Team:Wageningen UR/Model/Cpx Kinetics
Cpx Kinetics
Mantis will be able to sense biomarkers from samples harboring antigens of infectious diseases (e.g. blood) using modified bacteria. These bacteria are able to report the presence of an antigen by making 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). This way, visualization of antigen presence can be achieved. To design an optimal molecular system for Mantis, we have considered three possible visualization strategies (Figure 1). By combining insights from kinetic models of these systems together with data from the wet-lab, we aim to create Mantis with the best possible characteristics. We used Yellow Fluorescent Protein (YFP) as placeholder signal protein to study the kinetics of these systems. To learn more about the molecular workings of the systems, check out our Specific Visualization page!
Mantis is designed for use in remote areas as a point-of-care diagnostic to quickly determine whether members of the local population have a specific infection. For Mantis to function in these envisioned settings, it must fulfill several criteria:
- 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 demonstrate how we studied the kinetics of three different Cpx setups and give recommendations based on their speed, signal intensity, 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.
Construction of the mathematical models
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 the 'Assumptions' box below:
All three systems consist of two parts, a sensing module and a signaling module connected through the Cpx system (see Figure 1). The sensing module consists of an affinity body bound to CpxP. In absence of stress, the dimer CpxP [1] is linked to the outer periplasmic domain of CpxA. However, in case of membrane stress or when antigen protein is present in the periplasm CpxP will bind this protein instead using the affinity body, releasing CpxA. Consequently, CpxA will gain kinase activity leading to phosphorylation of CpxR, which can activate or repress the transcription of genes involved in protein folding or degradation. A protease, DegP, will then degrade the CpxP-protein complex. When the stimulus is removed, CpxP will bind CpxA once more. Inactivated CpxA has dephosphorylating activity, restoring dephosphorylated CpxR and switching the system off [2]. There are several ways in which phosphorylation of CpxR can lead to formation of a fluorescent signal, as shown in Figure 1. We chose to study interactions based on Bimolecular Fluorescent Complementation (BiFC) by binding split halves of YFP to CpxR and CpxA.
Keeping the aim of the model in mind, which is studying the kinetics of YFP signal production, we chose to make some simplifying assumptions. External influences such as membrane stress are therefore left out. Furthermore, the sensing and signal production using the Cpx system relies mostly on protein-protein interactions. Processes as protein production and degradation are assumed to happen on a different (much slower) time scale, therefore these were not explicitly modeled unless necessary.
In the Equations box below you can find more information on how the system components interact with each other in each of the three model setups.
The dynamics of all three systems were modeled using Ordinary Differential Equations (ODEs), which are detailed below.
In these tabs you can find the ODEs used in the models, as well as an explanation of the species and parameters. Click on the buttons inside the tabbed menu:
In this tab we show the ODEs used to model system 1, which depends on interaction between CpxA and CpxR to produce a fluorescent signal (see Figure 1.1). An explanation of the constants and variables used in the ODEs can be found below.
ODEs${dAg \over dt} = -k_{3}Ag \cdot CpxAP \cdot S$
${dCpxAP \over dt} = -k_{3}Ag \cdot CpxAP + k_{7}CpxP \cdot CpxA$
${dCpxRn \over dt} = -k_{4}CpxRn \cdot CpxA + k_{1}CpxRn^{P} \cdot CpxAP$
${dCpxRc \over dt} = -k_{5}CpxRc \cdot CpxA + k_{2}CpxRc^{P} \cdot CpxAP$
${dCpxRn^{P} \over dt} = -k_{1}CpxRn^{P} \cdot CpxAP
- k_{6}CpxRn^{P} \cdot dCpxRc^{P} + k_{4}CpxRn \cdot CpxA$
${dCpxRc^{P} \over dt} = -k_{2}CpxRc^{P} \cdot CpxAP - k_{6}CpxRn^{P} \cdot dCpxRc^{P} + k_{5}CpxRc \cdot CpxA$
${dCpxP-Ag \over dt} = \beta_{1}CpxP-Ag + k_{3}Ag \cdot CpxAP$
${dCpxA \over dt} = -k_{7}CpxP \cdot CpxA + k_{3}Ag \cdot CpxAP$
${dYFP \over dt} = k_{6}CpxRn^{P} \cdot dCpxRc^{P}$
${dCpxP \over dt} = -k_{7}CpxP \cdot CpxA + \alpha_{1} - \beta_{2}CpxP$
| Species | |
|---|---|
| $Ag$ | Antigen level in the periplasm |
| $CpxAP$ | CpxA - CpxP complex |
| $CpxRn$ | CpxR bound to the N-terminal half of split YFP |
| $CpxRc$ | CpxR bound to the C-terminal half of split YFP |
| $CpxRn^{P}$ | Phosphorylated CpxR bound to the N-terminal half of split YFP |
| $CpxRc^{P}$ | Phosphorylated CpxR bound to the C-terminal half of split YFP |
| $CpxP-Ag$ | CpxP bound to the antigen |
| $CpxA$ | CpxA level in the membrane |
| $YFP$ | Level of dimerized CpxR linked to maturated YFP halves |
| $CpxP$ | CpxP level in the periplasm Parameters |
| Parameters | |
|---|---|
| $S$ | External Signal constant |
| $\alpha_1$ | CpxP production rate |
| $k_{1}$ | formation rate of CpxRn upon dephosphorylation of CpxRnP by CpxAP |
| $k_{2}$ | formation rate of CpxRc upon dephosphorylation of CpxRcP by CpxAP |
| $k_{3}$ | formation rate of CpxP-Ag upon binding of Ag by CpxAP followed by release of CpxA |
| $k_{4}$ | formation rate of CpxRnP upon phosphorylation of CpxRn by CpxA |
| $k_{5}$ | formation rate of CpxRcP upon phosphorylation of CpxRc by CpxA |
| $k_{6}$ | formation & maturation rate of the full YFP upon dimerization of CpxRnP and CpxRcP |
| $k_{7}$ | formation rate of CpxAP upon binding of CpxA by CpxP |
| $\beta_1$ | degradation rate of CpxP-Ag |
| $\beta_2$ | degradation rate of CpxP |
In this tab we show the ODEs used to model system 2 which depends on dimerization of phosphorylated CpxR to produce a fluorescent signal (see Figure 1.2). An explanation of the parameters and species used in the ODEs can be found below.
ODEs${dAg \over dt} = -k_{1}Ag \cdot CpxAcP \cdot S$
${dCpxP \over dt} = \alpha_{1} - k_{4}CpxP \cdot CpxAc - \beta_{2}CpxP$
${dCpxAc \over dt} = k_{1}Ag \cdot CpxAcP - k_{3}CpxRn \cdot CpxAc - k_{4}CpxP \cdot CpxAc$
${dCpxRn \over dt} = k_{5}CpxRn^{P} \cdot CpxAcP - k_{2}CpxRn \cdot CpxAc - k_{3}CpxRn \cdot CpxAc$
${dCpxRn^{P} \over dt} = k_{2}CpxRn \cdot CpxAc - k_{5}CpxRn^{P} \cdot CpxAcP - k_{6}CpxRn^{P} \cdot CpxAcP$
${dCpxAcP \over dt} = k_{4}CpxP \cdot CpxAc - k_{1}Ag \cdot CpxAcP - k_{6}CpxRn^{P} \cdot CpxAcP$
${dCpxP-Ag \over dt} = k_{1}Ag \cdot CpxAcP - \beta_{1}CpxP-Ag$
${dCpxA-YFP \over dt} = k_{3}CpxRn \cdot CpxAc$
${dCpxAP-YFP \over dt} = k_{6}CpxRn^{P} \cdot CpxAcP$
${dYFP \over dt} = k_{6}CpxRn^{P} \cdot CpxAcP + k_{3}CpxRn \cdot CpxAc$
| Species | |
|---|---|
| $Ag$ | Antigen level in the periplasm |
| $CpxP$ | CpxP level in the periplasm |
| $CpxAc$ | CpxA bound to the C-terminal half of split YFP |
| $CpxRn$ | CpxR bound to the N-terminal half of split YFP |
| $CpxRn^{P}$ | Phosphorylated CpxR bound to the n-terminal half of split YFP |
| $CpxAcP$ | CpxAc - CpxP complex |
| $CpxP-Ag$ | CpxP bound to the antigen |
| $CpxA-YFP$ | Level of CpxR - CpxA complex linked to maturated YFP halves |
| $CpxAP-YFP$ | Level of CpxRP - CpxAP complex linked to maturated YFP halves |
| $YFP$ | Total level of maturated YFP |
| Parameters | |
|---|---|
| $S$ | External Signal constant |
| $\alpha_1$ | CpxP production rate |
| $k_{1}$ | formation rate of CpxP-Ag upon binding of Ag by CpxAcP followed by release of CpxAc |
| $k_{2}$ | formation rate of CpxRnP upon phosphorylation of CpxRn by CpxAc |
| $k_{3}$ | formation & maturation rate of the full YFP upon phosphorylation of CpxRn by CpxAcP |
| $k_{4}$ | formation rate of CpxAcP upon binding of CpxAc by CpxP |
| $k_{5}$ | rate of CpxRn upon dephosphorylation of CpxRnP by CpxAcP |
| $k_{6}$ | formation & maturation rate of the full YFP upon dephosphorylation of CpxRnP by CpxAcP |
| $\beta_1$ | degradation rate of CpxP-Ag |
| $\beta_2$ | degradation rate of CpxP |
Note: Reaction k6 results in YFP formation even in absence of antigen. This reaction is not desired (see section Inherent background)
In this tab we show the ODEs used to model system 3 which depends on interaction between CpxA and CpxR to produce a fluorescent signal (see Figure 1.3). An explanation of the constants and variables used in the ODEs can be found below.
ODEs${dCpxAP \over dt} = k_{1}CpxP \cdot CpxA + k_{8}CpxAcP \cdot CpxR^{P} + k_{10}CpxAnP \cdot CpxR^{P} - k_{2}CpxAP \cdot Ag$
${dCpxAcP \over dt} = k_{3}CpxAc \cdot CpxP - k_{5}CpxAcP \cdot Ag - k_{8}CpxAcP \cdot CpxR^{P}$
${dCpxAnP \over dt} = k_{4}CpxAn \cdot CpxP - k_{6}CpxAnP \cdot Ag - k_{10}CpxAnP \cdot CpxR^{P}$
${dCpxA \over dt} = k_{2}CpxAP \cdot Ag + k_{7}CpxR \cdot CpxAc + k_{9}CpxR \cdot CpxAn - k_{1}CpxP \cdot CpxA$
${dCpxAc \over dt} = k_{5}CpxAcP \cdot Ag - k_{3}CpxAc \cdot CpxP - k_{7}CpxR \cdot CpxAc$
${dCpxAn \over dt} = k_{6}CpxAnP \cdot Ag - k_{4}CpxAn \cdot CpxP - k_{9}CpxR \cdot CpxAn$
${dAg \over dt} = - k_{2}CpxAP \cdot Ag - k_{5}CpxAcP \cdot Ag - k_{6}CpxAnP \cdot Ag \cdot S$
${dCpxP-Ag \over dt} = k_{2}CpxAP \cdot Ag + k_{5}CpxAcP \cdot Ag + k_{6}CpxAnP \cdot Ag - \beta_{1}CpxP-Ag$
${dCpxR \over dt} = - k_{7}CpxR \cdot CpxAc + k_{8}CpxAcP \cdot CpxR^{P} - k_{9}CpxR \cdot CpxAn + k_{10}CpxAnP \cdot CpxR^{P} \\ - k_{11}CpxR \cdot CpxA + k_{12}CpxAP \cdot CpxR^{P}$
${dCpxR^{P} \over dt} = k_{7}CpxR \cdot CpxAc - k_{8}CpxAcP \cdot CpxR^{P} + k_{9}CpxR \cdot CpxAn - k_{10}CpxAnP \cdot CpxR^{P} \\ + k_{11}CpxR \cdot CpxA - k_{12}CpxAP \cdot CpxR^{P}$
${dYFPc \over dt} = k_{7}CpxR \cdot CpxAc + k_{8}CpxAcP \cdot CpxR^{P} - k_{13}YFPc \cdot YFPn$
${dYFPn \over dt} = k_{9}CpxR \cdot CpxAn - k_{10}CpxAnP \cdot CpxR^{P} - k_{13}YFPc \cdot YFPn$
${dYFP \over dt} = k_{13}YFPc \cdot YFPn$
${dCpxP \over dt} = \alpha_{1} - k_{1}CpxP \cdot CpxAc - k_{3}CpxP \cdot CpxAc - k_{4}CpxP \cdot CpxAn - \beta_{2}CpxP$
| Species | |
|---|---|
| $CpxAP$ | CpxA - CpxP complex |
| $CpxAcP$ | Level of CpxAP linked to the C-terminal of split YFP |
| $CpxAnP$ | Level of CpxAP linked to the N-terminal of split YFP |
| $CpxA$ | CpxA level in the periplamic membrane |
| $CpxAc$ | Level of CpxA linked to the C-terminal of split YFP |
| $CpxAn$ | Level of CpxA linked to the N-terminal of split YFP |
| $Ag$ | Antigen level in the periplasm |
| $CpxP-Ag$ | Level of CpxP - Ag complex in the periplasm |
| $CpxR$ | Level of CpxR bound to TEV protease |
| $CpxR^{P}$ | Level of phosphorylated CpxR bound to TEV protease |
| $YFPc$ | Level of C-terminal split YFP |
| $YFPn$ | Level of N-terminal split YFP |
| $YFP$ | Total level of maturated YFP |
| $CpxP$ | CpxP level in the periplasm |
| Parameters | |
|---|---|
| $S$ | External Signal constant |
| $\alpha_1$ | CpxP production rate |
| $k_{1}$ | formation rate of CpxAP upon binding of CpxA by CpxP |
| $k_{2}$ | formation rate of CpxP-Ag upon binding of Ag by CpxAP followed by release of CpxA |
| $k_{3}$ | formation rate of CpxAcP upon binding of CpxAc by CpxP |
| $k_{4}$ | formation rate of CpxAnP upon binding of CpxAn by CpxP |
| $k_{5}$ | formation rate of CpxP-Ag upon binding of Ag by CpxAcP followed by release of CpxAc |
| $k_{6}$ | formation rate of CpxP-Ag upon binding of Ag by CpxAnP followed by release of CpxAn |
| $k_{7}$ | formation rate of C-terminal split YFP, CpxA and CpxRP upon phosphorylation of CpxR by CpxAc |
| $k_{8}$ | formation rate of C-terminal split YFP, CpxAP and CpxR upon dephosphorylation of CpxRP by CpxAcP |
| $k_{9}$ | formation rate of N-terminal split YFP, CpxA and CpxRP upon phosphorylation of CpxR by CpxAn |
| $k_{10}$ | formation rate of N-terminal split YFP, CpxAP and CpxR upon dephosphorylation of CpxRP by CpxAnP |
| $k_{11}$ | formation rate of CpxRP upon phosphorylation of CpxR by CpxA |
| $k_{12}$ | formation rate of CpxR upon dephosphorylation of CpxRP by CpxAP |
| $k_{13}$ | formation & maturation rate of the full YFP in the cytoplasm |
| $k_{14}$ | formation rate of CpxR upon dephosphorylation of CpxRP by CpxAcP |
| $k_{15}$ | formation rate of CpxRP upon phosphorylation of CpxR by CpxAc |
| $k_{16}$ | formation rate of CpxR upon dephosphorylation of CpxRP by CpxAnP |
| $k_{17}$ | formation rate of CpxRP upon phosphorylation of CpxR by CpxAn |
| $\beta_1$ | degradation rate of CpxP-Ag |
| $\beta_2$ | degradation rate of CpxP |
Note: Reactions k8 and k10 result in YFP formation even in absence of antigen. This reaction is not desired (see section Inherent background)
The particular ways in which our systems are set up have never been studied before, 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:
The initial concentrations for the components CpxAP, CpxR, CpxR-P and CpxP was set to 1, and an initial concentration of all other species was set to 0. In system 3, the initial concentration of CpxAcP and CpxAnP was set to 1. We generated 500,000 sets of parameters for each of the three systems using the Latin hypercube sampling function in Matlab (lhsdesign()), with parameter values ranging between 0 and 1. These values were then converted to a log-uniform distribution ranging from 0.0001 to 10 to resemble a natural distribution of kinetic rates[3].
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. The first two criteria can be directly calculated from the models. 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 constructed 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
The YFP score is defined as the difference in YFP concentration. We compare the maximum concentrations when antigen is added compared to when antigen is absent. This is similar to comparing the positive and negative control values that can be measured by Mantis if its bacteria have the given parameters. The YFP score can be written as:
$Score_{YFP} = YFP_{max} - YFP_{NoAntigen}$
The speed score is defined as the speed at which the system reaches half of the maximum YFP increase from the point where antigen is added. This can be written as:
$Score_{speed} = { {1 \over 2} \cdot {(YFP_{max}-YFP_{NoAntigen})} \over \tau} = {{1 \over 2} Score_{YFP} \over \tau }$
The parameter sets are scored by simulating the model from 0 to 600 time units using the ode45() function of Matlab. Antigen is introduced to the system using the external signal factor $S$. $S$ represents the activity of the antigen at a given time, and is set from 0 to 1 at time unit 100, introducing antigen to the system.
The $YFP_{NoAntigen}$ concentration is the YFP concentration at time unit 100, The $YFP_{max}$ concentration the concentration at time unit 600. See Figure A. for a schematic representation of the score calculation.
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 sub-optimally. 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 under 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. The number of sets belonging to the 5% highest scoring sets for both the YFP and speed score, so the sets producing both a fast and intense signal (hereafter referred to as the 'union'), vary per system. System 1 produced the largest union of 8948 sets, followed by system 2 (556 sets). System 3 produced the smallest union, of 91 sets. This difference might indicate that system 1 is more likely to produce both a fast and intense signal given a random parameter set, although the other two systems can reach higher speeds.
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 (Figure B). Parameter k3, k4, k5 and k6 show clear shifts from the original value distribution.
Figure B 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 C). Parameter k1, k3, k5 and k6 show clear shifts from the original value distribution (black).
Figure C 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
The sets giving the fastest signal had low k5 and high k6 parameter values, compared to the sets reaching both a fast and high signal (union, yellow). This is related to the scoring function, as it this does not take into account the difference in YFP produced by induced CpxA (k3) and non-induced CpxA (k6); just the speed of overall YFP production. Even though a high k6 value decreases the net level of YFP reached after antigen addition, it makes YFP maturation faster. The sets resulting in both a high and fast signal (union) generally have low values of k6 and high values of k3, giving a high, fast and specific signal.
The prevalence of each parameter value was plotted for each individual parameter (Figure D). Parameter k1, k8, k10, k13, k14 and k16 show clear shifts from the original value distribution.
Figure D shows that the sets producing high (red), fast (blue) and both high and fast (yellow) signals share one general property:
- k13 - Relatively fast maturation of the released YFP halves
We can intuitively explain this result, as for a high and fast signal, high amounts of YFP need to maturate.
To improve this system setup in the laboratory to create faster and more intense signals, we have to make sure that the parameter set enables certain properties. Main properties for a fast system are:
- k8 - Relatively fast phosphorylation of CpxR and release of YFPc by CpxA-CpxP
- k10 - Relatively fast hosphorylation of CpxR and release of YFPn by CpxA-CpxP
- k14 - Relatively slow phosphorylation of CpxR without YFPc release by CpxA-YPFc
- k16 - Relatively slow phosphorylation of CpxR without YFPn release by CpxA-YFPn
For high signal producing sets however, this is the other way around. In addition, the k1 rate plays a role:
- k1 - Relatively slow phosphorylation of CpxR by free CpxA
- k8 - Relatively slow phosphorylation of CpxR and release of YFPc by CpxA-CpxP
- k10 - Relatively slow phosphorylation of CpxR and release of YFPn by CpxA-CpxP
- k14 - Relatively fast phosphorylation of CpxR without YFPc release by CpxA-YPFc
- k16 - Relatively fast phosphorylation of CpxR without YFPn release by CpxA-YFPn
A possible explanation of the roles of these parameters might be their influence on CpxR availability. For a fast signal, YFP halves need to be released quickly, with or without presence of antigen. Both CpxR phosphorylation and dephosphorylation is needed. For a high signal, only the YFP halves released in presence of antigen matter. This requires only CpxR phosphorylation. Rates k14 and k16 lower phosphorylated CpxR (CpxR-P) concentrations and higher CpxR concentrations. These parameters should be high for a specific and high signal. Rate k1 shifts these concentrations the other way around, and should thus be low. Rates k8 and k10 facilitate quick release of YFP halves in presence of CpxR-P. To save CpxR-P for these reactions, as no CpxR is phosphorylated under uninduced circumstances, CpxR-P consuming reactions as k14 and k16 should be lower. However, this signal is not sensitive.
What does this mean for Mantis?
All three systems are capable of reaching high YFP levels quickly, however not all three systems are equally sensitive. Systems 1 and 3 seem to have a trade-off: the sets producing a maximal YFP production speed do not also reach the maximal YFP concentration. This trade-off is not seen for the second system.
For all three systems, the maturation rate of YFP is most important for generating a fast and high signal!
Sensitivity analysis
When Mantis will be applied in the field, it is not known what concentration of pathogenic marker will be present in the sample of the patient. Mantis must thus be sensitive and be able to detect a wide range of antigen concentrations. To find out which setup is most sensitive, simulations were performed with a range of antigen concentrations.
Click on the buttons inside the tabbed menu:
In addition, we want to know under what relative concentrations of Cpx components Mantis functions best. We chose to run the model under different concentrations of CpxA and CpxR, as these key proteins are major players in production of YFP in the models and can be tested in the lab.
Click on the buttons inside the tabbed menu:
The model shows that both the second and third setup are limited in their YFP production. The CpxA concentration or the CpxR concentration becomes either too high or too low, which is mainly due to either the intrinsic background increasing too much or the CpxR availability becoming too low for the response to occur. In addition, this effect differs per parameter set (Figures I,J). The first setup seems not limited by these factors, since it is not directly dependent on CpxA - CpxR interactions.
Robustness analysis
A robust system retains reliable performance under adverse environmental circumstances. To test robustness of each setup, we calculated the variability of YFP signals when the model is simulated using a number of different parameter sets. This is displayed below as standard deviation of the mean YFP concentration over time when comparing simulations from the best 10, 100, 1000 and 10,000 sets.
From a biological perspective, this is equivalent to using Mantis under a range of different environmental conditions (that perturb all parameters in the system). If Mantis is not a robust tool the variability in YFP signals would increase quickly as more parameter sets are included in the analysis. However, if Mantis is a robust tool the YFP signal will show low variability even when large numbers of poorly performing parameter sets are simulated.
From Figure 4, it seems that for all three setups, and as expected, the deviation from the optimal set increases as we include increasingly more suboptimal parameter sets. However, the three systems appear to be robust, as:
- The system retains its qualitative behavior, meaning that there is little signal in absence of antigen and a clear signal when antigen has been added.
- The average of the signal does, in fact, not increase as more suboptimal sets are included.
- The increase in variability of the signal is limited.
The most stable signal is produced by system 1. As can be seen in Figure 4, system 1 shows a stable minimum signal (e.g. the signal produced in absence of antigen) according to the model. The signal produced after antigen addition however shows an increased variation when more non-ideal sets are incorporated.
Systems 2 and 3 on the other hand, show a stable variation in maximum signal as more suboptimal sets are included. The minimum signal however seems to increase slightly in these cases, which means that for all setups the signal produced after addition of antigen might be less detectable if there are large external influences during testing, but detectable nonetheless.
Integration of model with in vivo system
The models described above serve to predict the behavior of the eventual Mantis system in vivo. We managed to combine the findings from the computational models with those from the in vivo systems and make some suggestions on how to improve the signalling systems. Here we elaborate on the computational work done in this in vivo- in silico collaboration.
Sensitivity assay in vivo
To test the influence of CpxA and CpxR levels in vivo, constructs were created in which we coupled split YFP halves to CpxA and CpxR respectively and placed them under control of the inducible promoter. We transformed E. coli K12 with these constructs. The Cpx system was activated with the known activator KCl in different concentrations to mimic different antigen concentrations at a time-point of 20 minutes. An extensive overview of the performed experiments can be found here.
The KCl and CpxR and CpxA constructs allow for variable induction of the system and variable expression of CpxR and CpxA, similar to the change of initial concentration in the sensitivity assay (see above). The sensitivity assay showed that 1) signal production of system 2 and 3 could be inhibited by CpxA and CpxR concentrations, and 2) increase in antigen concentration can increase the signal speed and intensity. The first finding was confirmed in the wet-lab. Only system 1 managed to produce a clear signal. We then set to test hypothesis 2 using system 1: The strongest signal will be obtained when Cpx activation is maximized. This was confirmed by wet-lab experiments: antigen binding was mimicked by adding a known activator of the Cpx pathway, KCl. Wet-lab experiments showed that increasing the KCl concentration lead to increase of fluorescence intensity.
Fitting the computational model to the experimental data
In order to improve the in vivo system, we must know more about its characteristics. We assessed this by comparing the experimental data to the 500,000 parameter sets generated above. The sets were assessed on their similarity to the measured data in YFP production. The similarity to the wet-lab data was quantified as 'distance score'. A large distance score indicates a large difference in YFP production between the simulation and the wet-lab data. A parameter set was found which fitted the wet-lab data most (Figure 5).
The in vivo systems contained native CpxR, which is not taken into account by the model for system 1 described above. Therefore, an adapted model was constructed which included native CpxR. Comparison between these models showed that including native CpxR had little effect on the behavior of the model. The model containing native CpxR was used to find the best fitting parameter set.
To improve the current signaling system we want to know which parameters should be changed in order to increase the YFP signal and the response speed. To assess this, each parameter of the best fitting set was individually varied between 0.001 and 100, keeping the other parameters constant. The effect of this change on the maximum achievable YFP concentration and production speed was calculated (Figure 6). We only show the parameters that can potentially be altered in vivo.
This figure shows that the maximum YFP fluorescence has more possibility to be improved than the speed at which YFP increases. Both system properties could be profoundly improved by increasing the kinetic rates for antigen binding by the affibody and CpxR-YFPc phosphorylation (find all the used parameters here). However, given the iGEM time limits, it was not feasible to confirm this hypothesis in the lab. One viable option, though, to improve system performance was to use a fluorophore with a faster maturation time.
Implementation in vivo
We aim to improve the response time of our visualization system. As stated, our model shows that this can be done by using a faster maturing fluorescent protein. During our "Fluorescent Protein" project we tested a number of fluorescent proteins, of which mVenus showed one of the highest levels of brightness and a high percentage of fluorescence reconstitution. Furthermore mVenus is designed to have a fast and efficient maturation time [1], exactly what we need! We show here how this experiment was designed, and what conclusions we could make.
Conclusion
Taking into account all the simulations shown above, we find system 1 is most sensitive to antigen, most robust and can reach the highest fluorescent signal when the CpxR concentrations are increased. The fastest signal can potentially be produced by system 2, however the required reaction rates are not feasible in vivo. System 3 seems least feasible as a signaling module for Mantis, as it is limited under the wrong CpxR concentrations or parameter values. When comparing the wet-lab data to the sensitivity assay we found that the model could explain the behavior of the in vivo system. In addition, by fitting the wet-lab data to the model, we came up with several suggestions to improve signal production in the wet-lab. For more information on the interaction between the lab-data and these models, look here. More information on construction of the systems in vivo can be found here.
References
- X. Zhou, R. Keller, R. Volkmer, N. Krauss, P. Scheerer, and S. Hunke, “Structural basis for two-component system inhibition and pilus sensing by the auxiliary CpxP protein,” J. Biol. Chem., vol. 286, no. 11, pp. 9805–9814, 2011.
- D. D. Isaac, J. S. Pinkner, S. J. Hultgren, and T. J. Silhavy, “The extracytoplasmic adaptor protein CpxP is degraded with substrate by DegP.,” Proc. Natl. Acad. Sci. U. S. A., vol. 102, no. 49, pp. 17775–17779, 2005.
- Xiao, Xiao, et al. "On the use of log‐transformation vs. nonlinear regression for analyzing biological power laws." Ecology 92.10 (2011): 1887-1894.
