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Team:Peking/Modeling/Phototaxis/PDE
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Recent paper derived the K-S chemotaxis equation based on mean-field model<sup><a href="#ref8" title="Pathway-Based Mean-Field Model for Escherichia coli Chemotaxis. Guangwei Si, etc. Physical Review Letters">[8]</a></sup> | Recent paper derived the K-S chemotaxis equation based on mean-field model<sup><a href="#ref8" title="Pathway-Based Mean-Field Model for Escherichia coli Chemotaxis. Guangwei Si, etc. Physical Review Letters">[8]</a></sup> | ||
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[formula3] | [formula3] | ||
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and showed the linkage between the cells' population level motility factors and [CheY<sub>P</sub>] with<sup><a href="#ref8" title="Pathway-Based Mean-Field Model for Escherichia coli Chemotaxis. Guangwei Si, etc. Physical Review Letters">[8]</a>,<a href="#ref9" title="Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time. Lili Jiang, etc.">[9]</a></sup> | and showed the linkage between the cells' population level motility factors and [CheY<sub>P</sub>] with<sup><a href="#ref8" title="Pathway-Based Mean-Field Model for Escherichia coli Chemotaxis. Guangwei Si, etc. Physical Review Letters">[8]</a>,<a href="#ref9" title="Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time. Lili Jiang, etc.">[9]</a></sup> | ||
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[formula4] | [formula4] | ||
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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. Therefore, the previous equation becomes | 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. Therefore, the previous equation becomes | ||
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[formula5] | [formula5] | ||
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Theoretic analysis shows that the final state of this system would be | Theoretic analysis shows that the final state of this system would be | ||
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[formula6] | [formula6] | ||
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with closed boundary conditions. This result means that the population density in light areas is higher than in dark ones. | with closed boundary conditions. This result means that the population density in light areas is higher than in dark ones. | ||
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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. | 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. | ||
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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. | 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. | ||
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Revision as of 11:07, 24 September 2012
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[8]
and showed the linkage between the cells' population level motility factors and [CheYP] with[8],[9]
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. Therefore, the previous equation becomes
Theoretic analysis shows that the final state of this system would be
with closed boundary conditions. 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.
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.
