Stochastic Dynamics of Eukaryotic Flagellar Growth

  • Muruhan Rathinam
  • Yuriy Sverchkov
Special Issue: Gillespie and his Algorithms


We study the dynamics of flagellar growth in eukaryotes where intraflagellar transporters (IFT) play a crucial role. First we investigate a stochastic version of the original balance point model where a constant number of IFT particles move up and down the flagellum. The detailed model is a discrete event vector-valued Markov process occurring in continuous time. First the detailed stochastic model is compared and contrasted with a simple scalar ordinary differential equation (ODE) model of flagellar growth. Numerical simulations reveal that the steady-state mean value of the stochastic model is well approximated by the ODE model. Then we derive a scalar stochastic differential equation (SDE) as a first approximation and obtain a “small noise” approximation showing flagellar length to be Gaussian with mean and variance governed by simple ODEs. The accuracy of the small noise model is compared favorably with the numerical simulation results of the detailed model. Secondly, we derive a revised SDE for flagellar length following the revised balance point model proposed in 2009 in which IFT particles move in trains instead of in isolation. Small noise approximation of the revised SDE yields the same approximate Gaussian distribution for the flagellar length as the SDE corresponding to the original balance point model.


Eukaryotic flagellar length dynamics Stochastic balance point model Small noise approximation Gillespie’s approximation method 



We thank Hana El-Samad for bringing our attention to this interesting problem and for her valuable comments.


  1. Bressloff PC (2006) Stochastic model of intraflagellar transport. Phys Rev E 73(6):061916CrossRefGoogle Scholar
  2. Cole DG, Diener DR, Himelblau AL, Beech PL, Fuster JC, Rosenbaum JL (1998) Chlamydomonas kinesin-ii-dependent intraflagellar transport (IFT): IFT particles contain proteins required for ciliary assembly in Caenorhabditis elegans sensory neurons. J Cell Biol 141(4):993–1008CrossRefGoogle Scholar
  3. Dentler W (2005) Intraflagellar transport (IFT) during assembly and disassembly of chlamydomonas flagella. J Cell Biol 170(4):649–659. CrossRefGoogle Scholar
  4. Engel BD, Ludington WB, Marshall WF (2009) Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model. J Cell Biol 187(1):81–89CrossRefGoogle Scholar
  5. Ethier SN, Kurtz TG (2005) Markov processes: characterization and convergence, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  6. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361CrossRefGoogle Scholar
  7. Gillespie DT (2000) The chemical Langevin equation. J Chem Phys 113:297–306CrossRefGoogle Scholar
  8. Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115(4):1716–1733CrossRefGoogle Scholar
  9. Gross SP (2004) Hither and yon: a review of bi-directional microtubule-based transport. Phys Biol 1(2):R1CrossRefGoogle Scholar
  10. Iomini C, Babaev-Khaimov V, Sassaroli M, Piperno G (2001) Protein particles in chlamydomonas flagella undergo a transport cycle consisting of four phases. J Cell Biol 153(1):13–24CrossRefGoogle Scholar
  11. Kozminski KG, Johnson KA, Forscher P, Rosenbaum JL (1993) A motility in the eukaryotic flagellum unrelated to flagellar beating. Proc Natl Acad Sci USA 90(12):5519–5523CrossRefGoogle Scholar
  12. Lefebvre PA, Rosenbaum JL (1986) Regulation of the synthesis and assembly of ciliary and flagellar proteins during regeneration. An Rev Cell Biol 2(1):517–546CrossRefGoogle Scholar
  13. Ludington WB, Wemmer KA, Lechtreck KF, Witman GB, Marshall WF (2013) Avalanche-like behavior in ciliary import. Proc Natl Acad Sci USA 110(10):3925–3930CrossRefGoogle Scholar
  14. Ludington WB, Ishikawa H, Serebrenik YV, Ritter A, Hernandez-Lopez RA, Gunzenhauser J, Kannegaard E, Marshall WF (2015) A systematic comparison of mathematical models for inherent measurement of ciliary length: how a cell can measure length and volume. Biophys J 108(6):1361–1379CrossRefGoogle Scholar
  15. Marshall WF, Rosenbaum JL (2001) Intraflagellar transport balances continuous turnover of outer doublet microtubules. J Cell Biol 155(3):405–414CrossRefGoogle Scholar
  16. Marshall WF, Qin H, Brenni MR, Rosenbaum JL (2005) Flagellar length control system: testing a simple model based on intraflagellar transport and turnover. Mol Biol Cell 16(1):270–278CrossRefGoogle Scholar
  17. Nguyen RL, Tam LW, Lefebvre PA (2005) The lf1 gene of chlamydomonas reinhardtii encodes a novel protein required for flagellar length control. Genetics 169(3):1415–1424CrossRefGoogle Scholar
  18. Nishinari K, Okada Y, Schadschneider A, Chowdhury D (2005) Intracellular transport of single-headed molecular motors KIF1A. Phys Rev Lett 95(11):118101CrossRefGoogle Scholar
  19. Pedersen LB, Geimer S, Rosenbaum JL (2006) Dissecting the molecular mechanisms of intraflagellar transport in chlamydomonas. Curr Biol 16(5):450–459CrossRefGoogle Scholar
  20. Pigino G, Geimer S, Lanzavecchia S, Paccagnini E, Cantele F, Diener DR, Rosenbaum JL, Lupetti P (2009) Electron-tomographic analysis of intraflagellar transport particle trains in situ. J Cell Biol 187(1):135–148CrossRefGoogle Scholar
  21. Ross S (2002) Introduction to probability models, 8th edn. Academic Press, LondonGoogle Scholar
  22. Tam LW, Dentler WL, Lefebvre PA (2003) Defective flagellar assembly and length regulation in LF3 null mutants in chlamydomonas. J Cell Biol 163(3):597–607CrossRefGoogle Scholar
  23. Wemmer KA, Marshall WF (2007) Flagellar length control in chlamydomonas-a paradigm for organelle size regulation. Int Rev Cytol 260:175–212CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland Baltimore CountyBaltimoreUSA
  2. 2.Department of Biostatistics and Medical InformaticsUniversity of WisconsinMadisonUSA

Personalised recommendations