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}-\gamma_1 c_1+\gamma_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
\gamma_1: degradation and dilution rate of tRNA. Unless notice, "degradation rate" in this model means the combination of degradation rate and dilution rate.
\gamma_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
\gamma_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
\gamma_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: rate
\gamma_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
\gamma_5: degradation rate of trigger CI mRNA

Bistable

AND Gate 2

Output

Full Model

^Top