Advertisement

Remarks About Diffusion Mediated Transport: Thinking About Motion in Small Systems

  • S. Hastings
  • D. Kinderlehrer
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

We describe a dissipation principle/variational principle which may be useful in modeling motion in small viscous systems and provide brief illustrations to brownian motor or molecular rachet situations which are found in intracellular transport. Monge-Kantorovich mass transport and Wasserstein metric play an interesting role in these developments. Some properties of the system that ensure the presence of transport are discussed.

Keywords

Variational Principle Molecular Motor Period Interval Logarithmic Sobolev Inequality Brownian Motor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adjari, A. and Prost, J. (1992) Mouvement induit par un potentiel périodique de basse symétrie: dielectrophorese pulse, C. R. Acad. Sci. Paris t. 315, Serie II, 1653.Google Scholar
  2. [2]
    Astumian, R.D. (1997) Thermodynamics and kinetics of a Brownian motor, Science 276 (1997), 917–922.CrossRefGoogle Scholar
  3. [3]
    Benamou, J.-D. and Brenier, Y. (2000) A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math. 84, 375–393.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Chipot, M., Hastings, S., and Kinderlehrer, D., to appearGoogle Scholar
  5. [5]
    Chipot, M., D. Kinderlehrer, D. and Kowalczyk, M. (2003) A variational principle for molecular motors, Meccanica, 38, 505–518zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Dolbeault, J., Kinderlehrer, D., and Kowalczyk, M. Remarks about the flashing rachet, to appear Proc. PASI 2003Google Scholar
  7. [7]
    Heath, D., Kinderlehrer, D. and Kowalczyk, M. (2002) Discrete and continuous ratchets: from coin toss to molecular motor, Discrete and continuous dynamical systems Ser. B 2 no. 2, 153–167.zbMATHMathSciNetGoogle Scholar
  8. [8]
    Howard, J. (2001) Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associates, Inc., 2001.Google Scholar
  9. [9]
    Jordan, R., Kinderlehrer, D. and Otto, F. (1998) The variational formulation of the Fokker-Planck equation, SI AM J. Math. Anal. Vol. 29 no. 1, 1–17.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Kinderlehrer, D. and Kowalczyk, M (2002) Diffusion-mediated transport and the flashing ratchet, Arch. Rat. Mech. Anal. 161, 149–179.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Kinderlehrer, D. and Walkington, N. (1999) Approximation of parabolic equations based upon Wasserstein’s variational principle, Math. Model. Numer. Anal. (M2AN) 33 no. 4, 837–852.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Okada, Y. and Hirokawa, N. (1999) A processive single-headed motor: kinesin superfamily protein KIF1 A, Science Vol. 283, 19CrossRefGoogle Scholar
  13. [13]
    Okada, Y. and Hirokawa, N. (2000) Mechanism of the single headed processivity: diffusional anchoring between the K-loop of kinesin and the C terminus of tubulin, Proc. Nat. Acad. Sciences 7 no. 2, 640–645.CrossRefGoogle Scholar
  14. [14]
    Otto, F. (1998) Dynamics of labyrinthine pattern formation: a mean field theory, Arch. Rat. Mech. Anal. 141, 63–103zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Otto, F. (2001) The geometry of dissipative evolution equations: the porous medium equation, Comm. PDE 26, 101–174zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Otto, F. and Villani, C. (2000) Generalization of an inequality by Talagrand and links with the logarithmic Sobolev Inequality, J. Funct. Anal. 173, 361–400zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Parmeggiani, A., Jülicher, F., Adjari, A. and Prost, J. (1999) Energy transduction of isothermal ratchets: generic aspects and specific examples close and far from equilibrium, Phys. Rev. E, 60 no. 2, 2127–2140.CrossRefGoogle Scholar
  18. [18]
    Peskin, C.S. Ermentrout, G.B. and Oster, G.F. (1995) The correlation ratchet: a novel mechanism for generating directed motion by ATP hydrolysis, in Cell Mechanics and Cellular Engineering (V.C Mow et.al eds.), Springer, New YorkGoogle Scholar
  19. [19]
    Petrelli, L. and Tudorascu, A. Variational principle for general Fokker-Planck equations, to appearGoogle Scholar
  20. [20]
    Reimann, P. (2002) Brownian motors: noisy transport far from equilibrium, Phys. Rep. 361 nos. 2–4, 57–265.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Tudorascu, A. A one phase Stefan problem via Monge-Kantorovich theory, (CNA Report 03-CNA-007)Google Scholar
  22. [22]
    Vale, R.D. and Milligan, R.A. (2000) The way things move: looking under the hood of motor proteins, Science 288, 88–95.CrossRefGoogle Scholar
  23. [23]
    C. Villani (2003) Topics in optimal transportation, AMS Graduate Studies in Mathematics vol. 58, ProvidenceGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. Hastings
    • 1
  • D. Kinderlehrer
    • 2
  1. 1.Dept. of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Center for Nonlinear Analysis and Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations