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<p>In numerical simulation<br/>
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a computational molecule<br/>
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describes the space and time<br/>
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relationship of data.</p>
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Revision as of 11:48, 8 September 2015

In numerical simulation
a computational molecule
describes the space and time
relationship of data.

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:

Figure 1
computational molecule

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

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

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Internal model

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