Team:Wageningen UR/Model/QStest
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.
Cell signaling: Cells in the ON-state should be able to signal cells in the OFF-state to also transition to the ON-state.
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.
Simple 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.
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.
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.
${dAHL_{cell} \over dt} = \alpha_{1}LuxI + D_{AHL}(AHL_{ext}-AHL_{cell})+k_{-1}RA-k_1 LuxR \cdot AHL -\beta_{1}AHL_{cell}$
${dAHL_{ext} \over dt} = V_{ratio} \cdot D_{AHL} (AHL_{ext}-AHL_{cell}) - \beta_{2}AHL_{cell}$
${dRA \over dt} = k_{-1}LuxR \cdot AHL - k_{-2}RA^{2} - \beta_{3}RA$
${dLuxI \over dt} = \alpha_2 + \alpha_3 {{LuxR \cdot RA^2} \over {km_1 + RA^2}} + \alpha_4 + S \cdot \alpha_5 - \beta_4 LuxI$
Positive regulation of LuxR:
${dLuxR \over dt} = \alpha_6 + \alpha_7 {{LuxR \cdot RA^2} \over {km_1 + RA^2}} + \alpha_8 + S \cdot \alpha_9 - \beta_5 LuxR$
Negative regulation of LuxR:
${dLuxR \over dt} = \alpha_6 + \alpha_7 {{LuxR \cdot km_1} \over {km_1 + RA^2}} + \alpha_8 + S \cdot \alpha_9 - \beta_5 LuxR$
${dLysis \over dt} = \alpha_{10} + \alpha_{11} {{LuxR \cdot RA^2} \over {km_2 + RA^2}} - \beta_6 Lysis$
${dSplitFP_{cell} \over dt} = \alpha_{12} - \beta_7 SplitFP$
${dSplitFP_{N} \over dt} = -k_3 SplitFP_{N} \cdot dSplitFP_{C}$
${dSplitFP_{C} \over dt} = -k_3 SplitFP_{N} \cdot dSplitFP_{C}$
${dActiveFP \over dt} = k_3 SplitFP_{N} \cdot dSplitFP_{C}$
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 $RA
$\beta_4$ degradation rate of $LuxI$
$\beta_5$ degradation rate of $LuxR$
$\beta_5$ degradation rate of $Lysis$
$\beta_6$ degradation rate of $SplitFP_{cell}$
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Results
Global parameter sampling
A total of 500 000 parameter sets were sampled for the double positively regulated system.
Spatial Model
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Put a graph and a table placeholder example below.
Anim pariatur cliche reprehenderit, enim eiusmod high life accusamus terry richardson ad squid. 3 wolf moon officia aute, non cupidatat skateboard dolor brunch. Food truck quinoa nesciunt laborum eiusmod. Brunch 3 wolf moon tempor, sunt aliqua put a bird on it squid single-origin coffee nulla assumenda shoreditch et. Nihil anim keffiyeh helvetica, craft beer labore wes anderson cred nesciunt sapiente ea proident. Ad vegan excepteur butcher vice lomo. Leggings occaecat craft beer farm-to-table, raw denim aesthetic synth nesciunt you probably haven't heard of them accusamus labore sustainable VHS.
Anim pariatur cliche reprehenderit, enim eiusmod high life accusamus terry richardson ad squid. 3 wolf moon officia aute, non cupidatat skateboard dolor brunch. Food truck quinoa nesciunt laborum eiusmod. Brunch 3 wolf moon tempor, sunt aliqua put a bird on it squid single-origin coffee nulla assumenda shoreditch et. Nihil anim keffiyeh helvetica, craft beer labore wes anderson cred nesciunt sapiente ea proident. Ad vegan excepteur butcher vice lomo. Leggings occaecat craft beer farm-to-table, raw denim aesthetic synth nesciunt you probably haven't heard of them accusamus labore sustainable VHS.
InfoBox
And we also want to show a figure.
Conclusion
Conclusion/Discussion on the affibody experiment.
References
- Williams, Joshua W., et al. "Robust and sensitive control of a quorum‐sensing circuit by two interlocked feedback loops." Molecular systems biology 4.1 (2008): 234.
- Bousse, Luc. "Whole cell biosensors." Sensors and Actuators B: Chemical 34.1-3 (1996): 270-275.
- Bassler, Bonnie L., and Richard Losick. "Bacterially speaking." Cell 125.2 (2006): 237-246.
