×

zbMATH — the first resource for mathematics

A user’s guide to PDE models for chemotaxis. (English) Zbl 1161.92003
Summary: Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis [E. F. Keller and L. A. Segel, J. Theor. Biol 26, 399–415 (1970; Zbl 1170.92306); ibid. 30, 225–234 (1971; Zbl 1170.92307)] has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations.
One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist.
We explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

MSC:
92C17 Cell movement (chemotaxis, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65C20 Probabilistic models, generic numerical methods in probability and statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allegretto W., Xie H., Yang S.: Properties of solutions for a chemotaxis system. J. Math. Biol. 35, 949–966 (1997) · Zbl 0892.92009
[2] Alt W.: Biased random walk model for chemotaxis and related diffusion approximation. J. Math. Biol. 9, 147–177 (1980) · Zbl 0434.92001
[3] Alt W., Lauffenburger D.A.: Transient behavior of a chemotaxis system modelling certain types of tissue inflammation. J. Math. Biol. 24(6), 691–722 (1987) · Zbl 0609.92020
[4] Baker M.D., Wolanin P.M., Stock J.B.: Signal transduction in bacterial chemotaxis. Bioessays 28(1), 9–22 (2006)
[5] Balding D., McElwain D.L.: A mathematical model of tumour-induced capillary growth. J. Theor. Biol. 114(1), 53–73 (1985)
[6] Biler P.: Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8(2), 715–743 (1998) · Zbl 0913.35021
[7] Biler P.: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9(1), 347–359 (1999) · Zbl 0941.35009
[8] Boon J.P., Herpigny B.: Model for chemotactic bacterial bands. Bull. Math. Biol. 48(1), 1–19 (1986) · Zbl 0582.92012
[9] Budd C.J., Carretero-Gonzdflez R., Russell R.D.: Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202(2), 463–487 (2005) · Zbl 1063.65096
[10] Budick S.A., Dickinson M.H.: Free-flight responses of Drosophila melanogaster to attractive odors. J. Exp. Biol. 209(15), 3001–3017 (2006)
[11] Budrene E.O., Berg H.C.: Complex patterns formed by motile cells of Escherichia coli. Nature 349(6310), 630–633 (1991)
[12] Budrene E.O., Berg H.C.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376(6535), 49–53 (1995)
[13] Burger, M., Di Francesco, M., Dolak-Struss, Y.: The Keller-Segel model for chemotaxis: linear vs. nonlinear diffusion. SIAM J. Math. Anal. (2008) (to appear) · Zbl 1114.92008
[14] Byrne H.M., Cave G., McElwain D.L.: The effect of chemotaxis and chemokinesis on leukocyte locomotion: a new interpretation of experimental results. IMA J. Math. Appl. Med. Biol. 15(3), 235–256 (1998) · Zbl 0908.92018
[15] Byrne H.M., Owen M.R.: A new interpretation of the Keller–Segel model based on multiphase modelling. J. Math. Biol. 49, 604–626 (2004) · Zbl 1055.92004
[16] Chaplain M.A.: Mathematical modelling of angiogenesis. J. Neurooncol. 50(1–2), 37–51 (2000)
[17] Chaplain M.A.J., Stuart A.M.: A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168 (1993) · Zbl 0783.92019
[18] Childress S., Percus J.K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981) · Zbl 0481.92010
[19] Condeelis J., Singer R.H., Segall J.E.: The great escape: when cancer cells hijack the genes for chemotaxis and motility. Annu. Rev. Cell Dev. Biol. 21, 695–718 (2005)
[20] Corrias L., Perthame B., Zaag H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004) · Zbl 1115.35136
[21] Dahlquist F.W., Lovely P., Koshland D.E.: Quantitative analysis of bacterial migration in chemotaxis. Nat. New Biol. 236, 120–123 (1972)
[22] Dallon J.C., Othmer H.G.: A discrete cell model with adaptive signalling for aggregation of dictyostelium discoideum. Philos. Trans. R. Soc. B 352, 391–417 (1997)
[23] Dkhil F.: Singular limit of a degenerate chemotaxis-fisher equation. Hiroshima Math. J. 34, 101–115 (2004) · Zbl 1063.35094
[24] Dolak Y., Hillen T.: Cattaneo models for chemotaxis, numerical solution and pattern formation. J. Math. Biol. 46(2), 153–170 (2003) · Zbl 1020.92004
[25] Dolak Y., Schmeiser C.: The Keller-Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Math. 66, 286–308 (2005) · Zbl 1102.35011
[26] Dormann D., Weijer C.J.: Chemotactic cell movement during Dictyostelium development and gastrulation. Curr. Opin. Genet. Dev. 16(4), 367–373 (2006)
[27] Eberl H.J., Parker D.F., van Loosdrecht M.C.M.: A new deterministic spatio-temporal continuum model for biofilm development. J. Theor. Med. 3(3), 161–175 (2001) · Zbl 0985.92009
[28] Eisenbach M.: Chemotaxis. Imperial College Press, London (2004)
[29] Ford R.M., Lauffenburger D.A.: Measurement of bacterial random motility and chemotaxis coefficients: II. application of single cell based mathematical model. Biotechnol. Bioeng. 37, 661–672 (1991)
[30] Gajewski H., Zacharias K.: Global behavior of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 159, 77–114 (1998) · Zbl 0918.35064
[31] Gueron S., Liron N.: A model of herd grazing as a travelling wave, chemotaxis and stability. J. Math. Biol. 27(5), 595–608 (1989) · Zbl 0716.92026
[32] Henry M., Hilhorst D., Schätzle R.: Convergence to a viscocity solution for an advection- reaction-diffusion equation arising from a chemotaxis-growth model. Hiroshima Math. J. 29, 591–630 (1999) · Zbl 1157.35405
[33] Hildebrand E., Kaupp U.B.: Sperm chemotaxis: a primer. Ann. N. Y. Acad. Sci. 1061, 221–225 (2005)
[34] Hillen T.: A classification of spikes and plateaus. SIAM Rev. 49(1), 35–51 (2007) · Zbl 1160.35033
[35] Hillen T., Othmer H.G.: The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61(3), 751–775 (2000) · Zbl 1002.35120
[36] Hillen T., Painter K.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 280–301 (2001) · Zbl 0998.92006
[37] Hillen T., Painter K., Schmeiser C.: Global existence for chemotaxis with finite sampling radius. Discr. Cont. Dyn. Syst. B 7(1), 125–144 (2007) · Zbl 1116.92011
[38] Höfer T., Sherratt J.A., Maini P.K.: Dictyostelium discoideum: cellular self-organisation in an excitable biological medium. Proc. R. Soc. Lond. B. 259, 249–257 (1995)
[39] Horstmann D.: Lyapunov functions and L p -estimates for a class of reaction-diffusion systems. Coll. Math. 87, 113–127 (2001) · Zbl 0966.35022
[40] Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I. Jahresberichte DMV 105(3), 103–165 (2003) · Zbl 1071.35001
[41] Horstmann D., Stevens A.: A constructive approach to traveling waves in chemotaxis. J. Nonlin. Sci. 14(1), 1–25 (2004) · Zbl 1063.35071
[42] Jabbarzadeh E., Abrams C.F.: Chemotaxis and random motility in unsteady chemoattractant fields: a computational study. J. Theor. Biol. 235(2), 221–232 (2005)
[43] Kareiva P., Odell G.: Swarms of predators exhibit ’prey-taxis’ if individual predators use area-restricted search. Am. Nat. 130(2), 233–270 (1987)
[44] Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) · Zbl 1170.92306
[45] Keller E.F., Segel L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971) · Zbl 1170.92307
[46] Keller E.F., Segel L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 377–380 (1971) · Zbl 1170.92308
[47] Kennedy J.S., Marsh D.: Pheromone-regulated anemotaxis in flying moths. Science 184, 999–1001 (1974)
[48] Kim I.C.: Limits of chemotaxis growth model. Nonlinear Anal. 46, 817–834 (2001) · Zbl 1001.35052
[49] Kolokolnikov T., Erneux T., Wei J.: Mesa-type patterns in the one-dimensional Brusselator and their stability. Physica D 214, 63–77 (2006) · Zbl 1108.35088
[50] Kowalczyk R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–588 (2005) · Zbl 1065.35063
[51] Kuiper H.: A priori bounds and global existence for a strongly coupled quasilinear parabolic system modelling chemotaxis. Electron. J. Differ. Equ. 52, 1–18 (2001) · Zbl 0977.35061
[52] Kuiper H., Dung L.: Global attractors for cross-diffusion systems on domains of arbitrary dimensions. Rocky Mountain J. Math. 37(5), 1645–1668 (2007) · Zbl 1146.35017
[53] Landman K.A., Pettet G.J., Newgreen D.F.: Chemotactic cellular migration: smooth and discontinuous travelling wave solutions. SIAM J. Appl. Math. 63(5), 1666–1681 (2003) · Zbl 1044.34006
[54] Landman K.A., Pettet G.J., Newgreen D.F.: Mathematical models of cell colonization of uniformly growing domains. Bull. Math. Biol. 65(2), 235–262 (2003) · Zbl 1334.92058
[55] Lapidus I.R., Schiller R.: Model for the chemotactic response of a bacterial population. Biophys. J 16(7), 779–789 (1976)
[56] Larrivee B., Karsan A.: Signaling pathways induced by vascular endothelial growth factor (review). Int. J. Mol. Med. 5(5), 447–456 (2000)
[57] Lauffenburger D.A., Kennedy C.R.: Localized bacterial infection in a distributed model for tissue inflammation. J. Math. Biol. 16(2), 141–163 (1983) · Zbl 0537.92007
[58] Lee, J.M., Hillen, T., Lewis, M.A.: Continuous travelling waves for prey-taxis. Bull. Math. Biol. (2007) (in review) · Zbl 1142.92043
[59] Levine H.A., Sleeman B.D.: A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683–730 (1997) · Zbl 0874.35047
[60] Logan J.A., White B.J., Bentz P., Powell J.A.: Model analysis of spatial patterns in Mountain Pine Beetle outbreaks. Theor. Popul. Biol. 53(3), 236–255 (1998) · Zbl 0941.92030
[61] Luca M., Chavez-Ross A., Edelstein-Keshet L., Mogilner A.: Chemotactic signaling, microglia, and Alzheimer’s disease senile plaques: is there a connection? Bull. Math. Biol. 65(4), 693–730 (2003) · Zbl 1334.92077
[62] Maini P.K., Myerscough M.R., Winters K.H., Murray J.D.: Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53(5), 701–719 (1991) · Zbl 0725.92004
[63] Mantzaris N.V., Webb S., Othmer H.G.: Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49(2), 111–187 (2004) · Zbl 1109.92020
[64] Maree A.F., Hogeweg P.: How amoeboids self-organize into a fruiting body: multicellular coordination in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 98(7), 3879–3883 (2001)
[65] Mimura M., Tsujikawa T.: Aggregation pattern dynamics in a chemotaxis model including growth. Physica A 230, 499–543 (1996)
[66] Mittal N., Budrene E.O., Brenner M.P., Van Oudenaarden A.: Motility of Escherichia coli cells in clusters formed by chemotactic aggregation. Proc. Natl. Acad. Sci. USA 100(23), 13259–13263 (2003)
[67] Mori I., Ohshima Y.: Molecular neurogenetics of chemotaxis and thermotaxis in the nematode Caenorhabditis elegans. Bioessays 19(12), 1055–1064 (1997)
[68] Murray J.D.: Mathematical Biology II: Spatial Models and Biochemical Applications, 3rd edn. Springer, New York (2003) · Zbl 1006.92002
[69] Murray J.D., Myerscough M.R.: Pigmentation pattern formation on snakes. J. Theor. Biol. 149(3), 339–360 (1991)
[70] Myerscough M.R., Maini P.K., Painter K.J.: Pattern formation in a generalized chemotactic model. Bull. Math. Biol. 60(1), 1–26 (1998) · Zbl 1002.92511
[71] Nagai T., Senba T., Yoshida K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997) · Zbl 0901.35104
[72] Nanjundiah V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)
[73] Odell G.M., Keller E.F.: Traveling bands of chemotactic bacteria revisited. J. Theor. Biol. 56(1), 243–247 (1976)
[74] Osaki K., Tsujikawa T., Yagi A., Mimura M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 51, 119–144 (2002) · Zbl 1005.35023
[75] Osaki K., Yagi A.: Finite dimensional attractor for one-dimensional Keller-Segel equations. Funkcial. Ekvac. 44, 441–469 (2001) · Zbl 1145.37337
[76] Othmer H.G., Dunbar S.R., Alt W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988) · Zbl 0713.92018
[77] Othmer H.G., Hillen T.: The diffusion limit of transport equations II: chemotaxis equations. SIAM J. Appl. Math. 62(4), 1122–1250 (2002) · Zbl 1103.35098
[78] Othmer H.G., Stevens A.: Aggregation, blowup and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081 (1997) · Zbl 0990.35128
[79] Owen M.R., Sherratt J.A.: Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions. J. Theor. Biol. 189(1), 63–80 (1997)
[80] Painter K., Hillen T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Quart. 10(4), 501–543 (2002) · Zbl 1057.92013
[81] Painter K.J., Maini P.K., Othmer H.G.: Complex spatial patterns in a hybrid chemotaxis reaction-diffusion model. J. Math. Biol. 41(4), 285–314 (2000) · Zbl 0969.92003
[82] Painter K.J., Maini P.K., Othmer H.G.: Development and applications of a model for cellular response to multiple chemotactic cues. J. Math. Biol. 41(4), 285–314 (2000) · Zbl 0969.92003
[83] Painter K.J., Othmer H.G., Maini P.K.: Stripe formation in juvenile pomacanthus via chemotactic response to a reaction-diffusion mechanism. Proc. Natl. Acad. Sci. USA 96, 5549–5554 (1999)
[84] Palsson E., Othmer H.G.: A model for individual and collective cell movement in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 97(19), 10448–10453 (2000)
[85] Park H.T., Wu J., Rao Y.: Molecular control of neuronal migration. Bioessays 24(9), 821–827 (2002)
[86] Patlak C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953) · Zbl 1296.82044
[87] Perthame B.: Transport Equations in Biology. Birkhäuser, Basel (2007) · Zbl 1185.92006
[88] Perumpanani A.J., Sherratt J.A., Norbury J., Byrne H.M.: Biological inferences from a mathematical model for malignant invasion. Invas. Metastas. 16(4–5), 209–221 (1996)
[89] Post. K.: A non-linear parabolic system modeling chemotaxis with sensitivity functions (1999) · Zbl 1032.92006
[90] Potapov A., Hillen T.: Metastability in chemotaxis models. J. Dyn. Diff. Equ. 17, 293–330 (2005) · Zbl 1170.35460
[91] Rascle M., Ziti C.: Finite time blow up in some models of chemotaxis. J. Math. Biol. 33, 388–414 (1995) · Zbl 0814.92014
[92] Rivero M.A., Tranquillo R.T., Buettner H.M., Lauffenburger D.A.: Transport models for chemotactic cell populations based on individual cell behavior. Chem. Eng. Sci. 44, 1–17 (1989)
[93] Segel L.A.: Incorporation of receptor kinetics into a model for bacterial chemotaxis. J. Theor. Biol. 57(1), 23–42 (1976)
[94] Segel L.A.: A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32, 653–665 (1977) · Zbl 0356.92009
[95] Sherratt J.A.: Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model. Bull. Math. Biol. 56(1), 129–146 (1994) · Zbl 0789.92017
[96] Sherratt J.A., Sage E.H., Murray J.D.: Chemical control of eukaryotic cell movement: a new model. J. Theor. Biol. 162(1), 23–40 (1993)
[97] Stevens A.: The derivation of chemotaxis-equations as limit dynamics of moderately interacting stochastic many particle systems. SIAM J. Appl. Math. 61(1), 183–212 (2000) · Zbl 0963.60093
[98] Suzuki T.: Free Energy and Self-Interacting Particles. Birkhäuser, Boston (2005) · Zbl 1082.35006
[99] Tranquillo R.T., Lauffenburger D.A., Zigmond S.H.: A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations. J. Cell Biol. 106(2), 303–309 (1988)
[100] Tyson R., Lubkin S.R., Murray J.D.: A minimal mechanism for bacterial pattern formation. Proc. R. Soc. Lond. B 266, 299–304 (1999)
[101] Tyson R., Lubkin S.R., Murray J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Math. Biol. 38(4), 359–375 (1999) · Zbl 0921.92005
[102] Velazquez J.J.L.: Point dynamics for a singular limit of the Keller-Segel model. I. motion of the concentration regions. SIAM J. Appl. Math. 64(4), 1198–1223 (2004) · Zbl 1058.35021
[103] Velazquez J.J.L.: Point dynamics for a singular limit of the Keller-Segel model. II. formation of the concentration regions. SIAM J. Appl. Math. 64(4), 1198–1223 (2004) · Zbl 1058.35021
[104] Wang X.: Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics. SIAM J. Math. Ana. 31, 535–560 (2000) · Zbl 0990.92001
[105] Wang, Z.A., Hillen, T.: Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos, 17(037108) (2007), 13 pp · Zbl 1163.37383
[106] Wang Z.A., Hillen T.: Shock formation in a chemotaxis model. Math. Methods Appl. Sci. 31(1), 45–70 (2008) · Zbl 1147.35057
[107] Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal dependent sensitivity · Zbl 1205.35037
[108] Woodward D.E., Tyson R., Myerscough M.R., Murray J.D., Budrene E.O., Berg H.C.: Spatio-temporal patterns generated by Salmonella typhimurium. Biophys. J. 68(5), 2181–2189 (1995)
[109] Wrzosek D.: Long time behaviour of solutions to a chemotaxis model with volume filling effect. Proc. Roy. Soc. Edinb. Sect. A 136, 431–444 (2006) · Zbl 1104.35007
[110] Wrzosek, D.: Global attractor for a chemotaxis model with prevention of overcrowding. Nonlin. Ana. 59, 1293–1310, P2004 · Zbl 1065.35072
[111] Wu D.: Signaling mechanisms for regulation of chemotaxis. Cell Res. 15(1), 52–56 (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.