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International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals.org/systems_and_measurements/ coefficients Dn and Dc [4], [21]. The model parameter s is required to support the spatial and temporal scale for simulating systems and processes of the interest. The essential parameter χ0 controls the chemotactic response of the cells to the concentrations of the attractant and allows to reproduce the experimentally observed bands. Earlier, it was shown that r and φ are also essential for modeling the pattern formation in a colony of luminous E. coli [17]. VI. CONCLUSIONS The quasi-one dimensional spatiotemporal pattern formation along the three phase contact line in the fluid cultures of lux-gene engineered Escherichia coli can be simulated and studied on the basis of the Patlak-Keller-Segel model. The mathematical model (11)-(13) and the corresponding dimensionless model (15), (18), (19) of the bacterial selforganization in a circular container as detected by bioluminescence imaging may be successfully used to investigate the pattern formation in a colony of luminous E. coli. A constant function (χ(u,v) as well as h(n,c)) of the chemotactic sensitivity can be used for modeling the formation of the bioluminescence patterns in a colony of luminous E. coli (Figures 4 and 5). Oscillations and fluctuations similar to experimental ones (Figure 2) can be computationally simulated ignoring the signal-dependence as well as the density-dependence of the chemotactic sensitivity (Figure 3a). The local sampling and the linear diffusion can be successfully applied to modeling the formation of the bioluminescence patterns in a colony of luminous E. coli. The influence of the non-local gradient to the pattern formation can be partially compensated with the nonlinear diffusion (Figures 6, 7 and 8). However, the non-local sampling and the nonlinear diffusion do not yield in patterns more similar to the experimentally observed ones when compared to the patterns obtained by the corresponding model with the local sampling and the linear diffusion. The more precise and sophisticated two- and threedimensional computational models implying the formation of structures observed on bioluminescence images are now under development and testing. ACKNOWLEDGMENT The research by R. Baronas is funded by the European Social Fund under Measure VP1-3.1-ˇSMM-07-K ”Support to Research of Scientists and Other Researchers (Global Grant)”, Project ”Developing computational techniques, algorithms and tools for efficient simulation and optimization of biosensors of complex geometry”. REFERENCES [1] R. Baronas, “Simulation of bacterial self-organization in circular container along contact line as detected by bioluminescence imaging,” in SIMUL 2011: The Third International Conference on Advances in System Simulation, Barcelona, Spain, Oct. 2011, pp. 156–161. [2] M. Eisenbach, Chemotaxis. London: Imperial College Press, 2004. [3] T. C. Williams, Chemotaxis: Types, Clinical Significance, and Mathematical Models. New York: Nova Science, 2011. [4] J. D. Murray, Mathematical Biology: II. Spatial Models and Biomedical Applications, 3rd ed. Berlin: Springer, 2003. [5] E. F. Keller and L. A. Segel, “Model for chemotaxis,” J. Theor. Biol., vol. 30, no. 2, pp. 225–234, 1971. [6] T. Hillen and K. J. Painter, “A user’s guide to PDE models for chemotaxis,” J. Math. Biol., vol. 58, no. 1-2, pp. 183–217, 2009. [7] E. O. Budrene and H. C. Berg, “Dynamics of formation of symmetrical patterns by chemotactic bacteria,” Nature, vol. 376, no. 6535, pp. 49–53, 1995. [8] M. P. Brenner, L. S. Levitov, and E. O. Budrene, “Physical mechanisms for chemotactic pattern formation by bacteria,” Biophys. J., vol. 74, no. 4, pp. 1677–1693, 1998. [9] S. Sasaki et al., “Spatio-temporal control of bacterialsuspension luminescence using a PDMS cell,” J. Chem. Engineer. Japan, vol. 43, no. 11, pp. 960–965, 2010. [10] K. J. Painter and T. Hillen, “Spatio-temporal chaos in a chemotactic model,” Physica D, vol. 240, no. 4-5, pp. 363– 375, 2011. [11] R. ˇSimkus, “Bioluminescent monitoring of turbulent bioconvection,” Luminescence, vol. 21, no. 2, pp. 77–80, 2006. [12] R. ˇSimkus, V. Kirejev, R. Meˇskien˙e, and R. Meˇskys, “Torus generated by Escherichia coli,” Exp. Fluids, vol. 46, no. 2, pp. 365–369, 2009. [13] S. Daunert et al., “Genetically engineered whole-cell sensing systems: coupling biological recognition with reporter genes,” Chem. Rev., vol. 100, no. 7, pp. 2705–2738, 2000. [14] Y. Lei, W. Chen, and A. Mulchandani, “Microbial biosensors,” Anal. Chim. Acta, vol. 568, no. 1-2, pp. 200–210, 2006. [15] M. B. Gu, R. J. Mitchell, and B. C. Kim, “Whole-cell-based biosensors for environmental biomonitoring and application,” Adv. Biochem. Eng. Biotechnol., vol. 87, pp. 269–305, 2004. [16] R. ˇSimkus and R. Baronas, “Metabolic self-organization of bioluminescent Escherichia coli,” Luminescence, vol. 26, no. 6, pp. 716–721, 2011. [17] R. Baronas and R. ˇSimkus, “Modeling the bacterial selforganization in a circular container along the contact line as detected by bioluminescence imaging,” Nonlinear Anal. Model. Control, vol. 16, no. 3, pp. 270–282, 2011. [18] A. A. Samarskii, The Theory of Difference Schemes. New York-Basel: Marcel Dekker, 2001. [19] I. R. Lapidus and R. Schiller, “Model for the chemotactic response of a bacterial population,” Biophys. J., vol. 16, no. 7, pp. 779–789, 1976. 2012, © Copyright by authors, Published under agreement with <strong>IARIA</strong> - www.iaria.org 162
International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals.org/systems_and_measurements/ [20] P. K. Maini, M. R. Myerscough, K. H. Winters, and J. D. Murray, “Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation,” Bull. Math. Biol., vol. 53, no. 5, pp. 701–719, 1991. [21] R. Tyson, S. R. Lubkin, and J. D. Murray, “Model and analysis of chemotactic bacterial patterns in a liquid medium,” J. Math. Biol., vol. 38, no. 4, pp. 359–375, 1999. [22] R. Kowalczyk, “Preventing blow-up in a chemotaxis model,” J. Math. Anal. Appl., vol. 305, no. 2, pp. 566–588, 2005. [23] M. R. Myerscough, P. K. Maini, and K. J. Painter, “Pattern formation in a generalized chemotactic model,” Bull. Math. Biol., vol. 60, no. 1, pp. 1–26, 1998. [24] E. F. Keller and L. A. Segel, “Travelling bands of chemotactic bacteria: A theoretical analysis,” J. Theor. Biol., vol. 30, no. 2, pp. 235–248, 1971. [25] K. Painter and T. Hillen, “Volume-filling and quorum-sensing in models for chemosensitive movement,” Can. Appl. Math. Quart., vol. 10, no. 4, pp. 501–543, 2002. [26] J. J. L. Velazquez, “Point dynamics for a singular limit of the Keller-Segel model. I. Motion of the concentration regions,” SIAM J. Appl. Math., vol. 64, no. 4, pp. 1198–1223, 2004. [27] H. G. Othmer and T. Hillen, “The diffusion limit of transport equations II: Chemotaxis equations,” SIAM J. Appl. Math., vol. 62, no. 4, pp. 1122–1250, 2002. [28] T. Hillen, K. Painter, and C. Schmeiser, “Global existence for chemotaxis with finite sampling radius,” Discr. Cont. Dyn. Syst. B, vol. 7, no. 1, pp. 125–144, 2007. [29] R. Baronas, F. Ivanauskas, and J. Kulys, Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians, ser. Springer Series on Chemical Sensors and Biosensors, G. Urban, Ed. Dordrecht: Springer, 2010. [30] R. Čiegis and A. Bugajev, “Numerical approximation of one model of bacterial self-organization,” Nonlinear Anal. Model. Control, vol. 17, no. 3, pp. 253–270, 2012. [31] S. Leestma and L. Nyhoff, Pascal Programming and Problem Solving, 4th ed. New York: Prentice Hall, 1993. [32] R. Tyson, L. Stern, and R. J. LeVeque, “Fractional step methods applied to a chemotaxis model,” J. Math. Biol., vol. 41, no. 5, pp. 455–475, 2000. [33] P. Romanczuk, U. Erdmann, H. Engel, and L. Schimansky- Geier, “Beyond the Keller-Segel model. Microscopic modeling of bacterial colonies with chemotaxis,” Eur. Phys. J. Special Topics, vol. 157, no. 1, pp. 61–77, 2007. 2012, © Copyright by authors, Published under agreement with <strong>IARIA</strong> - www.iaria.org 163
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