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 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
To generate the video file above the system above has been discretized using a finite volume 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] | |
| $K_{c}$ | $8.5 \cdot 10^{-3}$ | $cm^2 \cdot cl/h$ | 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 |
2-D continuous model
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 $Kc = 1.5 * 10^{-1}$
References
| [1] | D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene, and H. C. Berg. Spatio-temporal patterns generated by Salmonella typhimurium. Biophysical journal, 68(5):2181-2189, May 1995. [ DOI | http ] |
| [2] | Benjamin Franz and Radek Erban. Hybrid modelling of individual movement and collective behaviour. Lecture Notes in Mathematics, 2071:129-157, 2013. [ http ] |
| [3] | Monica E Ortiz and Drew Endy. Supplement to- 1754-1611-6-16-s1.pdf, 2012. [ .pdf ] |
| [4] | A. B. Goryachev, D. J. Toh, and T. Lee. Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants. In BioSystems, volume 83, pages 178-187, 2006. [ DOI ] |
| [5] | A. L. Schaefer, B. L. Hanzelka, M. R. Parsek, and E. P. Greenberg. Detection, purification, and structural elucidation of the acylhomoserine lactone inducer of Vibrio fischeri luminescence and other related molecules. Bioluminescence and Chemiluminescence, Pt C, 305:288-301, 2000. |
| [6] | Tatsuya Umecky, Tomoyuki Kuga, and Toshitaka Funazukuri. Infinite Dilution Binary Diffusion Coefficients of Several α-Amino Acids in Water over a Temperature Range from (293.2 to 333.2) K with the Taylor Dispersion Technique. Journal of Chemical & Engineering Data, 51(5):1705-1710, September 2006. [ DOI ] |
| [7] | Xuejing Yu, Xingguo Wang, and Paul C. Engel. The specificity and kinetic mechanism of branched-chain amino acid aminotransferase from Escherichia coli studied with a new improved coupled assay procedure and the enzyme's potential for biocatalysis. FEBS Journal, 281(1):391-400, January 2014. [ DOI ] |
| [8] | Yasushi Ishihama, Thorsten Schmidt, Juri Rappsilber, Matthias Mann, F Ulrich Hartl, Michael J Kerner, and Dmitrij Frishman. Protein abundance profiling of the Escherichia coli cytosol. BMC genomics, 9:102, 2008. [ DOI ] |
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