# Team:Tsinghua-A/Modeling

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# Introduction to Model  Designed gene circuit in cell A and cell B

In our project, we designed a quorum-sensing oscillator which consists of two types of cells. The expression of the reporter genes (GFP of one cell type and GFP of another) of the cells of the same type can fluctuate synchronously and certain designs were made to adjust the phase and the period of oscillation. To understand the property of our system, we built a mathematical model based on ODEs (Ordinary Differential Equations) and DDEs (Delayed Differential Equations) to model and characterize this system. The simulation results helped us to deepen into further characteristics of the system.

# Original Full Model

Firstly we described the system thoroughly without leaving out any seemingly unimportant actions and factors. As a result, the description of the system contains every possible mass actions as well as some hill kinetics, Henri-Michaelis-Menten kinetics, and the parameters were got from literature. The model was represented and simulated in the Matlab toolbox SIMBIOLOGY, but too many parameters make it difficult to do further analyses, So here we only listed all 19 ODEs and a reletive parameter table( see attached pdf file).

# Simplified DDE Simplified DDEs

original model contains too many factors for analyzing the general property of system. To understand the essential characters of the oscillator, we simplify the original model according to certain appropriate assumptions, like Quasi-equilibrium for fast reactions.

After series of derivation based on those assumptions, we came up with the following set of DDEs (Delay Differential Equations) which contains only 6 equations, see the figure right. And it would be much more convenient for us to do some neccessary analyses and research into the mathematical essence of our model. Figure shows all variables are oscillating

We coded the system by DDE description in MATLAB and did simulation analysis accordingly. The result showed that the system could oscillate under certain parameters.

To further understand what parameters could make the system oscillate, we did bifurcation analysis on the Hill parameters. What we had to do was find the critical points where the system can nearly oscillate but a little disruption may lead to a steady state.

Depicting all those critical points, as shown in the figure, the system could oscillate when cellB's Hill parameters were located in the area named Bistable. 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' By adjusting certain parameters, we saw that the oscillation’s period and phase could be controlled properly, which is the most impressive character of our system. Here we present a figure that the oscillation phase was adjusted by adding araC, which could induce the pBad promoter, in cell type B. After adding araC to our system at certain time, the oscillation was interrupted at beginning, but could gradually recover and finally, the phase was changed.

# 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. 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.

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