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Quasi-steady State Reduction of Molecular Motor-Based Models of Directed Intermittent Search

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Abstract

We present a quasi-steady state reduction of a linear reaction-hyperbolic master equation describing the directed intermittent search for a hidden target by a motor-driven particle moving on a one-dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile nonsearch phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker–Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V, diffusivity D, and target absorption rate λ of the random search. The quasi-steady state reduction determines V, D, and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3-state model and show that our quasi-steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a “tug-of-war”, and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets.

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References

  • Baas, P.W., Deitch, J.S., Black, M.M., Banker, G.A., 1988. Polarity orientation of microtubules in hippocampal neurons: uniformity in the axon and nonuniformity in the dendrite. Proc. Natl. Acad. Sci. USA 85, 8335–8339.

    Article  Google Scholar 

  • Bannai, H., Inoue, T., Nakayama, T., Hattori, M., Mikoshiba, K., 2004. Kinesin dependent, rapid bidirectional transport of ER sub-compartment in dendrites of hippocampal neurons. J. Cell Sci. 117, 163–175.

    Article  Google Scholar 

  • Bean, A.J. (Ed.), 2007. Protein Trafficking in Neurons. Academic Press, San Diego.

    Google Scholar 

  • Bell, J.W., 1991. Searching Behaviour, the Behavioural Ecology of Finding Resources. Chapman and Hall, London.

    Google Scholar 

  • Benichou, O., Coppey, M., Moreau, M., Suet, P., Voituriez, R., 2005. Optimal search strategies for hidden targets. Phys. Rev. Lett. 94, 198101.

    Article  Google Scholar 

  • Benichou, O., Loverdo, C., Moreau, M., Voituriez, R., 2007. A minimal model of intermittent search in dimension two. J. Phys. A 19, 065141.

    Google Scholar 

  • Berg, O.G., Winter, R.B., von Hippel, P.H., 1981. Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory. Biochemistry 20, 6929–6948.

    Article  Google Scholar 

  • Bramham, C.R., Wells, D.G., 2007. Dendritic mRNA: transport, translation and function. Nat. Rev. Neurosci. 8, 776–789.

    Article  Google Scholar 

  • Bressloff, P., Newby, J., 2009. Directed intermittent search for hidden targets. New J. Phys. 11, 023033.

    Article  Google Scholar 

  • Brooks, E., 1999. Probabilistic methods for a linear reaction-hyperbolic system with constant coefficients. Ann. Appl. Probab. 9, 719–731.

    Article  MATH  MathSciNet  Google Scholar 

  • Brown, A., 2000. Slow axonal transport: stop and go traffic in the axon. Nat. Rev. Mol. Cell Biol. 1, 153–156.

    Article  Google Scholar 

  • De Vos, K.J., Grierson, A.J., Ackerley, S., Miller, C.C.J., 2008. Role of axonal transport in neurodegenerative diseases. Annu. Rev. Neurosci. 31, 151–173.

    Article  Google Scholar 

  • Dynes, J., Steward, O., 2007. Dynamics of bidirectional transport of ARC mRNA in neuronal dendrites. J. Comput. Neurol. 500, 433–447.

    Article  Google Scholar 

  • Friedman, A., Craciun, G., 2006. Approximate traveling waves in linear reaction-hyperbolic equations. SIAM J. Math. Anal. 38, 741–758.

    Article  MATH  MathSciNet  Google Scholar 

  • Friedman, A., Hu, B., 2007. Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems. Indiana Univ. Math. J. 56, 2133–2158.

    Article  MATH  MathSciNet  Google Scholar 

  • Gardiner, C.W., 2004. Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd edn. Springer, Berlin.

    MATH  Google Scholar 

  • Goldstein, A.Y.N., Wang, X., Schwarz, T.L., 2008. Axonal transport and the delivery of pre-synaptic components. Curr. Opin. Neurobiol. 18, 495–503.

    Article  Google Scholar 

  • Halford, S.E., Marko, J.F., 2004. How do site-specific DNA-binding proteins find their targets? Nucl. Acid Res. 32, 3040–3052.

    Article  Google Scholar 

  • Hirokawa, N., Takemura, R., 2005. Molecular motors and mechanisms of directional transport in neurons. Nat. Rev. Neurosci. 6, 201–214.

    Article  Google Scholar 

  • Howard, J., 2001. Mechanics of Motor Proteins and the Cytoskeleton. Sianuer, Sunderland.

    Google Scholar 

  • Kelleher, R.L., Govindarajan, A., Tonegawa, S., 2004. Translational regulatory mechanisms review in persistent forms of synaptic plasticity. Neuron 44, 59–73.

    Article  Google Scholar 

  • Kennedy, M.J., Ehlers, M.D., 2006. Organelles and trafficking machinery for postsynaptic plasticity. Annu. Rev. Neurosci. 29, 2325–2362.

    Article  Google Scholar 

  • Knowles, R., Sabry, J., Martone, M., Deerinck, T., Ellisman, M., Bassell, G., Kosik, K., 1996. Translocation of RNA granules in living neurons. J. Neurosci. 16, 7812–7820.

    Google Scholar 

  • Kural, C., Ki, H., Syed, S.D., Goshima, G., Gelfand, V.I., Selvin, P.R., 2005. Kinesin and Dynein move a peroxisome in vivo: a tug-of-war or coordinated movement. Science 308, 1469–1472.

    Article  Google Scholar 

  • Lamprecht, R., LeDoux, J., 2004. Structural plasticity and memory. Nat. Rev. Neurosci. 5, 45–54.

    Article  Google Scholar 

  • Liepelt, S., Lipowsky, R., 2007. Kinesin’s network of chemomechanical motor cycles. Phys. Rev. Lett. 98, 258102.

    Article  Google Scholar 

  • Loverdo, C., Benichou, O., Moreau, M., Voituriez, R., 2008. Enhanced reaction kinetics in biological cells. Natl. Phys. 4, 134–137.

    Article  Google Scholar 

  • Mattson, M.P., Gleichmann, M., Cheng, A., 2008. Mitochondria in neuroplasticity and neurological disorders. Neuron 60, 748–766.

    Article  Google Scholar 

  • Miller, K.E., Sheetz, M.P., 2004. Axonal mitochondrial transport and potential are correlated. J. Cell Sci. 117, 2791–2804.

    Article  Google Scholar 

  • Morris, R.L., Hollenbeck, P.J., 1993. The regulation of bidirectional mitochondrial transport is coordinated with axonal outgrowth. J. Cell Sci. 104, 917–927.

    Google Scholar 

  • Mueller, M.J.I., Klumpp, S., Lipowsky, R., 2008. Tug-of-war as a cooperative mechanism for bidirectional cargo transport by molecular motors. Proc. Natl. Acad. Sci. USA 105, 4609–4614.

    Article  Google Scholar 

  • Nakata, T., Terada, S., Hirokawa, N., 1998. Visualization of the dynamics of synaptic vesicle and plasma membrane proteins in living axons. J. Cell Biol. 160, 659–674.

    Article  Google Scholar 

  • Newby, J., Bressloff, P.C., 2009. Directed intermittent search for a hidden target on a dendritic tree. Phys. Rev. E 80, 021913.

    Article  Google Scholar 

  • Puthasnveettil, S.V., Monje, F.J., Miniaci, M.C., Choi, Y.-B., Karl, K.A., Khandros, E., Gawinowicz, M.A., Sheetz, M.P., Kandel, E.R., 2008. A new component in synaptic plasticity: upregulation of kinesin in the neurons of the gill-withdrwal reflex. Cell 135, 960–973.

    Article  Google Scholar 

  • Redner, S., 2001. A Guide to First Passage Time Processes. Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Reed, M.C., Venakides, S., Blum, J.J., 1990. Approximate traveling waves in linear reaction-hyperbolic equations. SIAM J. Appl. Math. 50, 167–180.

    Article  MATH  MathSciNet  Google Scholar 

  • Rook, M.S., Lu, M., Kosik, K.S., 2000. CaMKIIα 3′ untranslated regions-directed mRNA translocation in living neurons: Visualization by GFP linkage. J. Neurosci. 20, 6385–6393.

    Google Scholar 

  • Schnitzer, M.J., Visscher, K., Block, S.M., 2000. Force production by single kinesin motors. Nat. Cell. Biol. 2, 718.

    Article  Google Scholar 

  • Stokin, G.B., Goldstein, L.S.B., 2006. Axonal transport and Alzheimer’s disease. Annu. Rev. Biochem. 75, 607–627.

    Article  Google Scholar 

  • Sutton, M.A., Schuman, E.M., 2006. Dendritic protein synthesis, synaptic plasticity, and memory. Cell 127, 49–58.

    Article  Google Scholar 

  • Vale, R.D., 2003. The molecular motor toolbox for intracellular transport. Cell 112, 467–480.

    Article  Google Scholar 

  • Visscher, K., Block, S.M., 1999. Single kinesin molecules studied with a molecular force clamp. Nature 400, 184.

    Article  Google Scholar 

  • Viswanathan, G.M., Buldyrev, S.V., Havlin, S., da Luz, M.G.E., Raposo, E.P., Stanley, H.E., 1999. Optimizing the success of random searches. Nature 401, 911–914.

    Article  Google Scholar 

  • Waites, C., Craig, A., Garner, C., 2005. Mechanisms of vertebrate synaptogenesis. Annu. Rev. Neurosci. 28, 251–274.

    Article  Google Scholar 

  • Washbourne, P., Liu, X.-B., Jones, E.G., McAllister, A.K., 2004. Cycling of NMDA receptors during trafficking in neurons before synapse formation. J. Neurosci. 24, 8253–8264.

    Article  Google Scholar 

  • Welte, M.A., 2004. Bidirectional transport along microtubules. Curr. Biol. 14, 525.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Jay M. Newby & Paul C. Bressloff

  2. Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford, OX1 3LB, UK

    Paul C. Bressloff

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  1. Jay M. Newby
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  2. Paul C. Bressloff
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Correspondence to Paul C. Bressloff.

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Newby, J.M., Bressloff, P.C. Quasi-steady State Reduction of Molecular Motor-Based Models of Directed Intermittent Search. Bull. Math. Biol. 72, 1840–1866 (2010). https://doi.org/10.1007/s11538-010-9513-8

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  • DOI: https://doi.org/10.1007/s11538-010-9513-8

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