January  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  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.

Citation: Sabine Hittmeir, Laura Kanzler, Angelika Manhart, Christian Schmeiser. Kinetic modelling of colonies of myxobacteria. Kinetic & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

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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