Difference between revisions of "Team:Wageningen UR/Model/Cpx Kinetics"

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                                            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.
 
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                                            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.
 
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                                            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.
 
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                                            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.
 
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                                            <br> 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.
 
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Revision as of 16:45, 30 October 2017

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). To design an optimal system, 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 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 BiFC page!

Figure 1: Three different coupling setups of Mantis. System 1) Production of YFP signal relies on phosphorylation of two CpxR molecules by CpxA that senses the presence of antigen using CpxP. System 2) Signal production relies on interaction of CpxR and CpxA upon (de)phosphorylation of CpxR by CpxA. System 3) Signal production relies on the release of split YFP upon phosphorylation of CpxR by CpxA. The TEV protease bound to CpxR cleaves the split YFP from CpxA, allowing it to dimerize in the cytosol.
  • 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

All three systems consist of two parts, a sensing module and a signalling module connected through the Cpx system. 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, turning on/off 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]. Keeping the aim of the model in mind, studying the kinetics of YFP signal production, we chose for a simple model. External influences as membrane stress are therefore left out. Furthermore, the sensing and signal production using the Cpx system relies mostly on protein interactions. Processes as protein production and degradation are therefore left out unless necessary.

In the ODE boxes below you can find more info on the components (species) and their interactions come together in the three model setups.

Click on the buttons inside the tabbed menu:

System 1

London is the capital city of England.

System 2

Paris is the capital of France.

System 3

Tokyo is the capital of Japan.

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

${dAHL_{ext} \over dt} = V_{ratio} \cdot D_{AHL} (AHL_{ext}-AHL_{cell}) - \beta_{2}AHL_{cell}$

${dRA \over dt} = k_{-1}LuxR \cdot AHL - k_{-2}RA^{2} - \beta_{3}RA$

${dLuxI \over dt} = \alpha_2 + \alpha_3 {{LuxR \cdot RA^2} \over {km_1 + RA^2}} + \alpha_4 + S \cdot \alpha_5 - \beta_4 LuxI$

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:
${dLuxR \over dt} = \alpha_6 + \alpha_7 {{LuxR \cdot km_1} \over {km_1 + RA^2}} + \alpha_8 + S \cdot \alpha_9 - \beta_5 LuxR$

Species

$AHL_{cell}$ 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:

  1. Maximum YFP concentration,
  2. 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.

Figure 2: Overview of mechanism inherent to system 2 and 3 which results in a YFP signal without antigen being added. System 2: Upon dephosphorylation of CpxR, the two YFP halves might interact and maturate. System 3: Upon dephosphorylation of CpxR, the TEV protease might cleave a YFP halve from CpxA, which can maturate in the cytosol with another released YFP halve.

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:

  1. Maximum YFP concentration in the presence of antigen relative to YFP concentration in absence of antigen,
  2. 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: Placeholder! for proper pictures of parameter scores distribution.

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 3: Placeholder! for proper pictures of parameter scores 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 3: Placeholder! Distribution of each parameter value in the best scoring parameter sets. The original distribution of values over the sets is displayed in black. Parameter k1 - k6 represent respectively antigen binding by CpxP (k1), CpxR phosphorylation (k2), maturation of GFP upon interaction between CpxR and CpxA (k3), CpxA inhibition by CpxP (k4), CpxR dephosphorylation (k5) and maturation of GFP upon interaction between CpxR and the CpxA-CpxP complex. Alpha1, beta1 and beta2 represent respectively CpxP production, Antigen-CpxP degradation and CpxP degradation rates.

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

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 (fig x). Parameter k1, k8, k10, k13, k14 and k16 show clear shifts from the original value distribution.

Figure 3: Placeholder! Distribution of each parameter value in the best scoring parameter sets. The original distribution of values over the sets is displayed in black. Parameter k1 - k6 represent respectively antigen binding by CpxP (k1), CpxR phosphorylation (k2), maturation of GFP upon interaction between CpxR and CpxA (k3), CpxA inhibition by CpxP (k4), CpxR dephosphorylation (k5) and maturation of GFP upon interaction between CpxR and the CpxA-CpxP complex. Alpha1, beta1 and beta2 represent respectively CpxP production, Antigen-CpxP degradation and CpxP degradation rates.

Figure x 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 XpxR (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 specific.

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 have a trade-off: their maximal YFP production speed cannot be reached without decreasing their sensitivity and maximal YFP concentration. There is no trade-off for the second system, as it does not rely on CpxA-CpxR interaction.
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 run under a range of antigen concentrations.

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 majors players in production of YFP in the models and can be tested in the lab.

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. The second 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 x, 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, three systems appear to be robust, as:

  1. 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.
  2. The average of the signal does, in fact, not increase as more suboptimal sets are included.
  3. The increase in variability of the signal is limited.

The most stable signal is produced by system 1. As can be seen in figure x, 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. Still, even with a more variable minimum signal This 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.

Conclusion

Taking into account all simulations shown above, we find Setup 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 the setup 2, however the required reaction rates are not feasible in vitro. Setup 3 seems least feasible as signalling module for Mantis, as it is limited under the wrong CpxR concentrations or parameter values.

References

  1. Reference :)