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Modeling

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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 AHL, 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 start to increase. This triggers the type B cells to move away from the center. Movement will continue until the concentration of AHL is insufficient for the type B cells to move further.

Our Keller-Segel type model

$$\frac{\partial A}{\partial t} = D_a \bigtriangledown^2 A + \gamma A(1 - \frac{A}{k_{p}}),$$ $$\frac{\partial B}{\partial t} = D_b \bigtriangledown^2 B + \bigtriangledown (P(B,H,R) \bigtriangledown R)+ \gamma B(1 - \frac{B}{k_{p}}), $$ $$ \frac{\partial R}{\partial t} = D_r \bigtriangledown^2 R + k_r A - k_{lossH} R $$ $$\frac{\partial H}{\partial t} = D_h \bigtriangledown^2 H + k_h A - k_{lossR} H . $$

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, For finite volume methods to work we rewrite our equations as conservation laws. Then each grid point is assigned the area around it such that flux of cells or molecules leaving one cell enters another one. From discretizing the integrated conservation law the following expression is obtained in one dimension:

Discretized Keller-Segel type model

$$ A^{n+1}_j = A^n_j + (\triangle.* (D_a/(\triangle x)^2 \cdot ( A^n_{j-1} + A^n_{j+1} - 2.*A^n_j)) ... $$ $$ + \gamma \cdot A^n_j \cdot (1 - A^n_j / kp)) $$ $$B^{n+1}_j = B^n_j +\triangle t \cdot (1/ (\triangle x)^2 \cdot (D_b\cdot (B^n_{j-1} + B^n_{j+1} - 2B^n_j)\dots $$ $$ -(P^n_{j+\frac{1}{2}} \cdot (R^n_{j+1} - R^n_j) - P^n_{j-\frac{1}{2}} \cdot (R^n_j - R^n_{j-1}))) $$ $$ + \gamma \cdot B^n_j \cdot (1 - B^n_j / kp)) $$ $$ R^{n+1}_j = R^n_j + \triangle t \cdot( D_r \cdot (R^n_{j+1} + R^n_{j-1} - 2*R^n_j) /(\triangle x^2) \dots $$ $$ + kr \cdot A^n_j - k_r \cdot R^n_j) $$ $$ H^{n+1}_j = H^n_j + \triangle t \cdot ( D_h \cdot (H^n_{j+1} + H^n_{j-1} - 2*H^n_j) / (\triangle x)^2 \dots $$ $$ + k_h \cdot A^n_j - k_h \cdot H^n_j ) $$

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.
Zero flux and periodic boundary conditions have been implemented. The boundaries are the edges of the domain on which the equation system is solved. Here the domain ranges from zero to eight centimetres, which is the diameter of a Petri dish. With zero flux boundaries the first derivative is set to zero at the boundaries, which means that neither bacteria nor chemicals are allowed to pass trough the boundary. Periodic boundaries connect pairs of boundaries to each other, which means that cells leaving at the top of the boundary appear at the bottom, cells leaving at the left boundary reappear at the right and so on. In the continuous context these boundary conditions have been implemented to allow comparisons with the hybrid model, where these boundaries are also used. 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]
$k_{lossH}$	$ln(2)/48$	$h^{-1}$	from ^[5]	$ ph = 7$
$k_{lossR}$	$ln(2)/80$	$h^{-1}$	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. Above four simulation videos with Gaussian initial conditions can be observed. A fifth video shows a simulation using random initial data. The two last videos illustrate the effect of zero flux and periodic boundary conditions.

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 ]