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

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     The video above shows how the proposed method for pattern formation works. Two cell types A and B are interacting. Type
 
     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
 
     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  
+
     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 AHL concentrations  
+
     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 contiue until the concentration  
+
     start to increase. This triggers the type B cells to move away from the center. Movement will contiue until the concentration of OHHL is insufficient for the type B cells to move further.
    of AHL is insuficcent for the type B cells to move further.
+
 
      
 
      
 
</br>
 
</br>

Revision as of 16:09, 28 July 2015

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 contiue until the concentration of OHHL is insufficient 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

[1] Temporal Patterns Generated by Salmonella typhimurium, D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. 0.Budrene,l and H. C.Berg , Biophysical Journal Volume 68 May 1995 2181-2189

[2] Hybrid modelling of individual movement and collective behaviour, B. Franz and R. Erban, DISPERSAL, INDIVIDUAL MOVEMENT AND SPATIAL ECOLOGY: A MATHEMATICAL PERSPECTIVE Book Series: Lecture Notes in Mathematics Volume: 2071 Pages: 129-157

[3] Supplement 1 to Engineered cell-cell communication via DNA messaging, Monica E Ortiz and Drew Endy, Journal of Biological Engineering

[4] Protein abundance profiling of the Escherichia coli cytosol, Yasushi Ishihama1, Thorsten Schmidt, Juri Rappsilber, Matthias Mann, F Ulrich Hartl, Michael J Kerner and Dmitrij Frishman38, BMC Genomics 2008, 9:102

[5] The specificity and kinetic mechanism of branched-chain amino acid amino transferase from Escherichia coli studied with a new improved coupled assay procedure and the enzyme’s potential for biocatalysis, Xuejing Yu, Xingguo Wang and Paul C. Engel, the FEBS Journal

[6] Infinite Dilution Binary Diffusion Coefficients of Several $\alpha $-Amino Acids in Water over a Temperature Range from (293.2 to 333.2) K with the Taylor Dispersion Technique, Tatsuya Umecky, Tomoyuki Kuga, and Toshitaka Funazukuri, J. Chem. Eng. Data 2006,51,1705-1710

[7] The hydrolysis of unsubstituted N-acylhomoserine lactones to their homoserine metabolites: Analytical approaches using ultra performance liquid chromatography, Matthias Englmanna, Agnes Feketea, Christina Kuttlerb, Moritz Frommbergera, Xiaojing Lia, Istvan Gebefügia, Jenoe Feketee, Philippe Schmitt-Kopplina, Journal of Chromatography A, Volume 1160, Issues 1–2, 10 August 2007, Pages 184–193

Hybrid model

Coming Soon

Internal model

Coming Soon