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December  2015, 8(4): 651-666. doi: 10.3934/krm.2015.8.651

Confinement by biased velocity jumps: Aggregation of escherichia coli

Vincent Calvez 1, , Gaël Raoul 2, and  Christian Schmeiser 3,

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

Received  April 2014 Revised  May 2015 Published  July 2015

We investigate a one-dimensional linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint, it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.
Keywords: equilibrium, velocity-jump processes, chemotaxis, hypocoercivity., Kinetic equations.
Mathematics Subject Classification: Primary: 35B40, 35Q92, 92C1.
Citation: Vincent Calvez, Gaël Raoul, Christian Schmeiser. Confinement by biased velocity jumps: Aggregation of escherichia coli. Kinetic & Related Models, 2015, 8 (4) : 651-666. doi: 10.3934/krm.2015.8.651
References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. Google Scholar

[2]

W. Alt, Orientation of cells migrating in a chemotactic gradient,, Biological Growth and Spread, 38 (1980), 353. Google Scholar

[3]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations,, J. Math. Biol., 9 (1980), 147. doi: 10.1007/BF00275919. Google Scholar

[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. doi: 10.1081/PDE-100002246. Google Scholar

[5]

G. Bal, Couplage D'équations et Homogénéisation en Transport Neutronique,, Thèse de doctorat de l'Université Paris, (1997). Google Scholar

[6]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size,, Trans. AMS, 284 (1984), 617. doi: 10.1090/S0002-9947-1984-0743736-0. Google Scholar

[7]

H. Berg and D. Brown, Chemotaxis in Escheria coli analyzed by 3-dimensional tracking,, Nature, 239 (1972). Google Scholar

[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. Google Scholar

[9]

H. C. Berg, E. Coli in Motion,, Springer-Verlag, (2004). doi: 10.1007/b97370. Google Scholar

[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. doi: 10.1081/PDE-120021182. Google Scholar

[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. doi: 10.1137/110851687. Google Scholar

[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. doi: 10.1007/s00605-004-0234-7. Google Scholar

[13]

S. Chatterjee, R. A. da Silveira and Y. Kafri, Chemotaxis when bacteria remember: Drift versus diffusion,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1002283. Google Scholar

[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. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. Google Scholar

[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. doi: 10.1007/s00222-004-0389-9. Google Scholar

[16]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595. doi: 10.1007/s00285-005-0334-6. Google Scholar

[17]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, Trans. Amer. Math. Soc., 367 (2015), 3807. doi: 10.1090/S0002-9947-2015-06012-7. Google Scholar

[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. doi: 10.1137/040603565. Google Scholar

[19]

F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[20]

F. Filbet and C. Yang, Numerical simulations of kinetic models for chemotaxis,, SIAM J. Sci. Comput., 36 (2014). doi: 10.1137/130910208. Google Scholar

[21]

F. Golse, The Milne problem for the radiative transfer equations (with frequency dependence),, Trans. Amer. Math. Soc., 303 (1987), 125. doi: 10.1090/S0002-9947-1987-0896011-0. Google Scholar

[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. doi: 10.1016/0022-1236(88)90051-1. Google Scholar

[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. doi: 10.1016/j.jmaa.2011.10.039. Google Scholar

[24]

L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws,, SIMAI Springer Series, (2013). doi: 10.1007/978-88-470-2892-0. Google Scholar

[25]

T. Goudon and A. Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann equation,, Control, 9 (2003), 371. doi: 10.1051/cocv:2003018. Google Scholar

[26]

K. P. Hadeler, Reaction transport systems in biological modelling,, Mathematics inspired by biology, 1714 (1999), 95. doi: 10.1007/BFb0092376. Google Scholar

[27]

T. Hillen, On the $L^2$-moment closure of transport equations: The Cattaneo approximation,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961. Google Scholar

[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. doi: 10.3934/dcdsb.2005.5.319. Google Scholar

[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. doi: 10.1512/iumj.2006.55.2677. Google Scholar

[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. doi: 10.1007/s00205-003-0276-3. Google Scholar

[31]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations. II. Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[32]

F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101. doi: 10.1007/s00030-012-0155-4. Google Scholar

[33]

Y. Kafri and R. A. da Silveira, Steady-state chemotaxis in escherichia coli,, Phys. Rev. Lett., 100 (2008). Google Scholar

[34]

J. M. Newby and J. P. Keener, An asymptotic analysis of the spatially inhomogeneous velocity-jump process,, Multiscale Model. Simul., 9 (2011), 735. doi: 10.1137/10080676X. Google Scholar

[35]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[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. Google Scholar

[37]

J. T. Locsei, Persistence of direction increases the drift velocity of run and tumble chemotaxis,, J. Math. Biol., 55 (2007), 41. doi: 10.1007/s00285-007-0080-z. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/100803067. Google Scholar

[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. doi: 10.1016/j.compbiolchem.2009.06.003. Google Scholar

[41]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[42]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar

[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). doi: 10.1371/journal.pcbi.1000890. Google Scholar

[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. doi: 10.1073/pnas.1101996108. Google Scholar

[45]

M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis,, Phys. Rev. E, 48 (1993), 2553. doi: 10.1103/PhysRevE.48.2553. Google Scholar

[46]

N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis,, Kinet. Relat. Models, 3 (2010), 501. doi: 10.3934/krm.2010.3.501. Google Scholar

[47]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. Google Scholar

[48]

C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations,, Bull. Math. Biol., 73 (2011), 1695. doi: 10.1007/s11538-010-9586-4. Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. Google Scholar

[2]

W. Alt, Orientation of cells migrating in a chemotactic gradient,, Biological Growth and Spread, 38 (1980), 353. Google Scholar

[3]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations,, J. Math. Biol., 9 (1980), 147. doi: 10.1007/BF00275919. Google Scholar

[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. doi: 10.1081/PDE-100002246. Google Scholar

[5]

G. Bal, Couplage D'équations et Homogénéisation en Transport Neutronique,, Thèse de doctorat de l'Université Paris, (1997). Google Scholar

[6]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size,, Trans. AMS, 284 (1984), 617. doi: 10.1090/S0002-9947-1984-0743736-0. Google Scholar

[7]

H. Berg and D. Brown, Chemotaxis in Escheria coli analyzed by 3-dimensional tracking,, Nature, 239 (1972). Google Scholar

[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. Google Scholar

[9]

H. C. Berg, E. Coli in Motion,, Springer-Verlag, (2004). doi: 10.1007/b97370. Google Scholar

[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. doi: 10.1081/PDE-120021182. Google Scholar

[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. doi: 10.1137/110851687. Google Scholar

[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. doi: 10.1007/s00605-004-0234-7. Google Scholar

[13]

S. Chatterjee, R. A. da Silveira and Y. Kafri, Chemotaxis when bacteria remember: Drift versus diffusion,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1002283. Google Scholar

[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. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. Google Scholar

[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. doi: 10.1007/s00222-004-0389-9. Google Scholar

[16]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595. doi: 10.1007/s00285-005-0334-6. Google Scholar

[17]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, Trans. Amer. Math. Soc., 367 (2015), 3807. doi: 10.1090/S0002-9947-2015-06012-7. Google Scholar

[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. doi: 10.1137/040603565. Google Scholar

[19]

F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[20]

F. Filbet and C. Yang, Numerical simulations of kinetic models for chemotaxis,, SIAM J. Sci. Comput., 36 (2014). doi: 10.1137/130910208. Google Scholar

[21]

F. Golse, The Milne problem for the radiative transfer equations (with frequency dependence),, Trans. Amer. Math. Soc., 303 (1987), 125. doi: 10.1090/S0002-9947-1987-0896011-0. Google Scholar

[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. doi: 10.1016/0022-1236(88)90051-1. Google Scholar

[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. doi: 10.1016/j.jmaa.2011.10.039. Google Scholar

[24]

L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws,, SIMAI Springer Series, (2013). doi: 10.1007/978-88-470-2892-0. Google Scholar

[25]

T. Goudon and A. Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann equation,, Control, 9 (2003), 371. doi: 10.1051/cocv:2003018. Google Scholar

[26]

K. P. Hadeler, Reaction transport systems in biological modelling,, Mathematics inspired by biology, 1714 (1999), 95. doi: 10.1007/BFb0092376. Google Scholar

[27]

T. Hillen, On the $L^2$-moment closure of transport equations: The Cattaneo approximation,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961. Google Scholar

[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. doi: 10.3934/dcdsb.2005.5.319. Google Scholar

[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. doi: 10.1512/iumj.2006.55.2677. Google Scholar

[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. doi: 10.1007/s00205-003-0276-3. Google Scholar

[31]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations. II. Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[32]

F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101. doi: 10.1007/s00030-012-0155-4. Google Scholar

[33]

Y. Kafri and R. A. da Silveira, Steady-state chemotaxis in escherichia coli,, Phys. Rev. Lett., 100 (2008). Google Scholar

[34]

J. M. Newby and J. P. Keener, An asymptotic analysis of the spatially inhomogeneous velocity-jump process,, Multiscale Model. Simul., 9 (2011), 735. doi: 10.1137/10080676X. Google Scholar

[35]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[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. Google Scholar

[37]

J. T. Locsei, Persistence of direction increases the drift velocity of run and tumble chemotaxis,, J. Math. Biol., 55 (2007), 41. doi: 10.1007/s00285-007-0080-z. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/100803067. Google Scholar

[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. doi: 10.1016/j.compbiolchem.2009.06.003. Google Scholar

[41]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[42]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar

[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). doi: 10.1371/journal.pcbi.1000890. Google Scholar

[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. doi: 10.1073/pnas.1101996108. Google Scholar

[45]

M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis,, Phys. Rev. E, 48 (1993), 2553. doi: 10.1103/PhysRevE.48.2553. Google Scholar

[46]

N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis,, Kinet. Relat. Models, 3 (2010), 501. doi: 10.3934/krm.2010.3.501. Google Scholar

[47]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. Google Scholar

[48]

C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations,, Bull. Math. Biol., 73 (2011), 1695. doi: 10.1007/s11538-010-9586-4. Google Scholar

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