Team:Peking/Modeling/Phototaxis

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  <h3 id="title1">Summary</h3>
  <h3 id="title1">Summary</h3>
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Phototaxis refers to light-controlled motility, whose input is the space-distribution of light. We have constructed a simple phototaxis system coupling the <i>Luminesensor</i> with the expression level of CheZ protein. In order to verify our design, we used Mean-field PDE model. Later we managed to confirm these phenomena in Stochastic Simulation by tracing each cell.  
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We have constructed a simple phototaxis system coupling our <i>Luminesensor</i> with the expression level of cheZ protein. After gathering the principles and parameters of the chemotaxis system, we then simulated our phototaxis system in a stochastic way. Based on our simulations, we predicted the outcome of the two demonstrations of phototaxis and presented the mechanism of phototaxis in a quantificational way.
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  <h3 id="title2">Phototaxis System</h3>
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  <h3 id="title2">Phototaxis Pathway</h3>
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  <p>
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Our phototaxis system functions as <i>Stopping on Light and Running in Dark</i>. As the sketch of this phototaxis system shows (Figure 1), Light activates the <i>Luminesensor</i> which represses the expression of the CheZ protein. CheZ inactivates CheY<sub>P</sub>, which changes the rotation direction of the flagellum by protein-protein interaction and makes the bacteria tumbling, and reduces the tumbling frequency therefore. Bacteria moves slow with high tumbling frequency and <i>vice versa</i>.
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Our phototaxis system functions as <i>Stopping on Light and Running in Dark</i>. As the sketch of this phototaxis system shows (Fig. 1), Light activates the <i>Luminesensor</i> which represses the expression of the CheZ protein. CheZ inactivates CheY<sub>P</sub>, which changes the rotation direction of the flagellum by protein-protein interaction and makes the bacteria tumbling, and reduces the tumbling frequency therefore. Bacteria moves slow with high tumbling frequency and <i>vice versa</i>.
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  </p>
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<div class="floatC">
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  <div class="floatC">
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  <img src="/wiki/images/f/fc/Peking2012_Modeling_CheZ_Network.png" alt="" style="width:500px;"/>
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[fig 1: Phototaxis Circuit]
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  <p class="description"  style="width:260px;text-align:center;">Figure 1. Phototaxis Pathway</p>
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  <p class="description">Fig 1. Phototaxis Circuit</p>
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</div>
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  </div>
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<p>
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  <p>
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To simplify the calculation, we assume the CheZ component responses immediately. When light reaches the bacteria, the concentration of CheZ behaves as Hill Function:
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To simplify the calculation, we assume the CheZ component responses immediately. (well, I prefer it having some delay which can enhance the phototaxis though.) When light reaches the bacteria, the concentration of CheZ behaves as Hill Function:
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  <div class="floatC">
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  <img src="/wiki/images/e/e0/Peking2012_Formula002.png" alt="" style="width:300px;"/>
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[fig: CheZ Equation]
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</div>
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[CheZ](I) = [CheZ]<sub>0</sub> * I<sub>0</sub> / (I + I<sub>0</sub>)
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<p>where</p><ul><li>
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  </div>
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[CheZ] : the concentration of CheZ</li><li>
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  <p>where</p><ul><li>
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[CheZ]<sub>0</sub> : the superior limit of CheZ concentration</li><li>
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[CheZ] denotes the concentration of CheZ</li><li>
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I<sub>0</sub> : the critical illuminance</li><li>
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[CheZ]<sub>0</sub> denotes the superior limit of CheZ concentration</li><li>
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I : the current illuminance</li></ul>
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I<sub>0</sub> denotes the critical illuminance</li><li>
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<p>
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I denotes the current illuminance</li></ul>
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Then CheZ dephosphorylates CheY<sub>P</sub> into CheY while CheA phosphorylates CheY back. The typical time of dephosphorylation by CheZ is around 0.5 second and the typical time of phosphorylation by CheA (independent from light) is around 0.05 second.<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> By listing ODE equations, we can derive the equilibrium state of CheY<sub>P</sub> concentration as: <!--<a href="/Team:Peking/Modeling/Appendix/Phototaxis">(detail here)</a>-->
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  <p>
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Then CheZ dephosphorylates CheY<sub>P</sub> into CheY while CheA phosphorylates CheY back. The typical time of dephosphorylation by CheZ is around 0.5 second and the typical time of phosphorylation by CheA (independent from light) is around 0.05 second.<sup><a href="#ref1" title="Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer, Howard C. Berg, etc. PNAS">[1]</a></sup> By listing ODE equations, we can derive the equilibrium state of CheY<sub>P</sub> concentration as: <a href="/Team:Peking/Modeling/Detail_Phototaxis_1">(detail here)</a>
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<!--
<!--
CheY + CheAp -kY-> CheYp + CheA
CheY + CheAp -kY-> CheYp + CheA
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[CheY] = [CheY]t - [CheYp]
[CheY] = [CheY]t - [CheYp]
-->
-->
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  </p>
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</p>
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  <div class="floatC">
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<div class="floatC">
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[fig: CheYp Equation]
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   <img src="/wiki/images/d/d2/Peking2012_Formula001.png" alt="" style="width:500px;" />
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[CheYp(CheZ) = kY * CheAp * CheYt / ( kY * CheAp + kZ * CheZ + gamY )]
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   </div>
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  <p>where</p><ul><li>
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[CheY<sub>P</sub>] denotes the concentration of phosphorylated CheY</li><li>
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[CheA<sub>P</sub>] denotes the steady concentration of active CheA</li><li>
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[CheY<sub>T</sub>] denotes the total concentration of CheY</li><li>
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k<sub>Y</sub> denotes the rate constant of CheY phosphorylation</li><li>
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k<sub>Z</sub> denotes the rate constant of CheY dephosphorylation</li><li>
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gamma-Y denotes the decay rate constant of CheY<sub>P</sub></li></ul>
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  <p>
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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="Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate, Nikita Vladimirov, etc. PLoS Computational Biology">[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 <a href="/Team:Peking/Modeling/PoissonProcess">Possion Process</a> whose average frequency is determined by the concentration of CheY<sub>P</sub> with a Hill Function:<sup><a href="#ref3" title="An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells">[3]</a></sup>
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  </p>
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  <div class="floatC">
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[fig: CW Triggering Frequency Equation]
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[FreqCW(CheYp) = pow(CheYp/CheYpc,N)/TUMBLE_TIME]
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  </div>
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  <p>where</p><ul><li>
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FreqCW denotes the average frequency of CW (clockwise) rotation inducing</li><li>
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[CheY<sub>Pc</sub>] denotes the critical concentration of phosphorylated CheY in this Hill Function</li><li>
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N denotes the exponential rate of this Hill Function</li><li>
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TUMBLE_TIME denotes the average relaxing time in a tumbling inducing</li></ul>
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  </div>
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<p>where</p><ul><li>
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[CheY<sub>P</sub>] : the concentration of phosphorylated CheY</li><li>
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[CheA<sub>P</sub>] : the steady concentration of active CheA</li><li>
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[CheY]<sub>t</sub> : the total concentration of CheY</li><li>
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k<sub>Y</sub> : the rate constant of CheY phosphorylation</li><li>
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k<sub>Z</sub> : the rate constant of CheY dephosphorylation</li><li>
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&gamma;<sub>Y</sub> : the decay rate constant of CheY<sub>P</sub></li></ul>
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<p>
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CheY<sub>P</sub> can interact with 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="2. 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 <!--<a href="/Team:Peking/Modeling/Appendix/Stochastic">Possion Process</a>--> whose average frequency is determined by the concentration of CheY<sub>P</sub> with a Hill Function:<sup><a href="#ref3" title="3. Cluzel, P., et al.(2000). An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287: 1652: 1655">[3]</a></sup>
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</p>
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  <h3 id="title3">Phototaxis Simulation</h3>
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  <h3 id="title3">Review</h3>
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  <p>
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  <p>
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The relationship between average moving speed and lighting produces phototaxis function. Mechanism of speed change is related to the rotating direction of motors which determines the running state bias of the cell. The component of CheY<sub>P</sub> directly influences the motors and thus we showed the relationship between [CheY<sub>P</sub>] and illuminance to present the model basis.
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With the principles above, we construct our simulation system as following:
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</p>
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  </p>
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  <ul><li>
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(1) There are several bacteria cells in a room.
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  </li><li>
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(2) Cells can not run through the border of room.
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  </li><li>
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(3) The cells can divide in a random cell cycle in uniform distribution between 15min to 30min.
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  </li><li>
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(4) There are only two states of the cells --- running and tumbling.
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  </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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  </li><li>
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(6) Cells return running state after tumbling for a fixed time --- TUMBLE_TIME.
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  </li><li>
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(7) Cells run at a fixed speed --- v<sub>0</sub>.
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  </li><li>
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(8) In SPECS model, the running direction after tumbling is independent from previous direction;
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while in RapidCell model, the new running direction performs as:<sup><a href="#ref2" title="Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate, Nikita Vladimirov, etc. PLoS Computational Biology">[2]</a></sup>
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  </li></ul>
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  <div class="floatC">
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[fig: Tumbling angle distribution in new running]
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[rho(theta) = (1+cos(theta))*sin(theta)/2]
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[rho(theta,phi) = (1+cos(theta))/4pi]
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  </div>
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  <p>where</p><ul><li>
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theta denotes the tumbling angle (angle from origin direction to new direction)</li><li>
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rho(theta) denotes the probability density of tumbling angle in value</li><li>
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rho(theta,phi) denotes probability density of tumbling angle in the 3D space</li></ul>
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  </p><p>
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Parameters are shown as following:
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  <table><tr>
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    <td>Parameter</td><td>Value</td><td>Unit</td><td>Description</td><td>Source</td>
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  </tr><tr>
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    <td>v<sub>0</sub></td><td>20</td><td>um/s</td><td>running speed</td><td><sup><a href="#ref4" title="Chemotaxis in Escherichia coli analysed by Three-dimensional Tracking, Howard C.Berg, Douglas A.Brown, NATURE">[4]</a></sup></td>
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  </tr><tr>
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    <td>TUMBLE_TIME</td><td>0.43</td><td>s</td><td>time during a tumbling</td><td><sup><a href="#ref5" title="Real-Time Imaging of Fluorescent Flagellar Filaments, Linda Turner, etc. JOURNAL OF BACTERIOLOGY">[5]</a></sup></td>
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  </tr><tr>
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    <td>CELL_PERIOD</td><td>15~30</td><td>min</td><td>period of a cell cycle</td><td></td>
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  </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="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a></sup></td>
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  </tr><tr>
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    <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="Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer, Victor Sourjik and Howard C. Berg, PNAS">[1]</a></sup></td>
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  </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="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a></sup></td>
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  </tr><tr>
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    <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="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a></sup></td>
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  </tr><tr>
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    <td>k<sub>Z</sub></td><td>30/[CheZ]<sub>c</sub></td><td>(u mol/L)<sup>-1</sup> s<sup>-1</sup></td><td>dephosphorylation rate constant of CheY</td><td><sup><a href="#ref6" title="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a></sup></td>
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  </tr><tr>
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    <td>gamma<sub>Y</sub></td><td>0.1</td><td>s<sup>-1</sup></td><td>decay rate constant of CheY<sub>P</sub></td><td><sup><a href="#ref2" title="Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate, Nikita Vladimirov, etc. PLoS Computational Biology">[2]</a></sup></td>
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  </tr><tr>
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    <td>N</td><td>10.3</td><td>1</td><td>the exponential rate of Hill Function of CW (clockwise) bias</td><td><sup><a href="#ref7" title="An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells, Philippe Cluzel, etc. Science">[7]</a></sup></td>
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  </tr><tr>
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    <td>[CheY]<sub>Pc</sub></td><td>3.1</td><td>u mol/L</td><td>the critical concentration of phosphorylated CheY of Hill Function of CW (clockwise) bias</td><td><sup><a href="#ref7" title="An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells, Philippe Cluzel, etc. Science">[7]</a></sup></td>
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  </tr><tr>
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    <td>r<sub>A</sub></td><td>1/3</td><td>1</td><td>phosphorylation rate of CheA</td><td></td>
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  </tr></table>
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[table 1: Simulation Parameters]
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  <p class="description">Tab 1. </p>
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  <h3 id="title4">Result 1: Half-light-half-dark Room</h3>
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  <h3 id="title5">Reference</h3>
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  <p></p>
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  <p>
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<ul class="refer"><li id="ref1">
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Our first Demonstration is in a Half-light-half-dark plate, and we would like to see how cells behave differently in such a high contrast environment.
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1. Sourjik, V., <i>et al.</i>(2002) Binding of the <i>Escherichia coli</i> response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer. <i>Proc. Natl Acad. Sci. USA</i>, 99(20): 12669: 12674
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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:
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  </li><li id = "ref2">
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  </p>
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2. Vladimirov, N., Lovdok, L., Lebiedz, D., and Sourjik, V.(2008). Dependence of bacterial chemotaxis on gradient shape and adaptation rate. <i>PLoS Comput. Biol.</i>, 4: e1000242
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  <div>
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   </li><li id = "ref3">
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[fig 2: diffusion from center]
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3. Cluzel, P., Surette, M., and Leibler, S.(2000). An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. <i>Science</i>, 287: 1652: 1655
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[fig 3: initial uniform distribution]
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</li></ul>
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  <p class="description">Fig 2,3 </p>
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  </div>
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  <p>
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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 band will emerge at the border in light area. Our experiments show:
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  </p>
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  <div>
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[fig 4: diffusion from center (experiment)]
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[fig 5: initial uniform distribution (experiment)]
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  <p class="description">Fig 4,5 </p>
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  <p>
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which fit our predictions by modeling.
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<h3 id="title5">Result 2: Light Gradient Room</h3>
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Phototaxis is designed to move cells in a given direction. Just like diffusion (SPECS model in a large population can derive the diffusion equation<sup><a href="#ref8" title="A Pathway-based Mean-field Model for Escherichia coli Chemotaxis, Tailin Wu, etc. PACS">[8]</a></sup>), the movement order requires a gradient lighting field in the room. We set the lighting from 0 to 1 unit in 1 mm, then discovered the directed movement bias towards light area in this simulation.
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[fig 6: Gradient Lighting]
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  <p class="description">Fig 6 </p>
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Then we do this movement experiment in a much larger scale, and the bacteria successfully response with their motion.
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Latest revision as of 05:04, 26 October 2012

Summary

Phototaxis refers to light-controlled motility, whose input is the space-distribution of light. We have constructed a simple phototaxis system coupling the Luminesensor with the expression level of CheZ protein. In order to verify our design, we used Mean-field PDE model. Later we managed to confirm these phenomena in Stochastic Simulation by tracing each cell.

Phototaxis Pathway

Our phototaxis system functions as Stopping on Light and Running in Dark. As the sketch of this phototaxis system shows (Figure 1), Light activates the Luminesensor which represses the expression of the CheZ protein. CheZ inactivates CheYP, which changes the rotation direction of the flagellum by protein-protein interaction and makes the bacteria tumbling, and reduces the tumbling frequency therefore. Bacteria moves slow with high tumbling frequency and vice versa.

Figure 1. Phototaxis Pathway

To simplify the calculation, we assume the CheZ component responses immediately. When light reaches the bacteria, the concentration of CheZ behaves as Hill Function:

where

  • [CheZ] : the concentration of CheZ
  • [CheZ]0 : the superior limit of CheZ concentration
  • I0 : the critical illuminance
  • I : the current illuminance

Then CheZ dephosphorylates CheYP into CheY while CheA phosphorylates CheY back. The typical time of dephosphorylation by CheZ is around 0.5 second and the typical time of phosphorylation by CheA (independent from light) is around 0.05 second.[1] By listing ODE equations, we can derive the equilibrium state of CheYP concentration as:

where

  • [CheYP] : the concentration of phosphorylated CheY
  • [CheAP] : the steady concentration of active CheA
  • [CheY]t : the total concentration of CheY
  • kY : the rate constant of CheY phosphorylation
  • kZ : the rate constant of CheY dephosphorylation
  • γY : the decay rate constant of CheYP

CheYP can interact with 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,[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 whose average frequency is determined by the concentration of CheYP with a Hill Function:[3]

Review

The relationship between average moving speed and lighting produces phototaxis function. Mechanism of speed change is related to the rotating direction of motors which determines the running state bias of the cell. The component of CheYP directly influences the motors and thus we showed the relationship between [CheYP] and illuminance to present the model basis.

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
  • Totop Totop

Retrieved from "http://2012.igem.org/Team:Peking/Modeling/Phototaxis"