Journal of Mathematical Biology

, Volume 70, Issue 1–2, pp 1–44 | Cite as

Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

  • Chuan Xue


Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak–Keller–Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli  populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.


Multiscale analysis Bacterial chemotaxis Cell signaling  Keller–Segel Velocity jump process 

Mathematics Subject Classification (2000)

92B05 92C17 92D25 35Q80 41A60 60J75 



CX would like to thank Professor Hans Othmer and Dr. Radek Erban for insightful discussions and comments on the paper. CX would like to dedicate this paper to Professor Hans G. Othmer’s 70th birthday.


  1. Adler J (1966) Chemotaxis in bacteria. Science 153:708–716CrossRefGoogle Scholar
  2. Armitage JP, Pitta TP, Vigeant MA, Packer HL, Ford RM (1999) Transformations in flagellar structure of Rhodobacter sphaeroides and possible relationship to changes in swimming speed. J Bacteriol 181(16):4825–4833Google Scholar
  3. Aminzare Z, Sontag ED (2013) Remarks on a population-level model of chemotaxis: advection-diffusion approximation and simulations. arXiv:1302.2605 (preprint)Google Scholar
  4. Berg HC (1975) How bacteria swim. Sci Am 233:36–44CrossRefGoogle Scholar
  5. Berg HC (1983) Random walks in biology. Princeton University Press, PrincetonGoogle Scholar
  6. Berg HC (2000) Motile behavior of bacteria. Phys Today 53(1):24–29CrossRefGoogle Scholar
  7. Berg HC, Brown D (1972) Chemotaxis in Escherichia coli analyzed by three-dimensional tracking. Nature 239:502–507CrossRefGoogle Scholar
  8. Bray D, Levin MD, Lipkow K (2007) The chemotactic behavior of computer-based surrogate bacteria. Curr Biol 17(1):12–19CrossRefGoogle Scholar
  9. Briegel A, Li X, Bilwes AM, Hughes KT, Jensen GJ, Crane BR (2012) Bacterial chemoreceptor arrays are hexagonally packed trimers of receptor dimers networked by rings of kinase and coupling proteins. Proc Natl Acad Sci USA 109(10):3766–3771CrossRefGoogle Scholar
  10. Budrene EO, Berg HC (1991) Complex patterns formed by motile cells of Escherichia coli. Nature 349(6310):630–633CrossRefGoogle Scholar
  11. Budrene EO, Berg HC (1995) Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376(6535):49–53CrossRefGoogle Scholar
  12. Butler SM, Camilli A (2004) Both chemotaxis and net motility greatly influence the infectivity of Vibrio cholerae. Proc Natl Acad Sci USA 101(14):5018–5023. doi: 10.1073/pnas.0308052101 CrossRefGoogle Scholar
  13. Chen KC, Cummings PT, Ford RM (1998) Perturbation expansion of alt’s cell balance equations reduces to Segel’s one-dimensional equations for shallow chemoattractant gradients. SIAM J Appl Math 59(1):35–57CrossRefMathSciNetGoogle Scholar
  14. Cluzel P, Surette M, Leibler S (2000) An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science 287:1652–1655CrossRefGoogle Scholar
  15. Dallon JC, Othmer HG (1997) A discrete cell model with adaptive signalling for aggregation of Dictyostelium discoideum. Philos Trans R Soc Lond B Biol Sci 352(1351):391–417CrossRefGoogle Scholar
  16. Duffy KJ, Ford RM (1997) Turn angle and run time distributions characterize swimming behavior for Pseudomonas putida. J Bacteriol 179(4):1428–1430Google Scholar
  17. Erban R, Othmer HG (2004) From individual to collective behavior in bacterial chemotaxis. SIAM J Appl Math 65(2):361–391CrossRefMathSciNetzbMATHGoogle Scholar
  18. Erban R, Othmer H (2005) From signal transduction to spatial pattern formation in E. coli: a paradigm for multiscale modeling in biology. Multiscale Model Simul 3(2):362–394Google Scholar
  19. Franz B, Erban R (2013) Hybrid modelling of individual movement and collective behaviour. In: Dispersal, individual movement and spatial ecology. Springer, Berlin, pp 129–157Google Scholar
  20. Franz B, Xue C, Painter K, Erban R (2013) Travelling waves in hybrid chemotaxis models. Bull Math Biol. doi: 10.1007/s11538-013-9924-4
  21. Friedl P, Gilmour D (2009) Collective cell migration in morphogenesis, regeneration and cancer. Nat Rev Mol Cell Biol 10(7):445–457. doi: 10.1038/nrm2720 CrossRefGoogle Scholar
  22. Gyrya V, Aranson IS, Berlyand LV, Karpeev D (2010) A model of hydrodynamic interaction between swimming bacteria. Bull Math Biol 72(1):148–183. doi: 10.1007/s11538-009-9442-6 CrossRefMathSciNetzbMATHGoogle Scholar
  23. Hazelbauer GL (2012) Bacterial chemotaxis: the early years of molecular studies. Annu Rev Microbiol 66:285–303. doi: 10.1146/annurev-micro-092611-150120 CrossRefGoogle Scholar
  24. Hillen T, Othmer HG (2000) The diffusion limit of transport equations derived from velocity-jump processes. SIAM J Appl Math 61(3):751–775CrossRefMathSciNetzbMATHGoogle Scholar
  25. Hillen T, Painter KJ (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58(1–2):183–217. doi: 10.1007/s00285-008-0201-3 CrossRefMathSciNetzbMATHGoogle Scholar
  26. Hilpert M (2005) Lattice–Boltzmann model for bacterial chemotaxis. J Math Biol 51(3):302–332CrossRefMathSciNetzbMATHGoogle Scholar
  27. Horstmann D (2003) From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. Jahresbericht der DMV 105(3):103–165MathSciNetzbMATHGoogle Scholar
  28. Hugdahl MB, Beery JT, Doyle MP (1988) Chemotactic behavior of Campylobacter jejuni. Infect Immun 56(6):1560–1566Google Scholar
  29. Jin T, Xu X, Hereld D (2008) Chemotaxis, chemokine receptors and human disease. Cytokine 44(1):1–8. doi: 10.1016/j.cyto.2008.06.017 CrossRefGoogle Scholar
  30. Kalinin YV, Jiang L, Tu Y, Wu M (2009) Logarithmic sensing in Escherichia coli bacterial chemotaxis. Biophys J 96(6):2439–2448. doi: 10.1016/j.bpj.2008.10.027 CrossRefGoogle Scholar
  31. Kaya T, Koser H (2012) Direct upstream motility in Escherichia coli. Biophys J 102(7):1514–1523. doi: 10.1016/j.bpj.2012.03.001 CrossRefGoogle Scholar
  32. Keller EF, Segel LA (1970) Initiation of slime mold aggregation viewed as an instability. J Theor Biol 26:399–415CrossRefzbMATHGoogle Scholar
  33. Keller EF, Segel LA (1971a) Model for chemotaxis. J Theor Biol 30:225–234CrossRefzbMATHGoogle Scholar
  34. Keller EF, Segel LA (1971b) Traveling bands of chemotactic bacteria: a theoretical analysis. J Theor Biol 30:235–248CrossRefzbMATHGoogle Scholar
  35. Kim Y, Stolarska MA, Othmer HG (2011) The role of the microenvironment in tumor growth and invasion. Prog Biophys Mol Biol 106(2):353–379. doi: 10.1016/j.pbiomolbio.2011.06.006 CrossRefGoogle Scholar
  36. Kojadinovic M, Armitage JP, Tindall MJ, Wadhams GH (2013) Response kinetics in the complex chemotaxis signalling pathway of Rhodobacter sphaeroides. J R Soc Interface 10(81):20121001. doi: 10.1098/rsif.2012.1001 Google Scholar
  37. Koshland DE (1980) Bacterial chemotaxis as a model behavioral system. Raven Press, New YorkGoogle Scholar
  38. Liu J, Hu B, Morado DR, Jani S, Manson MD, Margolin W (2012) Molecular architecture of chemoreceptor arrays revealed by cryoelectron tomography of Escherichia coli minicells. Proc Natl Acad Sci USA 109(23):E1481–E1488. doi: 10.1073/pnas.1200781109 CrossRefGoogle Scholar
  39. Long W, Hilpert M (2008) Lattice–Boltzmann modeling of contaminant degradation by chemotactic bacteria: exploring the formation and movement of bacterial bands. Water Resour Res 44(9):W09415Google Scholar
  40. Marcos M, Fu HC, Powers TR, Stocker R (2012) Bacterial rheotaxis. Proc Natl Acad Sci USA 109(13):4780–4785. doi: 10.1073/pnas.1120955109 CrossRefGoogle Scholar
  41. Marx RB, Aitken MD (2000) A material-balance approach for modeling bacterial chemotaxis to a consumable substrate in the capillary assay. Biotechnol Bioeng 68(3):308–315CrossRefGoogle Scholar
  42. Nicolau JDVV, Armitage JP, Maini PK (2009) Directional persistence and the optimality of run-and-tumble chemotaxis. Comput Biol Chem 33(4):269–274. doi: 10.1016/j.compbiolchem.2009.06.003 CrossRefMathSciNetGoogle Scholar
  43. Othmer HG, Hillen T (2002) The diffusion limit of transport equations II: chemotaxis equations. SIAM J Appl Math 62:1222–1250CrossRefMathSciNetzbMATHGoogle Scholar
  44. Othmer H, Xue C (2013) The mathematical analysis of biological aggregation and dispersal: progress, problems and perspectives. In: Lewis M, Maini P, Petrovskii S (eds) Dispersal, individual movement and spatial ecology: a mathematical perspective. Springer, BerlinGoogle Scholar
  45. Othmer HG, Dunbar SR, Alt W (1988) Models of dispersal in biological systems. J Math Biol 26(3):263–298CrossRefMathSciNetzbMATHGoogle Scholar
  46. Othmer HG, Painter KJ, Umulis D, Xue C (2009) The intersection of theory and application in elucidating pattern formation in developmental biology. Math Model Nat Phenom 4(4):3–82CrossRefMathSciNetzbMATHGoogle Scholar
  47. Othmer HG, Xin X, Xue C (2013) Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation. Int J Mol Sci 14(5):9205–9248CrossRefGoogle Scholar
  48. O’Toole R, Lundberg S, Fredriksson SA, Jansson A, Nilsson B, Wolf-Watz H (1999) The chemotactic response of Vibrio anguillarum to fish intestinal mucus is mediated by a combination of multiple mucus components. J Bacteriol 181(14):4308–4317Google Scholar
  49. Pandey G, Jain RK (2002) Bacterial chemotaxis toward environmental pollutants: role in bioremediation. Appl Environ Microbiol 68(12):5789–5795CrossRefGoogle Scholar
  50. Papanicolaou GC (1975) Asymptotic analysis of transport processes. Bull Am Math Soc 81(2):330–393CrossRefMathSciNetzbMATHGoogle Scholar
  51. Patlak CS (1953) Random walk with persistence and external bias. Bull Math Biophys 15:311–338CrossRefMathSciNetzbMATHGoogle Scholar
  52. Pittman MS, Goodwin M, Kelly DJ (2001) Chemotaxis in the human gastric pathogen Helicobacter pylori: different roles for chew and the three chev paralogues, and evidence for chev2 phosphorylation. Microbiology 147(Pt 9):2493–2504Google Scholar
  53. Porter SL, Wadhams GH, Armitage JP (2008) Rhodobacter sphaeroides: complexity in chemotactic signalling. Trends Microbiol 16(6):251–260. doi: 10.1016/j.tim.2008.02.006 CrossRefGoogle Scholar
  54. Potomkin M, Gyrya V, Aranson I, Berlyand L (2013) Collision of microswimmers in a viscous fluid. Phys Rev E Stat Nonlin Soft Matter Phys 87(5–1):053005Google Scholar
  55. Rao CV, Kirby JR, Arkin AP (2004) Design and diversity in bacterial chemotaxis: a comparative study in Escherichia coli and Bacillus subtilis. PLoS Biol 2(2):E49. doi: 10.1371/journal.pbio.0020049 CrossRefGoogle Scholar
  56. Rao CV, Glekas GD, Ordal GW (2008) The three adaptation systems of Bacillus subtilis chemotaxis. Trends Microbiol 16(10):480–487. doi: 10.1016/j.tim.2008.07.003 CrossRefGoogle Scholar
  57. Rivero MA, Tranquillo RT, Buettner HM, Lauffenburger DA (1989) Transport models for chemotactic cell populations based on individual cell behavior. Chem Eng Sci 44(12):2881–2897CrossRefGoogle Scholar
  58. Ryan SD, Haines BM, Berlyand L, Ziebert F, Aranson IS (2011) Viscosity of bacterial suspensions: hydrodynamic interactions and self-induced noise. Phys Rev E Stat Nonlin Soft Matter Phys 82(5 Pt 1):050904Google Scholar
  59. Saragosti J, Calvez V, Bournaveas N, Perthame B, Buguin A, Silberzan P (2011) Directional persistence of chemotactic bacteria in a traveling concentration wave. Proc Natl Acad Sci USA 108(39):16235–16240CrossRefGoogle Scholar
  60. Simons JE, Milewski PA (2011) The volcano effect in bacterial chemotaxis. Math Comput Model 53(7–8):1374–1388CrossRefMathSciNetzbMATHGoogle Scholar
  61. Singh R, Paul D, Jain RK (2006) Biofilms: implications in bioremediation. Trends Microbiol 14(9):389–397. doi: 10.1016/j.tim.2006.07.001 CrossRefGoogle Scholar
  62. Sze CW, Zhang K, Kariu T, Pal U, Li C (2012) Borrelia burgdorferi needs chemotaxis to establish infection in mammals and to accomplish its enzootic cycle. Infect Immun 80(7):2485–2492. doi: 10.1128/IAI.00145-12 CrossRefGoogle Scholar
  63. Tindall MJ, Maini PK, Porter SL, Armitage JP (2008a) Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations. Bull Math Biol 70(6):1570–1607. doi: 10.1007/s11538-008-9322-5 CrossRefMathSciNetzbMATHGoogle Scholar
  64. Tindall MJ, Porter SL, Maini PK, Gaglia G, Armitage JP (2008b) Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell. Bull Math Biol 70(6):1525–1569. doi: 10.1007/s11538-008-9321-6 CrossRefMathSciNetzbMATHGoogle Scholar
  65. Tu Y (2013) Quantitative modeling of bacterial chemotaxis: Signal amplification and accurate adaptation. Annu Rev Biophys. doi: 10.1146/annurev-biophys-083012-130358
  66. Tu Y, Shimizu TS, Berg HC (2008) Modeling the chemotactic response of Escherichia coli to time-varying stimuli. Proc Natl Acad Sci USA 105(39):14855–14860. doi: 10.1073/pnas.0807569105 CrossRefGoogle Scholar
  67. Tyson R, Lubkin SR, Murray JD (1999a) A minimal mechanism for bacterial pattern formation. Proc R Soc Lond B 266:299–304CrossRefGoogle Scholar
  68. Tyson R, Lubkin SR, Murray JD (1999b) Model and analysis of chemotactic bacterial patterns in a liquid medium. J Math Biol 38:359–375CrossRefMathSciNetzbMATHGoogle Scholar
  69. Tyson JJ, Chen KC, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 15(2):221–231CrossRefGoogle Scholar
  70. Wang ZA (2013) Mathematics of traveling waves in chemotaxis. DCDS-B 18:601–641CrossRefGoogle Scholar
  71. Williams SM, Chen YT, Andermann TM, Carter JE, McGee DJ, Ottemann KM (2007) Helicobacter pylori chemotaxis modulates inflammation and bacterium-gastric epithelium interactions in infected mice. Infect Immun 75(8):3747–3757. doi: 10.1128/IAI.00082-07 CrossRefGoogle Scholar
  72. Woodward D, Tyson R, Myerscough M, Murray J, Budrene E, Berg H (1995) Spatio-temporal patterns generated by Salmonella typhimurium. Biophys J 68:2181–2189CrossRefGoogle Scholar
  73. Xin X (2010) Mathematical models of bacterial chemotaxis. Ph.D. thesis, University of MinnesotaGoogle Scholar
  74. Xin X, Othmer HG (2012) A “trimer of dimers”-based model for the chemotactic signal transduction network in bacterial chemotaxis. Bull Math Biol 74(10):2339–2382. doi: 10.1007/s11538-012-9756-7 CrossRefMathSciNetzbMATHGoogle Scholar
  75. Xue C, Othmer HG (2009) Multiscale models of taxis-driven patterning in bacterial populations. SIAM J Appl Math 70(1):133–167CrossRefMathSciNetzbMATHGoogle Scholar
  76. Xue C, Othmer HG, Erban R (2009) From individual to collective behavior of unicellular organisms: recent results and open problems. In: Multiscale phenomena in biology: proceedings of the 2nd conference on mathematics and biology. AIP conference proceedings, vol 1167(1), pp 3–14Google Scholar
  77. Xue C, Budrene EO, Othmer HG (2011), Radial and spiral stream formation in Proteus mirabilis colonies. PLoS Comput Biol 7(12):e1002332Google Scholar
  78. Zhu X, Si G, Deng N, Ouyang Q, Wu T, He Z, Jiang L, Luo C, Tu Y (2012) Frequency-dependent Escherichia coli chemotaxis behavior. Phys Rev Lett 108(12):128101CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA

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