Travelling Waves in Hybrid Chemotaxis Models

Bulletin of Mathematical Biology

Volumn 76, Issue 2, 2014, Pages 377-400

  Franz, Benjamin ^a   Xue, Chuan ^b   Painter, Kevin J ^c   Erban, Radek ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b OHIO STATE UNIVERSITY   (United States)

^c HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Bacterial chemotaxis; Hybrid model; Travelling wave

Indexed keywords

BACTERIAL PHENOMENA AND FUNCTIONS; BIOLOGICAL MODEL; CHEMOTAXIS; CYTOLOGY; ESCHERICHIA COLI; GROWTH, DEVELOPMENT AND AGING; MATHEMATICAL PHENOMENA; PHYSIOLOGY; SIGNAL TRANSDUCTION;

BACTERIAL PHYSIOLOGICAL PHENOMENA; CHEMOTAXIS; ESCHERICHIA COLI; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; SIGNAL TRANSDUCTION;

EID: 84893909321     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-013-9924-4     Document Type: Article

References (42)

1. Adler, J. (1966). Chemotaxis in bacteria. Science, 153, 708-716.
2. Berg, H. (1983). Random walks in biology. Princeton: Princeton University Press.
3. Berg, H. (1975). How bacteria swim. Sci. Am., 233, 36-44.
4. Berg, H., & Brown, D. (1972). Chemotaxis in Esterichia coli analysed by three-dimensional tracking. Nature, 239, 500-504.
5. Bourret, R., Borkovich, K., & Simon, M. (1991). Signal transduction pathways involving protein phosphorylation in prokaryotes. Annu. Rev. Biochem., 60, 401-441.
6. Brenner, M., Levitov, L., & Budrene, E. (1998). Physical mechanisms for chemotactic pattern formation by bacteria. Biophys. J., 74(4), 1677-1693.
7. Budrene, E., & Berg, H. (1991). Complex patterns formed by motile cells of Esterichia coli. Nature, 349, 630-633.
8. Budrene, E., & Berg, H. (1995). Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature, 376, 49-53.
9. Cercignani, C., Illner, R., & Pulvirenti, M. (1994). The mathematical theory of dilute gases. Applied mathematical sciences: Vol. 106. Berlin: Springer.
10. Chavanis, P. (2010). A stochastic Keller-Segel model of chemotaxis. Commun. Nonlinear Sci. Numer. Simul., 15, 60-70.
11. Erban, R. (2005). From individual to collective behaviour in biological systems. Ph. D. Thesis, University of Minnesota.
12. Erban, R., & Haskovec, J. (2012). From individual to collective behaviour of coupled velocity jump processes: a locust example. Kinet. Relat. Models, 5(4), 817-842.
13. Erban, R., & Othmer, H. (2004). From individual to collective behaviour in bacterial chemotaxis. SIAM J. Appl. Math., 65(2), 361-391.
14. Erban, R., & Othmer, H. (2005). From signal transduction to spatial pattern formation in E. coli: a paradigm for multi-scale modeling in biology. Multiscale Model. Simul., 3(2), 362-394.
15. Fisher, R. (1937). The wave of advance of advantageous genes. Ann. Eugen., 7, 355-369.
16. Franz, B., & Erban, R. (2013). Hybrid modelling of individual movement and collective behaviour. In M. Lewis, P. Maini, & S. Petrovskii (Eds.), Dispersal, individual movement and spatial ecology: a mathematical perspective. Berlin: Springer.
17. Gerisch, A., & Painter, K. (2010). Mathematical modelling of cell adhesion and its applications to developmental biology and cancer invasion. In A. Chauviere & L. Preziosi (Eds.), Cell mechanics: from single scale-based models to multiscale modeling (pp. 319-350). Boca Raton: CRC Press. Chapter 12.
18. Guo, Z., Sloot, P., & Tay, J. (2008). A hybrid agent-based approach for modeling microbiological systems. J. Theor. Biol., 255, 163-175.
19. Hillen, T., & Painter, K. (2009). A user's guide to PDE models for chemotaxis. J. Math. Biol., 58, 183-217.
20. Horstmann, D. (2004). From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II. Jahresber. Dtsch. Math.-Ver., 106, 51-69.
21. Keller, E., & Segel, L. (1971). Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol., 30, 235-248.
22. Kennedy, C., & Aris, R. (1980). Traveling waves in a simple population model involving growth and death. Bull. Math. Biol., 42, 397-429.
23. Landman, K., Petter, G., & Newgreen, D. (2003). Chemotactic cellular migration: smooth and discontinuous travelling wave solutions. SIAM J. Appl. Math., 63(5), 1666-1681.
24. Landman, K., Simpson, M., Slater, J., & Newgreen, D. (2005). Diffusive and chemotactic cellular migration: smooth and discontinuous travelling wave solutions. SIAM J. Appl. Math., 65, 1420-1442.
25. Landman, K. A., Simpson, M. J., & Newgreen, D. F. (2007). Mathematical and experimental insights into the development of the enteric nervous system and hirschsprung's disease. Dev. Growth Differ., 49, 277-286.
26. Li, T., & Wang, Z. (2011). Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis. J. Differ. Equ., 250, 1310-1333.
27. Li, T., & Wang, Z. (2012). Steadily propagating waves of a chemotaxis model. Math. Biosci., 240, 161-168.
28. Lui, R., & Wang, Z. (2010). Traveling wave solutions from microscopic to macroscopic chemotaxis models. J. Math. Biol., 61, 739-761.
29. Metcalf, M., Merkin, J., & Scott, S. (1994). Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 447, 155-174.
30. Murray, J. (2002). Mathematical biology. Berlin: Springer.
31. Nadin, G., Perthame, B., & Ryzhik, L. (2008). Traveling waves for the Keller-Segel system with Fisher birth term. Interfaces Free Bound., 10, 517-538.
32. Othmer, H., Dunbar, S., & Alt, W. (1988). Models of dispersal in biological systems. J. Math. Biol., 26, 263-298.
33. Othmer, H., & Schaap, P. (1998). Oscillatory cAMP signaling in the development of Dictyostelium discoideum. Comments Theor. Biol., 5, 175-282.
34. Reitz, R. D. (1981). A study of numerical methods for reaction-diffusion equations. SIAM J. Sci. Stat. Comput., 2, 95-106.
35. Saragosti, J., Calvez, V., Bournaveas, N., Buguin, A., Silberzan, P., & Perthame, B. (2010). Mathematical description of bacterial traveling pulses. PLoS Comput. Biol., 6, e1000890.
36. 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), 16235-16240.
37. Satnoianu, R., Maini, P., Garduno, F., & Armitage, J. (2001). Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete Contin. Dyn. Syst., Ser. B, 1, 339-362.
38. Wang, Z. A. (2013). Mathematics of traveling waves in chemotaxis-review paper. Discrete Contin. Dyn. Syst., Ser. B, 13, 601-641.
39. Witte, M. B., & Barbul, A. (1997). General principles of wound healing. Surg. Clin. North Am., 77, 509-528.
40. Xue, C., Budrene, E., & Othmer, H. (2011a). Radial and spiral streams in proteus mirabilis colonies. PLoS Comput. Biol., 7(12), e1002332.
41. Xue, C., Hwang, H., Painter, K., & Erban, R. (2011b). Travelling waves in hyperbolic chemotaxis equations. Bull. Math. Biol., 73(8), 1695-1733.
42. Xue, C., & Othmer, H. (2009). Multiscale models of taxis-driven patterning in bacterial populations. SIAM J. Appl. Math., 70(1), 133-167.