The separate modules: Timer plus SUMO and Kill Switch are combined to form the complete model of the system: Timer - SUMO - Kill Switch. For the final model, the kill switch module is converted in such a way so as the holin and antiholin to be activated by the Pci promoter.

Figure 1: Circuit of the kill switch

By this model, we are finding answers for the following questions:

  1. How many minutes does the cell lysis take from the point of induction?
  2. How much peptides are produced by the circuit?
  3. How much peptides are still uncleaved by the SUMO at the point of cell lysis?

Differential Equations


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]
cpconst 0.5 Transcription rate of Pconst #m/min Assumption
cci 1.79 Maximum transcription rate of Pci #m/min [3]
dmRNA 0.231 Degradation rate of mRNA min-1 [8]
dH 0.0348 Degradation rate of holin / Antiholin M/min [17]
dTET 0.1386 Degradation rate of TET min-1 [9]
dCI 0.042 Degradation rate of CI min-1 [9]
kb,HAH 0.3*10-4 Backward rate [17]
kf,HAH 11.7*10-4 Forward rate [17]
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]
sH 219 Length of Holin amino acids
sAH 103 Length of Antiholin amino acids


The variables used are the same as in the separated modules(TimerPlusSumo,KillSwitch ).The one added in the combined model can be seen below:

Variable Description
Pepm concentration of translated peptide
Pep concentration of transcribed Pep


In Figure 1 the results of the simulation are shown.

Figure 1: Simulation Results


In Figure 1 the behavior of the total circuit is seen, and the answers to the questions can be given:
  1. How many minutes does the cell lysis take from the point of induction?
    After 165 minutes the Holin concentration passes 190 molecules per cell and cell lysis occurs.
  2. How much peptides are produced by the circuit?
    The total concentration is 18000 peptides per cell.
  3. How much peptides are still uncleaved by the SUMO at the point of cell lysis?
    The models shows that there is almost no SUMO-peptide uncleaved.

Sensitivity analysis

To asses the validity of the found answers a sensitivity analysis is performed. In this case the numerical derivative of the solution is taken with respect to the parameters of the model. This derivative represents the change of the solution upon changing the specific parameter. If this change is large, the solution is very sensitive for this parameter. Also, since the solution is highly non-linear this numerical derivative is taken in both directions, as they differ greatly in this model. To compare the found values of the derivative, they are normalized by the nominal value (the solution, e.g. the lyse time). In this way a percentual change is found. Mathematically this can be expressed as:

Figure 2: Equation of the relative sensitivity of s with respect to the parameter pi, note that sn is the nominal value of s. As an example the equation of the lysis time is written out.

So, for the three answers the sensitivity analysis is done, yielding the follow results and conclusions:

  1. The lysis time is not significantly affected by most parameters, only two are important. The degradation rate of cI has the most influence, the relative sensitivity is around 80%. This is logical, because this is of major influence on the timer, as the timer is determined by the time cI takes to degrade. In the same way, the CcI, the expression rate of the cI promotor also influences the lysis time. The cpconst also has a significant influence, a relative sensitivity of around 40%, which is explained by the fact that the expression rate changes the speed of the production of anti-holin and thus the lysis time. What is interesting is that the dimerization parameters do not hold a major influence, as well as the degradation rates of holin and anti-holin.
  2. The total amount of produced peptide is influenced by almost every parameter. The most significant ones are the expression strength of the T7 promoter and the degradation rate of the mRNA, both having a relative sensitivity of over 100%. The T7 promoter strength directly influences the amount of produced SUMO-peptides, thereby the amount of peptides. The mRNA degradation rate influences all the degradations in the system, also the SUMO-peptide production. Furthermore, the parameters found in the previous item, affecting the lysis time, also have a firm influence, over 50%. This is explained by the fact that if the lysis time increases, the circuit has more time to produce the peptides.
  3. The amount of uncleaved peptide is also influenced by a lot of parameters, however the relative sensitivity is always below 200%. This might seem a lot, but the concentration is virtually zero thus these influences can be neglected. It is safe to say that except for very large changes, the conclusion will hold that there is almost no uncleaved peptide.