American Institute of Mathematical Sciences

Advanced
July  2006, 6(4): 711-734. doi: 10.3934/dcdsb.2006.6.711

Molecular motors, Brownian ratchets, and reflected diffusions

Amarjit Budhiraja 1, and  John Fricks 2,

1. 

Department of Statistics & Operations Research, University of North Carolina, Chapel Hill, NC 27599-3250, United States
2. 

Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514, United States

Received  January 2005 Revised  September 2005 Published  April 2006

Molecular motors are protein structures that play a central role in accomplishing mechanical work inside a cell. While chemical reactions fuel this work, it is not exactly known how this chemical-mechanical conversion occurs. Recent advances in microbiological techniques have enabled at least indirect observations of molecular motors which in turn have led to significant effort in the mathematical modeling of these motors in the hope of shedding light on the underlying mechanisms involved in intracellular transport. Kinesin which moves along microtubules that are spread throughout the cell is a prime example of the type of motors that are studied in this work. The motion is linked to the presence of a chemical, ATP, but how the ATP is involved in motion is not clearly understood. One commonly used model for the dynamics of kinesin in the biophysics literature is the Brownian ratchet mechanism. In this work, we give a precise mathematical formulation of a Brownian ratchet (or more generally a diffusion ratchet) via an infinite system of stochastic differential equations with reflection. This formulation is seen to arise in the weak limit of a natural discrete space model that is often used to describe motor dynamics in the literature. Expressions for asymptotic velocity and effective diffusivity of a biological motor modeled via a Brownian ratchet are obtained. Linearly progressive biomolecular motors often carry cargos via an elastic linkage. A two-dimensional coupled stochastic dynamical system is introduced to model the dynamics of the motor-cargo pair. By proving that an associated two dimensional Markov process has a unique stationary distribution, it is shown that the asymptotic velocity of a motor pulling a cargo is well defined as a certain Law of Large Number limit, and finally an expression for the asymptotic velocity in terms of the invariant measure of the Markov process is obtained.
Keywords: Flashing ratchet., Kinesin, Molecular motors, Brownian ratchets, Reflected diffusions, Asymptotic velocity.
Mathematics Subject Classification: Primary: 60G35, 92B05; Secondary: 93E1.
Citation: Amarjit Budhiraja, John Fricks. Molecular motors, Brownian ratchets, and reflected diffusions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 711-734. doi: 10.3934/dcdsb.2006.6.711
[1]

Dmitry Vorotnikov. The flashing ratchet and unidirectional transport of matter. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 963-971. doi: 10.3934/dcdsb.2011.16.963
[2]

David Heath, David Kinderlehrer, Michal Kowalczyk. Discrete and continuous ratchets: from coin toss to molecular motor. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 153-167. doi: 10.3934/dcdsb.2002.2.153
[3]

Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001
[4]

Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245
[5]

Tzong-Yow Lee. Asymptotic results for super-Brownian motions and semilinear differential equations. Electronic Research Announcements, 1998, 4: 56-62.

[6]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[7]

Hongyong Zhao, Qianjin Zhang, Linhe Zhu. The spatial dynamics of a zebrafish model with cross-diffusions. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1035-1054. doi: 10.3934/mbe.2017054
[8]

Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090
[9]

Le Li, Lihong Huang, Jianhong Wu. Flocking and invariance of velocity angles. Mathematical Biosciences & Engineering, 2016, 13 (2) : 369-380. doi: 10.3934/mbe.2015007
[10]

Han Wu, Changfan Zhang, Jing He, Kaihui Zhao. Distributed fault-tolerant consensus tracking for networked non-identical motors. Journal of Industrial & Management Optimization, 2017, 13 (2) : 917-929. doi: 10.3934/jimo.2016053
[11]

Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091
[12]

Irina Berezovik, Carlos García-Azpeitia, Wieslaw Krawcewicz. Symmetries of nonlinear vibrations in tetrahedral molecular configurations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2473-2491. doi: 10.3934/dcdsb.2018261
[13]

Hirotada Honda. On a model of target detection in molecular communication networks. Networks & Heterogeneous Media, 2019, 14 (4) : 633-657. doi: 10.3934/nhm.2019025
[14]

G. Wei, P. Clifford. Analysis and numerical approximation of a class of two-way diffusions. Communications on Pure & Applied Analysis, 2003, 2 (1) : 91-99. doi: 10.3934/cpaa.2003.2.91
[15]

Atsushi Yagi. Exponential attractors for competing species model with cross-diffusions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1091-1120. doi: 10.3934/dcds.2008.22.1091
[16]

Patrick Guidotti. A family of nonlinear diffusions connecting Perona-Malik to standard diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 581-590. doi: 10.3934/dcdss.2012.5.581
[17]

Kimun Ryu, Inkyung Ahn. On certain elliptic systems with nonlinear self-cross diffusions. Conference Publications, 2003, 2003 (Special) : 752-759. doi: 10.3934/proc.2003.2003.752
[18]

Changbing Hu. Stability of under-compressive waves with second and fourth order diffusions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 629-662. doi: 10.3934/dcds.2008.22.629
[19]

Takashi Suzuki. Brownian point vortices and dd-model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 161-176. doi: 10.3934/dcdss.2014.7.161
[20]

Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences