Team:Peking/Modeling/Phototaxis/Stochastic

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  <h3 id="title1">Summary</h3>
  <h3 id="title1">Summary</h3>
  <p>
  <p>
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After the macro view of our phototaxis system, we also traced the cells on the micro level. We built a colony and gave the cells phototaxis function (besides, we also considered cell division). We simulated this phototaxis system in high contrast lighting and gradient lighting environment respectively. From simulation results, we confirm our conclusion that the linkage between population density and light signal.
+
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.  
  </p>
  </p>
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</div>
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  <h3 id="title2">Stochastic Simulation</h3>
  <h3 id="title2">Stochastic Simulation</h3>
  <p>
  <p>
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As in previous introduction, it is the component of CheY<sub>P</sub> that directly interacts with the motors and thus influence the mobility of the cell. In detail, CheY<sub>P</sub> can interact the flagellar motor to induce CW (clockwise) rotation. When flagellar motors rotate CCW (counterclockwise), 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,<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> the flagellar motors return to CCW state and reconstruct the bundle to make the cell run. Since the CW state is triggered by CheY<sub>P</sub> 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<sub>P</sub> with:<sup><a href="#ref3" title="Cluzel, P., et al.(2000). An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287: 1652: 1655">[3]</a></sup>
+
It is the component of CheY<sub>P</sub> that directly interacts with the motors and thus influences the mobility of the cell. In details, CheY<sub>P</sub> 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, the bundle breaks and the cell keeps tumbling. After in CW state for about 0.43s,<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> the flagellar motors return to CCW state and reconstruct the bundle to make the cell run. Since the CW state is triggered by CheY<sub>P</sub> 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<sub>P</sub> with:<sup><a href="#ref3" title="Cluzel, P., et al.(2000). An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287: 1652: 1655">[3]</a></sup>
  </p>
  </p>
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  </div>
  </div>
  <p>where</p><ul><li>
  <p>where</p><ul><li>
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FreqCW : the average frequency of CW (clockwise) rotation inducing</li><li>
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FreqCW : the average frequency of CW (clockwise) rotation event</li><li>
[CheY<sub>P</sub>]<sub>c</sub> : the critical concentration of phosphorylated CheY in this Hill Function</li><li>
[CheY<sub>P</sub>]<sub>c</sub> : the critical concentration of phosphorylated CheY in this Hill Function</li><li>
H : the exponential rate, a constant</li><li>
H : the exponential rate, a constant</li><li>
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&tau;<sub>0</sub> : the average relaxing time in a tumbling inducing</li></ul>.
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&tau;<sub>0</sub> : the average relaxing time in a tumbling event</li></ul>.
</div>
</div>
<div class="PKU_context floatR">
<div class="PKU_context floatR">
  <h3 id="title3">Phototaxis Simulation</h3>
  <h3 id="title3">Phototaxis Simulation</h3>
  <p>
  <p>
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With the principles above, we construct our simulation system as following:
+
With the principles above, we constructed our simulation system as following:
  </p>
  </p>
  <ul><li>
  <ul><li>
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(1) There are several bacteria cells in a room.
+
(1) There are multiple  bacterial cells in the simulation space.
   </li><li>
   </li><li>
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(2) Cells can not run through the border of room.
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(2) Cells can not run through the border of space.
   </li><li>
   </li><li>
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(3) The cells can divide in a random cell cycle in uniform distribution between 15min to 30min.
+
(3) The cells division is a random process in uniform distribution. Its time scale is tens of minutes (15~30 min).
   </li><li>
   </li><li>
(4) There are only two states of the cells --- running and tumbling.
(4) There are only two states of the cells --- running and tumbling.
   </li><li>
   </li><li>
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(5) Cells trigger tumbling as a Poisson Process, the average frequency is set by [CheY<sub>P</sub>] with the equation above.
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(5) Tumbling event is a Poisson Process, whose average frequency is set by [CheY<sub>P</sub>] with the equation above.
   </li><li>
   </li><li>
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(6) Cells return running state after tumbling for a fixed time --- &tau;<sub>0</sub>.
+
(6) Cells fall back to running state from tumbling for a fixed time --- &tau;<sub>0</sub>.
   </li><li>
   </li><li>
(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>
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(8) In SPECS model, the running direction after tumbling is independent from previous direction;
+
(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>
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   <td>CELL_PERIOD</td><td>15~30</td><td>min</td><td>period of a cell cycle</td><td></td>
   <td>CELL_PERIOD</td><td>15~30</td><td>min</td><td>period of a cell cycle</td><td></td>
   </tr><tr>
   </tr><tr>
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   <td>[CheA]<sub>T</sub></td><td>5.3</td><td>u mol/L</td><td>total concentration of CheA</td><td><sup><a href="#ref6" title="Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309">[6]</a></sup></td>
+
   <td>[CheA]<sub>t</sub></td><td>5.3</td><td>u mol/L</td><td>total concentration of CheA</td><td><sup><a href="#ref6" title="Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309">[6]</a></sup></td>
   </tr><tr>
   </tr><tr>
   <td>[CheZ]<sub>c</sub></td><td>1.1</td><td>u mol/L</td><td>typical concentration of CheZ</td><td><sup><a href="#ref1" title="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">[1]</a></sup></td>
   <td>[CheZ]<sub>c</sub></td><td>1.1</td><td>u mol/L</td><td>typical concentration of CheZ</td><td><sup><a href="#ref1" title="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">[1]</a></sup></td>
   </tr><tr>
   </tr><tr>
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   <td>[CheY]<sub>T</sub></td><td>9.7</td><td>u mol/L</td><td>total concentration of CheY</td><td><sup><a href="#ref6" title="Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309">[6]</a></sup></td>
+
   <td>[CheY]<sub>t</sub></td><td>9.7</td><td>u mol/L</td><td>total concentration of CheY</td><td><sup><a href="#ref6" title="Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309">[6]</a></sup></td>
   </tr><tr>
   </tr><tr>
   <td>k<sub>Y</sub></td><td>100</td><td>(u mol/L)<sup>-1</sup> s<sup>-1</sup></td><td>phosphorylation rate constant of CheY</td><td><sup><a href="#ref6" title="Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309">[6]</a></sup></td>
   <td>k<sub>Y</sub></td><td>100</td><td>(u mol/L)<sup>-1</sup> s<sup>-1</sup></td><td>phosphorylation rate constant of CheY</td><td><sup><a href="#ref6" title="Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309">[6]</a></sup></td>
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</div>
<div class="PKU_context floatR">
<div class="PKU_context floatR">
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  <h3 id="title4">Result 1: High Contrast Room</h3>
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  <h3 id="title4">Result 1: High Contrast Space</h3>
  <p>
  <p>
-
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<sub>0</sub> = 0.5. Here goes the results:
+
Our first demonstration is phototaxis in a high contrast plate, and we would like to see how cells behave differently in lighting or dark environment. The light intensity of light space was set to 0.8 unit while the light intensity of dark space was set to 0.1 unit with I<sub>0</sub> = 0.5. Here goes the result:
  </p>
  </p>
  <div>
  <div>
   <img src="/wiki/images/8/8e/Peking2012_PhototaxisRapidCelldot.png" alt="" style="width:500px;"/>
   <img src="/wiki/images/8/8e/Peking2012_PhototaxisRapidCelldot.png" alt="" style="width:500px;"/>
-
   <p class="description">Figure 1. Phototaxis Simulation Result. Initialized with cells in the center of this room.</p>
+
   <p class="description">Figure 1. Phototaxis Simulation Result. Initialized with cells in the center of this space.</p>
  </div>
  </div>
-
<p>
 
-
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.
 
-
</p>
 
</div>
</div>
<div class="PKU_context floatR">
<div class="PKU_context floatR">
-
  <h3 id="title5">Result 2: Light Gradient Room</h3>
+
  <h3 id="title5">Result 2: Light-Gradient Space</h3>
  <p>
  <p>
-
Just like diffusion (SPECS model in a large population can derive the diffusion equation<sup><a href="#ref7" title="Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101">[7]</a></sup>), 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.
+
Just like diffusion (SPECS model in a large population can derive the diffusion equation<sup><a href="#ref7" title="Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101">[7]</a></sup>), a gradient light intensity should cause a population flow. We set the light intensity from 0 to 1 unit in 1 mm, then observed directed movement bias towards light area in this simulation.
  </p>
  </p>
  <div>
  <div>
   <img src="/wiki/images/d/d1/Peking2012_PhototaxisRapidCellUniGradInv.png" alt="" style="width:300px;"/>
   <img src="/wiki/images/d/d1/Peking2012_PhototaxisRapidCellUniGradInv.png" alt="" style="width:300px;"/>
   <img src="/wiki/images/e/e2/Peking2012_PhototaxisRapidCellUniGradLin.png" alt="" style="width:300px;"/>
   <img src="/wiki/images/e/e2/Peking2012_PhototaxisRapidCellUniGradLin.png" alt="" style="width:300px;"/>
-
   <p class="description">Figure 2. Phototaxis Simulation Result. Initialized with cells in a uniform distribution in light gradient.</p>
+
   <p class="description">Figure 2. Phototaxis Simulation Result. Initialized with cells in a uniform distribution under light gradient.</p>
  </div>
  </div>
</div>
</div>

Latest revision as of 05:05, 26 October 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 CheYP that directly interacts with the motors and thus influences the mobility of the cell. In details, CheYP 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, 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 CheYP 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 CheYP with:[3]

where

  • FreqCW : the average frequency of CW (clockwise) rotation event
  • [CheYP]c : the critical concentration of phosphorylated CheY in this Hill Function
  • H : the exponential rate, a constant
  • τ0 : 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 [CheYP] with the equation above.
  • (6) Cells fall back to running state from tumbling for a fixed time --- τ0.
  • (7) Cells run at a fixed speed --- v0.
  • (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:

ParameterValueUnitDescriptionSource
v020um/srunning speed[4]
τ00.43stime during a tumbling[5]
CELL_PERIOD15~30minperiod of a cell cycle
[CheA]t5.3u mol/Ltotal concentration of CheA[6]
[CheZ]c1.1u mol/Ltypical concentration of CheZ[1]
[CheY]t9.7u mol/Ltotal concentration of CheY[6]
kY100(u mol/L)-1 s-1phosphorylation rate constant of CheY[6]
kZ30/[CheZ]c(u mol/L)-1 s-1dephosphorylation rate constant of CheY[6]
gammaY0.1s-1decay rate constant of CheYP[2]
N10.31the exponential rate of Hill Function of CW (clockwise) bias[3]
[CheY]Pc3.1u mol/Lthe critical concentration of phosphorylated CheY of Hill Function of CW (clockwise) bias[3]
rA1/31phosphorylation rate of CheA

Result 1: High Contrast Space

Our first demonstration is phototaxis in a high contrast plate, and we would like to see how cells behave differently in lighting or dark environment. The light intensity of light space was set to 0.8 unit while the light intensity of dark space was set to 0.1 unit with I0 = 0.5. Here goes the result:

Figure 1. Phototaxis Simulation Result. Initialized with cells in the center of this space.

Result 2: Light-Gradient Space

Just like diffusion (SPECS model in a large population can derive the diffusion equation[7]), a gradient light intensity should cause a population flow. We set the light intensity from 0 to 1 unit in 1 mm, then observed directed movement bias towards light area in this simulation.

Figure 2. Phototaxis Simulation Result. Initialized with cells in a uniform distribution under 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
  • Totop Totop