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

Line 172: Line 172:
 
  <div class="part">
 
  <div class="part">
  
     <video id="video" preload="auto" tabindex="0" width="50%" controls="" type="video/ogg">
+
     <video id="video" preload="auto" tabindex="0" width="100%" controls="" type="video/ogg">
 
         <source type="video/ogg" src="https://static.igem.org/mediawiki/2015/d/d2/2dSim.ogg">
 
         <source type="video/ogg" src="https://static.igem.org/mediawiki/2015/d/d2/2dSim.ogg">
 
         Sorry, your browser does not support HTML5 video.
 
         Sorry, your browser does not support HTML5 video.

Revision as of 13:31, 4 September 2015

En alle beelden op tv
Van bloed en oorlog om ons heen
Werken daar ook niet echt aan mee

Dus ik neem heel bewust het besluit
De krant leg ik weg
En de tv gaat uit

Rood is al lang het rood niet meer,
Het rood van rode rozen
De kleur van liefde van weleer
Lijkt door de haat gekozen

Dat mooie rood was ooit voor mij
De kleur van passie en van wijn
Ik wil haar terug, die mooie tijd
Maar zij lijkt lang vervlogen

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 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. 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}$$cm^2/h$, which leads to earlier pattern formation.

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 ]

Hybrid model

Coming Soon

Internal model

Coming Soon

Contact

Address: Celestijnenlaan 200G room 00.08 - 3001 Heverlee
Telephone n°: +32(0)16 32 73 19