Team:UT-Tokyo/Data/Modeling/Model02

From 2011.igem.org

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=Aim=
=Aim=
We derived a simple relationship between Asp concentration and E.coli chemotaxis so that
We derived a simple relationship between Asp concentration and E.coli chemotaxis so that
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we were able to simulate macroscopic ¥(n~10^8) E.coli colony chemotacitc behavior.
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we were able to simulate macroscopic (n~10<html><sup>8</sup></html>) E.coli colony chemotacitc behavior in the entire simulation.
=Background=
=Background=
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We considered internal condition of E.coli (receptor methylation etc) by SPECS model<html><sup class="ref">[1]</sup></html>.
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SPECS <html><sup class="ref">[1]</sup></html> is one of the model simulating E.coli's chemotactic behavior.
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SPECS is a relatively accurate taxis model, and is suitable for simulation of approximately ten thousand E.coli for approximately 10 minutes.
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It considers internal condition of E.coli (receptor methylation etc) so it is a relatively strict taxis model.  It is suitable for semi-macroscopic simulation which has around ten thousand E.coli and the timescale is within a hour.
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This model, however, take too much time to reproduce our system, because the system includes much more E.coli(about one hundred of million) and takes a relatively long time (a few days).
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For our system, however, it takes too much time to replicate entire system, because the system includes much more E.coli(about one hundred of million) and the time scale is relatively long (a few days).
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So, by adapting SPECS model to our system, we have devised a new model that provides less calculation amount and enough accuracy.
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Therefore we devised a new model based on SPECS which requires less calculation amount.
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Concretely, we divided chemotactic motion of E.coli groups into two factors, that is, parallel translation and diffusion. Then we derived approximation equations that figure out the two respective values from Asp concentration and its gradient.
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=Method=
=Method=
 +
We approximated chemotactic motion of E.coli groups into two factors, that is, parallel translation and diffusion.
 +
Let 'V' represents a parallel translation velocity 'D' represents a diffusion coefficient of E.coli groups and 'A' represents L-Asp concentration.
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E.coli can detect A and its gradient ∇A.
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They have an internal circuit to translate these signal into motions of their flagellum and try to run toward Asp-richer region.
 +
So we thought V and D could be represented as a functions of A and ∇A.
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And then we used SPECS model to derive these functions.
 +
 +
Here we introduce SPECS briefly.
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An E.coli belongs to one of two mobile states: "Run" and "Tumble".
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Run state indicates that E.coli goes straight until the state changes.
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Tumble state indicates that E.coli is changing its direction and doesn't move around.
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E.coli changes its state "Run" to "Tumble" with probability P<html><sub>rt</sub></html> [1/sec].
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"Tumble" to "Run" with P<html><sub>tr</sub></html> [1/sec].
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These probability depend on the E.coli's internal state.
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The internal state is represented as two variables: "a" and "m".
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"a" indicates its kinase activity and "m" represents its receptor's methylation level.
 +
They vary depending on L-Asp concentration.
 +
In addition, SPECS concern the Brownian fluctuation.
 +
From these modeling, SPECS can replicate the internal circuit of E.coli which translates Asp concentration distribution into its mobile state.
 +
 +
We processed Monte Carlo simulation for 20000 E.coli which behavior was based on SPECS in some different Asp concentration and its gradient strength.  The shape of the Asp gradient we used were exponential.
 +
 +
[[File:utt_m2_eqn1.png]]
 +
 +
We approximated E.coli movement as the combination of parallel translation and diffusion.
 +
We recorded the time course of average position <x> and root-mean-square ¥<x^2> and then
 +
calculated average velocity
 +
 +
[[File:utt_m2_eqn2.png]]
 +
 +
and the diffusion constant.
 +
 +
[[File:utt_m2_eqn3.png]]
 +
 +
We derived the relationship between
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    Asp concentration and average velocity of E.coli
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    Asp concentration and diffusion constant of E.coli
 +
    gradient of Asp concentration and average velocity of E.coli
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    gradient of Asp concentration and diffusion constant of E.coli
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Finally we fitted approximated curve (represented as functions of A and ∇A) to these data so that we could use these functions in the entire simulation.
 +
 +
 +
 +
 +
 +
 +
 +
 +
 +
In this study, we used SPECS method to simulate E.coli chemotactic motion.
In this study, we used SPECS method to simulate E.coli chemotactic motion.
Here we introduce the brief overview of SPECS.
Here we introduce the brief overview of SPECS.

Revision as of 13:36, 5 October 2011

A simple model for E.coli chemotaxis

Aim

We derived a simple relationship between Asp concentration and E.coli chemotaxis so that we were able to simulate macroscopic (n~108) E.coli colony chemotacitc behavior in the entire simulation.

Background

SPECS [1] is one of the model simulating E.coli's chemotactic behavior. It considers internal condition of E.coli (receptor methylation etc) so it is a relatively strict taxis model. It is suitable for semi-macroscopic simulation which has around ten thousand E.coli and the timescale is within a hour. For our system, however, it takes too much time to replicate entire system, because the system includes much more E.coli(about one hundred of million) and the time scale is relatively long (a few days). Therefore we devised a new model based on SPECS which requires less calculation amount.

Method

We approximated chemotactic motion of E.coli groups into two factors, that is, parallel translation and diffusion.
Let 'V' represents a parallel translation velocity 'D' represents a diffusion coefficient of E.coli groups and 'A' represents L-Asp concentration.

E.coli can detect A and its gradient ∇A. They have an internal circuit to translate these signal into motions of their flagellum and try to run toward Asp-richer region. So we thought V and D could be represented as a functions of A and ∇A. And then we used SPECS model to derive these functions.

Here we introduce SPECS briefly. An E.coli belongs to one of two mobile states: "Run" and "Tumble". Run state indicates that E.coli goes straight until the state changes. Tumble state indicates that E.coli is changing its direction and doesn't move around. E.coli changes its state "Run" to "Tumble" with probability Prt [1/sec]. "Tumble" to "Run" with Ptr [1/sec]. These probability depend on the E.coli's internal state. The internal state is represented as two variables: "a" and "m". "a" indicates its kinase activity and "m" represents its receptor's methylation level. They vary depending on L-Asp concentration. In addition, SPECS concern the Brownian fluctuation. From these modeling, SPECS can replicate the internal circuit of E.coli which translates Asp concentration distribution into its mobile state.

We processed Monte Carlo simulation for 20000 E.coli which behavior was based on SPECS in some different Asp concentration and its gradient strength. The shape of the Asp gradient we used were exponential.

Utt m2 eqn1.png

We approximated E.coli movement as the combination of parallel translation and diffusion. We recorded the time course of average position <x> and root-mean-square ¥<x^2> and then calculated average velocity

Utt m2 eqn2.png

and the diffusion constant.

Utt m2 eqn3.png

We derived the relationship between 

   Asp concentration and average velocity of E.coli
   Asp concentration and diffusion constant of E.coli
   gradient of Asp concentration and average velocity of E.coli
   gradient of Asp concentration and diffusion constant of E.coli

Finally we fitted approximated curve (represented as functions of A and ∇A) to these data so that we could use these functions in the entire simulation.






In this study, we used SPECS method to simulate E.coli chemotactic motion. Here we introduce the brief overview of SPECS.

We processed Monte Carlo simulation for 20000 E.colis which behavior was based on SPECS in some different Asp concentration and its gradient. The gradient we used were exponential ¥[eqn 1]. We approximated E.colis movement as the combination of parallel translation and diffusion. We recorded the time course of average position <x> and root-mean-square <x2> and then calculated average velocity ¥[eqn 2] and the diffusion constant ¥[eqn 3].

For these results we derived the relationship between:

  • Asp concentration and average velocity of E.coli
  • Asp concentration and diffusion constant of E.coli
  • gradient of Asp concentration and average velocity of E.coli
  • gradient of Asp concentration and diffusion constant of E.coli

Finally we fitted approximated curve to these data.


Result

¥[fig1. relationship between Asp conc. and avg. velocity] ¥[fig2. relationship between gradient of Asp conc. and avg. velocity] ¥[fig3. relationship between Asp conc. and diffusion coef.] ¥[fig4. relationship between gradient of Asp conc. and diffusion coef.] fitted curveは次の式で与えられる。 ¥[eqn4](tanh(0.66*(log(A)+11.35))+1.0)*0.055*grad(log(A)) ¥[eqn5](tanh(0.65*(log(A)+11.5))+1.0)*(tanh(0.71*(grad(log(A))-2.05)+0.9)*0.0037+0.0047 ¥φ means Asp concentration.

References

  • [1] Jiang L, Ouyang Q, Tu Y (2010) Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time. PLoS Comput Biol 6(4): e1000735. doi:10.1371/journal.pcbi.1000735

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