Skip to main content
SpringerLink
Log in

Models of Assembly and Disassembly of Individual Microtubules: Stochastic and Averaged Equations

  • Published:
Journal of Biological Physics Aims and scope Submit manuscript

Abstract

In this paper we present solutions of the master equations for the microtubule length and show that the local probability for rescues or catastrophes can lead to bell-shaped length histograms. Conversely, as already known, non-local probabilities for these events result in exponential length histograms. We also derive master equations for a stabilizing cap and obtain a new boundary condition which provides an explanation of the results obtained in dilution and cutting experiments.

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

Similar content being viewed by others

References

  • Amos, L.A.: The microtubule lattice - 20 years on, Trends in Cell Biol. 48 (1995).

  • Atkins, P.W.: Physical chemistry, Oxford University Press, Oxford, 1990.

    Google Scholar 

  • Bayley, P.M.: What makes microtubules dynamic? J. Cell Sci. 95 (1990), 329–334.

    Google Scholar 

  • Bayley, P.M., Martin, S.R. and Sharma, K.K.: AIP Conf. Proc. 226 (1991), 187–199.

    Google Scholar 

  • Belmont, L.D., Hyman, A.A., Sawin, K.E. and Mitchison, T.J.: Real-time visualization of cell cycle-dependent changes in microtubule dynamics in cytoplasmatic extracts. Cell 62 (1990), 579–589.

    Google Scholar 

  • Billger, M.A., Bhatacharjee, G. and Williams Jr., R.C.: Dynamic instability of microtubules assembled from microtubule-associated protein-free tubulin: neither variability of growth and shortening rates nor ‘rescue’ requires microtubule-associated proteins, Biochemistry 35 (1996), 13656–13663.

    Google Scholar 

  • Bolterauer, H., Limbach, H.J. and Tuszynński, J.A.: From stochastic to coherent assembly of microtubules: models and new results, Bioelectrochem. and Bioenerg. 41 (1996), 71–76.

    Google Scholar 

  • Caplow, M., Shanks, J. and Ruhlen, R.: How taxol modulates microtubule disassembly, J. Biol. Chem. 269 (1994), 2399–2402.

    Google Scholar 

  • Carlier, M.F. and Pantaloni, D.: Biochemistry 17 (1978), 1908–1915.

    Google Scholar 

  • Carlier, M.F., Melki, R., Pantaloni, D., Hill, T.L. and Chen, Y.: Synchronous oscillations in microtubule polymerization, Proc. Natl. Acad. Sci. USA 84 (1987), 5257–5261.

    Google Scholar 

  • Cassimeris, L.U., Wadsworth, P. and Salmon, E.D.: Dynamics of microtubule depolymerisation in monocytes, J. Cell Biol. 102 (1986), 2023–2032.

    Google Scholar 

  • Cassimeris, L., Pryer, N.K. and Salmon, E.D.: Real-time observations of microtubule dynamic instability in living cells, J. Cell Biol. 107 (1988), 2223–2231.

    Google Scholar 

  • Chen, Y. and Hill, T.L.: Theoretical study on oscillations in microtubule polymerization, Proc. Natl. Acad. Sci. USA 84 (1987), 8419.

    Google Scholar 

  • Chen, Y. and Hill, T.L.:Monte Carlo study of the GTP cap in a five-start helix model of a microtubule, Proc. Natl. Acad. Sci. USA 82 (1985), 1131–1135.

    Google Scholar 

  • Chretien, D., Fuller, S.D. and Karsenti, E.: Structure of growing microtubule ends. Two-dimensional sheets close into tubes at variable rates, J. Cell Biol. 129 (1995), 1311.

    Google Scholar 

  • Drechsel, D.N., Hyman, A.A., Coob, M.H. and Kirschner, M.W.: Modulation of dynamic instability of microtubule assembly by the microtubule associated protein tau, Mol. Biol. Cell 3 (1992), 1141–1154.

    Google Scholar 

  • Engelborghs, Y. and Van Houtte, A.: Biophys. Biochem. 14 (1981), 195–202.

    Google Scholar 

  • Erickson, H.P. and O'Brien, E.T.: Microtubule dynamic instability and gtp hydrolysis, Annu. Rev. Biophys. Biomol. Struct. 21 (1992), 145–166.

    Google Scholar 

  • Flyvbjerg, H., Holy, T.E. and Leibler, S.: Stochastic dynamics of microtubules: A model for caps and catastrophes, Phys. Rev. Lett. 73 (1994), 2372.

    Google Scholar 

  • Flyvbjerg, H., Holy, T.E. and Leibler, S.: Phys. Rev. E 54 (1996), 5538.

    Google Scholar 

  • Gildersleeve, R.F., Cross, A.R., Cullen, K.E., Fagen, A.P. and Williams, R.C.: Microtubules grow and shorten at intrinsically variable rates, J. Biol. Chem. 267 (1992), 7995–8006.

    Google Scholar 

  • Gliksman, N.R., Skibbens, R.V. and Salmon, E.D.: How the transition frequencies of microtubule dynamic instability (nucleation, catastrophe, and rescue) regulate microtubule dynamics in interphase and mitosis: analysis using a Monte Carlo computer simulation, Mol. Biol. Cell 4 (1993), 1035–1050.

    Google Scholar 

  • Horio, T. and Hotani, H.: Visualization of the dynamic instability of individual microtubules, Nature 321 (1986), 605.

    Google Scholar 

  • Houchmandzadeh, B. and Vallade, M.: Collective oscillations in microtubule growth, Phys. Rev. E 53 (1996), 6320–6324.

    Google Scholar 

  • Jobs, E., Wolf, D.E. and Flyvbjerg, H.: Modeling microtubule oscillations, Phys. Rev. E 79 (1997), 519–522.

    Google Scholar 

  • Lange, G., Mandelkow, E.M., Jagla, A. and Mandelkow, E.: Tubulin oligomers and microtubule oscillations, Eur. J. Biochem. 178 (1988), 61–69.

    Google Scholar 

  • Mandelkow, E.M., Lange, G., Jagla, A., Spann, U. and Mandelkow, E.: Dynamics of the microtubule oscillator: Role of mucleotides and tubulin - MAP interactions, EMBO J. 7 (1988), 357–365.

    Google Scholar 

  • Mandelkow, E.M., Mandelkow, E. and Milligan: Microtubule dynamics and MT caps: a time resolved cryo-electron microscopy study, J. Cell Biol. 114 (1991), 977.

    Google Scholar 

  • Martin, S.R., Schilstra, M.J. and Bayley, P.M.: Dynamics instability of microtubules: Monte Carlo simulation and application to different types of microtubule lattice, Biophys. J. 65 (1993), 578–596.

    Google Scholar 

  • Marx, A. and Mandelkow, E.: A model of microtubule oscillations, Eur. Biophys. J. 22 (1994), 405–421.

    Google Scholar 

  • Marx, A., Jagla, A. and Mandelkow, E.: Microtubule assembly and oscillations induced by flash photolysis of caged-GTP, Eur. Biophys. J. 19 (1990), 1–9.

    Google Scholar 

  • Mejillano, M.R., Tolo, E.T., Williams Jr., R.C. and Himes, R.H.: The conversion of tubulin carboxyl groups to amides has a stabilizing effect on microtubules, Biochem. 31 (1992), 3478–3483.

    Google Scholar 

  • Melki, R., Carlier, M.-F. and Pantaloni, D.: Oscillations in microtubule polymerization: the rate of GTP regeneration on tubulin controls the period, EMBO J. 7 (1988), 2653–2659.

    Google Scholar 

  • Melki, R., Carlier, M.-F., Pantaloni, D. and Timasheff, S.N.: Cold depolymerization of microtubules to double rings: geometric stabilization of assemblies, Biochem. 28 (1989), 9143.

    Google Scholar 

  • Mitchison, T. and Kirschner, M.: Dynamic instability of microtubule growth, Nature 312 (1984a), 237–242.

    Google Scholar 

  • Mitchison, T. and Kirschner, M.: Microtubule assembly nucleated by isolated centrosomes, Nature 312 (1984b), 232–237.

    Google Scholar 

  • Obermann, H., Mandelkow, E.-M., Lange, G. and Mandelkow, E.: Microtubule oscillations, J. Biol. Chem. 265 (1990), 4382–4388.

    Google Scholar 

  • O'Brien, E.T., Salmon, E.D., Walker, R.A. and Erickson, H.P.: Effects of magnesium on the dynamic instability of individual microtubules, Biochem. 29 (1990), 6648.

    Google Scholar 

  • O'Brien, E.T., Erickson, H.P. and Salmon, E.D.: Calcium increases catastrophe frequency without slowing elongation, J. Cell Biol. 111 (1992), 388.

    Google Scholar 

  • Odde, D.J., Cassimeris, L. and Buettner, H.M.: Spectral analysis of microtubule assembly dynamics, AlChE J. 42 (1996), 1434–1442.

    Google Scholar 

  • Pantaloni, D. and Carlier, M.F.: Ann. N. Y. Acad. Sci. 466 (1986), 496–509.

    Google Scholar 

  • Pirollet, F., Job, D., Margolis, R.L. and Garel, J.R.: An oscillatory mode for microtubule assembly, EMBO J. 6 (1987), 3247–3252.

    Google Scholar 

  • Semenov, M.V.: New concepts of microtubule dynamics and microtubule motor movement and new model of chromosome movement in mitoses, J. theor. biol. 179 (1996), 91–117.

    Google Scholar 

  • Sept, D., Limbach, H.J., Bolterauer, H. and Tuszyńsky, J.A.: A chemical kinetics model for microtubule oscillations; J. theor. Biol. to be published (1999)

  • Shelden, E. and Wadsworth, P.: Observation and quantification of individual microtubule behaviour in vivo: microtubule dynamics are cell-type specific, J. Cell Biol. 120 (1993), 935–945.

    Google Scholar 

  • Simon, J.R., Parsons, S.F. and Salmon, E.D.: Buffer conditions and nontubulin factors critically affects the microtubule dynamic instability of sea urchin egg tubulin, Cell Motil. Cytoskeleton 21 (1992), 1–14.

    Google Scholar 

  • Trinczek, B., Marx, E.M., Mandelkow, E.M., Murphy, D.B. and Mandelkow, E.: Mol. Biol. Cell 4 (1993), 323.

    Google Scholar 

  • Tuszyński, J.A., Sept, D., Bolterauer, H. and Limbach, H.J., Stochastic Data Analysis for the Assembly and Disassembly of Microtubules in Vitro, Adv. Str. Biol. 5 (1998) 169–201.

    Google Scholar 

  • Vandecandelaere, A., Martin, S.R., Schilstra, M.J. and Bayley, P.M.: Effects of the tubulin-colchicine complex on microtubule dynamic instability, Biochemistry 33 (1994), 2792–2801.

    Google Scholar 

  • Vandecandelaere, A., Martin, S.R. and Bayley, P.M.: Regulation of microtubule dynamic instability by tubulin-GDp, Biochemistry 34 (1995), 1332–1343.

    Google Scholar 

  • Verde, F., Dogterom, M., Stelzer, E., Karsenti, E. and Leibler, S.: Control of microtubule dynamics and length by cyclin a-and cyclin b-dependent kinease in xenopus egg extracts, J. Cell biol. 118 (1992), 1097–1108.

    Google Scholar 

  • Wade, R.H., Pirollet, F., Margolis, R.L., Garel, J.R. and Job, D.: Monotonic versus oscillating microtubule assembly: a cryo-electron microscope study, Biol. Cell 65 (1989), 37–44.

    Google Scholar 

  • Walker, R.A., O'Brien, E.T., Pryer, N.K., Soboeiro, M.F., Voter, W.A., Erickson, H.P. and Salmon, E.D.: Dynamic instability of individual microtubules analyzed by video light microscopy: Rate constants and transition frequencies, J. Cell Biol. 107 (1988), 1437–1448.

    Google Scholar 

  • Walker, R.A., Pryer, N.K. and Salmon, E.D.: Dilution of individual microtubules observed in real time in vitro: evidence that a cap size is small and independent of elongation rate, J. Cell Biol. 114 (1991), 73.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Justus-Liebig Universität Gießen, Gießen, Germany

    H. Bolterauer, H.-J. Limbach & J.A. Tuszyński

Authors
  1. H. Bolterauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H.-J. Limbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.A. Tuszyński
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bolterauer, H., Limbach, HJ. & Tuszyński, J. Models of Assembly and Disassembly of Individual Microtubules: Stochastic and Averaged Equations. Journal of Biological Physics 25, 1–22 (1999). https://doi.org/10.1023/A:1005159215657

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005159215657

Search

Navigation