Team:Peking/Modeling/Phototaxis/PDE

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

where

v₀ : average running velocity
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). 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

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