Team:Wageningen UR/Project/ModelingProj1

Short repetition of the Hasty construct used as base for our system

For a better understanding of the equations shown below, Figure 1 shows a short repetition of the circuit we used as a base for our design, in our team commonly referred to as the "Hasty construct".

Fig.1: Basic oscillating genetic circuit as published by Danino & Hasty.

The genes luxI and aiiA can be expressed when a AHL-LuxR complex binds to the promoter region. The circuit uses both a positive and negative feedback loop to control the AHL concentration and therefore the expression of the resulting proteins LuxI and AiiA. In the positive loop, LuxI genererates more AHL, which, together with LuxR, can form more of the activating AHL-LuxR complex and therefore stimulates even more production of LuxI and consequently AHL. In the negative feedback loop, AiiA degrades the AHL produced. In our design, aiiA is controlled by the same promoter region as LuxI, in contrast to the original design by Danino et al, who used three copies of the same promoter.

Mathematical model of the construct

Since our bio bricked oscillatory system is based on the design published by Danino et al. in the paper [http://www.nature.com/nature/journal/v463/n7279/abs/nature08753.html “A synchronized quorum of genetic clocks”], our first model of the system is a reproduction of the mathematical model in the supplementary information accompanying the publication mentioned. In their simulations, Danino et al. used the following set of delay differential equations, which we used as base for our simulation tool written in matlab.

Above: Set of delay differential equations derived by Danino et al. to model the oscillatory behaviour of the design using a positive and negative feedback loop.

The production of AiiA and LuxI depend on the internal AHL concentration in the single cell. The steps from gene to protein (transcription, translation, maturation etc.) are not modeled separately, but taken together in the following Hill function:

in which tau represents the time step between the relevant internal AHL concentration and the corresponding AiiA and LuxI, so that