Team:RDFZ-China/Model
The whole model aims to find out the relationship between the quantity of PCA that obtained from tea and the production of the 5-HTP which promotes the synthesis of serotonin. 5-HTP is converted from tryptophan with the catalyzing of TPH1 enzyme. According to existing researches, TPH1 coding gene is the downstream gene of PcaU coding gene that is located on the lac operon. In order to find out the sensibility of PcaU sensor that represents the metabolic rate of tea in our body ,we replace TPH1 coding gene to GFP in our experiment since GFP is easier to measure in the lab. Therefore, our model is separated into two parts:
1. The model of the sensibility of PcaU sensor
2. The model of the 5-HTP production at certain concentration of TPH1
2. PCA sensor model
2.2 General assumption
1. In transcription, RNA polymerase only has a low binding force on DNA strand. Besides, PcaU, the transcription factor of downstream expression, also has a low binding force on DNA strand. When PcaU first binds DNA strand, RNA polymerase has a high binding force with DNA·PcaU complex, therefore, PcaU promotes the binding of RNA polymerase on DNA strand.
2. PCA, the small molecule converted from tea, serve as an inducer of the downstream expression. PCA can slightly change the shape of PcaU so that the binding force of RNA polymerase will be increased, but it will not affect other molecules.
3. For simplicity, we do not consider the polymerization of the PcaU molecule, we just consider PcaU as a monomer in our following model.
4. the expression of downstream protein (GFP) is proportional to the sum of all complexes that contain DNA·RNAP.
2.3 Essential reactions
The transcription of target DNA have three conditions that enable RNA polymerase to bind the target gene:
1. Transcription happened when RNA Polymerase binds on DNA strands alone.
In this condition, with no help from other molecules, RNA Polymerases are easily dissociated from DNA, so the dissociation constant of this reaction is a high value.
2. PcaU first binds on DNA strand to from DNA·PcaU complex:
Then RNA Polymerase binds to DNA·PcaU complex, starting the transcription:
3. PCA first binds with PcaU to form PCA·PcaU complex:
Then PCA·PcaU complex binds on DNA to form DNA·PCA·PcaU complex:
Then RNA Polymerase binds to DNA·PCA·PcaU complex, starting the transcription:
2.4 Parameter and symbol
Note:
1. In equation (2), (6) and (12), since the transcription factor (PcaU) will significantly increase the binding force of RNA polymerase on DNA, k6 will be much smaller than k2, so we reduce 100 times of k2 to obtain k6. Besides, PCA only change the binding force of PcaU on the promoter, it does not affect the binding force of RNA polymerase on DNA, so k6 equals to k12.
2. In equation (3), (4), (9) and (10), PCA only affects the dissociation rate constant of PcaU on DNA, but it does not change the binding of that. k10 is much smaller than k4, and k3 is equal to k9, so we reduce 100 times of k4 to obtain k10.
3. In equation (5) and (11), since PCA does not affect the binding of RNA polymerase, k5=k11.
2.5 Model process
From equation (1) to (12), according to the the law of mass action, the rate of a chemical reaction is proportional to the product of the masses of the reacting substances, with each mass raised to a power equal to the coefficient that occurs in the chemical equation, we can write down:
The concentration rate of DNA=
The concentration rate of RNAP=
The concentration rate of DNA·RNAP=
The concentration rate of PcaU=
The concentration rate of DNA·PcaU=
The concentration rate of DNA·RNAP·PcaU=
The concentration rate of PCA=
The concentration rate of PCA·PcaU=
The concentration rate of DNA·PCA·PcaU=
The concentration rate of DNA·RNAP·PCA·PcaU=
Based on the equations above, we develop an ODE program to simulate the varied concentration of each molecule (figure 1):
Figure 1: The simulation of varied concentration of molecules involved in the expression of GFP
We then uses the concentration of PCA as the independent variable to plot the curve separately (figure 2):
Figure 2: The concentration of DNA·RNAP complexes involved in transcription
The final concentration of GFP is equals to the expression of GFP by transcription and translation minors the degradation of GFP due to diluting. Since the expression of GFP is proportional to the sum of all DNA·RNAP complexes, and the degradation of GFP is proportional to the concentration of GFP, we can finally obtain the formula of the concentration of GFP:
α in equation (23) represents the expression rate of GFP.
We put equation (23) in our ODE program, it graphs (figure 3):
Figure 3: The relation between the expression of GFP and PCA concentration
Fitting results:
α P3P5B = 4.5974e+03
α PcaU = 8.3752e+03
α P3P5C = 1.6101e+03
Consequently, we can finally obtain the sensibility of PcaU sensor by the production of GFP at certain PCA concentration.
3. TPH1 modeling
3.3 Parameter and symbol
3.4 Essential equation
In this enzymatic reaction, tryptophan first binds on the active site of TPH1, forming tryptophan·TPH1 complex:
Then tryptophan is converted to 5-HTP, releasing TPH1:
3.5 Model method
TPH1 model aims to predict the production of 5-HTP at certain concentration of TPH1.
3.6 Model process
The concentration rate of Tryptophan=
The concentration rate of TPH1=
The concentration rate of Tryptophan·TPH1=
The concentration rate of 5-HTP=
We also develop a ODE program test our TPH1 model based on equation (27) - (30) (figure 4):
Figure 4: The reaction speed of the conversion of tryptophan to 5-HTP
In order to make this model more practical, we also calculate the relation between the synthesizing rate of 5-HTP in the TPH1 concentration gradient:
Figure 5: The rate of the synthesis of 5-HTP varied by the concentration of TPH1
Figure 6: The fitting graph of the rate of the synthesis of 5-HTP varied by the concentration of TPH1
These two graphs show that the synthesizing rate of 5-HTP will be slower when the concentration of TPH1 increases, which fits the data derived from our experiment in trend.
For summary, we can calculate the production of 5-HTP at certain TPH1 enzyme concentration.
4. Future plan
In the future, we plan to make some improvements to our model.
For PcaU sensor model, we calculated our model in basic chemical reactions that involved in small molecules, however, we do not have such a huge database to precisely measure the all parameters like binding or dissociation rate constants from our experiment. To make our model more practical, we plan to make some sensibility tests on all parameters. Since the environment and reactions in cells are varied and sophisticated, the whole system may be damaged by uncontrolled sources such as a slight change of the dissociation rate of two molecules. We can find changes in the whole system by adjusting every parameter and therefore predict which steps or reactions affect the whole system the most.
On the other hand, we also tend to link PcaU sensor model and TPH1 model. In order to make our project more safe, the concentration of 5-HTP must be limited below a certain value. As a result, we plan to design a negative feedback that the repressor will finally inhibit the expression of TPH1, preventing the overproduction of 5-HTP.
5. Source Code
Here are our source code for modeling:
The main program 1 of PcaU sensor
The main function of PcaU sensor
The main program 2 of PcaU sensor
The main program of TPH1 model
The main function of TPH1 model
References
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[3] Du M., Kodner S., Bai L. Enhancement of LacI binding in vivo. Nucleic Acids Res. 2019;47:9609–9618. doi: 10.1093/nar/gkz698.
[4] Kowlessur D, Kaufman S. Cloning and expression of recombinant human pineal tryptophan hydroxylase in Escherichia coli: purification and characterization of the cloned enzyme. Biochim Biophys Acta. 1999 Oct 12;1434(2):317-30. doi: 10.1016/s0167-4838(99)00184-3. PMID: 10525150.
[5] https://bionumbers.hms.harvard.edu/bionumber.aspx?id=106505&ver=7&trm=the+binding+rate+constant+of+a+substrate+on+an+enzyme&org=