the duration of the simulation. But the lysis is not caused by induction of the activator cells by the external signal. Spontaneous auto-activation and cell lysis are caused by the build-up of AHL over time. This build-up is caused by leaky expression of LuxR and LuxI and in the case of a system overly sensitive to AHL, this leaky expression will lead to auto-activation. Receiver cells will thus auto-activate using AHL they themselves produce, in the abscence of antigen.
Desired behavior: Activator cells produce AHL and this triggers the receiver cells to auto-activate as well. At the end of the simulation, all cells will have lysed and released their split fluorescent protein, resulting in a significant fluorescent signal. However, there is no spontaneous auto-activation. The fluorescent signal is caused by the presence of antigen. The increase of fluorescence over time of a simulation with the desired behavior is displayed in figure 2.

Global parameter sampling

A total of 1 million parameter sets were sampled for the system with positively regulated LuxR.

Figure 2: Time series of the fluorescence produced by a parameter set with the desired behavior. The external signal simulating the presence of antigen is turned on at t=600. At t=607, the first increase of fluorescence starts. This fluoresence is produced by the split fluorescent protein of the 10% activator cells in the model. At t=638, there is a second increase of fluorescence. At this time, the receiver cells lyse, releasing the bulk of the fluorescent protein.
Table 1: Results of simulating 1 million parameter sets using latin hypercube sampling. LuxR was positively regulated by the dimer of the LuxR-AHL complex
Behavior # parameter sets Percentage
No lysis 778326 77.8%
Activator lysis - -
Spontaneous lysis 10986 1.1%
Induced full lysis 21 0.002%
Table 2: Results of simulating 1 million parameter sets using latin hypercube sampling. LuxR was negatively regulated by the dimer of the LuxR-AHL complex
Behavior # parameter sets Percentage
No lysis 813045 81.3%
Activator lysis 146105 14.6%
Spontaneous lysis 40842 4.1%
Induced full lysis 8 0.001%

The variation in parameters for each behavior was visualized by generating a histogram showing how many parameter sets fell within a certain parameter range for each behavior.

Figure 2: Comparing parameter sets that lyse and those that do not lyse, low values for the degradation rates of the [LuxR-AHL] dimer, the [LuxR-AHL] complex, and LuxR are overrepresented in sets that lyse. Low degradation rates of species containing LuxR are associated with systems are more likely to lyse. The same is true for the LuxI protein, responsible for AHL production.

A similar trend is seen in the data of the negative regulation of LuxR.

Figure 3: Comparing parameter sets that lyse and those that do not lyse, low values for the degradation rates of the [LuxR-AHL] dimer, the [LuxR-AHL] complex, and LuxR are overrepresented in sets that lyse. Low degradation rates of species containing LuxR are associated with systems are more likely to lyse. The LuxI rate of parameter sets that lyse are uniformly distributed, suggesting the LuxI rate has no significant effect.

As the number of parameter sets that quorum sense and lyse in response to antigen were very low in both cases, it was impossible to say something about the difference between the parameters with the desired behavior and those that lysed spontaneosly. Therefore, the local parameter space around these parameter sets was explored. By scoring parameter sets in the local vicinity of the parameter sets discovered by the global search, parameter sets with the desired behavior could be compared to those that lysed spontaneously. While the two types of parameter sets were still quite similar, a different behavior could be seen for the extracellular degradation of AHL and the degradation rate of LuxR (figure 4 and 5).

Figure 4: Parameter sets with spontaneous self-activation have higher external AHL degradation rates while parameter sets with the desired behavior have lower external AHL degradation rates.
Figure 4: The paramter sets with high values of LuxR degradation very rarely have spontaneous lysis when compared to those with the desired behavior. There is a clear trend of a shift towards higher values of LuxR for the desired behavior.
Figure 6: Several high blue peaks are visible for low values of external AHL degradation rates. Of the parameter sets simulated that had high external AHL degradation rates, very few of them had the desired behavior.
Figure 7: The parameter sets with the desired behavior have a clear shift towards higher rates of LuxR degradation when compared to the spontaneous lysis parameter sets.

Spatial Model

As the assumption that AHL diffuses instantaneous breaks down when the system becomes larger, or when the number of activator cells becomes lower, we developed a spatial model that incorporates both diffusion of AHL, but also of the mixing of the split fluorescent proteins. To more realistically model the biological system, we used the spatial model with the following features:

  • Diffusion of the signaling moledule, AHL
  • Diffusion of fluorescent proteins
  • Cell movement
  • Asymmetric cell division
  • Variable growth rates

Im the ODE model the cell density had remained constant throughout the simulation. Now, in the spatial model cell density will increases exponentially.

This spatial model is comprised out of a 2d grid where each square is the size of a single bacterial cell. Squares are either external medium, or completely occupied by a single cell. The grid system gives every cell a definite position while at the same time discritizing space. The diffusion equation is approximated using the finite volume approximation, which relies on this discritization of space. Diffusion occurs through the face that each square has with its top, left, right, and bottom neighbor. In the finite volume method, all volume elements contain a certain concentration of AHL. This concentration is approximates the average concentration pf AHL in that volume element in the analytical solution. The finite volume method conserves the amount of AHL in the system; the flux that leaves a certain square automatically flows into an adjacent square. Diffusion rates from a cell to a medium square or from one medium square to another medium square have different diffusion coefficients, where the diffusion coefficient of AHL inside the external medium is much larger. When two cells are in adjacent squares, diffusion is still possible, but at a much slower rate.

Cell movement in this model is modeled by a random movement where cells have a certain probability to move to the four adjacent squares as well as the four diagonal squares. The model therefore does not realistically model chemotaxis or other forms of bacterial movement. When a cell tries to move into a square that is already occupied, movement is prevented and the cell simply stays in their current square. When a cell moves into a unoccupied square, the AHL concentration of the original square and the destination square get switched around, as squares occupied by cells have no external medium AHL values. Cells don't change in size in this model. A bacteria is always the size of exactly one square. All cells do have their own growth rate. Once the cell has become big enough to divide, the simulation simply creates a new cell in an adjacent square and distributes all cellular species among both daughter cells. Every species is independently and asymmetrically distributed by drawing a number from a normal distribution (µ=0.5, σ=0.2).Once the cells are divided, the new cell is awarded a growth rate independent of that of the parent cell and both cells have their growth rate clock reset to zero. Cells that lyse still release their AHL and fluorescent protein contents the same way as in the ODE model, except in the spatial method the square the lysing cell occupies becomes a external medium square with the same concentration of extracellular AHL as had been the cellular AHL concentration of that cell upon lysis.

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Results

From ODE to spatial

After developing the spatial mode, we first explored the differences between the spatial model and the original ODE model by taking parameter sets with the desired behavior from the ODE model and simulating them in the spatial model. The main difference between the spatial and the ODE model is neighbor-to-neighbor cell signaling. In the spatial model, AHL is being generated by an activator cell, generating a local cloud of higher AHL concentrations. This changes the sensitivity of the model to the ratio between activator cells and receiver cells. Under the right parameters, an activator cell can activate it's nearest neigbhors regardless of the total system size. In some simulations, a single cell activated by antigen can signal to and activate the entire population, as a wave of activation travels through the system.

Spatial model and aiiA

We took a spontaneously lysing parameter set from the ODE model and transfered it to the spatial model. Production and degradation terms for aiiA were added to the system of ODE's, as well as degradation of AHL by aiiA. Simulations were run both with and without antigen induction. With the right parameters associated with aiiA, a parameter set with spontaneous self-activation and lysis can be adjusted into a parameter set where cells fully activate and lyse only in response to antigen (table 3).

Table 3: A system with positive regulation of LuxR that self-activated spontaneously was modified by adding aiiA or increasing the degradation rate of LuxR, resulting in the desired behavior.
Score Animation, no antigen Animation, antigen
Original set 99.98 link link
Original Set & aiiA 0.22 link link
Original Set & increased LuxR deg. 0.24 link link
Table 3: A system with negatively regulation of LuxR that self-activated spontaneously was modified by adding aiiA or increasing the degradation rate of LuxR, resulting in the desired behavior.
Score Animation, no antigen Animation, antigen
Original set 99.97 link link
Original Set & aiiA 0.059 link link
Original Set & increased LuxR deg. 0.084 link link

By adding aiiA to our model, we can simulate a model where cells are able to communicate the presence of antigen, resulting in a population-wide signal. But in the absence of antigen, cells remain inactive and no fluoresence is generated.


Conclusion

In the global parameter space, only a very small amount of parameter sets have the desired behavior. However, global parameter sampling also naievely samples parameters while the native quorum sensing system evolved to carry out its biological function. Furthermore, from the lab we know that the lux operon is prone to self-activation, even at low cell densities. The desired behavior lies in the transition area when moving from spontaneous activation to no quorum sensing. By lowering the sensitivty of the quorum sensing system, a system will move towards that area of the parameter space where the desired behavior is found. By comparing the topologies of either positive LuxR regulation of negative LuxR regulation, we found that having positive LuxR regulation is more likely to result in the desired behavior. The explanation here is that in the case of negative regulation of LuxR, all cells start out with high levels of LuxR, making them very sensitive to AHL and putting them at risk to self-activate. In the formation of the LuxR-AHL complex, LuxR is more likely to be a limiting factor. This explains why the degradation rates of LuxR, the LuxR-AHL complex as well as the LuxR dimer showed such high parameter sensitivity. We hypothized that negative regulated LuxR system may have a faster response time and that they would be able to switch from a state where they would take up and sense AHL to a state where they would produce and release AHL. While we did find some evidence to suggest negatively regulated systems are faster, and indeed the parameter set that both had the desired behavior and was very quick in generating a fluorescent signal was one with negative LuxR regulation, we were not able to fully test this hypothesis.

We hypothesized that such a strategy may be attempted by introducing the aiiA enzyme. Therefore, we tested this hypothesis with our mathematical model and found that by introducing aiiA, the sensitivity of a system to AHL can be reduced while maintaining the ability of the system switch from an OFF-state to an ON-state, self-activate, and cell-signal to its neighbouring cells. Both for positive and negative regulation of LuxR, we were able to show that only by adding aiiA, the desired functionality can be obtained.

Similarly, we found that the degradation rate of LuxR is a parameter that plays a large role in determining the system properties. As the LuxR degradation is tunable in the lab, by adding a degradation tag to the protein, we also explored increasing the LuxR degradation rate. Here we find similar results, where for some spontanously self-activating parameters, we can obtain the desired behavior by only adding aiiA to the system. This was also succesful for both positive and negative regulation of LuxR.

References

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