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 circuits and devices for 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)
The following figure illustrates the basic elements of the simulation. The flux J between cells is computed by the difference of concentrations. Then, this flux is used to compute the amount of substance that will flow to the neighbour cell. The amount is proportional to the flux and the common surface between cells.
![Basic parameters of the diffusion simulation.](https://static.igem.org/mediawiki/2010/9/97/UPO-Model-diffusion.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 [Topp and Gallivan, 2007].
E. Coli has two main motion modes, which we will name:
- Random Walk
- Gradient climbing
Random walk mode
When no gradient of chemoattractant is present, E. Coli is in random walk mode. In this case, the bacteria performs smooth runs followed by tumbles.
Mathematically, we will model this as a Brownian motion:
![First Fick Law](https://static.igem.org/mediawiki/2010/b/bc/UPO-Model-Eq5.png)
where x is the position of the bacteria and v is the velocity. This (vector) velocity is randomly sampled from a normal distribution of zero mean and a certain covariance matrix that models the potential .
![First Fick Law](https://static.igem.org/mediawiki/2010/b/bd/UPO-Model-Eq6.png)
The bacteria will move for a time Δ trm with constant velocity v(t). At next time instant, a new velocity is (randomly selected) selected
Gradient climbing behaviourWhen a positive difference of concentration (a gradient) on the chemoattractant is detected, the bacteria enters into a new mode that we will call gradient climbing.
In this mode, the flagellar motor tends to move counterclockwise; as a result, the smooth runs last for more time, and the tumbling frequency decreases.
In order to model that, we use the same model than above, but a larger tumbling interval (smaller tumbling frequency) Δ tgc;[Topp and Gallivan, 2007] Topp, S. and Gallivan, J. (2007). Guiding Bacteria with Small Molecules and RNA. J. Am. Chem. Soc., 129:6807–6811.
Circuits and devices
![Simbiology model](https://static.igem.org/mediawiki/2010/8/8a/IGEMUPO-model-small.png)
Modeling Tools