Team:UPO-Sevilla/Modeling
From 2010.igem.org
Models
The members of the dry lab are simulating the different components of the full system. Three main components can be identified:- The difussion of the chemoattractant through the medium.
- The motion of the bacterias through the medium due to the gradient on the chemoattractant concentration.
- The chemoattractant generation within the bacteria.
Chemoattractant Diffusion
The basic equations for the diffusion of the chemoattractant in the medium are the Fick laws of diffusion, which govern the variation of the concentration of a substance within a medium.
The flux J (that is, the amount of substance that flows through a given surface per unit of time mol m-2s-1) is given by:
![First Fick Law](https://static.igem.org/mediawiki/2010/6/6f/UPO-Model-Eq1.png)
where φ is the concentration (mol m-3) in a given point. D is a constant called the diffusion coefficient, and that depends on the medium .
Basically, the equation states that the is directed towards places with lower concentration (thus the minus sign). If the concentration is constant in the space (∇φ=0) there is no flux.
If the flux is known, it is possible to determine the amount of substance that goes through a small surface S and a small amount of time dt
![First Fick Law](https://static.igem.org/mediawiki/2010/f/f8/UPO-Model-Eq2.png)
In order to simulate the diffusion, we define the environment and discretize it in very small cells. Each cell determines a given volume V, and has a surface S. At a given time instant, the cell has an amount of substance c (and then a concentration c\V).
If the cells and time step Δt are small, we can consider that the gradient of concentration can be approximated though the differences in concentration between a cell i and 4 (or 8) neighbors j. Thus:
![First Fick Law](https://static.igem.org/mediawiki/2010/e/ea/UPO-Model-Eq3.png)
and then, the amount of substance that diffusses from i to j:
![First Fick Law](https://static.igem.org/mediawiki/2010/6/6f/UPO-Model-Eq4.png)
Bacteria motion
The main actuator of E. Coli is a flagellar motor that can rotate clock and counterclockwise. Through a set of transmembrane receptors proteins, E. Coli is able to detect chemoattractants. Moreover, this detection influences the motion of the flagellar motor. E. Coli has two main motion modes, which we will name:- Random Walk
- Gradient climbing
![First Fick Law](https://static.igem.org/mediawiki/2010/b/bc/UPO-Model-Eq5.png)
![First Fick Law](https://static.igem.org/mediawiki/2010/b/bd/UPO-Model-Eq6.png)
Chemical reactions
Modeling Tools