Team:Peking/Modeling/Phototaxis/Stochastic
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- | (1) There are | + | (1) There are multiple bacterial cells in the simulation space. |
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- | (2) Cells can not run through the border of | + | (2) Cells can not run through the border of space. |
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- | (3) The cells | + | (3) The cells division is a random process in uniform distribution. Its time scale is tens of minutes (15~30 min). |
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(4) There are only two states of the cells --- running and tumbling. | (4) There are only two states of the cells --- running and tumbling. | ||
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- | (5) | + | (5) Tumbling event is a Poisson Process, whose average frequency is set by [CheY<sub>P</sub>] with the equation above. |
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- | (6) Cells | + | (6) Cells fall back to running state from tumbling for a fixed time --- τ<sub>0</sub>. |
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(7) Cells run at a fixed speed --- v<sub>0</sub>. | (7) Cells run at a fixed speed --- v<sub>0</sub>. | ||
</li><li> | </li><li> | ||
- | (8) In SPECS model, the running direction after tumbling is independent | + | (8) In SPECS model, the running direction after tumbling is independent of previous direction; |
while in RapidCell model, the new running direction performs as:<sup><a href="#ref2" title="Vladimirov, N., et al.(2008). Dependence of bacterial chemotaxis on gradient shape and adaptation rate. PLoS Comput. Biol., 4: e1000242">[2]</a></sup> | while in RapidCell model, the new running direction performs as:<sup><a href="#ref2" title="Vladimirov, N., et al.(2008). Dependence of bacterial chemotaxis on gradient shape and adaptation rate. PLoS Comput. Biol., 4: e1000242">[2]</a></sup> | ||
</li></ul> | </li></ul> |
Revision as of 17:59, 26 September 2012
Summary
We also traced the cells on the macroscopic level. We started from a colony where all cells are competent for phototaxis function (besides, we also considered cell division). We simulated this phototaxis system in high contrast lighting and gradient lighting environment, respectively.
Stochastic Simulation
It is the component of CheY_{P} that directly interacts with the motors and thus influences the mobility of the cell. In details, CheY_{P} can interact with the flagellar motor to induce CW (clockwise) rotation. When flagellar motors rotate CCW (counter-clockwise), they form a bundle to generate a force similar to a worm wheel. However, if some of the flagellar motors rotate CW (clockwise), the bundle breaks and the cell keeps tumbling. After in CW state for about 0.43s,^{[2]} the flagellar motors return to CCW state and reconstruct the bundle to make the cell run. Since the CW state is triggered by CheY_{P} molecule stochastically and is independent from its state history, this event is a typical Possion Process whose average frequency is determined by the concentration of CheY_{P} with:^{[3]}
where
- FreqCW : the average frequency of CW (clockwise) rotation event
- [CheY_{P}]_{c} : the critical concentration of phosphorylated CheY in this Hill Function
- H : the exponential rate, a constant
- τ_{0} : the average relaxing time in a tumbling event
Phototaxis Simulation
With the principles above, we constructed our simulation system as following:
- (1) There are multiple bacterial cells in the simulation space.
- (2) Cells can not run through the border of space.
- (3) The cells division is a random process in uniform distribution. Its time scale is tens of minutes (15~30 min).
- (4) There are only two states of the cells --- running and tumbling.
- (5) Tumbling event is a Poisson Process, whose average frequency is set by [CheY_{P}] with the equation above.
- (6) Cells fall back to running state from tumbling for a fixed time --- τ_{0}.
- (7) Cells run at a fixed speed --- v_{0}.
- (8) In SPECS model, the running direction after tumbling is independent of previous direction; while in RapidCell model, the new running direction performs as:^{[2]}
where
- θ : the tumbling angle (angle from origin direction to new direction)
- ρ(θ) : the probability density of tumbling angle in value
- ρ(θ,φ) : probability density of tumbling angle in the 3D space
Parameters are shown as following:
Parameter | Value | Unit | Description | Source |
v_{0} | 20 | um/s | running speed | ^{[4]} |
τ_{0} | 0.43 | s | time during a tumbling | ^{[5]} |
CELL_PERIOD | 15~30 | min | period of a cell cycle | |
[CheA]_{T} | 5.3 | u mol/L | total concentration of CheA | ^{[6]} |
[CheZ]_{c} | 1.1 | u mol/L | typical concentration of CheZ | ^{[1]} |
[CheY]_{T} | 9.7 | u mol/L | total concentration of CheY | ^{[6]} |
k_{Y} | 100 | (u mol/L)^{-1} s^{-1} | phosphorylation rate constant of CheY | ^{[6]} |
k_{Z} | 30/[CheZ]_{c} | (u mol/L)^{-1} s^{-1} | dephosphorylation rate constant of CheY | ^{[6]} |
gamma_{Y} | 0.1 | s^{-1} | decay rate constant of CheY_{P} | ^{[2]} |
N | 10.3 | 1 | the exponential rate of Hill Function of CW (clockwise) bias | ^{[3]} |
[CheY]_{Pc} | 3.1 | u mol/L | the critical concentration of phosphorylated CheY of Hill Function of CW (clockwise) bias | ^{[3]} |
r_{A} | 1/3 | 1 | phosphorylation rate of CheA |
Result 1: High Contrast Room
Our first Demonstration is in a high contrast plate, and we would like to see how cells behave differently in difference lighting environment. The lighting of light room is set to 0.8 unit while the dark is set to 0.1 unit with I_{0} = 0.5. Here goes the results:
Figure 1. Phototaxis Simulation Result. Initialized with cells in the center of this room.
Since the frequency of tumbling in light area is much higher than in dark area, the diffusion of population in light area is much smaller. If we initialize the room with cells in uniform distribution, a high population outline will emerge on the border in light area.
Result 2: Light Gradient Room
Just like diffusion (SPECS model in a large population can derive the diffusion equation^{[7]}), a gradient lighting should cause a population flow. We set the lighting from 0 to 1 unit in 1 mm, then discovered the directed movement bias towards light area in this simulation.
Figure 2. Phototaxis Simulation Result. Initialized with cells in a uniform distribution in light gradient.
Reference
- 1. Sourjik, V., et al.(2002) Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer. Proc. Natl Acad. Sci. USA, 99(20): 12669: 12674
- 2. Vladimirov, N., Lovdok, L., Lebiedz, D., and Sourjik, V.(2008). Dependence of bacterial chemotaxis on gradient shape and adaptation rate. PLoS Comput. Biol., 4: e1000242
- 3. Cluzel, P., Surette, M., and Leibler, S.(2000). An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287: 1652: 1655
- 4. Berg, H. C. & Brown, D. A.(1972) Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature, 239: 500: 504
- 5. Turner, L., Ryu, W. S. & Berg, H. C.(2000) Real-time imaging of fluorescent flagellar filaments. J. Bacteriol., 182: 2793: 2801
- 6. Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309
- 7. Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101 Philippe Cluzel. PNAS