Team:KU Leuven/Modeling/Top
1-D continuous model
The video above shows how the proposed method for pattern formation works. Two cell types A and B are interacting. Type
A cells produce a repellent called leucine which causes the cells of type B to move away. At the same time type A cells
also produce OO-AHL, which is required by the cells of type B to move. Initially colonies of the two cell types are placed
at the center of the dish. As molecule production within the type A cells kicks in, the repellent and AHL concentrations
start to increase. This triggers the type B cells to move away from the center. Movement will contiue until the concentration
of AHL is insuficcent for the type B cells to move further.
The Keller segel model type model we used is given by the following equation system:
$$\frac{\partial A}{\partial t} = D_a \bigtriangledown^2 A + \gamma A(1 - \frac{A}{k_{p}}),$$
$$\frac{\partial B}{\partial t} = D_b \bigtriangledown^2 B + \bigtriangledown (P(B,H,R) \bigtriangledown R)+ \gamma B(1 - \frac{B}{k_{p}}), $$
$$ \frac{\partial R}{\partial t} = D_r \bigtriangledown^2 B + k_r A - k_{lossH} R $$
$$\frac{\partial H}{\partial t} = D_h \bigtriangledown^2 B + k_h A - k_{lossR} H . $$
With:
$$ P(B,H,R) = \frac{-B K_{c} H}{R}. $$
The model has been derived while looking at [1] and [2]
The terms that appear can be grouped into four categories. Every equation has a diffusion term given by
$D_x \bigtriangledown^2 X$, diffusion smoothes peaks by spreading them out in space. The two equations related to cell
densities contain logistic growth terms of the form $\gamma X(1-\frac{X}{k_x})$, which model the cell growth during
simulation time. Finally the second equation describing the moving cells comes with a variable coefficient Poisson term
$\bigtriangledown (P \bigtriangledown X)$ which describes the cell movement. Last but not least: the two bottom equations.
They model concentrations, contain linear production and degradation terms, which look like $kX$.
To generate the video file above the system above has been discretized using a finite element approach in conjunction,
with an explicit euler scheme. Finally simulation has been done using the parameters given in the table below:
| Parameter | Value | Unit | Source | Comment |
|---|---|---|---|---|
| $D_a$ | $0.072 \cdot 10^{-3}$ | $cm^2/h$ | following [1] | |
| $D_b$ | $2.376 \cdot 10^{-3}$ | $cm^2/h$ | following [1] | |
| $D_r$ | $26.46 \cdot 10^{-3}$ | $cm^2/h$ | as found in [6] | $298.2 K$ |
| $D_h$ | $50 \cdot 10^{-3}$ | $cm^2/h$ | from [3] | |
| $K_{c}$ | $8.5 \cdot 10^{-3}$ | $cm^2/h$ | guessed | |
| $\gamma$ | $10^{-5}$ | $h^{-1}$ | from [1] | |
| $k_p$ | $1.0 \cdot 10^2$ | $cl^{-1}$ | from [1] | |
| $k_r$ | $1.584\cdot 10^{-4}$ | $nmol/h$ | computed using [4] and [5] | |
| $k_h$ | $1.5 \cdot 1.584 \cdot 10^{-4}$ | $nmol/h$ | guessed | |
| $k_{lossH}$ | $19.36 \cdot 10^{-4}$ | $nmol/h$ | from [7] | $ ph = 7$ |
| $k_{lossR}$ | $10^{-5}$ | $nmol/h$ | guessed |
2-D continuous model
Using the equation system and described above, the model may also be simulated in two dimensions. Once more a finite volume approach has been taken in connection with an explicit Euler scheme. All parameters have been kept constant with the one exception of the chemotatctic sensitivity $K_c$. Which has been inreased to $Kc = 1.5 * 10^{-1}$, which leads to earlier pattern formation.
References
