February  2021, 14(1): 1-24. doi: 10.3934/krm.2020046

Kinetic modelling of colonies of myxobacteria

1. 

University of Vienna, Faculty for Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

2. 

University College London, Dept. of Mathematics, 25 Gordon Street, WC1H 0AY London, UK

* Corresponding author: Christian Schmeiser

Received  January 2020 Revised  August 2020 Published  February 2021 Early access  September 2020

A new kinetic model for the dynamics of myxobacteria colonies on flat surfaces is derived formally, and first analytical and numerical results are presented. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal. We investigate two different versions: a) realistic rod-shaped bacteria and b) artificial circular shaped bacteria called Maxwellian myxos in reference to the similar simplification of the gas dynamics Boltzmann equation for Maxwellian molecules. The sum of the corresponding collision operators produces relaxation towards nematically aligned equilibria, i.e. two groups of bacteria polarized in opposite directions.

For the spatially homogeneous model a global existence and uniqueness result is proved as well as exponential decay to equilibrium for special initial conditions and for Maxwellian myxos. Only partial results are available for the rod-shaped case. These results are illustrated by numerical simulations, and a formal discussion of the macroscopic limit is presented.

 

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Citation: Sabine Hittmeir, Laura Kanzler, Angelika Manhart, Christian Schmeiser. Kinetic modelling of colonies of myxobacteria. Kinetic and Related Models, 2021, 14 (1) : 1-24. doi: 10.3934/krm.2020046
References:
[1]

R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J., 58 (2009), 999-1022.  doi: 10.1512/iumj.2009.58.3506.

[2]

R. AlonsoV. BaglandY. Cheng and B. Lods, One-dimensional dissipative Boltzmann equation: Measure solutions, cooling rate, and self-similar profile, SIAM J. Math. Anal., 50 (2018), 1278-1321.  doi: 10.1137/17M1136791.

[3]

R. J. Alonso and B. Lods, Two proofs of Haff's law for dissipative gases: The use of entropy and the weakly inelastic regime, J. Math. Anal. Appl., 397 (2013), 260-275.  doi: 10.1016/j.jmaa.2012.07.045.

[4]

I. S. Aranson and L. S. Tsimring, Pattern formation of microtubules and motors: Inelastic interaction of polar rods,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 71 (2005), 050901.

[5]

A. Baskaran and M. C. Marchetti, Enhanced diffusion and ordering of self-propelled rods,, Phys. Rev. Lett., 101 (2008), 268101. doi: 10.1103/PhysRevLett.101.268101.

[6]

A. Baskaran and M. C. Marchetti, Nonequilibrium statistical mechanics of self propelled hard rods,, J. Stat. Mech., 2010 (2010), P04019. doi: 10.1103/PhysRevE.77.011920.

[7]

D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN, 35 (2001), 899-905.  doi: 10.1051/m2an:2001141.

[8]

E. Ben-Naim and P. L. Krapivsky, Alignment of rods and partition of integers,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 73 (2006), 031109. doi: 10.1103/PhysRevE.73.031109.

[9]

E. Bertin, M. Droz and G. Gregoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis,, J. Phys. A: Math. Theor., 42 (2006), 445001. doi: 10.1088/1751-8113/42/44/445001.

[10]

A. V. BobylevJ. A.Carrillo and I. M. Gamba, On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions, J. Stat. Phys., 98 (2000), 743-773.  doi: 10.1023/A:1018627625800.

[11]

A. V. Bobylev and C. Cercignani, Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions,, J. Stat. Phys., 110 (2003), 333–375. doi: 10.1023/A:1021031031038.

[12]

L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, WTB Wissenschaftliche Taschenbücher book series, 68, In Kinetische Theorie II, pp 115-225 doi: 10.1007/978-3-322-84986-1_3.

[13]

E. CarlenM. C. CarvalhoP. Degond and B. Wennberg, A Boltzmann model for rod alignment and schooling fish, Nonlinearity, 28 (2015), 1783-1804.  doi: 10.1088/0951-7715/28/6/1783.

[14]

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198. 

[15]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[16]

P. DegondA. Frouvelle and G. Raoul, Local stability of perfect alignment for a spatially homogeneous kinetic model, J. Stat. Phys., 157 (2014), 84-112.  doi: 10.1007/s10955-014-1062-3.

[17]

P. DegondA. Manhart and H. Yu, A continuum model of nematic alignment of self-propelled particles, DCDS-B, 22 (2017), 1295-1327.  doi: 10.3934/dcdsb.2017063.

[18]

P. DegondA. Manhart and H. Yu, An age-structured continuum model for myxobacteria, M3AS, 28 (2018), 1737-1770.  doi: 10.1142/S0218202518400043.

[19]

P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech., 134 (1983), 401-30.  doi: 10.1017/S0022112083003419.

[20]

J. Hodgkin and D. Kaiser, Genetics of gliding motility in Myxococcus xanthus (Myxobacterales): two gene systems control movement, Mol. Gen. Genet., 171 (1979), 177-191.  doi: 10.1007/BF00270004.

[21]

O. A. Igoshin, A. Mogilner, R. D. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling,, \emphPNAS, 98 (2001), 14913-14918. doi: 10.1073/pnas.221579598.

[22]

O. A. Igoshin and G. Oster, Rippling of myxobacteria, Math. Biosci., 188 (2004), 221-233.  doi: 10.1016/j.mbs.2003.04.001.

[23]

O. A. IgoshinR. WelchD. Kaiser and G. Oster, Waves and aggregation patterns in myxobacteria, PNAS, 101 (2004), 4256-4261.  doi: 10.1073/pnas.0400704101.

[24]

P.-E. Jabin and T. Rey, Hydrodynamic limit of granular gases to pressureless Euler in dimension 1, Quart. Appl. Math., 75 (2017), 155-179.  doi: 10.1090/qam/1442.

[25]

Y. JiangO. Sozinova and M. Alber, On modeling complex collective behavior in myxobacteria, Adv. in Complex Syst., 9 (2006), 353-367.  doi: 10.1142/S0219525906000860.

[26]

L. Jelsbak and L. Sogaard-Andersen, The cell surface-associated intercellular C-signal induces behavioral changes in individual Myxococcus xanthus cells during fruiting body morphogenesis, PNAS, 96 (1999), 5031-5036.  doi: 10.1073/pnas.96.9.5031.

[27]

S. K. Kim and D. Kaiser, C-factor: A cell-cell signaling protein required for fruiting body morphogenesis of M. xanthus, Cell, 61 (1990), 19-26.  doi: 10.1016/0092-8674(90)90211-V.

[28]

O. E. Lanford, Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111. 

[29]

E. M. F. MaurielloT. MignotZ. Yang and D. R. Zusman, Gliding motility revisited: How do the myxobacteria move without flagella?, Microbiol. Mol. Biol. Rev., 74 (2010), 229-249.  doi: 10.1128/MMBR.00043-09.

[30]

S. MischlerC. Mouhot and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅰ: The Cauchy problem, J. Stat. Phys., 124 (2006), 655-702.  doi: 10.1007/s10955-006-9096-9.

[31]

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8.

[32]

B. Nan and D. R. Zusman, Uncovering the mystery of gliding motility in the myxobacteria, Annu. Rev. Genet., 45 (2011), 21-39.  doi: 10.1146/annurev-genet-110410-132547.

[33]

B. Sager and D. Kaiser, Intercellular C-signaling and the traveling waves of Myxococcus, Genes Dev., 8 (1994), 2793-2804.  doi: 10.1101/gad.8.23.2793.

[34]

G. Toscani, Hydrodynamics from the Dissipative Boltzmann Equation, , in: G. Capriz, P.M. Mariano, P. Giovine (eds), Mathematical Models of Granular Matter, Lect. Notes in Math. 1937, Springer, Berlin–Heidelberg, 2008. doi: 10.1007/978-3-540-78277-3_3.

[35]

I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Functional Anal., 270 (2016), 1922-1970.  doi: 10.1016/j.jfa.2015.09.025.

[36]

C. Villani, Topics in Optimal Transportation, , Graduate Studies in Math. 58, AMS, 2003. doi: 10.1090/gsm/058.

[37]

D. Wall and D. Kaiser, Type Ⅳ pili and cell motility, Mol. Microbiol., 32 (1999), 1-10.  doi: 10.1046/j.1365-2958.1999.01339.x.

[38]

R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, PNAS, 98 (2001), 14907-14912.  doi: 10.1073/pnas.261574598.

[39]

C. WolgemuthE. HoiczykD. Kaiser and G. Oster, How myxobacteria glide, Curr. Biol., 12 (2002), 369-377.  doi: 10.1016/S0960-9822(02)00716-9.

show all references

References:
[1]

R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J., 58 (2009), 999-1022.  doi: 10.1512/iumj.2009.58.3506.

[2]

R. AlonsoV. BaglandY. Cheng and B. Lods, One-dimensional dissipative Boltzmann equation: Measure solutions, cooling rate, and self-similar profile, SIAM J. Math. Anal., 50 (2018), 1278-1321.  doi: 10.1137/17M1136791.

[3]

R. J. Alonso and B. Lods, Two proofs of Haff's law for dissipative gases: The use of entropy and the weakly inelastic regime, J. Math. Anal. Appl., 397 (2013), 260-275.  doi: 10.1016/j.jmaa.2012.07.045.

[4]

I. S. Aranson and L. S. Tsimring, Pattern formation of microtubules and motors: Inelastic interaction of polar rods,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 71 (2005), 050901.

[5]

A. Baskaran and M. C. Marchetti, Enhanced diffusion and ordering of self-propelled rods,, Phys. Rev. Lett., 101 (2008), 268101. doi: 10.1103/PhysRevLett.101.268101.

[6]

A. Baskaran and M. C. Marchetti, Nonequilibrium statistical mechanics of self propelled hard rods,, J. Stat. Mech., 2010 (2010), P04019. doi: 10.1103/PhysRevE.77.011920.

[7]

D. Benedetto and M. Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN, 35 (2001), 899-905.  doi: 10.1051/m2an:2001141.

[8]

E. Ben-Naim and P. L. Krapivsky, Alignment of rods and partition of integers,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 73 (2006), 031109. doi: 10.1103/PhysRevE.73.031109.

[9]

E. Bertin, M. Droz and G. Gregoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis,, J. Phys. A: Math. Theor., 42 (2006), 445001. doi: 10.1088/1751-8113/42/44/445001.

[10]

A. V. BobylevJ. A.Carrillo and I. M. Gamba, On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions, J. Stat. Phys., 98 (2000), 743-773.  doi: 10.1023/A:1018627625800.

[11]

A. V. Bobylev and C. Cercignani, Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions,, J. Stat. Phys., 110 (2003), 333–375. doi: 10.1023/A:1021031031038.

[12]

L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, WTB Wissenschaftliche Taschenbücher book series, 68, In Kinetische Theorie II, pp 115-225 doi: 10.1007/978-3-322-84986-1_3.

[13]

E. CarlenM. C. CarvalhoP. Degond and B. Wennberg, A Boltzmann model for rod alignment and schooling fish, Nonlinearity, 28 (2015), 1783-1804.  doi: 10.1088/0951-7715/28/6/1783.

[14]

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198. 

[15]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[16]

P. DegondA. Frouvelle and G. Raoul, Local stability of perfect alignment for a spatially homogeneous kinetic model, J. Stat. Phys., 157 (2014), 84-112.  doi: 10.1007/s10955-014-1062-3.

[17]

P. DegondA. Manhart and H. Yu, A continuum model of nematic alignment of self-propelled particles, DCDS-B, 22 (2017), 1295-1327.  doi: 10.3934/dcdsb.2017063.

[18]

P. DegondA. Manhart and H. Yu, An age-structured continuum model for myxobacteria, M3AS, 28 (2018), 1737-1770.  doi: 10.1142/S0218202518400043.

[19]

P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech., 134 (1983), 401-30.  doi: 10.1017/S0022112083003419.

[20]

J. Hodgkin and D. Kaiser, Genetics of gliding motility in Myxococcus xanthus (Myxobacterales): two gene systems control movement, Mol. Gen. Genet., 171 (1979), 177-191.  doi: 10.1007/BF00270004.

[21]

O. A. Igoshin, A. Mogilner, R. D. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling,, \emphPNAS, 98 (2001), 14913-14918. doi: 10.1073/pnas.221579598.

[22]

O. A. Igoshin and G. Oster, Rippling of myxobacteria, Math. Biosci., 188 (2004), 221-233.  doi: 10.1016/j.mbs.2003.04.001.

[23]

O. A. IgoshinR. WelchD. Kaiser and G. Oster, Waves and aggregation patterns in myxobacteria, PNAS, 101 (2004), 4256-4261.  doi: 10.1073/pnas.0400704101.

[24]

P.-E. Jabin and T. Rey, Hydrodynamic limit of granular gases to pressureless Euler in dimension 1, Quart. Appl. Math., 75 (2017), 155-179.  doi: 10.1090/qam/1442.

[25]

Y. JiangO. Sozinova and M. Alber, On modeling complex collective behavior in myxobacteria, Adv. in Complex Syst., 9 (2006), 353-367.  doi: 10.1142/S0219525906000860.

[26]

L. Jelsbak and L. Sogaard-Andersen, The cell surface-associated intercellular C-signal induces behavioral changes in individual Myxococcus xanthus cells during fruiting body morphogenesis, PNAS, 96 (1999), 5031-5036.  doi: 10.1073/pnas.96.9.5031.

[27]

S. K. Kim and D. Kaiser, C-factor: A cell-cell signaling protein required for fruiting body morphogenesis of M. xanthus, Cell, 61 (1990), 19-26.  doi: 10.1016/0092-8674(90)90211-V.

[28]

O. E. Lanford, Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111. 

[29]

E. M. F. MaurielloT. MignotZ. Yang and D. R. Zusman, Gliding motility revisited: How do the myxobacteria move without flagella?, Microbiol. Mol. Biol. Rev., 74 (2010), 229-249.  doi: 10.1128/MMBR.00043-09.

[30]

S. MischlerC. Mouhot and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅰ: The Cauchy problem, J. Stat. Phys., 124 (2006), 655-702.  doi: 10.1007/s10955-006-9096-9.

[31]

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8.

[32]

B. Nan and D. R. Zusman, Uncovering the mystery of gliding motility in the myxobacteria, Annu. Rev. Genet., 45 (2011), 21-39.  doi: 10.1146/annurev-genet-110410-132547.

[33]

B. Sager and D. Kaiser, Intercellular C-signaling and the traveling waves of Myxococcus, Genes Dev., 8 (1994), 2793-2804.  doi: 10.1101/gad.8.23.2793.

[34]

G. Toscani, Hydrodynamics from the Dissipative Boltzmann Equation, , in: G. Capriz, P.M. Mariano, P. Giovine (eds), Mathematical Models of Granular Matter, Lect. Notes in Math. 1937, Springer, Berlin–Heidelberg, 2008. doi: 10.1007/978-3-540-78277-3_3.

[35]

I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Functional Anal., 270 (2016), 1922-1970.  doi: 10.1016/j.jfa.2015.09.025.

[36]

C. Villani, Topics in Optimal Transportation, , Graduate Studies in Math. 58, AMS, 2003. doi: 10.1090/gsm/058.

[37]

D. Wall and D. Kaiser, Type Ⅳ pili and cell motility, Mol. Microbiol., 32 (1999), 1-10.  doi: 10.1046/j.1365-2958.1999.01339.x.

[38]

R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, PNAS, 98 (2001), 14907-14912.  doi: 10.1073/pnas.261574598.

[39]

C. WolgemuthE. HoiczykD. Kaiser and G. Oster, How myxobacteria glide, Curr. Biol., 12 (2002), 369-377.  doi: 10.1016/S0960-9822(02)00716-9.

Figure 1.  Graphic illustration of the collision rules. (a): Alignment collisions with two-step geometric algorithm to regularize it. (b): Already invertible reversal collisions
Figure 2.  Support of two group data (solid lines, purple)
Figure 3.  Two group initial conditions with the same mass in $ \mathbb{T}^1_{+} $ and $ \mathbb{T}^1_{-} $; rod shaped bacteria. Left: uniform distributions within $ \mathbb{T}^1_{+} $ and $ \mathbb{T}^1_{-} $. Right: vacuum around $ \pm\pi/2 $
Figure 4.  Left: initial condition with uniform distribution in $ \mathbb{T}^1_{+} $ and vacuum everywhere else. Right: initially two concentrated patches at a distance somewhat bigger than $ \pi/2 $ (yellow at the left end). Outer stripes created by reversal, then fill-in by alignment, followed by concentration towards opposite directions. The mean angles $ \bar \varphi_+ $ (red line) and $ \bar \varphi_- $ (dotted red line) in the two groups change significantly
Figure 6.  Instability of constant positive steady states. Left: random initial perturbation, leading to an unpredictable equilibrium direction. Right: initial perturbation at one direction, which eventually becomes the equilibrium direction. Note that this differs from the simulations in Figure 4, left, by the fact that a positive state is perturbed, and therefore reversal collisions are active
Figure 5.  Left: The evolution of the inverse square root of the variance $ V[f] $ from the simulation depicted on the left side of Figure 4, supporting the validity of Haff's law for rod shaped myxos. Right: Semi-log plot of $ V[f] $ for a simulation with the same initial data, but for Maxwellian myxos, demonstrating exponential decay to equilibrium as shown in Lemma 4.2 a)
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