Team:Peking/Modeling/Phototaxis/PDE

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Introduction

Based on the previous model basis, we are about to view the magic of phototaxis in a macro way. What we expected is to make light as a pointer to send information to the cells. If we give a bright area, we will expect to see cells gather together to this area. In order to judge whether this wish comes true, we contructed a simulation platform for dynamic system on a plane and tracked the process of population variance based on Mean-field approximation.

Result from Mean-field Model

Recent 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 [CheY_P] with^[1],[2]

where

v₀ : average running speed
z_θ : 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). Therefore, the previous equation becomes

where

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

Theoretic analysis shows that the final state of this system would 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

The equation to this system is a PDE (Partial Differential Equation). The simulation should be done with FDM (Finite Difference Method). To reduce the error caused by anisotropic mesh, we prefer using hexagonal mesh to quadratic mesh which is normally used.

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 high population density.

Figure 2. Simulation in Hexagonal-coordinate environment.

Figure 3. 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 a narrow line emerging 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.

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

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    <img src="/wiki/images/8/8c/Peking2012_Formula004.png" alt="" style="width:200px;"/>
   </div>
+ <p>
+ <p>where</p><ul><li>
+&rho; : population density</li><li>
+&mu; : population diffusion factor (cell motility)</li><li>
+g : growth rate</li></ul>
   <p>
 Theoretic analysis shows that the final state of this system would be
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   </div>
   <p>
-with closed boundary conditions. This result means that the population density in light areas is higher than in dark ones.
+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.
   </p>
 </div>