Team:UNILausanne/Model/extra
How to get $\alpha$ and $\beta$ values in the ODE model of the repressilator
Obtaining $\alpha$ value
In the scheme of the central dogma $k_1$ is transcription rate $k_2$ translation rate and $d_1$ and $d_2$ the degradation rates (or decay) for mRNA and protein respectively.
$\alpha$ can be obtainde as follows: $$\alpha = \frac{k_1*k_2}{d_1*d_2}$$ And then be reescaled dividing by $K_M$ So, using the values provided in Elowitz and Leiber 2000 (Box 1) we can obtain $\alpha=216$ (we want to thank Michael B.Elowitz for kindly helping us to understand this).
Provided values:
$d_1=\frac{\log_e{2}}{mRNA\ half-life}=\frac{\log_e{2}}{120(s)}$
$d_2=\frac{\log_e{2}}{protein\ half-life}=\frac{\log_e{2}}{600(s)}$
$k_1= 0.5 \frac{transcripts}{s}$
$k_2= 20 \frac{proteins}{transcript}* mRNA\ decay = \frac{20*\log_e{2}}{120}$
$k_M = 40 \frac{monomers}{cell}$
Thus we can obtain $\alpha$:
$$\alpha = \frac{\frac{k_1*k_2}{d_1*d_2}}{K_M}= \frac{\frac{0.5*\frac{20*\log_e{2}}{120}}{\frac{\log_e{2}}{120}*\frac{\log_e{2}}{600}}}{40}=216.4$$
obtaining $\beta$ value
As defined in Elowitz et Leiber 2000 (Box 1): $$\beta = \frac{protein\ decay}{mRNA\ decay}=\frac{\frac{\log_e{2}}{600 (s)}} {\frac{\log_e{2}}{120(s)}} = 0.2 $$
Note that this would be equivalent to calculate $\beta$ as mRNA half-life divided by protein half-life which is the inverse of the y axis of Figure 2c in Elowitz et Leiber 2000.
Sequences used in the Salis RBS predictor (Kill switch model)
We used the version 2.1. of the RBS calculator to predict translation rates (Salis lab). The sequences we used are the following ones:
B0030 (in red) + ccdB (in blue)
atactagaattaaagaggagaaatactagatgcagtttaaggtttacacctataaaagagagagccgttatcgtctgtttgtggatgtacagagtgatattattgacacgcccgggcgacggatggtgatccccctggccagtgcacgtctgctgtcagataaagtctcccgtgaactttacccggtggtgcatatcggggatgaaagctggcgcatgatgaccaccgatatggccagtgtgccggtctccgttatcggggaagaagtggctgatctcagccaccgcgaaaatgacatcaaaaacgccattaacctgatgttctggggaatataa
Obtained translation rate: 17798.33 arbitrary units
2F thermosensor (in orange) + ccdA (in green)
aggaagaaaaatataaatcttcctcaaagtcggtgacagataacaggagtaagtaatgaagcagcgtattacagtgacagttgacagcgacagctatcagttgctcaaggcatatgatgtcaatatctccggtctggtaagcacaaccatgcagaatgaagcccgtcgtctgcgtgccgaacgctggaaagcggaaaatcaggaagggatggctgaggtcgcccggtttattgaaatgaacggctcttttgctgacgagaacagggactggtga
Obtained translation rate: 7354.70 arbitrary units
The rescaling we did (obtained rate/10000) was arbitrary and has no reasoning behind further than obtaining values close to what has been reported in literature [40].
