Team:Peking/Modeling/Phototaxis/Stochastic

Stochastic Simulation

As in previous introduction, it is the component of CheY_P that directly influence the motors and thus influence the mobility of the cell. In detail, CheY_P 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 CheY_P 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 CheY_P 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
• [CheY_Pc] 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 [CheY_P] 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:

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:

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:

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.

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