IGEM:Tsinghua/2007/Projects/RAP

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Synthetic oscillator in a single operon that simulates the natural oscillations.
Referring to the expression oscillators, the Elowitz oscillator is the first and the only typically successful one so far, in which three operons express three different transcriptional repressors, lacI, lambda C1 and tetR, respectively. The three transcriptional repressors in the Elowitz oscillators inhibit one another to generate oscillations. However, this is a relatively complex system, and protein accumulation has been observed at the single-cell scale during the oscillating cycles. While the natural oscillations usually works in pulses including a stage when the signal is reset to zero. For example, in the process of the active potential, the cardiac cycles and the muscle contraction, the neural/electric signal is triggered at the state of &quot;zero&quot;, and then amplified until a feedback inhibiting mechanism is reinforced to blocked the amplification. At last, as the inhibition dominates, the signal falls to the state of &quot;zero&quot;, which can be thought as the &quot;reset&quot;. To understand and simulate the natural oscillations, we propose a method to allow the oscillator being reset to &quot;zero&quot; after each cycle and find that it is an still simpler method to generate oscillations. A fast-degrading DNA polymerase and a fast-degrading transcriptional repressor are engaged in this system. This oscillation works in four stages: (1)Triggering: the RNA polymerase gene is expressed by constructive promoters. (2)Amplication: the RNA transcripts the gene coding itself. (3) Inhibiting: the RNA transcripts a transcriptional repressor, which has longer degrading half-life than the RNA polymerase, that blocks the transcription of the RNA polymerase gene. (4) Resetting: the transcription of the RNA polymerase is blocked and until the transcriptional repressor degrades to a concentration below a specific level. And we found that this oscillator can be constructed in a single operon.

Fast-degrading T7 RNA polymerase.
Fast-degrading transcriptional repressors are available, such as the lacI, lambda C1 and tetR, because those protein are not conservative in their C-terminus and ready to be fused with a degrading marker. While the fast-degrading RNA polymerase is the key in this project. Most RNA polymerases consist of multiple subunits, except for T odd number RNA polymerase, SP6 RNA polymerase and so on. Those RNA polymerases are very conservative at their C-terminus and can not be engineered to fuse a C-terminus degrading marker. Fortunately their N-terminus are not that conservative and can be engineered to get some N-terminus fusions. Some degrading marker of E. coli can be identified on the N-terminus, such as the degrading marker recognized by Lon and other heat shock proteases. UmuD degrades quickly in E. coli and its N-terminal 30 amino residues were reported to work as a N-terminal degrading marker. Therefore, we try to construct a fast degrading T7 RNA polymerase with UmuD N-terminal degrading marker.

Model and simulation
To describe the system we are going to construct, we model the mRNA levels and protein levels respectively. In the following description, we use lower-case of the first letter to indicate mRNA levels whereas up-case to indicate protein levels. The system is described by following differential equations: where, We assume that the kinetics of mRNAs is governed by two respects: its spontaneous degradation and transcriptions which produce them specifically. Spontaneous degradation is assumed to be a ‘first-order’ reaction which means the rate of mRNA degradation is proportional to its current level and therefore produces a constant half life time. Here, for convenience, we assume that all mRNAs have the same half life time.

The rate of transcription is calculated by multiplying 3 items: the elongation rate, the ratio of DNA elements which occupied by transcription factors and the total copy numbers of DNA elements. The key point here is to estimate occupation ratios. We use classical M-C equation (eq. 7 and 8) to describe the behavior of protein binding. In these equations, Hill coefficient indicates the cooperation during the binding process. Please note that in calculating occupation ratio of LacI, the protein level is used after divided by 2 due to LacI binds to lacO as tetramers. In our model, we conveniently assume that LacI monomers form tetramers tightly soon after its production.

It is a little more complex to model the transcription of T7 polymerase which is regulated by a constitutive promoter, a T7 promoter and a lac operon. The transcription is considered as following conditions: The transcription ceases when LacI binding to lacO, no matter whether T7 or host polymerases are bound. Without the repression by LacI, T7 polymerase, once successfully binds to its promoter, dominates the transcription. The host polymerase functions only if neither LacI nor T7 polymerase bound. In eq. 9 and 10, number of activated DNA (i.e. DNA which form complex with T7 polymerase but not LacI) and number of free DNA (i.e. DNA which binds to neither T7 polymerase nor LacI) are calculated respectively.

The kinetics of proteins is also governed by two respects: spontaneous degradation and translation. Like the case assumed in mRNA modeling, spontaneous degradation of proteins are taken as first-order reaction. The translation rate is assumed to be the same among different proteins.

To make our model more realistic, we modify our model to a stochastic one. In this model, all the parameters, except copy numbers of plasmids, are assumed to obey to a norm distribution and a ‘sigma’ value (which indicates the peak width) is set for each parameter.

The numerical solution is made using following parameter values:

Copy number of the repressor plasmid: n_lac_plasmid=30 copy

Copy number of the amplifier plasmid: n_t7_plasmid=5 copy

Transcription rate of T7 polymerase: r_transcribe_T7=600 mRNA/min

Transcription rate of host polymerase: r_transcribe_P=30 mRNA/min

Translation rate: r_translate=10 proteins/min

Half life time of mRNA: τ(mRNA)=3min

Half life time of LacI: τ(LacI)=10min

Half life time of T7: τ(T7)=10min

Half life time of EGFP: τ(EGFP)=40min

Dissociation constant of LacI: K(LacI)=10 tetramers per cell

Dissociation constant of T7 polymerase: K(T7)=10 monomers per cell

Hill coefficient of LacI: hill_LacI=1;|

Hill coefficient of T7: hill_T7=1;

In stochastic model, sigma for all parameters equal to 10 percents of their average values respectively.

FIGURE1:T7&LacI（protein） The protein levels of T7 RNAP(Red) and LacI(Blue) are simulated by MATLAB with parameters described above. (Note: The scale of LacI molecules is 1,000 fold of that of T7 RNAP)

FIGURE2:T7&LacI_mRNA The mRNA levels of T7 RNAP(Red) and LacI(Blue) are simulated by MATLAB with parameters described above. (Note: The scale of LacI molecules is 1,000 fold of that of T7 RNAP)

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Plasmids
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Reagents
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Flowchart


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