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Bulletin of Mathematical Biology

, Volume 76, Issue 8, pp 1917–1940 | Cite as

Microtubule Patterning in the Presence of Stationary Motor Distributions

  • Diana WhiteEmail author
  • Gerda de Vries
  • Adriana Dawes
Original Article

Abstract

In this paper, we construct a novel nonlocal transport model that describes the evolution of microtubules (MTs) as they interact with stationary distributions of motor proteins. An advection term accounts for directed MT transport (sliding due to motor protein action), and an integral term accounts for reorientation of MTs due to their interactions with cross-linking motor proteins. Simulations of our model show how MT patterns depend on boundary constraints, as well as model parameters that represent motor speed, cross-linking capability (motor activity), and directionality. In large domains, and using motor parameter values consistent with experimentally-derived values, we find that patterns such as asters, vortices, and bundles are able to persist. In vivo, MTs take on aster patterns during interphase and they form bundles in neurons and polarized epithelial cells. Vortex patterns have not been observed in vivo, however, are found in in vitro experiments. In constrained domains, we find that similar patterns form (asters, bundles, and vortices). However, we also find that when two opposing motors are present, anti-parallel bundles are able to form, resembling the mitotic spindle during cell division. This model demonstrates how MT sliding and MT reorientation are sufficient to produce experimentally observed patterns.

Keywords

Microtubules Motor proteins Pattern formation  Non-local model 

Supplementary material

Supplementary material 1 (mp4 7239 KB)

Supplementary material 2 (mp4 4579 KB)

Supplementary material 3 (mp4 5584 KB)

References

  1. Aranson I, Tsimring L (2006) Theory of self-assembly of microtubules and motors. Phys Rev E 74(3):031915MathSciNetCrossRefGoogle Scholar
  2. Dogterom M, Surrey T (2013) Microtubule organization in vitro. Curr Opin Cell Biol 25:23–29CrossRefGoogle Scholar
  3. Gibbons F, Chauwin J-F, Despósito M, José J (2001) A dynamical model of kinesin-microtubule motility assays. Biophys J 80:2515–2526CrossRefGoogle Scholar
  4. Hentrich C, Surrey T (2010) Microtubule organization by the antagonistic mitotic motors kinesin-5 and kinesin-14. J Cell Biol 189(3):465–480CrossRefGoogle Scholar
  5. Hillen T (2006) M5 mesoscopic and macroscopic models for mesenchymal motion. J Math Biol 53:585–615MathSciNetCrossRefzbMATHGoogle Scholar
  6. Hillen T, White D (2014) Existence and uniqueness for a coupled PDE model for motor induced microtubule organization. Submitted to SIAM J Appl MathGoogle Scholar
  7. Howard J (2001) Mechanics of motor proteins and the cytoskeleton. Sinauer, SunderlandGoogle Scholar
  8. Humphrey D, Duggan C, Saha D, Smith D, Ka J (2002) Active fluidization of polymer networks through molecular motors. Lett Nat 416:413–416CrossRefGoogle Scholar
  9. Janson M, Loughlin R, Loiodice I, Fu C, Brunner D, Nedelec F, Tran P (2007) Crosslinkers and motors organize dynamic microtubules to form stable bipolar arrays in fission yeast. Cell 128:357–368CrossRefGoogle Scholar
  10. Jia Z, Karpeev D, Aranson I, Bates P (2008) Simulation studies of self-organization of microtubules and molecular motors. Phys Rev E 77:051905CrossRefGoogle Scholar
  11. Karp G (1996) Cell Mol Biol. Wiley, New YorkGoogle Scholar
  12. Kim J, Park Y, Kahng B, Lee HY (2003) Self-organized patterns in mixtures of microtubules and motor proteins. J Korean Phys Soc 42(1):162–166Google Scholar
  13. Kirschner M, Mitchison K (1984) Dynamic instability of microtubule growth. Nature 312:237–242CrossRefGoogle Scholar
  14. Lee HY, Kardar M (2001) Macroscopic equations for pattern formation in mixtures of microtubules and molecular motors. Am Phys Soc 64:056113Google Scholar
  15. Luo W, Yu C-H, Lieu Z, Allard J, Mogilner A, Sheetz M, Bershadsky A (2013) Analysis of the local organization and dynamics of cellular actin networks. J Cell Biol 202:10571073Google Scholar
  16. Miller C, Ermentrout G, Davidson L (2012) Rotational model for actin filament alignment by myosin. J Theor Biol 300:344359MathSciNetCrossRefGoogle Scholar
  17. Mitchison K, Kirschner M (1986) Beyond self-assembly: from microtubules to morphogenesis. Cell 45:329–342CrossRefGoogle Scholar
  18. Nedéléc F, Surrey T (2001) Dynamics of microtubule aster formation by motor complexes. Phys Scale Cell 4:841–847Google Scholar
  19. Nedéléc F, Surrey T, Maggs AC, Leibler S (1997) Self-organization of microtubules and motors. Nature 389:305–308CrossRefGoogle Scholar
  20. Othmer H (2010) Notes on space- and velocity-jump models of biological movement. Theor Biol 10:913–917Google Scholar
  21. Othmer H, Dunbar S, Alt W (1988) Models of dispersal in biological systems. J Math Biol 26:263–298MathSciNetCrossRefzbMATHGoogle Scholar
  22. Perthame B (2007) Transport equations in biology. Birkhäuser, BerlinzbMATHGoogle Scholar
  23. Porter D (1990) Integral equations: a practical treatment from spectral theory to applications. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  24. Reymann A-C, Martiel J-L, Cambier T, Blanchoin L, Boujemaa-Paterski R, Théry M (2010) Nucleation geometry governs ordered actin network structures. Nat Mater 9:827–832CrossRefGoogle Scholar
  25. Sharp D, Rogers G, Scholey J (2000) Microtubule motors in mitosis. Nature 407:41–47CrossRefGoogle Scholar
  26. Smith D, Ziebert F, Humphrey D, Duggan C, Steinbeck M, Zimmermann W, Ka J (2007) Molecular motor-induced instabilities and cross linkers determine biopolymer organization. Biophys J 93:44454452Google Scholar
  27. Surrey T, Nedéléc F, Leibler S, Karsenti E (2001) Physical properties determining self-organization of motors and microtubules. Science 292:1167–1171CrossRefGoogle Scholar
  28. Tao L, Mogliner A, Civelekoglu-Scholey G, Wollman R, Evans J, Stahlberg H, Scholey J (2006) A homotetrameric kinesin-5, KLP61F, bundles microtubules and antagonizes Ncd in motility assays. Curr Biol 16:2293–2302CrossRefGoogle Scholar
  29. Vale R, Malik F, Brown D (1992) Directional instability of microtubule transport in the presence of kinesin and dynein, two opposite polarity motor proteins. J Cell Biol 119:1589–1596CrossRefGoogle Scholar
  30. Vignaud T, Blanchoin L, Thery M (2001) Directed cytoskeleton self-organization. Synth Cell Biol 22:671–682Google Scholar
  31. Wade R (2009) On and around microtubules: an overview. Mol Biotechnol 43:177–191CrossRefGoogle Scholar
  32. Waterman-Storer C, Salmon E (1997) Microtubule dynamics: treadmilling comes around again. Curr Biol 7:369–372CrossRefGoogle Scholar
  33. Waterman-Storer C, Salmon E (1999) Microtubules: strange polymers inside the cell. Bioelectrochem Bioenerg 48:285–295CrossRefGoogle Scholar
  34. White D, de Vries G, Martin J, Dawes A (2014) Microtubule patterning in the presence of moving motors. In preparationGoogle Scholar
  35. Yokota E, Sonobe S, Igarashi H, Shimmen T (1995) Plant microtubules can be translocated by a dynein ATPase from sea urchin in vitro. Plant Cell Physiol 36(8):1563–1569Google Scholar
  36. Yuko M-K (2001) Shaping microtubules into diverse patterns: molecular connections for setting up both ends. Cytoskeleton 68:603–618Google Scholar
  37. Zhou J, Giannakakou P (2005) Targeting microtubules for cancer chemotherapy. Curr Med Chem 5:65–71Google Scholar

Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.University of AlbertaEdmontonCanada
  2. 2.Ohio State UniversityColumbusUSA

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