Skip to main content
SpringerLink
Log in

Bidirectional sliding of two parallel microtubules generated by multiple identical motors

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of this study is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Aldous D, Fill JA (2002) Reversible Markov chains and random walks on graphs. http://www.stat.berkeley.edu/~aldous/RWG/book.html

  • Anderson W (1991) Continuous-time Markov chains: an applications-oriented approach. Applied probability. Springer. ISBN 9780387973692

  • Atzberger PJ, Peskin CS (2006) A Brownian dynamics model of kinesin in three dimensions incorporating the force-extension profile of the coiled-coil cargo tether. Bull Math Biol 68(1):131–160

    Article  MathSciNet  MATH  Google Scholar 

  • Bell G (1978) Models for the specific adhesion of cells to cells. Science 200(4342):618–627

    Article  Google Scholar 

  • Bhat D, Gopalakrishnan M (2016) Transport of organelles by elastically coupled motor proteins. Eur Phys J E Soft Matter 39(7):71

    Article  Google Scholar 

  • Bouzat S (2016) Models for microtubule cargo transport coupling the Langevin equation to stochastic stepping motor dynamics: caring about fluctuations. Phys Rev E 93:012401

    Article  MathSciNet  Google Scholar 

  • Craig EM, Yeung HT, Rao AN, Baas PW (2017) Polarity sorting of axonal microtubules: a computational study. Mol Biol Cell 28(23):3271–3285

    Article  Google Scholar 

  • del Castillo U, Winding M, Lu W, Gelfand VI (2015) Interplay between kinesin-1 and cortical dynein during axonal outgrowth and microtubule organization in Drosophila neurons. Elife 4:e10140

    Article  Google Scholar 

  • Eugene S, Xue W-F, Robert P, Doumic-Jauffret M (2016) Insights into the variability of nucleated amyloid polymerization by a minimalistic model of stochastic protein assembly. J Chem Phys 144(17):12

    Article  Google Scholar 

  • Fink G, Hajdo L, Skowronek KJ, Reuther C, Kasprzak AA, Diez S (2009) The mitotic kinesin-14 Ncd drives directional microtubule-microtubule sliding. Nat Cell Biol 11(6):717–723

    Article  Google Scholar 

  • Grimmett G, Grimmett P, Stirzaker D, Stirzaker M, Grimmett S (2001) Probability and random processes. OUP, Oxford. ISBN 9780198572220

  • Gross SP, Welte MA, Block SM, Wieschaus EF (2002) Coordination of opposite-polarity microtubule motors. J Cell Biol 156(4):715–724

    Article  Google Scholar 

  • Gross SP, Vershinin M, Shubeita GT (2007) Cargo transport: two motors are sometimes better than one. Curr Biol 17(12):R478–R486

    Article  Google Scholar 

  • Huisinga W, Meyn S, Schütte C (2004) Phase transitions and metastability in Markovian and molecular systems. ANNAP 14(1):419–458

    MathSciNet  MATH  Google Scholar 

  • Ikuta J, Kamisetty NK, Shintaku H, Kotera H, Kon T, Yokokawa R (2014) Tug-of-war of microtubule filaments at the boundary of a kinesin- and dynein-patterned surface. Sci Rep 4:5281

    Article  Google Scholar 

  • Klumpp S, Lipowsky R (2005) Cooperative cargo transport by several molecular motors. Proc Natl Acad Sci USA 102(48):17284–17289

    Article  Google Scholar 

  • Kunwar A, Tripathy SK, Xu J, Mattson MK, Anand P, Sigua R, Vershinin M, McKenney RJ, Yu CC, Mogilner A, Gross SP (2011) Mechanical stochastic tug-of-war models cannot explain bidirectional lipid-droplet transport. Proc Natl Acad Sci 108(47):18960–18965

    Article  Google Scholar 

  • Kural C, Kim H, Syed S, Goshima G, Gelfand VI, Selvin PR (2005) Kinesin and dynein move a peroxisome in vivo: a tug-of-war or coordinated movement? Science 308(5727):1469–1472

    Article  Google Scholar 

  • Lee RH, Mitchell CS (2015) Axonal transport cargo motor count versus average transport velocity: is fast versus slow transport really single versus multiple motor transport? J Theor Biol 370:39–44

    Article  Google Scholar 

  • Levin DA, Peres Y, Wilmer EL (2009) Markov chains and mixing times. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Lu W, Gelfand VI (2017) Moonlighting motors: kinesin, dynein, and cell polarity. Trends Cell Biol 27(7):505–514

    Article  Google Scholar 

  • Lu W, Fox P, Lakonishok M, Davidson MW, Gelfand VI (2013) Initial neurite outgrowth in Drosophila neurons is driven by kinesin-powered microtubule sliding. Curr Biol 23(11):1018–1023

    Article  Google Scholar 

  • Ludecke A, Seidel AM, Braun M, Lansky Z, Diez S (2018) Diffusive tail anchorage determines velocity and force produced by kinesin-14 between crosslinked microtubules. Nat Commun 9(1):2214

    Article  Google Scholar 

  • McKinley SA, Athreya A, Fricks J, Kramer PR (2012) Asymptotic analysis of microtubule-based transport by multiple identical molecular motors. J Theor Biol 305:54–69

    Article  MathSciNet  MATH  Google Scholar 

  • Miclo L (2015) An absorbing eigentime identity. Markov Proc Relat Fields 21(2):249–262

    MathSciNet  MATH  Google Scholar 

  • Miles CE, Keener JP (2017) Bidirectionality from cargo thermal fluctuations in motor-mediated transport. J Theor Biol 424:37–48

    Article  MathSciNet  MATH  Google Scholar 

  • Müller MJI, Klumpp S, Lipowsky R (2008) Tug-of-war as a cooperative mechanism for bidirectional cargo transport by molecular motors. Proc Natl Acad Sci 105(12):4609–4614

    Article  Google Scholar 

  • Nascimento AA, Roland JT, Gelfand VI (2003) Pigment cells: a model for the study of organelle transport. Annu Rev Cell Dev Biol 19:469–491

    Article  Google Scholar 

  • Newby JM, Bressloff PC (2010) Quasi-steady state reduction of molecular motor-based models of directed intermittent search. Bull Math Biol 72(7):1840–1866

    Article  MathSciNet  MATH  Google Scholar 

  • Norris JR (1997) Markov chains. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  • Oelz DB, Del Castillo U, Gelfand VI, Mogilner A (2018) Microtubule dynamics, kinesin-1 sliding, and dynein action drive growth of cell processes. Biophys J 115(8):1614–1624

    Article  Google Scholar 

  • Patel SR, Richardson JL, Schulze H, Kahle E, Galjart N, Drabek K, Shivdasani RA, Hartwig JH, Italiano JE (2005) Differential roles of microtubule assembly and sliding in proplatelet formation by megakaryocytes. Blood 106(13):4076–4085

    Article  Google Scholar 

  • Rogers SL, Gelfand VI (2000) Membrane trafficking, organelle transport, and the cytoskeleton. Curr Opin Cell Biol 12(1):57–62

    Article  Google Scholar 

  • Saito N, Kaneko K (2017) Embedding dual function into molecular motors through collective motion. Sci Rep 7:44288

    Article  Google Scholar 

  • Saloff-Coste L (1997) Lectures on finite Markov chains. In: Bernard P (ed) Lectures on probability theory and statistics. Lecture notes in mathematics, vol 1665. Springer, Berlin, pp 301–413

  • Sharp DJ, Brown HM, Kwon M, Rogers GC, Holland G, Scholey JM (2000) Functional coordination of three mitotic motors in Drosophila embryos. Mol Biol Cell 11(1):241–253

    Article  Google Scholar 

  • Shimamoto Y, Forth S, Kapoor TM (2015) Measuring pushing and braking forces generated by ensembles of kinesin-5 crosslinking two microtubules. Dev Cell 34(6):669–681

    Article  Google Scholar 

  • Svoboda K, Block SM (1994) Force and velocity measured for single kinesin molecules. Cell 77(5):773–784

    Article  Google Scholar 

  • Visscher K, Schnitzer MJ, Block SM (1999) Single kinesin molecules studied with a molecular force clamp. Nature 400(6740):184–189

    Article  Google Scholar 

  • Wollman R, Civelekoglu-Scholey G, Scholey JM, Mogilner A (2008) Reverse engineering of force integration during mitosis in the Drosophila embryo. Mol Syst Biol 4:195

    Article  Google Scholar 

  • Zhang Y, Fisher ME (2010) Dynamics of the tug-of-war model for cellular transport. Phys Rev E Stat Nonlinear Soft Matter Phys 82(1 Pt 1):011923

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California Irvine, Irvine, CA, USA

    Jun Allard

  2. Inria, UPMC Univ Paris 06, Lab. J.L. Lions UMR CNRS 7598, Sorbonne Universités, Paris, France

    Marie Doumic

  3. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Alex Mogilner

  4. School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia

    Dietmar Oelz

Authors
  1. Jun Allard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marie Doumic
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alex Mogilner
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dietmar Oelz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dietmar Oelz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study has been supported by NIH Grant R01 GM123068, NSF Grant DMS 1715455 and NSF Grant DMS 1763272 to JA; furthermore by ERC Starting Grant SKIPPERAD (number 306321) to MD, by ARC Discovery Project DP180102956 to DO and by NIH Grant GM121971 to AM.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allard, J., Doumic, M., Mogilner, A. et al. Bidirectional sliding of two parallel microtubules generated by multiple identical motors. J. Math. Biol. 79, 571–594 (2019). https://doi.org/10.1007/s00285-019-01369-w

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-019-01369-w

Keywords

Mathematics Subject Classification

Search

Navigation