Team:PKU Beijing/Modeling/ODE

Our design this year consists of four modules. For each module, we constructs ODEs(Ordinary Differential Equations) to describe the biological process. In this page, we will demonstrate all of our equations, the corresponding biological reactions, and parameters related. For parameters we used, please go to parameters page. For modeling result of the deterministic model, please go to result page.

AND Gate 1

The AND Gate 1 module works like this: Sal activates the transcription of T7 RNA polymerase with amber mutation while AraC activates the production of the tRNA. The AND gate part works as T7 RNA polymerase mRNA is translated, which works only when both Sal and AraC present. After the AND gate, T7 RNA polymerase protein activates the expression of CI(Trigger CI), which will push the state of the bi-stable module from CI434 to CI.

• Synthesis of tRNA

Biological Process
AraC activates the transcription of supD gene, which will produce tRNA. tRNA interacts with animo acids to produce Aa-tRNA, which will be used in the translation process of T7 RNA polymerase. After the translation, tRNA in Aa-tRNA will be released, which will contributes to the enrichment of its concentration. The AND Gate 2 will do the similar effect on the concentration of tRNA. Meanwhile, tRNA and Aa-tRNA degrade in a certain rate. The degradation of tRNA will decrease its concentration, while Aa-tRNA's degradation will produce more tRNA molecules considering the fact that the bond between tRNA and aminoacyl is weak.

Equation
<math>\frac{\mathrm{d}c_1}{\mathrm{d}t}=k_1\frac{(s_1/K_1)^n_1}{1+(s_1/K_1)^n_1}-γ_1 c_1+γ_2' c_2-l_2 c_1+2\frac{\mathrm{d}c_4}{\mathrm{d}t}+2\frac{\mathrm{d}c_{11}}{\mathrm{d}t}</math>

Parameters
c_1: concentration of tRNA
k_1: maxinum transcription rate of tRNA
s_1: concentration of AraC, the stimulus
K_1: microscope dissociation constant
n_1: Hill co-effiency
γ_1: degradation and dilution rate of tRNA. Unless notice, "degradation rate" in this model means the combination of degradation rate and dilution rate.
γ_2': degradation rate of Aa-tRNA. This process DOES NOT consist of dilution, which will not break down the bond between tRNA and aminoacyl.
c_2: concentration of Aa-tRNA
k_2: rate of transformation from tRNA to Aa-tRNA.
c_4: T7 RNA polymerase, product of AND gate 1.
c_{11}: T3 RNA polymerase(P2), product of AND gate 2.

• Synthesis of Aa-tRNA

Biological Process
Aa-tRNA is produced by tRNA and amino acids. Suppose the amino acids are of large quantity in a cell, their concentration can be regarded as constant, which means that the production rate of Aa-tRNA can be describe by multiplying concentration of tRNA(c_1) by production rate(k_2). Aa-tRNA will be consumed in two AND gate while it keeps degrading in the cells.

Equation

Parameters
γ_2: degradation rate of Aa-tRNA

• Synthesis of T7 RNA polymerase mRNA

Biological Process
Sal activates the transcription of T7 RNA polymerase.

Equation

Parameters
c_3: concentration of T7 RNA polymerase mRNA
k_3: maximum transcription rate of T7 RNA polymerase
s_2: concentration of Sal
K_3: microscope dissociation constant
n_3: Hill co-effiency
γ_3: degradation rate of T7 RNA polymerase mRNA

• AND Gate 1

Biological Process
T7 RNA polymerase mRNA has two amber mutation. Only when Aa-tRNA synthesized from above reactions presents, can the translation process continues. Equation is adapted from J Christopher Anderson, et. al., Environmental signal integration by a modular AND gate, Molecular Systems Biology 3:133, supplementary information.

Equation

Parameters
k_4: maximum translation rate of T7 mRNA polymerase
k_s, γ_0: rate
γ_4: degradation rate of T7 mRNA polymerase

• Synthesis of trigger CI mRNA

Biological Process
T7 RNA polymerase activates the transcription of CI. The translation of trigger CI mRNA is described together with the translation of bi-stable CI mRNA.

Equation

Parameters
c_5: concentration of trigger CI mRNA
k_5: maximum transcription rate
K_5: microscope dissociation constant
n_5: Hill co-effiency
γ_5: degradation rate of trigger CI mRNA

Bistable

Bistable module was initially constructed by Chunbo Lou, a team member from PKU 2007 Team, also an instructor of our team this year. Here's the mechanism of the bi-stable module. CI(trigger CI and bi-stable CI) both activates the CI promoter and represses the CI434 promoter, while CI434 represses the CI promoter. Initially, the bi-stable remains CI434 state. With trigger CI presents, the synthesis of CI is encouraged and the synthesis of CI434 is repressed. If the trigger is strong enough, the bi-stable will jump to the CI state which means the dog creates a link between food and bell.

• Synthesis of bi-stable CI mRNA

Biological Process
CI(trigger and bi-stable) promotes the transcription of CI, while CI434(from bi-stable) repressed this process.

Equation

Parameters
c_6: concentration of bi-stable CI mRNA
k_6 and k_6': transcription rates
c_8: concentration of CI protein(trigger and bi-stable)
K_6: microscope dissociation constant between CI promoter and CI
n_6: Hill co-effiency between CI promoter and CI
K_6': microscope dissociation constant between CI promoter and CI434
n_6': Hill co-effiency between CI promoter and CI434
γ_6: degradation rate of bi-stable CI mRNA

• Synthesis of CI(trigger and bi-stable)

Biological Process
Trigger CI mRNA(from AND Gate 1 module) and bi-stable CI mRNA(from bi-stable module) are translated into CI protein.

Equation

Parameters
k_8: translation rate of trigger CI mRNA
k_8': translation rate of bi-stable CI mRNA
γ_8: degradation rate of CI protein

• Synthesis of CI434 mRNA

Biological Process
CI protein represses the transcription of CI434 mRNA.

Equation

Parameters
c_9: concentration of CI434 mRNA
k_9: maximum transcription rate of CI434 mRNA
K_9: dissociation constant
n_9: Hill co-effiency
γ_9: degradation rate of CI434 mRNA

• Synthesis of CI434

Biological Process
CI434 mRNA is translated into CI434 protein.

Equation

Parameters
c_10: concentration of CI434
k_10: translation rate of CI434
γ_10: degradation rate of CI434

AND Gate 2

T3 RNA polymerase(P2) is in exact same position of bi-stable CI from bi-stable module. CI activates its expression while CI434 represses it. The T3 RNA polymerase(P2) mRNA also has amber mutation, which can be translated into protein only with Aa-tRNA from AND gate 1 module presents. This is another AND gate, works exactly the same as the one in AND gate 1 module.

• Synthesis of T3 RNA polymerase(P2) mRNA

Biological Process
CI activates the translation of T3 RNA polymerase while CI434 represses it.

Equation

Parameters
c_7: concentration of T3 RNA polymerase(P2) mRNA
k_7 and k_7': transcription rate
K_7: microscope dissociation constant between CI promoter and CI, equals to K_6
K_7': microscope dissociatino constant between CI promoter and CI434, equals to K_6'
n_7: Hill co-effiency between CI promoter and CI, equals to n_6
n_7': Hill co-effiency between CI promoter and CI434, equals to n_6'
γ_7: degradation of T3 RNA polymerase(P2) mRNA

• AND gate

Biological Process
Similar to the previous AND gate, this AND gate consumes T3 RNA polymerase(P2) mRNA and Aa-tRNA to synthesize T3 RNA polymerase(P2) protein.

Equation

Parameters
k_11: maximum transcription rate
k_s',γ_0': rates
γ_11: degradation rate of T3 mRNA polymerase(P2) protein

OR Gate and Output

No matter whether the dog has been trained, it will definitely react when food presents. To achieve this phenomenon, we construct an OR gate. This module works like this: both T3 RNA polymerase(P2) and Sal can activates the expression of GFP, which will be considered as the final output.

• Synthesis of GFP mRNA

Biological Process
Both Sal and T3 RNA polymerase(P2) activate the transcription of GFP.

Equation

Parameters
c_12: concentration of GFP mRNA k_12: maximum transcription rate from Sal K_12: microscope dissociation constant from Sal n_12: Hill co-effiency from Sal k_12': maximum translation rate from T3 RNA polymerase(P2) K_12': microscope dissociation constant from T3 RNA polymerase(P2) n_12': Hill co-effiency from T3 RNA polymerase(P2) γ_12: degradation rate of GFP mRNA

• Synthesis of GFP

Biological Process
GFP mRNA is translated into GFP protein, the final output.

Equation

Parameters
c_13: concentration of GFP
k_13: translation rate of GFP mRNA
γ_13: degradation rate of GFP

Full Model

Here's the full ODEs we have constructed.

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