# Team:UT-Tokyo/Data/Modeling/Model02

### From 2011.igem.org

# Model02

# iGEM UT-Tokyo

# 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~10^{8}) *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 the motion of *E.coli* groups doing a process of chemotaxis into two factors, that is, parallel translation and diffusion.

Let 'V' represent the parallel translation velocity, 'D' represent the diffusion coefficient of *E.coli* groups and 'A' represent L-Asp concentration.
*E.coli* can detect A and the gradient ∇A.
They have an internal circuit to translate these signals into motions of their flagella and try to move towards regions with more Asp.
So we thought V and D could be represented as a function of A and ∇A.
And then we used the 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 from "Run" to "Tumble" with probability P_{rt} [1/sec].
From "Tumble" to "Run" with P_{tr} [1/sec].
These probabilities depend on the *E.coli* internal state.
The internal state is represented by two variables: "a" and "m".
"a" indicates kinase activity and "m" represents the receptor methylation level of the cell.
These vary depending on L-Asp concentration.
In addition, SPECS takes into account Brownian fluctuation.
Using this model, SPECS tries to replicate the internal circuit of *E.coli* which translates an Asp concentration distribution into a mobile state.

We processed a Monte Carlo simulation for 20000 *E.coli* behaving according to SPECS when placed in fields containing different concentrations and gradients of Asp. The shape of the Asp gradient used was exponential according to the following relationship:

where A0 represents the concentration of Asp at x = 0 and x0 a parameter that represents the slope of the gradient.

We approximated *E.coli* movement by a combination of parallel translation and diffusion.
We recorded the average position <x> and root-mean-square <x^{2}> over time and then
calculated the average velocity:

and the diffusion constant:

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 curves (represented as functions of A and ∇A) to these data so that we could derive functions we needed in the entire system simulation.

## Result

The fitted curves are represented by the following equations.

## 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