Difference between revisions of "Team:KU Leuven/Modeling/Top"

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 OHHL, 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 OHHL concentrations start to increase. This triggers the type B cells to move away from the center. Movement will continue until the concentration of OHHL is insufficient for the type B cells to move further.
The Keller-Segel type model we used is given by the following equation system: $$\frac{\partial A}{\partial t} = D_a \bigtriangledown^2 A + \gamma A(1 - \frac{A}{k_{p}}),$$ $$\frac{\partial B}{\partial t} = D_b \bigtriangledown^2 B + \bigtriangledown (P(B,H,R) \bigtriangledown R)+ \gamma B(1 - \frac{B}{k_{p}}), $$ $$ \frac{\partial R}{\partial t} = D_r \bigtriangledown^2 R + k_r A - k_{lossH} R $$ $$\frac{\partial H}{\partial t} = D_h \bigtriangledown^2 H + k_h A - k_{lossR} H . $$ 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 volume approach in conjunction, with an explicit Euler scheme:

The image above shows the dependency of data in time and space. The computational molecule used in this case uses only data of the previous time level $t_n$ to compute data at the next time level $t_{n+1}$. A scheme with a space time dependency like the one shown above is called an explicit 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]
$K_{c}$	$8.5 \cdot 10^{-3}$	$cm^2 \cdot cl/h$	estimated
$\gamma$	$10^{-5}$	$h^{-1}$	from [1]
$k_p$	$1.0 \cdot 10^2$	$cl^{-1}$	from [1]
$k_h$	$17.9 \cdot 10^{-4}$	$fmol/h$	computed from [4] and [8]
$k_r$	$5.4199\cdot 10^{-4}$	$fmol/h$	computed from [7] and [8]
$k_{lossH}$	$1/48$	$h^{-1}$	from [5]	$ ph = 7$
$k_{lossR}$	$1/80$	$h^{-1}$	estimated

Initial condition 1 Initial condition 2 Initial condition 3 Initial condition 4 Random initial data

Using the equation system as 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 chemotactic sensitivity $K_c$. Which has been increased to $Kc = 1.5 * 10^{-1}$$cm^2/h$, which leads to earlier pattern formation.

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