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Confinement by biased velocity jumps: Aggregation of escherichia coli
1. | CNRS UMR 5669 "Unité de Mathématiques Pures et Appliquées", and project-team Inria NUMED, Ecole Normale Supérieure de Lyon, Lyon, France |
2. | CNRS UMR 7641 "Centre de Mathématiques Appliquées", Ecole Polytechnique, Palaiseau, France |
3. | Fakultät für Mathematik, Universität Wien, Austria |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
W. Alt, Orientation of cells migrating in a chemotactic gradient, Biological Growth and Spread, (conf. proc., Heidelberg, 1979), Springer, Berlin, 38 (1980), 353-366. |
[3] |
W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.
doi: 10.1007/BF00275919. |
[4] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[5] |
G. Bal, Couplage D'équations et Homogénéisation en Transport Neutronique, Thèse de doctorat de l'Université Paris, 1997. |
[6] |
C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. AMS, 284 (1984), 617-649.
doi: 10.1090/S0002-9947-1984-0743736-0. |
[7] |
H. Berg and D. Brown, Chemotaxis in Escheria coli analyzed by 3-dimensional tracking, Nature, 239 (1972), p500. |
[8] |
H. Berg and L. Turner, Chemotaxis of bacteria in glass-capillary arrays - Escheria coli, motility, microchannel plate and light-scattering, Biophys. J., 58 (1990), 919-930. |
[9] |
H. C. Berg, E. Coli in Motion, Springer-Verlag, New York, 2004.
doi: 10.1007/b97370. |
[10] |
M. J. Cáceres, J. A. Carrillo and T. Goudon, Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. PDE, 28 (2003), 969-989.
doi: 10.1081/PDE-120021182. |
[11] |
J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336-361.
doi: 10.1137/110851687. |
[12] |
F. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.
doi: 10.1007/s00605-004-0234-7. |
[13] |
S. Chatterjee, R. A. da Silveira and Y. Kafri, Chemotaxis when bacteria remember: Drift versus diffusion, PLoS Comput. Biol., 7 (2011), e1002283, 2pp.
doi: 10.1371/journal.pcbi.1002283. |
[14] |
L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.
doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. |
[15] |
L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[16] |
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
doi: 10.1007/s00285-005-0334-6. |
[17] |
J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7. |
[18] |
R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Model. Simul., 3 (2005), 362-394.
doi: 10.1137/040603565. |
[19] |
F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
doi: 10.1007/s00285-004-0286-2. |
[20] |
F. Filbet and C. Yang, Numerical simulations of kinetic models for chemotaxis, SIAM J. Sci. Comput., 36 (2014), B348-B366, arXiv:1303.2445 (2013).
doi: 10.1137/130910208. |
[21] |
F. Golse, The Milne problem for the radiative transfer equations (with frequency dependence), Trans. Amer. Math. Soc., 303 (1987), 125-143.
doi: 10.1090/S0002-9947-1987-0896011-0. |
[22] |
F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.
doi: 10.1016/0022-1236(88)90051-1. |
[23] |
L. Gosse, Asymptotic-preserving and well-balanced schemes for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes, J. Math. Anal. Appl., 388 (2012), 964-983.
doi: 10.1016/j.jmaa.2011.10.039. |
[24] |
L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws, SIMAI Springer Series, Springer, Milan, 2013.
doi: 10.1007/978-88-470-2892-0. |
[25] |
T. Goudon and A. Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann equation, Control, Optimisation and Calculus of Variations, 9 (2003), 371-398.
doi: 10.1051/cocv:2003018. |
[26] |
K. P. Hadeler, Reaction transport systems in biological modelling, Mathematics inspired by biology, (Martina Franca, 1997), Lecture Notes in Math. Springer, Berlin, 1714 (1999), 95-150.
doi: 10.1007/BFb0092376. |
[27] |
T. Hillen, On the $L^2$-moment closure of transport equations: The Cattaneo approximation, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 961-982.
doi: 10.3934/dcdsb.2004.4.961. |
[28] |
H. J. Hwang, K. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.
doi: 10.3934/dcdsb.2005.5.319. |
[29] |
H. J. Hwang, K. Kang and A. Stevens, Global existence of classical solutions for a hyperbolic chemotaxis model and its parabolic limit, Indiana Univ. Math. J., 55 (2006), 289-316.
doi: 10.1512/iumj.2006.55.2677. |
[30] |
F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
doi: 10.1007/s00205-003-0276-3. |
[31] |
T. Hillen and H. G. Othmer, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[32] |
F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[33] |
Y. Kafri and R. A. da Silveira, Steady-state chemotaxis in escherichia coli, Phys. Rev. Lett., 100 (2008), 238101. |
[34] |
J. M. Newby and J. P. Keener, An asymptotic analysis of the spatially inhomogeneous velocity-jump process, Multiscale Model. Simul., 9 (2011), 735-765.
doi: 10.1137/10080676X. |
[35] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[36] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.) (in Russian), 3 (1948), 3-95 (English translation: Amer. Math. Soc. Translation 1950 (26)). |
[37] |
J. T. Locsei, Persistence of direction increases the drift velocity of run and tumble chemotaxis, J. Math. Biol., 55 (2007), 41-60.
doi: 10.1007/s00285-007-0080-z. |
[38] |
N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of Escheria coli cells in clusters formed by chemotactic aggregation, PNAS, 100 (2003), 13259-13263. |
[39] |
R. Natalini and M. Ribot, Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis, SIAM J. Numer. Anal., 50 (2012), 883-905.
doi: 10.1137/100803067. |
[40] |
D. V. Nicolau Jr., J. P. Armitage and P. K. Maini, Directional persistence and the optimality of run-and-tumble chemotaxis, Comp. Biol. and Chem., 33 (2009), 269-274.
doi: 10.1016/j.compbiolchem.2009.06.003. |
[41] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[42] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[43] |
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[44] |
J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.
doi: 10.1073/pnas.1101996108. |
[45] |
M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E, 48 (1993), 2553-2568.
doi: 10.1103/PhysRevE.48.2553. |
[46] |
N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis, Kinet. Relat. Models, 3 (2010), 501-528.
doi: 10.3934/krm.2010.3.501. |
[47] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[48] |
C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.
doi: 10.1007/s11538-010-9586-4. |
show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
W. Alt, Orientation of cells migrating in a chemotactic gradient, Biological Growth and Spread, (conf. proc., Heidelberg, 1979), Springer, Berlin, 38 (1980), 353-366. |
[3] |
W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.
doi: 10.1007/BF00275919. |
[4] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[5] |
G. Bal, Couplage D'équations et Homogénéisation en Transport Neutronique, Thèse de doctorat de l'Université Paris, 1997. |
[6] |
C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. AMS, 284 (1984), 617-649.
doi: 10.1090/S0002-9947-1984-0743736-0. |
[7] |
H. Berg and D. Brown, Chemotaxis in Escheria coli analyzed by 3-dimensional tracking, Nature, 239 (1972), p500. |
[8] |
H. Berg and L. Turner, Chemotaxis of bacteria in glass-capillary arrays - Escheria coli, motility, microchannel plate and light-scattering, Biophys. J., 58 (1990), 919-930. |
[9] |
H. C. Berg, E. Coli in Motion, Springer-Verlag, New York, 2004.
doi: 10.1007/b97370. |
[10] |
M. J. Cáceres, J. A. Carrillo and T. Goudon, Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. PDE, 28 (2003), 969-989.
doi: 10.1081/PDE-120021182. |
[11] |
J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336-361.
doi: 10.1137/110851687. |
[12] |
F. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.
doi: 10.1007/s00605-004-0234-7. |
[13] |
S. Chatterjee, R. A. da Silveira and Y. Kafri, Chemotaxis when bacteria remember: Drift versus diffusion, PLoS Comput. Biol., 7 (2011), e1002283, 2pp.
doi: 10.1371/journal.pcbi.1002283. |
[14] |
L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.
doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. |
[15] |
L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[16] |
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
doi: 10.1007/s00285-005-0334-6. |
[17] |
J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7. |
[18] |
R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Model. Simul., 3 (2005), 362-394.
doi: 10.1137/040603565. |
[19] |
F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
doi: 10.1007/s00285-004-0286-2. |
[20] |
F. Filbet and C. Yang, Numerical simulations of kinetic models for chemotaxis, SIAM J. Sci. Comput., 36 (2014), B348-B366, arXiv:1303.2445 (2013).
doi: 10.1137/130910208. |
[21] |
F. Golse, The Milne problem for the radiative transfer equations (with frequency dependence), Trans. Amer. Math. Soc., 303 (1987), 125-143.
doi: 10.1090/S0002-9947-1987-0896011-0. |
[22] |
F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.
doi: 10.1016/0022-1236(88)90051-1. |
[23] |
L. Gosse, Asymptotic-preserving and well-balanced schemes for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes, J. Math. Anal. Appl., 388 (2012), 964-983.
doi: 10.1016/j.jmaa.2011.10.039. |
[24] |
L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws, SIMAI Springer Series, Springer, Milan, 2013.
doi: 10.1007/978-88-470-2892-0. |
[25] |
T. Goudon and A. Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann equation, Control, Optimisation and Calculus of Variations, 9 (2003), 371-398.
doi: 10.1051/cocv:2003018. |
[26] |
K. P. Hadeler, Reaction transport systems in biological modelling, Mathematics inspired by biology, (Martina Franca, 1997), Lecture Notes in Math. Springer, Berlin, 1714 (1999), 95-150.
doi: 10.1007/BFb0092376. |
[27] |
T. Hillen, On the $L^2$-moment closure of transport equations: The Cattaneo approximation, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 961-982.
doi: 10.3934/dcdsb.2004.4.961. |
[28] |
H. J. Hwang, K. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.
doi: 10.3934/dcdsb.2005.5.319. |
[29] |
H. J. Hwang, K. Kang and A. Stevens, Global existence of classical solutions for a hyperbolic chemotaxis model and its parabolic limit, Indiana Univ. Math. J., 55 (2006), 289-316.
doi: 10.1512/iumj.2006.55.2677. |
[30] |
F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
doi: 10.1007/s00205-003-0276-3. |
[31] |
T. Hillen and H. G. Othmer, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[32] |
F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[33] |
Y. Kafri and R. A. da Silveira, Steady-state chemotaxis in escherichia coli, Phys. Rev. Lett., 100 (2008), 238101. |
[34] |
J. M. Newby and J. P. Keener, An asymptotic analysis of the spatially inhomogeneous velocity-jump process, Multiscale Model. Simul., 9 (2011), 735-765.
doi: 10.1137/10080676X. |
[35] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[36] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.) (in Russian), 3 (1948), 3-95 (English translation: Amer. Math. Soc. Translation 1950 (26)). |
[37] |
J. T. Locsei, Persistence of direction increases the drift velocity of run and tumble chemotaxis, J. Math. Biol., 55 (2007), 41-60.
doi: 10.1007/s00285-007-0080-z. |
[38] |
N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of Escheria coli cells in clusters formed by chemotactic aggregation, PNAS, 100 (2003), 13259-13263. |
[39] |
R. Natalini and M. Ribot, Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis, SIAM J. Numer. Anal., 50 (2012), 883-905.
doi: 10.1137/100803067. |
[40] |
D. V. Nicolau Jr., J. P. Armitage and P. K. Maini, Directional persistence and the optimality of run-and-tumble chemotaxis, Comp. Biol. and Chem., 33 (2009), 269-274.
doi: 10.1016/j.compbiolchem.2009.06.003. |
[41] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[42] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[43] |
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[44] |
J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.
doi: 10.1073/pnas.1101996108. |
[45] |
M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E, 48 (1993), 2553-2568.
doi: 10.1103/PhysRevE.48.2553. |
[46] |
N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis, Kinet. Relat. Models, 3 (2010), 501-528.
doi: 10.3934/krm.2010.3.501. |
[47] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[48] |
C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.
doi: 10.1007/s11538-010-9586-4. |
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