Team:UT-Tokyo/Data/Modeling/Model02

From 2011.igem.org

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=[[Team:UT-Tokyo/Data/Modeling/Model02/applet|Interactive demo]]=
=[[Team:UT-Tokyo/Data/Modeling/Model02/applet|Interactive demo]]=
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=Aim=
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==Aim==
We tried to derive a simple relationship between Asp concentration and ''E.coli'' chemotactic behavior so that
We tried to derive a simple relationship between Asp concentration and ''E.coli'' chemotactic behavior so that
it become possible to simulate macroscopic (n~10<html><sup>8</sup></html>) ''E.coli''-colony chemotactic behavior in the entire simulation.
it become possible to simulate macroscopic (n~10<html><sup>8</sup></html>) ''E.coli''-colony chemotactic behavior in the entire simulation.
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=Background=
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==Background==
SPECS <html><sup class="ref">[1]</sup></html> is one of the models simulating ''E.coli'' chemotactic behavior, and it considers internal condition of ''E.coli'' (receptor methylation etc).  It is suitable for semi-macroscopic simulation which has around ten thousand E.coli and the timescale is within a hour.
SPECS <html><sup class="ref">[1]</sup></html> is one of the models simulating ''E.coli'' chemotactic behavior, and it considers internal condition of ''E.coli'' (receptor methylation etc).  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).
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.
Therefore we devised a new model based on SPECS which requires less calculation amount.
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=Method=
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==Method==
We approximated chemotactic motion of ''E.coli'' groups into two factors, that is, parallel translation and diffusion.
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.
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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We approximated ''E.coli'' movement as the combination of parallel translation and diffusion.
We approximated ''E.coli'' movement as the combination of parallel translation and diffusion.
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We recorded the time course of average position <x> and root-mean-square \<x^2> and then
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We recorded the time course of average position <x> and root-mean-square <x<html><sup>2</sup></html>> and then
calculated average velocity  
calculated average velocity  
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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.
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.
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=Result=
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==Result==
{{:Team:UT-Tokyo/Templates/Image|file=utt_m2_fig1.png|caption=Figure 1. relationship between Asp conc. and avg. velocity}}
{{:Team:UT-Tokyo/Templates/Image|file=utt_m2_fig1.png|caption=Figure 1. relationship between Asp conc. and avg. velocity}}
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<div id="references">
<div id="references">
<ul>
<ul>
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   <li id="ref_1">[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</li>
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   <li id="ref_1">[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</li>
</ul>
</ul>
</div>
</div>
</html>
</html>
{{:Team:UT-Tokyo/Templates/EndContent}}
{{:Team:UT-Tokyo/Templates/EndContent}}

Revision as of 00:31, 6 October 2011

Modeling/Model2: A simple model for E.coli chemotaxis

Interactive demo

Aim

We tried to derive a simple relationship between Asp concentration and E.coli chemotactic behavior so that it become possible to simulate macroscopic (n~108) E.coli-colony chemotactic behavior in the entire simulation.

Background

SPECS [1] is one of the models simulating E.coli chemotactic behavior, and it considers internal condition of E.coli (receptor methylation etc). 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 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 <x2> 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.

Result

Figure 1. relationship between Asp conc. and avg. velocity
Figure 1. relationship between Asp conc. and avg. velocity

Figure 2. relationship between gradient of Asp conc. and avg. velocity
Figure 2. relationship between gradient of Asp conc. and avg. velocity

Figure 3. relationship between Asp conc. and diffusion coef
Figure 3. relationship between Asp conc. and diffusion coef

Figure 4. relationship between gradient of Asp conc. and diffusion coef
Figure 4. relationship between gradient of Asp conc. and diffusion coef

The fitted curves are represented by following equation.

Utt m2 eqn4.png

Utt m2 eqn5.png

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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