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

show all references

##### 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. 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. [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. [5] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates,, J. Phys. Soc. Jpn., 58 (1989), 3875. [6] H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient,, J. Phys. Soc. Jpn., 60 (1991), 88. [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. [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. doi: 10.1016/S0092-8240(80)80057-7. [10] E. Keller, Mathematical aspects of bacterial chemotaxis,, Antibiotics and Chemotherapy, 19 (1974), 79. [11] F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Micro. Biol., 16 (1988), 115. [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. [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. [14] D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations,, Micro. Ecol., 7 (1981), 207. [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. [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. [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. [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. [19] M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modeling of bacterial colony patterns,, Physica. A. Stat. Mech. Appl., 282 (2000), 283. [20] J. D. Murray, "Murray JD,", $1^{st}$, (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. [22] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. [23] P. T. Saunders and M. J. Bazin, On the stability of food chains,, J. Theor. Biol., 52 (1975), 121. [24] R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats,, Geobiology, 8 (2010), 179. [25] J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili,, PNAS, 98 (2001), 6901. [26] L. Simonsen, Dynamics of plasmid transfer on surfaces,, J. General Microbiology, 136 (1990), 1001. [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). [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. [29] J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems,", 2 1998., 2 (1998). [30] P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria,, Current Biology, 8 (1998), 408. [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. [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.
 [1] Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 [2] Hisashi Okamoto, Takashi Sakajo, Marcus Wunsch. Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3155-3170. doi: 10.3934/dcds.2014.34.3155 [3] Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417 [4] Roger Lui, Hirokazu Ninomiya. Traveling wave solutions for a bacteria system with density-suppressed motility. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 931-940. doi: 10.3934/dcdsb.2018213 [5] Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 [6] Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265 [7] Bingtuan Li. Some remarks on traveling wave solutions in competition models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 389-399. doi: 10.3934/dcdsb.2009.12.389 [8] Marek Fila, Juan-Luis Vázquez, Michael Winkler. A continuum of extinction rates for the fast diffusion equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1129-1147. doi: 10.3934/cpaa.2011.10.1129 [9] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [10] Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075 [11] Yoshikazu Giga, Robert V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 509-535. doi: 10.3934/dcds.2011.30.509 [12] Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319 [13] Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119 [14] Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116 [15] Wei-Ming Ni, Masaharu Taniguchi. Traveling fronts of pyramidal shapes in competition-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 379-395. doi: 10.3934/nhm.2013.8.379 [16] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [17] Matthew S. Mizuhara, Peng Zhang. Uniqueness and traveling waves in a cell motility model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2811-2835. doi: 10.3934/dcdsb.2018315 [18] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019061 [19] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 [20] Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

2018 Impact Factor: 1.313