Team:Peking/Modeling/Phototaxis

We have constructed a simple phototaxis system coupling our Luminesensor with the expression level of cheZ protein. After gathering the principles and parameters of the chemotaxis system, we then simulated our phototaxis system in a stochastic way. Based on our simulations, we predicted the outcome of the two demonstrations of phototaxis and presented the mechanism of phototaxis in a quantificational way.

Our phototaxis system functions as Stopping on Light and Running in Dark. As the sketch of this phototaxis system shows (Fig. 1), Light activates the Luminesensor which represses the expression of the CheZ protein. CheZ inactivates CheY_P, which changes the rotation direction of the flagellum by protein-protein interaction and makes the bacteria tumbling, and reduces the tumbling frequency therefore. Bacteria moves slow with high tumbling frequency and vice versa.

To simplify the calculation, we assume the CheZ component responses immediately. (well, I prefer it having some delay which can enhance the phototaxis though.) When light reaches the bacteria, the concentration of CheZ behaves as Hill Function:

where

• [CheZ] denotes the concentration of CheZ
• [CheZ]₀ denotes the superior limit of CheZ concentration
• I₀ denotes the critical illuminance
• I denotes the current illuminance

Then CheZ dephosphorylates CheY_P into CheY while CheA phosphorylates CheY back. The typical time of dephosphorylation by CheZ is around 0.5 second and the typical time of phosphorylation by CheA (independent from light) is around 0.05 second.^[1] By listing ODE equations, we can derive the equilibrium state of CheY_P concentration as: (detail here)

where

• [CheY_P] denotes the concentration of phosphorylated CheY
• [CheA_P] denotes the steady concentration of active CheA
• [CheY_T] denotes the total concentration of CheY
• k_Y denotes the rate constant of CheY phosphorylation
• k_Z denotes the rate constant of CheY dephosphorylation
• gamma-Y denotes the decay rate constant of CheY_P

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]

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

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]

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:

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.

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.