Team:OUC-China/Model
MOTIVATION
With the rapid development of synthetic biology, there is an increasing requirement of an accurate quantitative regulation of gene expression. Moreover, researchers can provide insight into how the quantitative regulation system can be improved with the application of mathematical modeling.
Briefly, our team was in the devotion to developing a brand new method of accurate quantitative regulation at post-transcriptional level by means of utilizing the effect of inhibiting degradation of mRNA stem loop. More specifically, we hoped to realize a gradient amount of expression at protein or mRNA level by quantitatively manipulating the free energy (∆G) of designed stem loops downstream. However, the designed system is quite complicated than one can think—it consists of a variety of reactions and complex physical chemical process. To validate the effectiveness of initial designs and realize quantitative manipulating, we constructed a series of mathematical expression and built a mathematical modeling to simulate the theoretic curves of mRNA and protein. Thus, we could mutually authenticate the theoretic curves with the experimental ones and prove the feasibility and potential capacity of our designs’ functionality.
INTRODUCTION
Firstly, we analyzed and constructed the whole metabolic network of the system. The whole network could be separated into several independent processes: (1)mRNA transcription, (2) mRNA translation, (3) mRNA Cleavage by RNase E, (4) mRNA decay by RNase II. The complete metabolic network was listed in [ ]. To determine the parameters of these reactions, we respectively sought for Michaelis-Menten Kinetics, Partition Function and Empirical Formula (literature material) to respectively calculate the parameters’ value.
Secondly, our designed stem loops shared a gradient free energy. In order to simulate the theoretic curves of mRNA and protein and add the free energy (∆𝐺) into our model, we expect to determine the mathematical expression between the values of free energy (∆𝐺) of stem loops with the corresponding parameters—the constant of decay. To solve this problem, we sought for the Hansch-Fujita Equation which was a regression models widely used in the chemical and biological sciences. We also combined this sub model with some data from literature and experiments due to some theoretical defects.
Finally, we got the simulated results and evaluated it with statistical method, with the expectation to improve our model and our designs and make it more realistic and practical.
Metabolic reaction networks
(1)$ \phi \xrightarrow{{K}_{r}} [mRNA] $Transcription
(2)$ [mRNA] \xrightarrow{{K}_{p_1}} [mRNA] + [{Protein}_{1}] + [{Protein}_{2}] $Primary translation
(3)$ [mRNA] \xrightarrow{{K}_{d_1}} [{mRNA}_{1}] + [{mRNA}_{2}] $Cleavage by RNase E
(4)$ [{mRNA}_{1}] \xrightarrow{{K}_{{p}_{11}}} [{mRNA}_{1}] + [{Protein}_{1}] $Secondary translation of GFP mRNA
(5)$ [{mRNA}_{2}] \xrightarrow{{K}_{{p}_{12}}} [{mRNA}_{2}] + [{Protein}_{2}] $Secondary translation of mCherry mRNA
(6)$ [mRNA] \xrightarrow{{K}_{d_0}} \phi $Primary decay by RNase II
(7)$ [{mRNA}_{1}] \xrightarrow{{K}_{{d}_{11}}} \phi $Secondary decay of GFP mRNA by RNase II
(8)$ [{mRNA}_{2}] \xrightarrow{{K}_{{d}_{12}}} \phi $Secondary decay of mCherry mRNA by RNase II
(9)$ [{Protein}_{1}] \xrightarrow{{K}_{{d}_{p_1}}} \phi $Decay of GFP
(10)$ [{Protein}_{2}] \xrightarrow{{K}_{{d}_{p_2}}} \phi $Decay of mCherry
Symbol Description
| Symbol | Definition | Units |
|---|---|---|
| $[mRNA]$ | The concentration of transcription product designed | |
| $[{mRNA}_{1}]$ | The concentration of GFP mRNA | |
| $[{mRNA}_{2}]$ | The concentration of GFP mRNA | |
| $[{Protein}_{1}]$ | The concentration of GFP | |
| $[{Protein}_{2}]$ | The concentration of mCherry | |
| $K_r$ | The constant of Transcription | |
| ${K}_{p_1}$ | The constant of Primary translation | |
| ${K}_{d_1}$ | The constant of Cleavage by RNase E | |
| ${K}_{{p}_{11}}$ | The constant of Secondary translation of GFP mRNA | |
| ${K}_{{p}_{12}}$ | The constant of Secondary translation of mCherry mRNA | |
| ${K}_{{d}_{0}}$ | The constant of Primary decay by RNase II | |
| ${K}_{{d}_{11}}$ | The constant of Secondary decay of GFP mRNA by RNase II | |
| ${K}_{{d}_{12}}$ | The constant of Secondary decay of mCherry mRNA by RNase II | |
| ${K}_{{d}_{p_1}}$ | The constant of Decay of GFP | |
| ${K}_{{d}_{p_2}}$ | The constant of Decay of mCherry |
MODEL DEVELOPMENT
1) Michaelis-Menten Kinetics
To determine the reaction constant $K_r$ in (1), we seek for Michaelis-Menten kinetics. Further, we take the pilot process of DNA transcription into consideration for the sake of validating the accuracy of calculation and finally got satisfactory results.
| Symbol | Definition | Units |
|---|---|---|
| $[AraC]$ | The concentration of dissociative repressor protein - AraC | |
| $[AraB]$ | The concentration of arabinose | |
| $[AraC·AraB]$ | The concentration of complex - [AraC·AraB] | |
| ${[AraC]_T}$ | The sum of the concentration of both dissociative repressor protein - Arac and complex - AraC·AraB | |
| $K_i$,i = 1, 2, 3 | reaction rate constant | |
| $k_m$ | Michaelis constant | |
| v | transcription rate |
In our circuit design, we chose araBAD promoter, which will be combined with repressor protein - AraC and the latter represses transcription of mRNA without arabinose. Then, Arabinose of reagent addition will bind to AraC and form the AraB·AraC compound, allowing transcription to occur.
[Hypothesis] We make an assumption that AraC is always in large concentration and that its binding to arabinose happens on a faster time scale to transcription. Therefore, we do not need to consider the individual concentrations of arabinose and AraC, instead we just need to include the concentration of the complex AraC · AraB.
The process boils down to following formula:
$$Arac + AraB \overset{k_1}{\underset{k_2}{\rightleftharpoons}} AraC·AraB \xrightarrow{{k}_{3}} mRNA + Arac$$according to law of mass action,
$$ { {d[Arac·AraB]} \over {dt} } = {{k_1}·{({{[AraC]}_T}-{[AraC·AraB]})} - {k_2}·{[AraC·AraB]} - {k_3}·{[AraC·AraB]} }$$Because $ { {d[Arac·AraB]} \over {dt} } = 0 $
Thus $ { { {k_2} + {k_3}} \over {k1} } = { { {({{[AraC]}_T}-{[AraC·AraB]})}·[AraB] } \over {[AraC·AraB]} } $
Define $ {k_m} = { { {k_2} + {k_3}} \over {k1} } $, then
$$ { [AraC·AraB] } = { { { { [AraC] }_T } · [AraB] } \over { {k_m} + [AraB] } } $$Then the transcription rate can be confirmed like following mathematical expression:
$$ {v} = {k_3}·{ [AraC·AraB] } = {k_3} · { {{[AraC]}_T} } · { {[AraB]} \over { {k_m} + [AraB] } } $$Further, according to the theory of order of reaction, transcription rate can be convert into reaction rate constant.
$$ {K_r} = { v \over { [AraC·AraB] } }$$2) Partition Function
To determine the reaction rate constant ${K}_{{p}_{1}}$ , ${K}_{{p}_{11}}$ , ${K}_{{p}_{12}}$ , we seek for partition function. Inspired by the references [ ], we applied the partition function to the dynamic description of the translation process, in which we can obtain the probability that the ribosomes bind to RNA and then fortunately succeeded in converting the probability into translation rate. Further, we can calculate the reaction rate constant ${K}_{{p}_{1}}$ , ${K}_{{p}_{11}}$ , ${K}_{{p}_{12}}$ by utilizing the concentration data of (2),(4),(5) from our wet lab experiments.
NATIVE VS DESIGNED STEM-LOOPS
By using the native stem loop, we have confirmed that in E.coli, the stem loop at the 3’termini can indeed influence the quantitative expression of its upstream gene. Next we aimed to design nonnative stem loops to verify the precise correlation between the △G and the quantitative expression. But only if this mechanism is determined by △G can we design the stem-loops quantitatively. Thus we need to explore whether the protecting efficiency of the stem loops is determined by its Gibbs free energy or by other factors such as certain specific sequence.
Then we designed 3 stem loops that have the same free energy as a native one (△G=-38.7kcal/mol)[4] but with different base sequence and measured their relative expression of the up and down stream genes on protein and mRNA level.
THE PRECISE CORRELATION
We designed a series of stem loops of gradient free energy to explore the relationship between free energy and quantitative expression. And measured the relative expression of the up and down stream genes on both mRNA and protein level. The result are as follows:
FURTHER VERIFICATION
After we got the relationship between free energy and quantitative expression, we wanted to test our result in the tri-fluorescent reporter system.and we constructed the tri-fluorescent reporter system as follows:
The result are as follows:
[1] Carrier, T. A., & Keasling, J. D. (1997). Engineering mRNA stability in E. coli by the addition of synthetic hairpins using a 5′ cassette system.Biotechnology and bioengineering, 55(3), 577-580.
[2] Smolke, C. D., & Keasling, J. D. (2002). Effect of gene location, mRNA secondary structures, and RNase sites on expression of two genes in an engineered operon. Biotechnol Bioeng, 80(7), 762-776. doi: 10.1002/bit.10434
[3] Nojima, Takahiko, et al. "Controlling the expression ratio of two proteins by inserting a terminator between the two genes." Nucleic Acids Symposium Series. Vol. 50. No. 1. Oxford University Press, 2006.
[4] Nilsson, P., & Uhtin, B. E. (1991). Differential decay of a polycistronic Escherichia coli transcript is initiated by RNaseE‐dependent endonucleolytic processing. Molecular microbiology, 5(7), 1791-1799.
Cistrons Concerto
Thanks:
1.Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences
2.NEW ENGLAND Biolabs
Contact us:
E-mail: oucigem@163.com
Designed and built by @ Jasmine Chen and @ Zexin Jiao
We are OUC-iGEM
