# American Institute of Mathematical Sciences

2013, 10(2): 399-424. doi: 10.3934/mbe.2013.10.399

## Competition of motile and immotile bacterial strains in a petri dish

 1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada, Canada

Received  September 2012 Revised  November 2012 Published  January 2013

Bacterial competition is an important component in many practical applications such as plant roots colonization and medicine (especially in dental plaque). Bacterial motility has two types of mechanisms --- directed movement (chemotaxis) and undirected movement. We study undirected bacterial movement mathematically and numerically which is rarely considered in literature. To study bacterial competition in a petri dish, we modify and extend the model used in Wei et al. (2011) to obtain a group of more general and realistic PDE models. We explicitly consider the nutrients and incorporate two bacterial strains characterized by motility. We use different nutrient media such as agar and liquid in the theoretical framework to discuss the results of competition. The consistency of our numerical simulations and experimental data suggest the importance of modeling undirected motility in bacteria. In agar the motile strain has a higher total density than the immotile strain, while in liquid both strains have similar total densities. Furthermore, we find that in agar as bacterial motility increases, the extinction time of the motile bacteria decreases without competition but increases in competition. In addition, we show the existence of traveling-wave solutions mathematically and numerically.
Citation: Silogini Thanarajah, Hao Wang. Competition of motile and immotile bacterial strains in a petri dish. Mathematical Biosciences & Engineering, 2013, 10 (2) : 399-424. doi: 10.3934/mbe.2013.10.399
##### References:
 [1] S.Asei, B. Byers, A. Eng, N. James and J. Leto, "Bacterial Chemostat Model," 2007. [2] P. K. Brazhnik and J. J Tyson, On traveling wave solutions of fisher's equation in two spatial dimensions, SIAM. J. Appl. Math., 60 (2000), 371-391. doi: 10.1137/S0036139997325497. [3] I. Chang, E. S. Gilbert, N. Eliashberg and J. D. Keasling, A three-dimensional stochastic simulation of biofilm growth and transport-related factors that affect structure, Micro. Bio., 149 (2003), 2859-2871. [4] M. Fontes and D. Kaiser, Myxococcus cells respond to elastic forces in their substances, Proceedings of the National Academy of Sciences of the United States of America, 96 (1999), 8052-8057. [5] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Jpn., 58 (1989), 3875-3878. [6] H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient, J. Phys. Soc. Jpn., 60 (1991), 88-94. [7] M. E. Hibbing, C. Fuqua, M. R. Parsek and B. S. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nature Reviews Microbiology, 8 (2010), 15-25. [8] D. P. Hzder, R. Hemmerbach and M. Lebert, Gravity and the bacterial unicellular organisms, Developmental and Cell Biology Series, 40 (2005). [9] C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. of Math. Biol., 42 (1980), 397-429. doi: 10.1016/S0092-8240(80)80057-7. [10] E. Keller, Mathematical aspects of bacterial chemotaxis, Antibiotics and Chemotherapy, 19 (1974), 79-93. [11] F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Micro. Biol., 16 (1988), 115-131. [12] E. Khain, L. M. Sander and A. M. Stein, A model for glioma growth, Research Article, 11 (2005), 53-57. doi: 10.1002/cplx.20108. [13] S. M. Krone, R. Lu, R. Fox, H. Suzuki and E. M. Top, Modelling the spatial dynamics of plasmid transfer and persistence, Micro. Biol., 153 (2007), 2803-2816. [14] D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations, Micro. Ecol., 7 (1981), 207-227. [15] D. Lauffenburger, R. Aris and K. H. Keller, Effects of cell motility and chemotaxis on growth of bacterial populations, Biophys. J., 40 (1982), 209-219. [16] D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Bio. Tech. and Bio. Eng., xxv (1983), 2103-2125. [17] M. Matsushita, J. Wakitaa, H. Itoha, K. Watanabea, T. Araia, T. Matsuyamab, H. Sakaguchic and M. Mimurad, Formation of colony patterns by a bacterial cell population, Physica A: Statistical Mechanics and Its Applications, 274 (1999), 190-199. [18] M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamazaki and T. Matsuyama, Colony formation in bacteria: Experiments and modeling, Biofilms, 1 (2004), 305-317. [19] M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modeling of bacterial colony patterns, Physica. A. Stat. Mech. Appl., 282 (2000), 283-303. [20] J. D. Murray, "Murray JD," $1^{st}$, $3^{rd}$ edition, USA, 2002. [21] K. Nowaczyk, A. Juszczak A and F. Domka, Microbiological oxidation of the waste ferrous sulphate, Polish Journal of Environmental Studies, 6 (1999), 409-416. [22] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. [23] P. T. Saunders and M. J. Bazin, On the stability of food chains, J. Theor. Biol., 52 (1975), 121-142. [24] R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats, Geobiology, 8 (2010), 179-190. [25] J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili, PNAS, 98 (2001), 6901-6904. [26] L. Simonsen, Dynamics of plasmid transfer on surfaces, J. General Microbiology, 136 (1990), 1001-1007. [27] R. Tokita, T. Katoh, Y. Maeda, J. I. Wakita, M. Sano, T. Matsuyama and M. Matsushita, Pattern formation of bacterial colonies by Escherichia coli, J. Phys. Soc. Jpn., 78 (2009), 074005 (6 pages). [28] Y. Wei, X. Wang, J. Liu, L. Nememan, A. H. Singh, H. Howie and B. R. Levin, The populatiion and evolutionary of bacteria in physically structured habitats: The adaptive virtues of motility, PNAS, 108 (2011), 4047-4052. [29] J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems," 2 1998. [30] P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria, Current Biology, 8 (1998), 408-411. [31] A. Ishihara, J. E. Segall, S. M. Block and H. L Berg, Coordination of flagella on filgmentous cells of Escherichia Coli, J. Bacteriology, 155 (1983), 228-237. [32] B. L. Taylor and D. E. Koshlard, Reversal of flafella rotation in Monotrichous and Peritrichous bacteria: Generation of changes in direction, J. Bacteriology, 119 (1974), 640-642.

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##### References:
 [1] S.Asei, B. Byers, A. Eng, N. James and J. Leto, "Bacterial Chemostat Model," 2007. [2] P. K. Brazhnik and J. J Tyson, On traveling wave solutions of fisher's equation in two spatial dimensions, SIAM. J. Appl. Math., 60 (2000), 371-391. doi: 10.1137/S0036139997325497. [3] I. Chang, E. S. Gilbert, N. Eliashberg and J. D. Keasling, A three-dimensional stochastic simulation of biofilm growth and transport-related factors that affect structure, Micro. Bio., 149 (2003), 2859-2871. [4] M. Fontes and D. Kaiser, Myxococcus cells respond to elastic forces in their substances, Proceedings of the National Academy of Sciences of the United States of America, 96 (1999), 8052-8057. [5] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Jpn., 58 (1989), 3875-3878. [6] H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient, J. Phys. Soc. Jpn., 60 (1991), 88-94. [7] M. E. Hibbing, C. Fuqua, M. R. Parsek and B. S. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nature Reviews Microbiology, 8 (2010), 15-25. [8] D. P. Hzder, R. Hemmerbach and M. Lebert, Gravity and the bacterial unicellular organisms, Developmental and Cell Biology Series, 40 (2005). [9] C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. of Math. Biol., 42 (1980), 397-429. doi: 10.1016/S0092-8240(80)80057-7. [10] E. Keller, Mathematical aspects of bacterial chemotaxis, Antibiotics and Chemotherapy, 19 (1974), 79-93. [11] F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Micro. Biol., 16 (1988), 115-131. [12] E. Khain, L. M. Sander and A. M. Stein, A model for glioma growth, Research Article, 11 (2005), 53-57. doi: 10.1002/cplx.20108. [13] S. M. Krone, R. Lu, R. Fox, H. Suzuki and E. M. Top, Modelling the spatial dynamics of plasmid transfer and persistence, Micro. Biol., 153 (2007), 2803-2816. [14] D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations, Micro. Ecol., 7 (1981), 207-227. [15] D. Lauffenburger, R. Aris and K. H. Keller, Effects of cell motility and chemotaxis on growth of bacterial populations, Biophys. J., 40 (1982), 209-219. [16] D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Bio. Tech. and Bio. Eng., xxv (1983), 2103-2125. [17] M. Matsushita, J. Wakitaa, H. Itoha, K. Watanabea, T. Araia, T. Matsuyamab, H. Sakaguchic and M. Mimurad, Formation of colony patterns by a bacterial cell population, Physica A: Statistical Mechanics and Its Applications, 274 (1999), 190-199. [18] M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamazaki and T. Matsuyama, Colony formation in bacteria: Experiments and modeling, Biofilms, 1 (2004), 305-317. [19] M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modeling of bacterial colony patterns, Physica. A. Stat. Mech. Appl., 282 (2000), 283-303. [20] J. D. Murray, "Murray JD," $1^{st}$, $3^{rd}$ edition, USA, 2002. [21] K. Nowaczyk, A. Juszczak A and F. Domka, Microbiological oxidation of the waste ferrous sulphate, Polish Journal of Environmental Studies, 6 (1999), 409-416. [22] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. [23] P. T. Saunders and M. J. Bazin, On the stability of food chains, J. Theor. Biol., 52 (1975), 121-142. [24] R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats, Geobiology, 8 (2010), 179-190. [25] J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili, PNAS, 98 (2001), 6901-6904. [26] L. Simonsen, Dynamics of plasmid transfer on surfaces, J. General Microbiology, 136 (1990), 1001-1007. [27] R. Tokita, T. Katoh, Y. Maeda, J. I. Wakita, M. Sano, T. Matsuyama and M. Matsushita, Pattern formation of bacterial colonies by Escherichia coli, J. Phys. Soc. Jpn., 78 (2009), 074005 (6 pages). [28] Y. Wei, X. Wang, J. Liu, L. Nememan, A. H. Singh, H. Howie and B. R. Levin, The populatiion and evolutionary of bacteria in physically structured habitats: The adaptive virtues of motility, PNAS, 108 (2011), 4047-4052. [29] J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems," 2 1998. [30] P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria, Current Biology, 8 (1998), 408-411. [31] A. Ishihara, J. E. Segall, S. M. Block and H. L Berg, Coordination of flagella on filgmentous cells of Escherichia Coli, J. Bacteriology, 155 (1983), 228-237. [32] B. L. Taylor and D. E. Koshlard, Reversal of flafella rotation in Monotrichous and Peritrichous bacteria: Generation of changes in direction, J. Bacteriology, 119 (1974), 640-642.
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