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

By this model we are finding answers to the following questions:

  1. At what time point does the peptide+SUMO start to be produced?
  2. After how many minutes does the Ulp-1 cleave the peptide?

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.


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]


Variable Description
TETm concentration of translated TET
PSUm concentration of translated peptide plus SUMO
CIm concentration of translated Ci
Ulpm concentration of translated Ulp
TET concentration of transcribed TET
PSU concentration of transcribed peptide plus SUMO
CI concentration of transcribed Ci
Ulp concentration of transcribed Ulp


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


In the graph of Figure 2 the behavior of the circuit can be seen and the answers to the questions can be given:

  1. The peptide+SUMO is being produced at time point zero, reaching the maximum number of molecules, 3400 molecules, at time point 70.
  2. The cleaving of the SUMO can be observed from the start probably due to the fact that the promoter is leaky. After 80 minutes more Ulp-1 is produced, leading to an increased cleaving of the SUMO.