Team:Peking/Modeling/Phototaxis

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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.

Phototaxis System

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.

[fig 1: Phototaxis Circuit]

Fig 1. Phototaxis Circuit

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:

[fig: CheZ Equation] [CheZ](I) = [CheZ]₀ * I₀ / (I + I₀)

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.^{<a href="#ref1" title="Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer, Howard C. Berg, etc. PNAS">[1]</a>} By listing ODE equations, we can derive the equilibrium state of CheY_P concentration as: <a href="/Team:Peking/Modeling/Detail_Phototaxis_1">(detail here)</a>

[fig: CheYp Equation] [CheYp(CheZ) = kY * CheAp * CheYt / ( kY * CheAp + kZ * CheZ + gamY )]

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,^{<a href="#ref2" title="Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate, Nikita Vladimirov, etc. PLoS Computational Biology">[2]</a>} 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 <a href="/Team:Peking/Modeling/PoissonProcess">Possion Process</a> whose average frequency is determined by the concentration of CheY_P with a Hill Function:^{<a href="#ref3" title="An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells">[3]</a>}

[fig: CW Triggering Frequency Equation] [FreqCW(CheYp) = pow(CheYp/CheYpc,N)/TUMBLE_TIME]

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:^{<a href="#ref2" title="Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate, Nikita Vladimirov, etc. PLoS Computational Biology">[2]</a>}

[fig: Tumbling angle distribution in new running] [rho(theta) = (1+cos(theta))*sin(theta)/2] [rho(theta,phi) = (1+cos(theta))/4pi]

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

</p>

Parameters are shown as following:

Parameter	Value	Unit	Description	Source
v₀	20	um/s	running speed	^{<a href="#ref4" title="Chemotaxis in Escherichia coli analysed by Three-dimensional Tracking, Howard C.Berg, Douglas A.Brown, NATURE">[4]</a>}
TUMBLE_TIME	0.43	s	time during a tumbling	^{<a href="#ref5" title="Real-Time Imaging of Fluorescent Flagellar Filaments, Linda Turner, etc. JOURNAL OF BACTERIOLOGY">[5]</a>}
CELL_PERIOD	15~30	min	period of a cell cycle
[CheA]_T	5.3	u mol/L	total concentration of CheA	^{<a href="#ref6" title="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a>}
[CheZ]_c	1.1	u mol/L	typical concentration of CheZ	^{<a href="#ref1" title="Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer, Victor Sourjik and Howard C. Berg, PNAS">[1]</a>}
[CheY]_T	9.7	u mol/L	total concentration of CheY	^{<a href="#ref6" title="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a>}
k_Y	100	(u mol/L)^-1 s^-1	phosphorylation rate constant of CheY	^{<a href="#ref6" title="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a>}
k_Z	30/[CheZ]_c	(u mol/L)^-1 s^-1	dephosphorylation rate constant of CheY	^{<a href="#ref6" title="Relationship between cellular response and behavioral variability in bacterial chemotaxis, Thierry Emonet, Philippe Cluzel. PNAS">[6]</a>}
gamma_Y	0.1	s^-1	decay rate constant of CheY_P	^{<a href="#ref2" title="Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate, Nikita Vladimirov, etc. PLoS Computational Biology">[2]</a>}
N	10.3	1	the exponential rate of Hill Function of CW (clockwise) bias	^{<a href="#ref7" title="An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells, Philippe Cluzel, etc. Science">[7]</a>}
[CheY]_Pc	3.1	u mol/L	the critical concentration of phosphorylated CheY of Hill Function of CW (clockwise) bias	^{<a href="#ref7" title="An Ultrasensitive Bacterial Motor Revealed by Monitoring Signaling Proteins in Single Cells, Philippe Cluzel, etc. Science">[7]</a>}
r_A	1/3	1	phosphorylation rate of CheA

[table 1: Simulation Parameters]

 <p class="description">Tab 1. </p>

</div>

Result 1: Half-light-half-dark Room

<p>

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:

</p>

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

 <p class="description">Fig 2,3 </p>

<p>

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:

</p>

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

 <p class="description">Fig 4,5 </p>

<p>

which fit our predictions by modeling.

</p>

Result 2: Light Gradient Room

<p>

Phototaxis is designed to move cells in a given direction. Just like diffusion (SPECS model in a large population can derive the diffusion equation^{<a href="#ref8" title="A Pathway-based Mean-field Model for Escherichia coli Chemotaxis, Tailin Wu, etc. PACS">[8]</a>}), 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.

</p>

[fig 6: Gradient Lighting]

 <p class="description">Fig 6 </p>

<p>

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

</p>

Team:Peking/Modeling/Phototaxis

From 2012.igem.org

Contents

Summary

Phototaxis System

Phototaxis Simulation

Result 1: Half-light-half-dark Room

Result 2: Light Gradient Room