Team:Peking/Modeling/Phototaxis/Stochastic

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Summary

After the macro view of our phototaxis system, we also traced the cells in a micro way to confirm the influence of light. We built a colony and gave the cells phototaxis function and considered cell division. In this stochastic simulation, we used two series of light sources to see this phototaxis system. Then we confirm our conclusion that the linkage between population density and light signal.

Stochastic Simulation

As in previous introduction, it is the component of CheYP that directly influence the motors and thus influence the mobility of the cell. In detail, CheYP 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,[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 Possion Process whose average frequency is determined by the concentration of CheYP with a Hill Function:[3]

[FreqCW(CheYp) = pow(CheYp/CheYpc,N)/TUMBLE_TIME]

fig: CW Triggering Frequency Equation

where

  • FreqCW denotes the average frequency of CW (clockwise) rotation inducing
  • [CheYPc] denotes the critical concentration of phosphorylated CheY in this Hill Function
  • N denotes the exponential rate of this Hill Function
  • TUMBLE_TIME denotes the average relaxing time in a tumbling inducing

Phototaxis Simulation

With the principles above, we construct our simulation system as following:

  • (1) There are several bacteria cells in a room.
  • (2) Cells can not run through the border of room.
  • (3) The cells can divide in a random cell cycle in uniform distribution between 15min to 30min.
  • (4) There are only two states of the cells --- running and tumbling.
  • (5) Cells trigger tumbling as a Poisson Process, the average frequency is set by [CheYP] with the equation above.
  • (6) Cells return running state after tumbling for a fixed time --- TUMBLE_TIME.
  • (7) Cells run at a fixed speed --- v0.
  • (8) In SPECS model, the running direction after tumbling is independent from previous direction; while in RapidCell model, the new running direction performs as:[2]
[rho(theta) = (1+cos(theta))*sin(theta)/2] [rho(theta,phi) = (1+cos(theta))/4pi]

fig: Tumbling angle distribution in new running

where

  • theta denotes the tumbling angle (angle from origin direction to new direction)
  • rho(theta) denotes the probability density of tumbling angle in value
  • rho(theta,phi) denotes probability density of tumbling angle in the 3D space

Parameters are shown as following:

ParameterValueUnitDescriptionSource
v020um/srunning speed[4]
TUMBLE_TIME0.43stime during a tumbling[5]
CELL_PERIOD15~30minperiod of a cell cycle
[CheA]T5.3u mol/Ltotal concentration of CheA[6]
[CheZ]c1.1u mol/Ltypical concentration of CheZ[1]
[CheY]T9.7u mol/Ltotal concentration of CheY[6]
kY100(u mol/L)-1 s-1phosphorylation rate constant of CheY[6]
kZ30/[CheZ]c(u mol/L)-1 s-1dephosphorylation rate constant of CheY[6]
gammaY0.1s-1decay rate constant of CheYP[2]
N10.31the exponential rate of Hill Function of CW (clockwise) bias[7]
[CheY]Pc3.1u mol/Lthe critical concentration of phosphorylated CheY of Hill Function of CW (clockwise) bias[7]
rA1/31phosphorylation rate of CheA

Tab 1. Simulation Parameters

Result 1: Half-light-half-dark Room

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. The lighting of light room is set to 0.8 unit while the dark is set to 0.1 unit with I0 = 0.5. Here goes the results:

[fig 2: diffusion from center] [fig 3: initial uniform distribution]

Fig 2,3

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:

[fig 4: diffusion from center (experiment)] [fig 5: initial uniform distribution (experiment)]

Fig 4,5

which fit our predictions by modeling.

Result 2: Light Gradient Room

Phototaxis is designed to move cells in a given direction. Just like diffusion (SPECS model in a large population can derive the diffusion equation[8]), 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.

[fig 6: Gradient Lighting]

Fig 6

Then we do this movement experiment in a much larger scale, and the bacteria successfully response with their motion.

  • Totop Totop

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