l o a d i n g . . .

Model

Detection Module

Overview

To make the psychoactive ingredient deactivated, first we need to build a sensitive detection model that is able to detect the sudden, high presence of Δ9-THC inside human body, thus triggering the expression of downstream degradation enzyme. Moreover, it must not do so after our Degradation Module works and the concentration of Δ9-THC is low enough to be harmless to human.

Considering the complexity of the system, it's quite difficult to decide which reaction happens first and what's next, and in reality, these reactions usually don't follow a fixed order. Hence stochastic simulation seems to be a better choice to accurately simulate what would happen when Detection Module is activated. But, on the other hand, such stochasticity may bring fluctuations to our simulation results and make them extremely hard to converge, thus leading to terrible coherence with experiment results. So here we propose 2 models for detection simulating. First, a stochastic model based on Gillespie's algorithm is established to simulate the Δ9-THC take-in situation. Second, a deterministic model based on ordinary differential equations (ODEs) is proposed. By implementing both models, we are able to make comparison between them, and further to optimize the parameters for each of them.

Stochastic Gillespie-based Model

Description

The stochastic model is established mostly based on Kierzek AM et al.'s [1] and Hoyle RB et al.'s work [2], and at its core is Gillespie's algorithm, which, generally, chooses one reaction among all according to its "propensity" and lets it happen in a random period of time. The "propensity", or propensity function more exactly, is basically determined by the concentration of the reactants for the corresponding reaction.

Generally, the detection pathway is designed based on Two-component regulatory system. As is stated in [3], PmrA in the modified PmrB-PmrA system we adopted belongs to the OmpR/PhoB family, the largest RR family. So it follows the discovery in [1] that OmpR/PhoB family RRs act as dimers, and also formation of HK dimers is necessary for autophosphatase reactions. So what this stochastic model differs from the ODE-based deterministic model discussed later, is that we have more theoretical bases and parameters found in literatures to establish a more detailed, more thorough simulation system for the former.

In our detection system, the PmrB-PmrA Two Component System (TCS) is composed of PmrB, the histidine kinase (HK) and PmrA, response regulator (RR). The external signal, namely Δ9-THC, binds to the antibody on our modified PmrB (exactly, PmrB dimer) specifically, and activates its autophosphorylation process by changing the propensity of its dephosphorylation to a lower level. Then RRs in our system, namely PmrA, bind to the phosphorylated PmrB dimers, and take away their phosphate group, thus phosphorylating themselves. After that, they also form dimers. The phosphorylated PmrA dimers function as catalyst for the expression of the downstream reporter gene, making it several times more than the basal expression amount. Listed in the table below are the propensity functions, constants and descriptions for the reactions included in this system.

Most of the constants are obtained from [1] and [2], except that c15, the constant for dephosphorylation of PmrB which varies as strength of external signal (Δ9-THC here) changes, is estimated under the 2 different conditions where there is Δ9-THC intake (0.5) and no Δ9-THC (150) respectively.

Gillespie's Algorithm

  1. Initialize the time and the status of the system .
  2. With status of the system x at time t, calculate all the propensity functions for each reaction , and let .
  3. Generate 2 random real numbers τ, μ that satisfy the PDF below.

    More specifically
    1. Choose 2 numbers r1,r2 that are distributed evenly over [0,1].
    2. Let
    3. Let μ be an integer that satisfies.
  4. Let , and update the system's status according to reaction , i.e., .
  5. Repeat Step 2~5 until t reaches the preset simulation time limit.

Results

Fig 1: Detection of Δ9-THC intake

We set simulation time limit to 86400s, which is 24 hours, the time where THC is taken in is 21600s (6hr) and the time it is fully degraded is set to 43200s (12hr). The simulation result is shown below in a MATLAB GUI application designed by ourselves. It's pretty apparent that concentration of our reporter protein, phycocyanin, as well as the multiple enzymes for degradation of Δ9-THC, starts to increase immediately and peaks around 1hr later, and goes way much more beyond average (about 3 times the average) in about half an hour.

The reason why we merely set 2 time points to represent the degradation process is that, here we only need to simulate how the detection system works and validate its sensitivity to Δ9-THC. Later after the design and implementation of Degradation Module is discussed, we will combine these 2 closely related modules and test their feasibility together.

References

  1. Kierzek AM, Zhou L, Wanner BL. Stochastic kinetic model of two component system signalling reveals all-or-none, graded and mixed mode stochastic switching responses. Mol Biosyst. 2010 Mar;6(3):531-42. doi: 10.1039/b906951h. Epub 2009 Dec 2. PMID: 20174681.
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  2. Hoyle RB, Avitabile D, Kierzek AM. Equation-free analysis of two-component system signalling model reveals the emergence of co-existing phenotypes in the absence of multistationarity. PLoS Comput Biol. 2012;8(6):e1002396. doi: 10.1371/journal.pcbi.1002396. Epub 2012 Jun 28. PMID: 22761552; PMCID: PMC3386199.
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  3. Minh-Phuong Nguyen, Joo-Mi Yoon, Man-Ho Cho, and Sang-Won Lee. Prokaryotic 2-component systems and the OmpR/PhoB superfamily. Canadian Journal of Microbiology. 61(11): 799-810.
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ODE-based Model

Description

Construction of the PmrA-PmrB TCS

The basis of the Detect part is the PmrA-PmrB Two-component System to detect and react with the Δ9-THC and promote the expression of the gene to produce the target proteins and enzymes. So the first step is to express PmrA and PmrB properly, the main components of the system. Since there are too many parameters needed to be estimated if we adapt the actual, complicated reaction system in stochastic model to ODEs, we made some reasonable simplifications on the reactions based on the literature we have found.

For the PmrA gene, the mRNAPmrA molecules and the PmrA protein are constitutively expressed when the following reactions happen:

Similarly, the mRNAPmrB molecules and the PmrB protein are constitutively expressed when the following reactions happen:

Where:

⋅ PPmrAPmrA and PPmrB represent the DNA which will express to produce the mRNAPmrA and mRNAPmrB
⋅ CA and CB represent the copy plasmid number for gene PmrA and the gene PmrB
⋅ kA and kA represent the transcription rate of mRNAPmrA and mRNAPmrB
⋅ PA and PB represent the effective translation rate of the mRNAPmrA and mRNAPmrB
⋅ dmA and dmB represent the degradation rate of mRNAdPmrA and mRNAdPmrB
⋅ ddA and ddB represent the degradation rate of PmrA and PmrB
⋅ μ represents the specific growth rate

Reaction of Δ9-THC

After building up the PmrA-PmrB system, when the environment is containing high Δ9-THC and the Δ9-THC is detected by the PmrB sensor. In response to the signals, the PmrB combines with the Δ9-THC and autophosphorylates.

Then it transfers the phosphory group to its cognate response regulator PmrA.

Not only the DNA (PPmrC) has its basal expression to produce the target enzyme and the target protein. In the meantime, the phosphorylated PmrA (PmrA∼P) acts as transcription factor and binds to the DNA (PPmrC) and promotes the expression of the gene to produce Target Enzyme and Target Protein.


ODE MODEL

We adopt the Law of Mass Action, Hill Functions and the Michaelis-Menten equations to build the ODE equations in the Detection Model. We have considered the same assumption in Hill Function for the PmrA, PmrB, Target Protein and Target Enzyme: the mRNA is transcripted much faster than the Protein translation, so we assume that d[mRNA]/dt≈0.

For the concentration of each substance in our model is effected by several procedures, we need to combine the equations in different processes together to derive the ODE Model to describe the change of the concentration.

For PmrA, in (3) and (4) processes, we consider Δ9-THC∼PmrB as an enzyme, and adopt the Michaelis-Menten equations. Then combine with the Law of Mass Action and Hill Functions in (1) .

For PmrB, the Law of Mass Action and Hill Functions are adopted in (2) and (3) processes.

For Δ9-THC , we adopt the the the Law of Mass Action in (3) and (6). Note that since here we merely discuss the detection process, we don't consider the degradation effect by related enzymes.

For Δ9-THC∼PmrB, the Law of Mass Action is adopted in (3).

For Target Protein, we adopt the Law of Mass Action and Hill Functions in (5) process.

For Target Enzyme, we adopt the Law of Mass Action and Hill Functions in (5) process. Also, the assumption in (6) is considered here so that the degradation of Δ9-THC in (6) will not effect the concentration of the Target Enzyme.

For PmrA∼P, we use the Law of Mass Action and Michaelis-Menten equations in (4) process. Then we combine it with the Law of Mass Action in (5) process.

Assumptions

During Target Enzyme catalyzing the degradation of the Δ9-THC, we make an assumption that only Δ9-THC degrades and the Target Enzyme does not degrade.

For the PmrA, PmrB, Target Protein and Target Enzyme expressed: the mRNA is expressed much faster than the protein production, we assume that d[mRNA]/dt≈0.

For the Target Protein basal expression, very little amount of Target Enzyme will be expressed even though there is no Δ9-THC in the environment. We assume that this cannot be spotted by naked eye.

Parameters

Conclusions

Using the previously described parameters and ODEs, we made computational simulations and visualized the results to show how concentrations of different substances vary over time. In order to obtain a steady state of the system that the stochastic model fails to offer, we conducted the simulation in a situation where there is no Δ9-THC taken in.

We initialized the concentration of THC to 0, and then stimulated the substance concentrations after vaccination.

Fig 2: Concentration of Δ9-THC, Δ9-THC~PmrB, PmrA~P

We found that the concentration of Δ9-THC∼PmrB, PmrA∼P is nearly equal to 0 all the time despite the negligible fluctuations.

Fig 3: Concentration of Target Proteins

Fig 4: Concentration of Target Enzymes

The concentrations of Target Protein and Target Enzyme are close to zero and fluctuate in a very tiny range ([0,10-7μM]) .

Fig 5: Concentration of PmrA and PmrB

After these analyses, now it's safe to draw the conclusions that:

  1. The PmrA-PmrB TCS reaches equilibrium at about 40000 seconds after vaccination.
  2. Before ingesting Δ9-THC, concentration of Target Protein is so low that it cannot be observed with an unaided eye.
  3. The results have done a great job making up for the lack of convergence in the stochastic model discussed before, and they prove that the PmrA-PmrB TCS is able to keep stable when no psychoactive ingredient is taken in, providing a more convincing evidence of the feasibility and credibility of our detection pathway.

References

  1. Kierzek AM, Zhou L, Wanner BL. Stochastic kinetic model of two component system signalling reveals all-or-none, graded and mixed mode stochastic switching responses. Mol Biosyst. 2010 Mar;6(3):531-42. doi: 10.1039/b906951h. Epub 2009 Dec 2. PMID: 20174681.
    Full text    

  2. Din MO, Danino T, Prindle A, Skalak M, Selimkhanov J, Allen K, Julio E, Atolia E, Tsimring LS, Bhatia SN, Hasty J. Synchronized cycles of bacterial lysis for in vivo delivery. Nature. 2016 Aug 4;536(7614):81-85. doi: 10.1038/nature18930. Epub 2016 Jul 20. PMID: 27437587; PMCID: PMC5048415.
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