Team:HokkaidoU Japan/Promoter/Modeling
From 2013.igem.org
Maestro E. coli
Promoter
![](https://static.igem.org/mediawiki/2013/e/ea/HokkaidoU2013_Maestro_Header.png)
Modeling
We tried to theoretically predict the strength distribution of 4096 promoters, which were artificially created by random mutation. We followed these 3 steps, referring the previous study[1][2].
- Calculate the binding energy of each promoter and σ-factor using the sequence
- Convert the binding energy to the probability that RNAP binds promoter using the method of statistical mechanics
- Utilizing the binding probability as the transcription efficiency
STEP 1: Calculation of Binding Energy
First, we found the binding energy of RNAP and our promoters. As we mutated only -35 region, we only use this region for calculations. Here we define the binding energy $\varepsilon$ as the energy released by RNAP’s binding to promoter. Simply saying, the higher is the binding energy, the stronger is the binding. We referred the data in Kenney, et al.[3] to calculate each binding energy.
The distribution of computed 4096 promoters' binding energies is shown below. The horizontal axis stands for $\varepsilon$ (at $0.05 k_BT$ intervals) and the vertical axis sample number.
![](https://static.igem.org/mediawiki/2013/b/bb/HokkaidoU2013_promoter_Modeling_fig1.png)
![](https://static.igem.org/mediawiki/2013/1/16/HokkaidoU2013_promoter_Modeling_fig2.png)
STEP 2: Conversion from Binding Energy to Binding Probability
Next, we estimated the binding probability. On this step, we used the method of statistical mechanics. So we assumed the following.
- The cell is a closed system
- There are $P$ RNAPs bound somewhere on DNA
- The number of bases is $N$ (bp) and $1$ of $N$ bases is +1 position of the promoter
The principle of statistical mechanics is very easy; any state emerges with the same probability. So we counted up the number of state. A state stands for every information of all the particles in the system, so the number is enormous. $W$ represents this number. Here $W$ can be separated as the following. \[ W=W_{\mathrm{unbound}}+W_{\mathrm{bound}} \] $W_{\mathrm{bound}}$ represents the number of state where the promoter is occupied and $W_{\mathrm{unbound}}$ unoccupied.
The purpose of this step is to find the ratio $W_{\mathrm{unbound}}:W_{\mathrm{bound}}$. Concerning the position of RNAP, \begin{align*} W_{\mathrm{unbound}}:W_{\mathrm{bound}}&=\frac{(N-1)!}{P!(N-P-1)!}\times W_{\mathrm{R}}(E):1 \times \frac{(N-1)!}{(P-1)!(N-P)!}\times W_{\mathrm{R}}(E+\varepsilon) \\ &=1:\frac{P}{N-P} \times \frac{W_{\mathrm{R}}(E+\varepsilon)}{W_{\mathrm{R}}(E)} \end{align*} where $W_{\mathrm{R}}$ represents the number of state in reservoir system (a system excluding the imformation of RNAP's position). $W_{\mathrm{R}}$ is a function of internal energy. Then, we converted $W_{\mathrm{R}}$ to entropy $S$ using the conversion formula: $S \equiv k_B \ln{W}$ ($k_B$ stands for Boltzmann constant, $\approx 1.38\times 10^{-23} \mathrm{J\cdot K^{-1}}$). \begin{align*} &=1:\frac{P}{N-P} \times \frac{\exp\left(\frac{S(E+\varepsilon)}{k_B}\right)}{\exp\left(\frac{S(E)}{k_B}\right)} \\ &=1:\frac{P}{N-P} \times \exp\left(\frac{S(E+\varepsilon)-S(E)}{k_B}\right) \\ &\approx 1:\frac{P}{N} \times \exp\left(\frac{\varepsilon \frac{\partial S}{\partial E}}{k_B}\right) \end{align*} Entropy $S$ and energy $E$ is connected as temperature $T$ as the following. \[ \frac{\partial S}{\partial E} \equiv \frac{1}{T} \] So, \[ W_{\mathrm{unbound}}:W_{\mathrm{bound}} \approx 1:\frac{P}{N} \times \exp\left(\frac{\varepsilon}{k_BT}\right) \] This is a final form of this calculation. Approximately the binding energy of -35 region is exponentially proportional to the binding probability.
STEP 3: Conclusion
The last step is to convert the binding probability to the transcription efficiency. Let us assume these suppositions.
- RNAP bound to promoter promptly initiate transcription
- There is no "traffic jam" of RNAPs on DNA (i. e., RNAP's transcription initiation is rate-limiting)
These assumptions mean that we can directly use the value of binding probability as transcription energy in an arbitrary unit. In this way, we get following conclusive result.
![](https://static.igem.org/mediawiki/2013/d/d3/HokkaidoU2013_promoter_Modeling_fig4.png)
As you can see in this figure, the strengths of our promoter families vary about 1000 fold!
- Rob Phillips, Jane Kondev and Julie Theriot. (2008). Physical Biology of the Cell. Garland Science.
- Brewster, et al. (2012). Tuning promoter strength through RNA polymerase binding site design in Escherichia coli. PLoS computational biology.
- Kenney, et al. (2010). Using deep sequencing to characterize the biophysical mechanism of a transcriptional regulatory sequence. Proceeding of the National Academy of Sciences of the United States of America.