Team:Peking/Modeling/Phototaxis/PDE

From 2012.igem.org

(Difference between revisions)
m
m
 
(4 intermediate revisions not shown)
Line 8: Line 8:
</script>
</script>
<div class="PKU_context floatR first">
<div class="PKU_context floatR first">
-
  <h3 id="title1">Introduction</h3>
+
  <h3 id="title1">Intention</h3>
  <p>
  <p>
-
Based on the previous model basis, we are about to view the magic of phototaxis on the macro level. We expected to make light as a pointer. If we give a bright area, cells should gather together to this area. In order to verify this expectation, we contructed a simulation platform for dynamic system on a plane and tracked the process of population variance based on Mean-field approximation.
+
Based on the previous model basis, we are to view Phototaxis on the macroscopic level. We expected to use light as a pointer; if we shine light on an area, cells should gather together to that specific spot. Based on mean-field approximation, we contructed a simulation platform for dynamic system on a plane and tracked the process of population variance. In order to reduce the error caused by anisotropic structure in traditional quadratic mesh, we prefer establishing the hexagonal mesh in the simulation enviroment, since the number of neighbors of a chunk unit in the hexagonal mesh is larger than that of the traditional quadratic mesh.
  </p>
  </p>
</div>
</div>
Line 16: Line 16:
  <h3 id="title2">Result from Mean-field Model</h3>
  <h3 id="title2">Result from Mean-field Model</h3>
  <p>
  <p>
-
Recent paper derived the K-S chemotaxis equation based on mean-field model<sup><a href="#ref1" title="Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101">[1]</a></sup>
+
A recently published paper derived the K-S chemotaxis equation based on mean-field model<sup><a href="#ref1" title="Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101">[1]</a></sup>
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
Line 32: Line 32:
&tau; : average time in a running</li></ul>
&tau; : average time in a running</li></ul>
  <p>
  <p>
-
Since f<sub>0</sub> only relates to the chemical signal in chemotaxis system, we consider it constant in our phototaxis system. Besides, we would like to add the growth function to the equation to approach the real situation. Due to the light to the system, the &mu;<sub>0</sub> is not constant any more (thus we denote &mu;<sub>0</sub> as &mu;). After some derivation, the previous equation should become
+
Since f<sub>0</sub> only relates to the chemical signal in chemotaxis system, we consider it constant in our phototaxis system. Besides, we would like to add the growth function to the equation to approach the real situation. Due to the light to the system, the &mu;<sub>0</sub> is not constant any more (thus we denote &mu;<sub>0</sub> as &mu;). After some derivation, the previous equation should become:
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
Line 43: Line 43:
g : growth rate</li></ul>
g : growth rate</li></ul>
  <p>
  <p>
-
Theoretic analysis shows that the equilibrium state of the density distribution should be
+
Theoretic analysis shows that the equilibrium state of the density distribution should be:
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
Line 55: Line 55:
  <h3 id="title3">Simulation on Hexagonal Mesh</h3>
  <h3 id="title3">Simulation on Hexagonal Mesh</h3>
  <p>
  <p>
-
This PDE (Partial Differential Equation) system is usually simulated in FDM (Finite Difference Method). To reduce the error caused by anisotropic mesh, we prefer using <!--<a href="/Team:Peking/Modeling/Appendix/Hexagonal">-->hexagonal mesh<!--</a>--> to quadratic mesh.
+
This PDE (Partial Differential Equation) system is usually simulated in FDM (Finite Difference Method). We constructed a FDM simulation environment in C++ with hexagonal mesh and simulated this cellular movement regulated by light intensity (so-called Phototaxis) in this environment. On the boundary of the lighting area, the simulation shows that there should be high population density.
-
<br /> <br />
+
-
We constructed a FDM simulation environment in C++ for hexagonal mesh and simulated this system with it. On the boundary of the lighting area, the simulation shows that there should be high population density.
+
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
Line 77: Line 75:
  <h3 id="title4">Conclusion</h3>
  <h3 id="title4">Conclusion</h3>
  <p>
  <p>
-
This simulation shows that our system will link the mobility of cells with the light signal. The result shows that there will be an outline with high density population on the boundary of lighting area as a temporary state. With this special property, this system has potential to be an edge-detection system to light in the future.
+
This simulation shows that our system will link the mobility of cells with the light signal. The result shows that there will be an outline with high density population on the boundary of lighting area as a temporary state. With this special property, this system has potential to be an edge-detection system to light in the future. In addtion, this hexagonal-mesh simulation environment would be a useful prototype for future simulation of 2D dynamic systems.
  </p>
  </p>
</div>
</div>

Latest revision as of 05:05, 26 October 2012

Intention

Based on the previous model basis, we are to view Phototaxis on the macroscopic level. We expected to use light as a pointer; if we shine light on an area, cells should gather together to that specific spot. Based on mean-field approximation, we contructed a simulation platform for dynamic system on a plane and tracked the process of population variance. In order to reduce the error caused by anisotropic structure in traditional quadratic mesh, we prefer establishing the hexagonal mesh in the simulation enviroment, since the number of neighbors of a chunk unit in the hexagonal mesh is larger than that of the traditional quadratic mesh.

Result from Mean-field Model

A recently published paper derived the K-S chemotaxis equation based on mean-field model[1]

and showed the linkage between the cells' population level motility factor μ0 and [CheYP] with[1],[2]

where

  • v0 : average running velocity
  • zθ : a const rate
  • τ : average time in a running

Since f0 only relates to the chemical signal in chemotaxis system, we consider it constant in our phototaxis system. Besides, we would like to add the growth function to the equation to approach the real situation. Due to the light to the system, the μ0 is not constant any more (thus we denote μ0 as μ). After some derivation, the previous equation should become:

where

  • ρ : population density
  • μ : population diffusion factor (cell motility)
  • g : growth rate

Theoretic analysis shows that the equilibrium state of the density distribution should be:

with closed boundary conditions. The constant is independent from position. This result means that the population density in light areas is higher than in dark ones.

Simulation on Hexagonal Mesh

This PDE (Partial Differential Equation) system is usually simulated in FDM (Finite Difference Method). We constructed a FDM simulation environment in C++ with hexagonal mesh and simulated this cellular movement regulated by light intensity (so-called Phototaxis) in this environment. On the boundary of the lighting area, the simulation shows that there should be high population density.

Figure 1. Simulation in Hexagonal-coordinate environment. The circle area is illuminated with light. The cell density increases on the border of the circle area.

We also light a pattern of Chinese Huabiao, and an outline of Huabiao appeared.

Figure 2. Simulation of cellular movement regulated by light. Here we use the pattern of Chinese Huabiao to regulate the cellular movement, and we see the cell's population begins to show a rough shape of the light pattern we give.

Actually, this is a temporary state phenomenum of this system. Simulation indicates that it will cost a tremendously long time to reach the final state, while temporary states are usually seen like the figure above.

Conclusion

This simulation shows that our system will link the mobility of cells with the light signal. The result shows that there will be an outline with high density population on the boundary of lighting area as a temporary state. With this special property, this system has potential to be an edge-detection system to light in the future. In addtion, this hexagonal-mesh simulation environment would be a useful prototype for future simulation of 2D dynamic systems.

Reference

  • 1.Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101
  • 2.Jiang, L., Ouyang, Q., Tu, Y.(2010) Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time. PLoS Comput. Biol., 6: e1000735
  • Totop Totop

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