Quorum sensing and cell lysis

We developed a mathematical model of the signal amplification module of the Mantis system. Using quorum sensing and a lytic mechanism, the goal of this module is to take a weak antigen signal only detected by a minority of the biosensor cells, and to turn this into a population-wide response. We explored the role of various reaction rates on the behavior of the system. Our mathematical model shows that the QS system can spontaneously activate, be insensitive to AHL, or function properly. In our most complete model, 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.

System can be insensitive to AHL
System can spontaneously auto-activate
Adjust LuxR degradation
Add aiiA enzyme
Lab-tunable parameters be used to obtain desired functionality

Introduction

The Mantis biosensor can use an optional computational module. A detection module can be directly linked to a reporter module. When using the quorum sensing (QS) system, detection of an antigen will trigger the producing of the QS signaling molecule AHL. Only when AHL levels are sufficiently high, cells will activate their reporter module.

Adding a computational module to Mantis can make it more sensitive to very low levels of antigen, making it able to diagnose diseases that are difficult to detect. As the QS does depend on transcription and translation, the QS system will make the diagnostic test slower. But different circumstances require diagnostics that are either very rapid, but not very sensitive, or are very sensitive, but take more time until a result is produced. Therefore, the computational module can be very useful to whole-cell biosensors.

The signaling molecule 3-oxo-hexanoyl homoserine lactone (AHL) is central to QS. Together with the two proteins proteins LuxR and LuxI their allow bacteria in nature produce a signaling molecule and activate genes when the bacteria reach a high enough cell density. AHL can freely diffuse across the cellular membrane. Inside the cell, it can bind with LuxR, dimerize, and become an active transcription factor. The LuxR-AHL dimer RA₂ can result in positive and negative regulation of genes. The pLuxR promoter is positively regulated while pLuxL is negatively regulated by RA₂.

Figure 1: A general overview of the QS module of Mantis. Some cells will detect antigen and start producing AHL. Other cells will detect AHL and the entire population will start QS, triggering cell lysis. The release of cellular components upon lysis allows the two protein of the interdependent component system to generate a fluorescent signal.

In our Mantis module, QS is coupled to a lytic mechanism. Initially, there will be two bacterial populations. Each population has contains a different protein that is part of a two component interdependent system. Together, proteins from each population can generate a fluorescent signal. THe cellular membranes will prevent the proteins from interacting, so initually there will be no fluorescence.

Figure 2: The molecular components of the Mantis QS and lytic system. Antigen will activate QS (top left). QS will initiate cell lysis (top right). Cell lysis will result in a fluorescent signal (bottom).

Once some cells detect antigen, they will start producing AHL. This will also activate in the vicinity of cells that detect antigen, eventually causing all cells in Mantis to start QS. The active form of LuxR will activate the lytic mechanism, resulting in lysis of all cells engaging in QS. Once cell contents are lysed, the two complimentary parts of the interdependent component system will start to interact. This will result in a fluorescent signal, which will give a read-out on the Mantis device.

In the QS system, LuxI positively regulates it’s own expression. This can lead to positive feedback, where more and more AHL is produced. Cells with this positive feedback motif can therefore become activated. This is important, as the lytic mechanism should either be active or completely inactive.

To achieve this goal, the temporal model was first developed. The temporal model was used to sample the global parameter space. The second model that we used is the spatiotemporal model. This model was used to test hypotheses and to guide wet-lab experiments.

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

Bistability: Upon activation by an external signal, cells must be able to switch from an OFF-state to an ON-state. This requires that the time-dependent system of ordinary differential equations (ODEs) has more than one equilibrium state solution. 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 remains unclear if the native lux box (BBa_K546000) 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 FP. But they should do so only after signaling cells still in the OFF-state.

Temporal model

The main goal of the simple 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.

Assumptions

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.

Equations

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 $SplitFP_{cell}$

Methods

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.trapezoid() 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.

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

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

Results

Global parameter sampling

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

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 both the external AHL degradation rate and LuxR degradation rate seemed to be interesting parameters, we took one spontaneously activating parameter set and in this set varied both these two interesting parameter values from 0 to 10. The scores for these 10 times 10 parameter sets can be visualized in a heatmap (figure 6). This heatmap shows the previously described types of behavior and how the desired behavior is the transition area between spontaneous lysis and lysis only of the activator cells and that by reducing the sensitivity of the system to AHL, for example by increasing the degradation rate of LuxR. This heatmap also shows that the external AHL degradation controls the thickness of the transition area and that for high values the transition area of desired system behavior disappears.

Figure 4: Heatmap of the scoring function when varying both the external AHL degradation rate and the LuxR degradation rate. This shows the transition area of desired behavior in between spontaneous lysis and lysis of only the activator cells.

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 parameter 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.

test

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 molecule, AHL
Diffusion of fluorescent proteins
Cell movement
Asymmetric cell division
Variable growth rates

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