Advertisement

Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process

  • Ramón G. PlazaEmail author
Article
  • 114 Downloads

Abstract

This paper is devoted to the justification of the macroscopic, mean-field nutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva et al. (Phys A 392:5644–5662, 2013) to model the complex spatio-temporal dynamics exhibited by the bacterium Bacillus subtilis during experiments run in vitro. This justification is based on a microscopic description of the movement of individual cells whose changes in velocity (in both speed and orientation) obey a velocity jump process governed by a transport equation of Boltzmann type. For that purpose, the asymptotic method introduced by Hillen and Othmer (SIAM J Appl Math 61:751–775, 2000; SIAM J Appl Math 62:1222–1250, 2002) is applied, which consists of the computation of the leading order term in a regular Hilbert expansion for the solution to the transport equation, under an appropriate parabolic scaling and a first order perturbation of the turning rate of Schnitzer type (Schnitzer in Phys Rev E 48:2553–2568, 1993). The resulting parabolic limit equation at leading order for the bacterial cell density recovers the degenerate nonlinear cross diffusion term and the associated chemotactic drift appearing in the original system of equations. Although the bacterium B. subtilis is used as a prototype, the method and results apply in more generality.

Keywords

Chemotaxis Degenerate diffusion Velocity jump processes Transport equations 

Mathematics Subject Classification

92C17 60J75 35K65 

Notes

Acknowledgements

The author warmly thanks Thomas Hillen and Michael Winkler for enlightening conversations. This research was partially supported by DGAPA-UNAM, program PAPIME, Grant PE-104116.

References

  1. Alt W (1980) Biased random walk models for chemotaxis and related diffusion approximations. J Math Biol 9(2):147–177MathSciNetzbMATHGoogle Scholar
  2. Aronson DG (1980) Density-dependent interaction-diffusion systems. In: Stewart WE, Ray WH, Conley CC (eds) Dynamics and modelling of reactive systems (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wisconsin, 1979), Publication of the Mathematics Research Center, University of Wisconsin, vol 44. Academic Press, New York, London, pp 161–176Google Scholar
  3. Arouh S, Levine H (2000) Nutrient chemotaxis suppression of a diffusive instability in bacterial colony dynamics. Phys Rev E 62(1):1444–1447Google Scholar
  4. Ben-Jacob E, Levine H (2006) Self-engineering capabilities of bacteria. J R Soc Interface 3(6):197–214Google Scholar
  5. Ben-Jacob E, Cohen I, Levine H (2000) Cooperative self-organization of microorganisms. Adv Phys 49(4):395–554Google Scholar
  6. Berg HC (1983) Random walks in biology. Princeton University Press, PrincetonGoogle Scholar
  7. Berg HC, Brown DA (1972) Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239:500–504Google Scholar
  8. Block SM, Segall JE, Berg HC (1983) Adaptation kinetics in bacterial chemotaxis. J Bacteriol 154(1):312–323Google Scholar
  9. Butanda JA, Málaga C, Plaza RG (2017) On the stabilizing effect of chemotaxis on bacterial aggregation patterns. Appl Math Nonlinear Sci 2(1):157–172zbMATHMathSciNetGoogle Scholar
  10. Chalub FACC, Markowich PA, Perthame B, Schmeiser C (2004) Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh Math 142(1–2):123–141MathSciNetzbMATHGoogle Scholar
  11. Chalub F, Dolak-Struss Y, Markowich P, Oelz D, Schmeiser C, Soreff A (2006) Model hierarchies for cell aggregation by chemotaxis. Math Models Methods Appl Sci 16(7, suppl):1173–1197MathSciNetzbMATHGoogle Scholar
  12. Codling EA, Hill NA, Pitchford JW, Simpson SD (2004) Random walk models for the movement and recruitment of reef fish larvae. Mar Ecol Prog Ser 279:215–224Google Scholar
  13. Cohen I, Czirók A, Ben-Jacob E (1996) Chemotactic-based adaptive self organization during colonial development. Phys A 233(3–4):678–698Google Scholar
  14. Dai S, Du Q (2016) Weak solutions for the Cahn–Hilliard equation with degenerate mobility. Arch Ration Mech Anal 219(3):1161–1184MathSciNetzbMATHGoogle Scholar
  15. Ellis RS (1973) Chapman–Enskog–Hilbert expansion for a Markovian model of the Boltzmann equation. Commun Pure Appl Math 26(3):327–359MathSciNetzbMATHGoogle Scholar
  16. Ellis RJ (2001) Macromolecular crowding: obvious but underappreciated. Trends Biochem Sci 26(10):597–604Google Scholar
  17. Erban R, Othmer HG (2004) From individual to collective behavior in bacterial chemotaxis. SIAM J Appl Math 65(2):361–391MathSciNetzbMATHGoogle Scholar
  18. Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York, London, SydneyzbMATHGoogle Scholar
  19. Galanti M, Fanelli D, Piazza F (2016) Macroscopic transport equations in many-body systems from microscopic exclusion processes in disordered media: a review. Front Phys 4:33Google Scholar
  20. Gilding BH, Kersner R (1996) A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations. Proc R Soc Edinb Sect A 126(4):739–767MathSciNetzbMATHGoogle Scholar
  21. Golding I, Kozlovsky Y, Cohen I, Ben-Jacob E (1998) Studies of bacterial branching growth using reaction-diffusion models for colonial development. Phys A 260(3–4):510–554Google Scholar
  22. Golse F, Lions PL, Bt Perthame, Sentis R (1988) Regularity of the moments of the solution of a transport equation. J Funct Anal 76(1):110–125MathSciNetzbMATHGoogle Scholar
  23. Gurtin ME, MacCamy RC (1977) On the diffusion of biological populations. Math Biosci 33(1–2):35–49MathSciNetzbMATHGoogle Scholar
  24. Habetler GJ, Matkowsky BJ (1975) Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation. J Math Phys 16(4):846–854MathSciNetzbMATHGoogle Scholar
  25. Hillen T (2003) Transport equations with resting phases. Eur J Appl Math 14(5):613–636MathSciNetzbMATHGoogle Scholar
  26. Hillen T (2004) On the $L^2$-moment closure of transport equations: the Cattaneo approximation. Discrete Cont Dyn Syst Ser B 4(4):961–982MathSciNetzbMATHGoogle Scholar
  27. Hillen T, Othmer HG (2000) The diffusion limit of transport equations derived from velocity-jump processes. SIAM J Appl Math 61(3):751–775MathSciNetzbMATHGoogle Scholar
  28. Hillen T, Painter KJ (2013) Transport and anisotropic diffusion models for movement in oriented habitats. In: Lewis MA, Maini PK, Petrovskii SV (eds) Dispersal, individual movement and spatial ecology. Lecture notes in mathematics, vol 2071. Springer, Heidelberg, pp 177–222Google Scholar
  29. Hillesdon AJ, Pedley TJ, Kessler JO (1995) The development of concentration gradients in a suspension of chemotactic bacteria. Bull Math Biol 57(2):299–344zbMATHGoogle Scholar
  30. Ito M, Terahara N, Fujinami S, Krulwich TA (2005) Properties of motility in bacillus subtilis powered by the H+-coupled MotAB flagellar stator, Na+-coupled MotPS or hybrid stators MotAS or MotPB. J Mol Biol 352(2):396–408Google Scholar
  31. Kato T (1980) Perturbation theory for linear operators. Classics in mathematics, 2nd edn. Springer, New YorkGoogle Scholar
  32. Kawasaki K, Mochizuki A, Matsushita M, Umeda T, Shigesada N (1997) Modeling spatio-temporal patterns generated by Bacillus subtilis. J Theor Biol 188(2):177–185Google Scholar
  33. Kearns DB, Losick R (2003) Swarming motility in undomesticated Bacillus subtilis. Mol Microbiol 49(3):581–590Google Scholar
  34. Keller EF, Segel LA (1971a) Model for chemotaxis. J Theor Biol 30(2):225–234zbMATHGoogle Scholar
  35. Keller EF, Segel LA (1971b) Traveling bands of chemotactic bacteria: a theoretical analysis. J Theor Biol 30(2):235–248zbMATHGoogle Scholar
  36. Lapidus RI, Schiller R (1976) Model for the chemotactic response of a bacterial population. Biophys J 16(7):779–789Google Scholar
  37. Larsen EW, Keller JB (1974) Asymptotic solution of neutron transport problems for small mean free paths. J Math Phys 15(1):75–81MathSciNetGoogle Scholar
  38. Lemou M, Mieussens L (2008) A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit. SIAM J Sci Comput 31(1):334–368MathSciNetzbMATHGoogle Scholar
  39. Leyva JF, Málaga C, Plaza RG (2013) The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion. Phys A 392(22):5644–5662MathSciNetzbMATHGoogle Scholar
  40. Lods B (2005) Semigroup generation properties of streaming operators with noncontractive boundary conditions. Math Comput Modell 42(13):1441–1462MathSciNetzbMATHGoogle Scholar
  41. Méndez V, Campos D, Pagonabarraga I, Fedotov S (2012) Density-dependent dispersal and population aggregation patterns. J Theor Biol 309:113–120MathSciNetGoogle Scholar
  42. Menolascina F, Rusconi R, Fernandez VI, Smriga S, Aminzare Z, Sontag ED, Stocker R (2017) Logarithmic sensing in Bacillus subtilis aerotaxis. NPJ Syst Biol Appl 3:16036Google Scholar
  43. Mesibov R, Ordal GW, Adler J (1973) The range of attractant concentrations for bacterial chemotaxis and the threshold and size of response over this range. J Gen Physiol 62(2):203–223Google Scholar
  44. Murray JD (2002) Mathematical biology I. An introduction, interdisciplinary applied mathematics, vol 17, 3rd edn. Springer, New YorkGoogle Scholar
  45. Myers JH, Krebs CJ (1974) Population cycles in rodents. Sci Am 230:38–46Google Scholar
  46. Ohgiwari M, Matsushita M, Matsuyama T (1992) Morphological changes in growth phenomena of bacterial colony patterns. J Phys Soc Jpn 61(3):816–822Google Scholar
  47. Othmer HG, Hillen T (2002) The diffusion limit of transport equations. II. Chemotaxis equations. SIAM J Appl Math 62(4):1222–1250MathSciNetzbMATHGoogle Scholar
  48. Othmer HG, Xue C (2013) The mathematical analysis of biological aggregation and dispersal: progress, problems and perspectives. In: Lewis MA, Maini PK, Petrovskii SV (eds) Dispersal, individual movement and spatial ecology, Lecture notes in mathematics, vol 2071. Springer, Heidelberg, pp 79–127Google Scholar
  49. Othmer HG, Dunbar SR, Alt W (1988) Models of dispersal in biological systems. J Math Biol 26(3):263–298MathSciNetzbMATHGoogle Scholar
  50. Palczewski A (1992) Velocity averaging for boundary value problems. In: Boffi VC, Bampi F, Toscani G (eds) Nonlinear kinetic theory and mathematical aspects of hyperbolic systems (Rapallo, 1992), series advanced mathematical in applied science, vol 9. World Science Publishing, River Edge, pp 179–186Google Scholar
  51. Patlak CS (1953) Random walk with persistence and external bias. Bull Math Biophys 15:311–338MathSciNetzbMATHGoogle Scholar
  52. 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):2881–2897Google Scholar
  53. Sánchez-Garduño F, Maini PK, Kappos ME (1996) A review on travelling wave solutions of one-dimensional reaction-diffusion equations with non-linear diffusion term. Forma 11(1):45–59MathSciNetzbMATHGoogle Scholar
  54. Schnitzer MJ (1993) Theory of continuum random walks and application to chemotaxis. Phys Rev E (3) 48(4):2553–2568MathSciNetGoogle Scholar
  55. Sengers BG, Please CP, Oreffo RO (2007) Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration. J R Soc Interface 4(17):1107–1117Google Scholar
  56. Sherratt JA (2010) On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion. Math Model Nat Phenom 5(5):64–79MathSciNetzbMATHGoogle Scholar
  57. Shigesada N, Kawasaki K, Teramoto E (1979) Spatial segregation of interacting species. J Theor Biol 79(1):83–99MathSciNetGoogle Scholar
  58. Stroock DW (1974) Some stochastic processes which arise from a model of the motion of a bacterium. Z Wahrscheinlichkeitstheorie und Verw Gebiete 28:303–315MathSciNetzbMATHGoogle Scholar
  59. Tadmor E, Tao T (2007) Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs. Commun Pure Appl Math 60(10):1488–1521MathSciNetzbMATHGoogle Scholar
  60. Winkler M (2014) How far can chemotactic cross-diffusion enforce exceeding carrying capacities? J Nonlinear Sci 24(5):809–855MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoMexico CityMexico

Personalised recommendations