Team:USTC-China/Project/modeling

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Revision as of 07:22, 2 October 2011

results

We have been working hard to model 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.

图片文件夹1

Following pictures show how cheZ concentrations change on time under conditions of different AHL and theophylline concentrations.

图片文件夹2

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

图片文件夹2

Following pictures show how a single cell moves and how cheZ concentration changes on time under conditions of 10nM AHL and gradient of theophylline concentration (30,600) in one single round of simulation.

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Following pictures show 100 cells move under 10nM AHL and gradient of theophylline concentration (25,500).

图片+flash

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 numbers of the cells grow under the rate of k=0.97*exp(-t/5), the unit of time is hour.

For no gradient of theophylline
图片
For theophylline gradient (25,500)
图片
results odes parameters and references

odes and equations

Odes and equations</p>

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

Regulatory ci mRNA

dC1/dt=k1/(1+(A/K1)^n1 )-γ1c1

Regulatory ci protein

dC2/dt=k2*c1-γ2c2

Bistable ci mRNA

dC3/dt=(k3(c4/K3)^n3)/(1+(c4/K3)^n3+(c6/(K_3^' ))^(n_3^' ) )-γ3c3

Total ci protein

dC4/dt=k4*c1+k_4^'*c3-γ4c4

Bistable ci434 mRNA

dC5/dt=k5/(1+(c4/K5)^n5 )-γ5c5

Bistable ci434 protein

dC6/dt=k6*c5-γ6c6 cheZ mRNA

dC7/dt=k7/(1+(c4/K5)^n5 )-γ7c7

cheZ protein

dC8/dt=k8*c7*(3*(theo/K7)^n7)/(1+(a2/K7)^n7 )-γ8*c8

lasI mRNA

dC9/dt=(k9(c4/K3)^n3)/(1+(c4/K3)^n3+(c6/(K_3^' ))^(n_3^' ) )-γ9c9

lasI protein

dC10/dt=k10*c9-γ10*c10

AHL produced

dC11/dt=(k11*c10-γ11*C11)

The real AHL concentration is no 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,

Aij=da*〖C11〗_j*e^(-dij^2*diffu_a)

Tumbling frequency of a cell is determined by its cheYp concentration

Ptumble=[cheYp]^ny/(〖5.6667(SetYp)〗^ny+[cheYp]^ny )

CheYp concentration is related to cheZ concentration

(d[cheYp])/dt=k_y*[Tp]([cheY]_0-[cheYp])-k_(-y) [cheZ]*[cheYp])

For the reaction is so fast, we assume that current cheZ concentration determines current cheYp concentration

[cheYp]=1140.0/(5*[cheZ]+57)

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.

dxi/dt=v*cos⁡(ai)

dyi/dt=v*sin⁡(ai)

For the gradient of theophylline,

〖[theophylline]〗_x=(1mM)*e^(-(x-edge)^2/diffu_t)

results odes parameters and references

parameters and references

Name	Value	Ref.
k1, max transcription rate of regulatory ci mRNA	0.0933 nM/s	1
k2, translation rate of regulatory ci protein	0.0072 /s	1
k3, max transcription rate of bistable ci mRNA	0.0933 nM/s	1
k4, translation rate of regulatory ci protein	0.0072 /s	1
k'4, translation rate of bistable ci protein	0.048 /s	1
k5, max transcription rate of ci434 mRNA	0.0987 nM/s	1
k6, translation rate of ci434 protein	0.0845 /s	1
k7, max transcription rate of cheZ mRNA	0.0834 nM/s	1, 2
k8, translation rate of cheZ protein	0.1869 /s	1, 2, 4, 7
k9, max transcription rate of lasI mRNA	0.014 nM/s	3
k10, translation rate of lasI protein	0.016 /s	3
k11, AHL synthesis rate	0.06	3
K1, Kd between AHL and Plux promoter	1.6 nM	10
K3, Kd between ci protein and bistable ci promoter	40 nM	1
K'3, Kd between ci434 protein and bistable ci promoter	50 nM	1
K5, Kd between ci protein and bistable ci434 promoter	40 nM	1
K7, Kd between theophylline and RNA aptamer	210 nM	4
n1, Hill co-effiency of AHL and Plux promter	1.6	10
n3, Hill co-effiency of ci protein and bistable ci promoter	4	1
n'3, Hill co-effiency of ci434 protein and bistable ci promoter	2	1
n5, Hill co-effiency of ci protein and ci434 promoter	4	1
n7, Hill co-effiency of theophylline and RNA aptamer	3	7
γ1, Degradation rate of regulatory ci mRNA	0.00434 /s	1
γ2, Degradation rate of regulatory ci protein	30.000935 /s	1
γ3, Degradation rate of bistbale ci mRNA	0.00434 /s	1
γ4, Degradation rate of total ci protein	30.000935 /s	1
γ5, Degradation rate of ci434 mRNA	0.00434 /s	1
γ6, Degradation rate of ci434 protein	0.000935 /s	1
γ7, Degradation rate of cheZ mRNA	0.00434 /s	1
γ8, Degradation rate of cheZ protein	0.00434 /s	1
γ9, Degradation rate of lasI mRNA	0.006 /s	3
γ10, Degradation rate of lasI protein	0.0001 /s	3
γ11, Degradation rate of AHL	0.00038 /s	3
diffu_a, diffusion constant of AHL	10^6 /mm^2	assumption
diffu_t, diffusion constant of theophylline	500, 600 /mm^2	assumption
da, diffusivity constant of AHL	0.23 /s	3
[SetYp], wild type ecoli cheYp concentration	4.4 uM	8
ny, Hill co-effiency of cheYp and cell tumble rate	5.5	9
ky, cheYp phosphorylation rate	310^7 /(MS)	8
k-y, cheYp dephosphorylation rate	510^5 /(MS)	8
[Tp], concentration of wild type chemotaxis phosphorylated receptor Tar	510^5 /(MS)	8
v, cell moving speed	0.025 mm/s	2, assumption
k, cell number growth rate	0.97 /h	6