Team:Wageningen UR/Model/Cpx Kinetics
Cpx kinetics
Mantis makes use of the Cpx signaling system native to E. coli to connect the sensing module (affinity body) to the reporter module (split fluorophore or chromoprotein). However, how this coupling is set up may vary. The several system setups can be seen in Figure 1. We used YFP as placeholder signal protein to study these systems.To learn more about the molecular workings of the systems, check out our BiFC page!
- Speed: When a sample is added to Mantis, the signal should be detectable as soon as possible. This way the patient can be informed immediately, without need for return visits to the healthcare center
- Signal intensity: The produced YFP signal should be clearly detectable
- Sensitivity: Mantis should detect both high and low levels of antigen
- Robustness: Mantis is applied in remote areas, where external factors as temperature and humidity can't be controlled. Mantis should be able to perform in such non-ideal circumstances
Here, we show how we studied the kinetics of three different Cpx setups and give recommendations based on their speed, maximum signal, sensitivity and robustness.
In other parts of this website, we show how the affinity body is fused to the Cpx system, how the bimolecular fluorescence complementation (BiFC) is connected to CpxA and how the split fluorescent proteins perform. Additionally, we show how interaction between this model and the wet-lab data lead to increased Mantis performance. How this project comes together with the other parts of Mantis can be seen here.
project page or result overview page?
Constructing the mathematical model: digitalizing biology
To gain insight into how Mantis works and how to improve the system, we must get an idea of how the different components interact with each other. For this purpose we constructed three models, one for each system in Figure 1. As mentioned above, Mantis relies on several proteins to sense antigens and produce a signal. The main players are antigen, CpxP, CpxA and CpxR. Read more about their interactions in this box:
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.
Click on the buttons inside the tabbed menu:
System 1
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System 2
Paris is the capital of France.
System 3
Tokyo is the capital of Japan.
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.
In this part, we show the equations for all three systems. Click on the boxes to find the equations and explanations of all parameters that were used in this study.
${dAHL_{cell} \over dt} = \alpha_{1}LuxI + D_{AHL}(AHL_{ext}-AHL_{cell})+k_{-1}RA-k_1 LuxR \cdot AHL -\beta_{1}AHL_{cell}$
${dAHL_{ext} \over dt} = V_{ratio} \cdot D_{AHL} (AHL_{ext}-AHL_{cell}) - \beta_{2}AHL_{cell}$
${dRA \over dt} = k_{-1}LuxR \cdot AHL - k_{-2}RA^{2} - \beta_{3}RA$
${dLuxI \over dt} = \alpha_2 + \alpha_3 {{LuxR \cdot RA^2} \over {km_1 + RA^2}} + \alpha_4 + S \cdot \alpha_5 - \beta_4 LuxI$
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:
${dLuxR \over dt} = \alpha_6 + \alpha_7 {{LuxR \cdot km_1} \over {km_1 + RA^2}} + \alpha_8 + S \cdot \alpha_9 - \beta_5 LuxR$
$AHL_{cell}$ Signaling molecule AHL inside the cell
$AHL_{ext}$ Signaling molecule AHL inside the external medium
Parameters$\alpha_1$ AHL production rate of LuxI
$\alpha_2$ Leaky production rate of LuxI from pLuxA
$\alpha_{11}$ 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
$\beta_1$ degradation rate of $AHL_{cell}$
$\beta_2$ degradation rate of $AHL_{ext}$
The particular ways in which our systems are set up have never been studied before in, so we don’t know the details on how fast each interaction will be nor what the ideal protein concentrations might be. To find out the optimal settings, we generated 500,000 random sets of parameter values. Read more about how we did this and the initial conditions below:
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Selecting optimal system parameters
To determine what parameters the ideal performing Mantis must have, we score each parameter set based on the above described criteria: speed, signal intensity, sensitivity and robustness. For all parameter sets we generated in the previous step, we simulate what happens to each of the system components over time. Using the optimal parameters, Mantis should have system properties that fulfill the criteria mentioned above. In order to quantify what constitutes a good system, we construct a scoring function that can capture this desirable system behavior.
This links back to two of the key aspects of Mantis: signal production and speed. We want a highly fluorescent signal that is produced quickly after the antigen has been sensed in the sample. Each simulation reaches a certain maximum level of YFP at a certain rate, both are dependent on the parameter values. We constructed two objective scores based on these system properties:
- Maximum YFP concentration,
- Speed at which this concentration is reached.
Inherent background
The second and third system rely on interaction between CpxA and CpxR upon phosphorylation or dephosphorylation by CpxA. However, dephosphorylation takes place when CpxA is still inhibited by CpxP. This way maturation of the split YFP halves can occur without antigen presence. This means Mantis can produce a signal even though the sample does not contain disease markers.
In order to minimize this background signal which is inherent to the second and third systems, we adapted our objective scores to take this unwanted effect into account. The objective functions now become:
- Maximum YFP concentration in the presence of antigen relative to YFP concentration in absence of antigen,
- Speed at which this maximum YFP concentration is reached.
Important parameter properties
Mantis should perform well in the field, which means the circumstances can range widely in terms of temperature and humidity e.g., possibly causing the cells to perform less than ideal. By looking at the good scoring sets and comparing their parameters, we can learn what parameters are essential for high performance and should be highly controlled and what parameters are not relevant for performance.
Figure 3 shows that all systems are capable of quickly producing high YFP concentrations when having the right parameter set. The extent of the speed scores are different for all three systems though, with system 1 being able to reach a maximum production speed of 0.35 and system 2 a speed of 3.0.
To determine what properties Mantis should have to get a high fluorescent signal and/or a fast signal, we studied the parameter values of the best scoring YFP and speed sets.
Determining key parameter values for performance
The distribution of each parameter value was plotted for each individual parameter (fig x). Parameter k3, k4, k5 and k6 show clear shifts from the original value distribution.
Figure x shows that the sets producing high (red), fast (blue) and both high and fast (yellow) signals make use of these main properties:
- k3 - Relatively fast antigen binding
- k4 - Relatively fast phosphorylation of CpxR-YFPc
- k5 - Relatively fast phosphorylation of CpxR-YFPn
- k6 - Relatively fast dimerization of CpxR and maturation of YFP
- Low CpxP levels due to low CpxP production and high CpxP degradation
These properties are found for both the sets that produce a fast signal, sets that produce a high signal, and sets that can do both (which we will refer to as the “union” of these sets). The fast sets have relatively higher k3, k4 and k5 values, whereas the sets resulting in an intense signal have a more moderate increase from the original distribution. These parameters appear to not be vital in order to obtain a high YFP signal. This shows the robustness of this first system.
The distribution of each parameter value in our second system was plotted for each individual parameter (Figure x). Parameter k1, k3, k5 and k6 show clear shifts from the original value distribution (black).
Figure x shows that the sets producing high (red), fast (blue) and both high and fast (yellow) signals make use of these main properties:
- k1 - Relatively fast antigen binding
- k3 - Relatively fast GFP maturation upon interaction with CpxA
- k5 - Relatively fast CpxR dephosphorylation
- k4 - Relatively slow CpxA inhibition by CpxP
- k6 - Relatively slow GFP maturation upon interaction with the CpxA-CpxP complex
These properties are found for both the sets that produce a fast signal, sets that produce a high signal, and sets that can do both (which we will refer to as the “union” of these sets). The fast sets have relatively higher k3, k4 and k5 values, whereas the sets resulting in an intense signal have a more moderate increase from the original distribution. These parameters appear to not be vital in order to obtain a high YFP signal. This shows the robustness of this first system.
System 3
Tokyo is the capital of Japan.
title of Spatial Model
Infobox below
InfoBox
Conclusion
Conclusion/Discussion on the Cpx modeling.
References
- Reference :)
