Team:TU-Delft/Timer Plus Sumo

Timer Plus Sumo

In this section the system of Figure 1 is modeled. The structure of the timer has two repressing promoters (PcI and Ptet) and the input is the T7 promoter and the output is the protease Ulp-1. This Ulp-1 cleaves off the SUMO from the produced SUMO-peptide.

Figure 1: Circuit of the timer including sumo cleaving

Differential Equations

The above circuit can be represented by the following differential equations. We assume a binary behavior of the T7 promoter. In the presence of IPTG, the T7 promoter will be active. So, we make the assumption that the T7 is binary variable with two possible states: either active 1 or inactive 0.

Parameters

For the stated parameters of the differential equations many values can be found in the literature. The degradation rates are relatively unknown, while the transcription rates vary widely. Many of these transcription rates resulted in our model in infeasible results, e.g. concentrations of millions of molecules per cell. For the PT7 a relative low transcription rate was thus selected in order to get the values in a reasonable range. The other promoters in the model we found as a fraction of the transcription rate of PT7 in the literature.

Parameter	Value	Description	Units	Reference
ca	1020	Translation rate per amino acid	min-1#a-1	[7]
cT7	4.16	Maximum transcription rate of T7	#m/min	[2]
cptet	2.79	Maximum transcription rate of Ptet	#m/min	[4]
cci	1.79	Maximum transcription rate of Pci	#m/min	[3]
dmRNA	0.231	Degradation rate of mRNA	min-1	[8]
dTET	0.1386	Degradation rate of TET	min-1	[9]
dCI	0.042	Degradation rate of CI	min-1	[9]
dPEP	2.1*10-3	Degradation rate of the peptide	min-1	Assumed three times slower same as GFP
dPSU	6.3*10-3	Degradation rate of the peptide plus SUMO	min-1	Assumed the same as GFP
dUlp	1.263*10-2	Degradation rate of Ulp	min-1	Assumed twice the rate of GFP
lt7	0.002	Leakage factor of T7	-	Assumption
lptet	0.002	Leakage factor of Ptet	-	Assumption
lci	0.002	Leakage factor of Pci	-	Assumption
ktet	6	Dissociation constant of Ptet	#m	[10]
kci	20	Dissociation constant of Pci	#m	[10]
kcUlp	3	Turnover rate of Ulp	min-1	[6]
nci	3	Hills coefficient	-	[11]
ntet	3	Hills coefficient	-	[11]
s	0 or 1	Activation/Inactivation of T7 promoter	Binary	Assumption
sci	228	Length of CI	amino acids	[12]
sPSU	18 + 110	Length of peptide plus SUMO	amino acids	[12]
sTET	206	Length of TET	amino acids	[13]
sUlp	233	Length of Ulp1	amino acids	[13]

Results

Here the results are given of the simulation upon activating the T7 promoter. For starting conditions the steady state values of the concentrations are used when T7 is switched off.

Figure 2: Simulation Results

A second simulation is run, in which only SUMO-peptide is produced as there is no Ulp. This will allow some confirmation of the values with the literature.

Figure 3: Simulation Results without Ulp

Discussion

In the graph of Figure 2 the effect of the timer is clearly observed, at first the peptide+SUMO is produced, and after approximatly 80 minutes more Ulp is produced leading to an increased cleaving of the SUMO. Also note that there is cleaving of the SUMO already from the start, as the promoters are somewhat leaky.

In Figure 3 the circuit is modeled without Ulp present, so the SUMO is not cleaved off. This is done to be able to found experimental data. In [], an expression of a SUMO fusion protein is done with the T7 promoter. These results were 0.5 μ is produced in 2L of culture. Calculating this back using an OD of 1, this results in