# 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., (). Google Scholar [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. doi: 10.1137/S0036139997325497. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates,, J. Phys. Soc. Jpn., 58 (1989), 3875. Google Scholar [6] H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient,, J. Phys. Soc. Jpn., 60 (1991), 88. Google Scholar [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. Google Scholar [8] D. P. Hzder, R. Hemmerbach and M. Lebert, Gravity and the bacterial unicellular organisms,, Developmental and Cell Biology Series, 40 (2005). Google Scholar [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. doi: 10.1016/S0092-8240(80)80057-7. Google Scholar [10] E. Keller, Mathematical aspects of bacterial chemotaxis,, Antibiotics and Chemotherapy, 19 (1974), 79. Google Scholar [11] F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Micro. Biol., 16 (1988), 115. Google Scholar [12] E. Khain, L. M. Sander and A. M. Stein, A model for glioma growth,, Research Article, 11 (2005), 53. doi: 10.1002/cplx.20108. Google Scholar [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. Google Scholar [14] D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations,, Micro. Ecol., 7 (1981), 207. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modeling of bacterial colony patterns,, Physica. A. Stat. Mech. Appl., 282 (2000), 283. Google Scholar [20] J. D. Murray, "Murray JD,", $1^{st}$, (2002). Google Scholar [21] K. Nowaczyk, A. Juszczak A and F. Domka, Microbiological oxidation of the waste ferrous sulphate,, Polish Journal of Environmental Studies, 6 (1999), 409. Google Scholar [22] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. Google Scholar [23] P. T. Saunders and M. J. Bazin, On the stability of food chains,, J. Theor. Biol., 52 (1975), 121. Google Scholar [24] R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats,, Geobiology, 8 (2010), 179. Google Scholar [25] J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili,, PNAS, 98 (2001), 6901. Google Scholar [26] L. Simonsen, Dynamics of plasmid transfer on surfaces,, J. General Microbiology, 136 (1990), 1001. Google Scholar [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). Google Scholar [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. Google Scholar [29] J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems,", 2 1998., 2 (1998). Google Scholar [30] P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria,, Current Biology, 8 (1998), 408. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] S.Asei, B. Byers, A. Eng, N. James and J. Leto, "Bacterial Chemostat Model,", 2007., (). Google Scholar [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. doi: 10.1137/S0036139997325497. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates,, J. Phys. Soc. Jpn., 58 (1989), 3875. Google Scholar [6] H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient,, J. Phys. Soc. Jpn., 60 (1991), 88. Google Scholar [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. Google Scholar [8] D. P. Hzder, R. Hemmerbach and M. Lebert, Gravity and the bacterial unicellular organisms,, Developmental and Cell Biology Series, 40 (2005). Google Scholar [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. doi: 10.1016/S0092-8240(80)80057-7. Google Scholar [10] E. Keller, Mathematical aspects of bacterial chemotaxis,, Antibiotics and Chemotherapy, 19 (1974), 79. Google Scholar [11] F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Micro. Biol., 16 (1988), 115. Google Scholar [12] E. Khain, L. M. Sander and A. M. Stein, A model for glioma growth,, Research Article, 11 (2005), 53. doi: 10.1002/cplx.20108. Google Scholar [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. Google Scholar [14] D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations,, Micro. Ecol., 7 (1981), 207. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modeling of bacterial colony patterns,, Physica. A. Stat. Mech. Appl., 282 (2000), 283. Google Scholar [20] J. D. Murray, "Murray JD,", $1^{st}$, (2002). Google Scholar [21] K. Nowaczyk, A. Juszczak A and F. Domka, Microbiological oxidation of the waste ferrous sulphate,, Polish Journal of Environmental Studies, 6 (1999), 409. Google Scholar [22] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. Google Scholar [23] P. T. Saunders and M. J. Bazin, On the stability of food chains,, J. Theor. Biol., 52 (1975), 121. Google Scholar [24] R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats,, Geobiology, 8 (2010), 179. Google Scholar [25] J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili,, PNAS, 98 (2001), 6901. Google Scholar [26] L. Simonsen, Dynamics of plasmid transfer on surfaces,, J. General Microbiology, 136 (1990), 1001. Google Scholar [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). Google Scholar [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. Google Scholar [29] J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems,", 2 1998., 2 (1998). Google Scholar [30] P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria,, Current Biology, 8 (1998), 408. Google Scholar [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. Google Scholar [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. Google Scholar
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