Bulletin of Mathematical Biology

, Volume 79, Issue 9, pp 1923–1978 | Cite as

Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors

  • Cole Zmurchok
  • Tim Small
  • Michael J. Ward
  • Leah Edelstein-Keshet
Original Article

Abstract

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction–advection–diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.

Keywords

Quasi-steady-state Molecular motors Intracellular transport Nonlinear kinetics 

References

  1. Baas PW, Deitch JS, Black MM, Banker GA (1988) Polarity orientation of microtubules in hippocampal neurons: uniformity in the axon and nonuniformity in the dendrite. Proc Natl Acad Sci 85(21):8335–8339CrossRefGoogle Scholar
  2. Bhat D, Gopalakrishnan M (2012) Effectiveness of a dynein team in a tug of war helped by reduced load sensitivity of detachment: evidence from the study of bidirectional endosome transport in D. discoideum. Phys Biol 9(4):46003CrossRefGoogle Scholar
  3. Blasius TL, Reed N, Slepchenko BM, Verhey KJ (2013) Recycling of kinesin-1 motors by diffusion after transport. PLoS One 8(9):e76081CrossRefGoogle Scholar
  4. Bressloff P, Newby J (2013) Stochastic models of intracellular transport. Rev Mod Phys 85(1):135–196CrossRefGoogle Scholar
  5. Bressloff PC, Newby JM (2011) Quasi-steady-state analysis of two-dimensional random intermittent search processes. Phys Rev E 83(6):061139CrossRefGoogle Scholar
  6. Burton PR (1988) Dendrites of mitral cell neurons contain microtubules of opposite polarity. Brain Res 473(1):107–115CrossRefGoogle Scholar
  7. Chowdhury D, Schadschneider A, Nishinari K (2005) Traffic phenomena in biology: from molecular motors to organisms. Traffic Granul Flow 2(4):223–238Google Scholar
  8. Ciandrini L, Romano MC, Parmeggiani A (2014) Stepping and crowding of molecular motors: statistical kinetics from an exclusion process perspective. Biophys J 107(5):1176–1184CrossRefGoogle Scholar
  9. Dauvergne D, Edelstein-Keshet L (2015) Application of quasi-steady state methods to molecular motor transport on microtubules in fungal hyphae. J Theor Biol 379:47–58MathSciNetCrossRefMATHGoogle Scholar
  10. Dixit R, Ross JL, Goldman YE, Holzbaur ELF (2008) Differential regulation of dynein. Science 319(February):8–11Google Scholar
  11. Fink G, Steinberg G (2006) Dynein-dependent motility of microtubules and nucleation sites supports polarization of the tubulin array in the fungus Ustilago maydis. Mol Biol Cell 17(7):3242–3253CrossRefGoogle Scholar
  12. Gou J, Edelstein-Keshet L, Allard J (2014) Mathematical model with spatially uniform regulation explains long-range bidirectional transport of early endosomes in fungal hyphae. Mol Biol Cell 25(16):2408–2415CrossRefGoogle Scholar
  13. Hendricks AG, Perlson E, Ross JL, Schroeder HW, Tokito M, Holzbaur ELF (2010) Motor coordination via a tug-of-war mechanism drives bidirectional vesicle transport. Curr Biol 20(8):697–702CrossRefGoogle Scholar
  14. Klumpp S, Lipowsky R (2005) Cooperative cargo transport by several molecular motors. Proc Natl Acad Sci USA 102(48):17284–17289CrossRefGoogle Scholar
  15. Leduc C, Padberg-Gehle K, Varga V, Helbing D, Diez S, Howard J (2012) Molecular crowding creates traffic jams of kinesin motors on microtubules. Proc Natl Acad Sci USA 109(16):6100–6105CrossRefGoogle Scholar
  16. Mallik R, Rai AK, Barak P, Rai A, Kunwar A (2013) Teamwork in microtubule motors. Trends Cell Biol 23(11):575–582CrossRefGoogle Scholar
  17. Mattila PK, Lappalainen P (2008) Filopodia: molecular architecture and cellular functions. Nat Rev Mol Cell Biol 9(6):446–454CrossRefGoogle Scholar
  18. McVicker DP, Chrin LR, Berger CL (2011) The nucleotide-binding state of microtubules modulates kinesin processivity and the ability of Tau to inhibit kinesin-mediated transport. J Biol Chem 286(50):42873–42880CrossRefGoogle Scholar
  19. Müller MJI, Klumpp S, Lipowsky R (2008) Motility states of molecular motors engaged in a stochastic tug-of-war. J Stat Phys 133(6):1059–1081MathSciNetCrossRefMATHGoogle Scholar
  20. Nambiar R, McConnell RE, Tyska MJ (2010) Myosin motor function: the ins and outs of actin-based membrane protrusions. Cell Mol Life Sci 67(8):1239–1254CrossRefGoogle Scholar
  21. Newby J, Bressloff PC (2010a) Local synaptic signaling enhances the stochastic transport of motor-driven cargo in neurons. Phys Biol 7(3):036004CrossRefGoogle Scholar
  22. Newby JM, Bressloff PC (2010b) Quasi-steady state reduction of molecular motor-based models of directed intermittent search. Bull Math Biol 72(7):1840–1866MathSciNetCrossRefMATHGoogle Scholar
  23. Parmeggiani A, Franosch T, Frey E (2004) Totally asymmetric simple exclusion process with langmuir kinetics. Phys Rev E 70(4):046101MathSciNetCrossRefGoogle Scholar
  24. Reed NA, Cai D, Blasius TL, Jih GT, Meyhofer E, Gaertig J, Verhey KJ (2006) Microtubule acetylation promotes kinesin-1 binding and transport. Curr Biol 16(21):2166–2172CrossRefGoogle Scholar
  25. Reichenbach T, Frey E, Franosch T (2007) Traffic jams induced by rare switching events in two-lane transport. New J Phys 9(6):159CrossRefGoogle Scholar
  26. Rzadzinska AK, Schneider ME, Davies C, Riordan GP, Kachar B (2004) Stereocilia functional architecture and self-renewal. J Cell Biol 164(6):887–897CrossRefGoogle Scholar
  27. Schneider ME, Dose AC, Salles FT, Chang W, Erickson FL, Burnside B, Kachar B (2006) A new compartment at stereocilia tips defined by spatial and temporal patterns of myosin IIIa expression. J Neurosci 26(40):10243–10252CrossRefGoogle Scholar
  28. Schuster M, Kilaru S, Fink G, Collemare J, Roger Y, Steinberg G (2011a) Kinesin-3 and dynein cooperate in long-range retrograde endosome motility along a nonuniform microtubule array. Mol Biol Cell 22(19):3645–3657CrossRefGoogle Scholar
  29. Schuster M, Lipowsky R, Assmann M-A, Lenz P, Steinberg G (2011b) Transient binding of dynein controls bidirectional long-range motility of early endosomes. Proc Natl Acad Sci USA 108(9):3618–3623CrossRefGoogle Scholar
  30. Schwander M, Kachar B, Müller U (2010) Review series: the cell biology of hearing. J. Cell Biol 190(1):9–20CrossRefGoogle Scholar
  31. Shubeita GT (2012) Intracellular transport: relating single-molecule properties to in vivo function. Compr Biophys 4:287–297CrossRefGoogle Scholar
  32. Smith DA, Simmons RM (2001) Models of motor-assisted transport of intracellular particles. Biophys J 80(1):45–68CrossRefGoogle Scholar
  33. Steinberg G (2011) Motors in fungal morphogenesis: cooperation versus competition. Curr Opin Microbiol 14(6):660–667CrossRefGoogle Scholar
  34. Steinberg G, Wedlich-Soldner R, Brill M, Schulz I (2001) Microtubules in the fungal pathogen Ustilago maydis are highly dynamic and determine cell polarity. J Cell Sci 114(Pt 3):609–622Google Scholar
  35. Stone MC, Roegiers F, Rolls MM (2008) Microtubules have opposite orientation in axons and dendrites of drosophila neurons. Mol Biol Cell 19(10):4122–4129CrossRefGoogle Scholar
  36. Yochelis A, Gov NS (2016) Reaction–diffusion–advection approach to propagating aggregates of molecular motors. Phys D 318–319:1–15Google Scholar
  37. Yochelis A, Ebrahim S, Millis B, Cui R, Kachar B, Naoz M, Gov NS (2015) Self-organization of waves and pulse trains by molecular motors in cellular protrusions. Sci Rep 5:13521CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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