August  2016, 36(8): 4553-4567. doi: 10.3934/dcds.2016.36.4553

A drift-diffusion model for molecular motor transport in anisotropic filament bundles

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, United States

2. 

Courant Institute of Mathematical Sciences and Department of Biology, New York University, 251 Mercer St, New York, NY 10012, United States

Received  May 2015 Revised  October 2015 Published  March 2016

In this study we consider the density of motor proteins in filament bundles with polarity graded in space. We start with a microscopic model that includes information on motor binding site positions along specific filaments and on their polarities. We assume that filament length is small compared to the characteristic length scale of the bundle polarity pattern. This leads to a separation of scales between molecular motor movement within the bundle and along single fibers which we exploit to derive a drift-diffusion equation as a first order perturbation equation. The resulting drift-diffusion model reveals that drift dominates in unidirectional bundles while diffusion dominates in isotropic bundles. In general, however, those two modes of transport are balanced according to the polarity and thickness of the filament bundle. The model makes testable predictions on the dependence of the molecular motor density on filament density and polarity.
Citation: Dietmar Oelz, Alex Mogilner. A drift-diffusion model for molecular motor transport in anisotropic filament bundles. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4553-4567. doi: 10.3934/dcds.2016.36.4553
References:
[1]

P. Baas, C. Nadar and K. Myers, Axonal transport of microtubules: The long and short of it, Traffic, 7 (2006), 490-498. doi: 10.1111/j.1600-0854.2006.00392.x.

[2]

P. Bressloff and J. Newby, Stochastic models of intracellular transport, Reviews of Modern Physics, 85 (2013), 135-196. doi: 10.1103/RevModPhys.85.135.

[3]

A. Friedman and G. Craciun, A model of intracellular transport of particles in an axon, Journal of Mathematical Biology, 51 (2005), 217-246. doi: 10.1007/s00285-004-0285-3.

[4]

K. O. Friedrichs and P. D. Lax, Boundary value problems for first order operators, Communications on Pure and Applied Mathematics, 18 (1965), 355-388. doi: 10.1002/cpa.3160180127.

[5]

W. Hancock, Bidirectional cargo transport: Moving beyond tug of war, Nature Reviews Molecular Cell Biology, 15 (2014), 615-628. doi: 10.1038/nrm3853.

[6]

T. Hillen and H. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM Journal on Applied Mathematics, 61 (2000), 751-775. doi: 10.1137/S0036139999358167.

[7]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[8]

M. Kneussel and W. Wagner, Myosin motors at neuronal synapses: Drivers of membrane transport and actin dynamics, Nature Reviews Neuroscience, 14 (2013), 233-247. doi: 10.1038/nrn3445.

[9]

A. Kunwar, S. Tripathy, J. Xu, M. Mattson, P. Anand, R. Sigua, M. Vershinin, R. McKenney, C. Yu, A. Mogilner and S. Gross, Mechanical stochastic tug-of-war models cannot explain bidirectional lipid-droplet transport, Proceedings of the National Academy of Sciences of the United States of America, 108 (2011), 18960-18965. doi: 10.1073/pnas.1107841108.

[10]

A. Kuznetsov, Modelling active transport in drosophila unipolar motor neurons, Computer Methods in Biomechanics and Biomedical Engineering, 14 (2011), 1117-1131. doi: 10.1080/10255842.2010.515983.

[11]

A. Kuznetsov and K. Hooman, Modeling traffic jams in intracellular transport in axons, International Journal of Heat and Mass Transfer, 51 (2008), 5695-5699, Biomedical-Related Special Issue. doi: 10.1016/j.ijheatmasstransfer.2008.04.022.

[12]

I. Maly, Diffusion approximation of the stochastic process of microtubule assembly, Bulletin of Mathematical Biology, 64 (2002), 213-238. doi: 10.1006/bulm.2001.0265.

[13]

D. Smith and R. Simmons, Models of motor-assisted transport of intracellular particles, Biophysical Journal, 80 (2001), 45-68. doi: 10.1016/S0006-3495(01)75994-2.

[14]

M. Stone, F. Roegiers and M. Rolls, Microtubules have opposite orientation in axons and dendrites of drosophila neurons, Molecular Biology of the Cell, 19 (2008), 4122-4129. doi: 10.1091/mbc.E07-10-1079.

[15]

R. Vale, The molecular motor toolbox for intracellular transport, Cell, 112 (2003), 467-480. doi: 10.1016/S0092-8674(03)00111-9.

[16]

W. J. Walter, V. Beránek, E. Fischermeier and S. Diez, Tubulin acetylation alone does not affect kinesin-1 velocity and run length in vitro, PLoS ONE, 7 (2012), e42218.

[17]

F. Wanka and E. Van Zoelen, Cellular organelle transport and positioning by plasma streaming, Cellular and Molecular Biology Letters, 8 (2003), 1035-1045.

show all references

References:
[1]

P. Baas, C. Nadar and K. Myers, Axonal transport of microtubules: The long and short of it, Traffic, 7 (2006), 490-498. doi: 10.1111/j.1600-0854.2006.00392.x.

[2]

P. Bressloff and J. Newby, Stochastic models of intracellular transport, Reviews of Modern Physics, 85 (2013), 135-196. doi: 10.1103/RevModPhys.85.135.

[3]

A. Friedman and G. Craciun, A model of intracellular transport of particles in an axon, Journal of Mathematical Biology, 51 (2005), 217-246. doi: 10.1007/s00285-004-0285-3.

[4]

K. O. Friedrichs and P. D. Lax, Boundary value problems for first order operators, Communications on Pure and Applied Mathematics, 18 (1965), 355-388. doi: 10.1002/cpa.3160180127.

[5]

W. Hancock, Bidirectional cargo transport: Moving beyond tug of war, Nature Reviews Molecular Cell Biology, 15 (2014), 615-628. doi: 10.1038/nrm3853.

[6]

T. Hillen and H. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM Journal on Applied Mathematics, 61 (2000), 751-775. doi: 10.1137/S0036139999358167.

[7]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[8]

M. Kneussel and W. Wagner, Myosin motors at neuronal synapses: Drivers of membrane transport and actin dynamics, Nature Reviews Neuroscience, 14 (2013), 233-247. doi: 10.1038/nrn3445.

[9]

A. Kunwar, S. Tripathy, J. Xu, M. Mattson, P. Anand, R. Sigua, M. Vershinin, R. McKenney, C. Yu, A. Mogilner and S. Gross, Mechanical stochastic tug-of-war models cannot explain bidirectional lipid-droplet transport, Proceedings of the National Academy of Sciences of the United States of America, 108 (2011), 18960-18965. doi: 10.1073/pnas.1107841108.

[10]

A. Kuznetsov, Modelling active transport in drosophila unipolar motor neurons, Computer Methods in Biomechanics and Biomedical Engineering, 14 (2011), 1117-1131. doi: 10.1080/10255842.2010.515983.

[11]

A. Kuznetsov and K. Hooman, Modeling traffic jams in intracellular transport in axons, International Journal of Heat and Mass Transfer, 51 (2008), 5695-5699, Biomedical-Related Special Issue. doi: 10.1016/j.ijheatmasstransfer.2008.04.022.

[12]

I. Maly, Diffusion approximation of the stochastic process of microtubule assembly, Bulletin of Mathematical Biology, 64 (2002), 213-238. doi: 10.1006/bulm.2001.0265.

[13]

D. Smith and R. Simmons, Models of motor-assisted transport of intracellular particles, Biophysical Journal, 80 (2001), 45-68. doi: 10.1016/S0006-3495(01)75994-2.

[14]

M. Stone, F. Roegiers and M. Rolls, Microtubules have opposite orientation in axons and dendrites of drosophila neurons, Molecular Biology of the Cell, 19 (2008), 4122-4129. doi: 10.1091/mbc.E07-10-1079.

[15]

R. Vale, The molecular motor toolbox for intracellular transport, Cell, 112 (2003), 467-480. doi: 10.1016/S0092-8674(03)00111-9.

[16]

W. J. Walter, V. Beránek, E. Fischermeier and S. Diez, Tubulin acetylation alone does not affect kinesin-1 velocity and run length in vitro, PLoS ONE, 7 (2012), e42218.

[17]

F. Wanka and E. Van Zoelen, Cellular organelle transport and positioning by plasma streaming, Cellular and Molecular Biology Letters, 8 (2003), 1035-1045.

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