The figure right is the phase trajectory of two signal molecules in environment, the more it looks like a circle, the more steadily our system will oscillate.

Our system oscillates when parameters were selected in the area named 'Bistable'

Dimensionless Model

In order to make a further analysis on stability of the system, and sensitivity of parameters, we further simplified the model to make them dimensionless. In addition, we tried to introduce feedback to our system and made a brief analysis on different types of feedback we introduced. Some analyses were similar to the simplified DDE model, and you can see more details by clicking read more

Quorum Sensing Effect

What we have analyzed so far is focused on two-cell oscillation. Quorum-sensing oscillator is not simply a matter of expansion in magnitude, but a matter of robustness in allowing difference of each individual cell. Moreover, we test the adjustment of phase and period of oscillation in this part.

As we all know, no two things in this world are exactly the same, so do cells. The major differences between individual cells that we take into consideration include:

●Each cell's activity of promoter is varied, so each cell has different rate to generate AHL.

●The initial amount of AHL may be disproportionally distributed among cells.

The rate of generating AHL is closely related to parameter m and n. Therefore, we introduce randomness to both parameters by letting them obey normal distribution, that is:

m(i)= μ1+N(0,σ1);

n(i)= μ2+N(0,σ2);

μ1 and μ2 are the average ability of generating 30C6HSL and 3012CHSL, and normal distribution-- N(0,σ)--describes the fluctuations of AHL generating rate in individual cell. We then expanded our equations from 2 cells to a population of cells. Each cell share a mutual environment in which we assume that AHL in environment is proportionally distributed.

Figures were simulated out
under varied initual AHL concentration

The figures indicate that our system can oscillate synchronically being able to tolerate differences at certain range among a population of cells.

We also tested whether the oscillation is dependent on initial distribution of AHL by changing the initial amount drastically by letting them follow uniform distribution. That is:

Initial(i)= U(0,20);

Based on this distribution restraining the initial AHL concentration in each cell, we simulated out a figure as follows.

The results demonstratively give evidence proving that our system can start to oscillate synchronically given variant initial starting status.

References

[1] Uri Alon, (2007). Network motifs: theory and experimental approaches. Nature.

[2] Chunbo Lou, Xili Liu, Ming Ni, et al. (2010). Synthesizing a novel genetic sequential logic circuit: a push-on push-off switch. Molecular Systems Biology.

[3] Tal Danino, Octavio Mondragon-Palomino, Lev Tsimring & Jeff Hasty (2010). A synchronized quorum of genetic clocks. Nature.

[4] Marcel Tigges, Tatiana T. Marquez-Lago, Jorg Stelling & Martin Fussenegger (2009). A tunable synthetic mammalian oscillator. Nature.

[5] Sergi Regot, Javio Macia el al. (2010). Distributed biological computation with multicellular engineered networks. Nature.

[6] Martin Fussenegger, (2010). Synchronized bacterial clocks. Nature.

[7] Andrew H Babiskin and Christina D Smolke, (2011). A synthetic library of RNA control modules for predictable tuning of gene expression in yeast. Molecular Systems Biology.

[8] Santhosh Palani and Casim A Sarkar, (2011). Synthetic conversion of a graded receptor signal into a tunable, reversible switch. Molecular Systems Biology.

[9] Nancy Kopell, (2002). Synchronizing genetic relaxation oscillation by intercell signaling. PNS