Team:Peking/Modeling/Phototaxis/Stochastic

Stochastic Simulation

It is the component of CheY_P that directly interacts with the motors and thus influences the mobility of the cell. In details, CheY_P can interact with the flagellar motor to induce CW (clockwise) rotation. When flagellar motors rotate CCW (counter-clockwise), they form a bundle to generate a force similar to a worm wheel. However, if some of the flagellar motors rotate CW, 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:^[3]

where

• FreqCW : the average frequency of CW (clockwise) rotation event
• [CheY_P]_c : the critical concentration of phosphorylated CheY in this Hill Function
• H : the exponential rate, a constant
• τ₀ : the average relaxing time in a tumbling event

Phototaxis Simulation

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

• (1) There are multiple bacterial cells in the simulation space.
• (2) Cells can not run through the border of space.
• (3) The cells division is a random process in uniform distribution. Its time scale is tens of minutes (15~30 min).
• (4) There are only two states of the cells --- running and tumbling.
• (5) Tumbling event is a Poisson Process, whose average frequency is set by [CheY_P] with the equation above.
• (6) Cells fall back to running state from tumbling for a fixed time --- τ₀.
• (7) Cells run at a fixed speed --- v₀.
• (8) In SPECS model, the running direction after tumbling is independent of previous direction; while in RapidCell model, the new running direction performs as:^[2]

where

• θ : the tumbling angle (angle from origin direction to new direction)
• ρ(θ) : the probability density of tumbling angle in value
• ρ(θ,φ) : probability density of tumbling angle in the 3D space

Parameters are shown as following:

Result 1: High Contrast Space

Our first demonstration is phototaxis in a high contrast plate, and we would like to see how cells behave differently in lighting or dark environment. The light intensity of light space was set to 0.8 unit while the light intensity of dark space was set to 0.1 unit with I₀ = 0.5. Here goes the result:

Figure 1. Phototaxis Simulation Result. Initialized with cells in the center of this space.

Result 2: Light-Gradient Space

Just like diffusion (SPECS model in a large population can derive the diffusion equation^[7]), a gradient light intensity should cause a population flow. We set the light intensity from 0 to 1 unit in 1 mm, then observed directed movement bias towards light area in this simulation.

Figure 2. Phototaxis Simulation Result. Initialized with cells in a uniform distribution under light gradient.

Reference

• 1. Sourjik, V., et al.(2002) Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer. Proc. Natl Acad. Sci. USA, 99(20): 12669: 12674
• 2. Vladimirov, N., Lovdok, L., Lebiedz, D., and Sourjik, V.(2008). Dependence of bacterial chemotaxis on gradient shape and adaptation rate. PLoS Comput. Biol., 4: e1000242
• 3. Cluzel, P., Surette, M., and Leibler, S.(2000). An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287: 1652: 1655
• 4. Berg, H. C. & Brown, D. A.(1972) Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature, 239: 500: 504
• 5. Turner, L., Ryu, W. S. & Berg, H. C.(2000) Real-time imaging of fluorescent flagellar filaments. J. Bacteriol., 182: 2793: 2801
• 6. Emonet, T., Cluzel, P.(2008) Relationship between cellular response and behavioral variability in bacterial chemotaxis. Proc. Natl Acad. Sci. USA, 105(9): 3304: 3309
• 7. Si, G., Wu, T., Ouyang, Q., Tu, Y.(2012) Pathway-based Mean-field Model for Escherichia coli Chemotaxis. Phys. Rev. Lett., 109: 048101 Philippe Cluzel. PNAS