Confinement by biased velocity jumps: Aggregation of <em> escherichia coli</em>

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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

Abstract
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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 and Related Models, 2015, 8 (4) : 651-666. doi: 10.3934/krm.2015.8.651

References:

[1]	J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716.
[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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