We decided to follow a deterministic approach for the mathematical representation of our genetic circuits. The simulations were made using MATLAB’s Simulink, according to the following general expressions for the genetic inhibition and activation elements present in the ODE systems. These expressions were adapted from Mendes, P, et al., (2003).
The effect of an inhibitor was modeled as follows:
where α is the maximum transcription rate of a gene, K is the dissociation constant of the inhibitor, I represents its concentration and n is its Hill coefficient.
In the other hand, the effect of an activator was expressed as follows:
where α is also the maximum transcription rate of a gene, K is the dissociation constant of the activator, A represents its concentration and n is its Hill coefficient.
The maximum transcription rate is used instead of the basal transcription rate under the assumption that inducible promoters should not present significant basal activity and that repressible promoters are active at this maximum rate in the absence of an inhibitor. The maximum transcription rate was calculated according to the Team Beijing iGEM 2009 (http://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters).
The general form of the differential equations used to model the change of the mRNA concentration of a gene is as follows:
This general form represents an hypothetical gene, whose transcription is affected by an inhibitor I and an activator A. Different Hill Coefficients (n1 and n2) are used for the inhibitor and activator. The equation includes also the degradation rate, µ, for which we used the same numerical value for all the genes included in our circuits, cell division rate (1/30 min) plus substance degradation rate (1/4.4 min), as suggested by the Team Beijing iGEM 2009.
For the protein concentration differential equations we took into consideration the maximum translation rate, αT , (also calculated as suggested also by the Team Beijing iGEM), the degradation rate, µT , and the rate of postranslational modifications, specifically, phosphorylations of some transcription factors.
Because all the genes in our circuits have been modified with a LVA tag, unless it is otherwise stated, all the protein degradation rates were considered to be equal to the cell division rate (1/30 min) plus the substance degradation rate (1/40 min, as suggested by Team Leuven iGEM 2008, http://2008.igem.org/Team:KULeuven/Model/Inverter)
The general form for the differential equation of protein concentration change the following:
- Mendes, P, et al., (2003), Artiﬁcial gene networks for objective comparison of analysis algorithms, BIOINFORMATICS, Vol. 19 Suppl. 2, pages ii122–ii129
- Team Beijing iGEM 2009 (http://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters).
- Team Leuven iGEM 2008,http://2008.igem.org/Team:KULeuven/Model/Inverter)