Modeling

1.Results
 2.Odes and equations
 3.Parameters
 4.References

Results

We have been working hard to model our system.

Before presenting our results, we would like to say that lasR protein is assumed to be always over production in our system, although it is not in our original design, it really helps a lot to simplify our modeling work.

Suggestions and questions are welcomed, for contact, you can email to PJH_panda(jhpanda@mail.ustc.edu.cn) or any other member of our team.

All codes are written in Python26(http://www.python.org/getit/releases/2.6/), you can download our codes here File:IGEM USTC 2011.zip, you will need Scipy(http://www.scipy.org/Download), Numpy(http://www.scipy.org/Download), and Matplotlib(http://matplotlib.sourceforge.net/) to run our codes.

Now see what we have done

Firstly, we would like to present some basic character of our system

In a single cell, the system acts like a fast switch upon AHL concentration, on the other hand, cheZ ouput is a slow switch of theophylline concentration.

We can see that cheZ concentration rises rapidly when AHL concentration rises from 3nM to 4nM. For the switch is so fast that we can use quorum system in our design. CheZ output rises relatively rapid when theophylline concentration is around 210nM.

For the system without quorum part, If AHL concentration is low, regulate ci protein accumulates, the toggle switch is in the ci state, cheZ production is very low, so the cell can rarely move. And when AHL concentration is high, not enough regulate ci protein is produced, the toggle switch is in the ci434 state, cheZ production is high, so the cell can spend less time tumbling, and will move to the direction where cheZ concentration rises.

Following pictures show how steady-state cheZ concentrations change under conditions of different AHL and theophylline concentrations （t = 64800）.

Following pictures show the switch's performances under different max transcription rate of ci mRNA (t = 43200). 1 means 0.1*k1, 2 means 0.2*k1, and 3 means 0.3*k1

Following pictures show how cheZ concentrations change on time under conditions of different AHL and theophylline concentrations (t = 43200).

Following pictures show how a single cell moves on time under conditions of different AHL and theophylline concentrations in one single round of simulation (t = 72000).

Following pictures show how a single cell moves and how cheZ concentration changes by time under conditions of different AHL and gradient of theophylline concentration (30,600) in one single round of simulation (t = 72000).

AHL = 10 nM

AHL = 3 nM

Secondly, we want to see the movement of 100 cells (or more) without quorum part under 10nM AHL and theophylline gradient (25,500)

Assume that 100 cells are around the center for 0.25mm distance at most. Following pictures shows the position of the 100 cells without quorum part under 10nM AHL and gradient of theophylline concentration (25,500) after t = 86400. And a little movie is shown

             320px

Thirdly, after the previous two modelling part, we believe that if we add quorum part into our system, the results will be acceptable.

For the whole system with quorum part, in the beginning, we assume that the concentrations of all related macromolecules are zero. Cells begin to divide, cell number is growing, and AHL is accumulating, when AHL concentration is high enough, toggle switch in some of the cells turns to the ci434 state, and these cells is moving out. After some time, AHL concentration in these cells will drop, and the toggle switch will most likely turn to the ci state, cells stop moving.

For this part we assume that initial cell number is 20, and numbers of the cells grow under the rate of k=0.97*exp(-t/5), the unit of time is hour.

In the beginning, max transcription rate of regulatory ci mRNA is set as k1. However, we found cheZ concentration keeps dropping during simulation time. So we set it to be 0.3*k1, this will not change the basic character of bistable switch. From the first part of our modeling, AHL acts as a fast switch between 3nM and 4nM. If we set max transcription rate of regulatory ci mRNA as 0.3*k1, the switch is between 1nM and 2nM.

For no gradient of theophylline

             320px

For theophylline gradient (25,500)

             320px

To be settled

There are some points should be improved in our modeling.

1. Noises

The strength of the same promoter in different cells will be a little different. Also, their translation abilities will be a little different. So, intrinsic noises must be considered to better describe our system. In our wet lab, before considering quorum sensing, we found that, because of noises around, cell clones contain bistable part grown on our plate produced two fluorescence randomly. Cell clones without regulatory part tend to be green, cell clones with regulatory part tend to be red. However this can not be seen in our current model. And if noises are considered in our model, we want to move aptamer-cheZ part to the ci (the red) side， as in our construction (Quorumsensing).

2. AHL diffusivity

In our current model, we have been trying hard to not find another more appropriate equation to describe AHL diffusivity. In our current model, the equation is so simplified that if one cell has more than 20 cells' distance below 1.5 um, the AHL concentration will be larger than AHL produced by one cell. Although, 20 cells for 1.5um will be difficult, for a large number of cells, the situation where AHL concentration larger than AHL produced may arise.

3. Application

For our project is not only about movement, but also about AIIS, so we think it necessary to take this part into our modelling.

Results  Odes  Parameters  References

Odes

Odes and equations

     C1, regulate ci mRNA concentration

     C2, regulate ci protein concentration

     C3, bistable ci mRNA concentration

     C4, total ci protein concentration

     C5, bistable ci434 mRNA concentration

     C6, bistable ci434 protein concentration

     C7, cheZ mRNA concentration

     C8, cheZ protein concentration

     C9, I mRNA concentration

     C10, I protein concentration

     C11, AHL produced

     A, AHL concentration

For regulatory ci mRNA and protein

For Bistable Part

For cheZ mRNA and protein

For Quorum Sensing Part

The real AHL concentration is not the AHL produced. Considering about fast diffusivity of AHL, we assume that external AHL concentration is zero, although it is not quite exact, but it can describe the character of AHL well. That is, for cell i, the compact of cell j is,

Tumbling frequency of a cell is determined by its cheYp concentration

CheYp concentration is related to cheZ concentration, and for the reaction is so fast, we assume that current cheZ concentration determines current cheYp concentration

We assume that cells move at the speed of 0.025um/s on our semisolid plate. For the movement of a single cell, if the cell tumbles, it will stay of the current second, and change its direction later, if not, the cell keeps on moving in one direction.

For the gradient of theophylline,

For cell number growth, here, we assume that the growth rate drops by time, the unit of time is hour

Results  Odes  Parameters  References

Parameters

Parameters