×

Transport in a molecular motor system. (English) Zbl 1077.35060

Summary: Intracellular transport in eukarya is attributed to motor proteins that transduce chemical energy into directed mechanical energy. This suggests that, in nonequilibrium systems, fluctuations may be oriented or organized to do work. Here we seek to understand how this is manifested by quantitative mathematical portrayals of these systems.

MSC:

35K57 Reaction-diffusion equations
34D23 Global stability of solutions to ordinary differential equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92C37 Cell biology
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] A. Ajdari and J. Prost , Mouvement induit par un potentiel périodique de basse symétrie : diélectrophorèse pulse . C. R. Acad. Sci. Paris II 315 ( 1992 ) 1653.
[2] R.D. Astumian , Thermodynamics and kinetics of a Brownian motor . Science 276 ( 1997 ) 917 - 922 .
[3] J.-D. Benamou and Y. Brenier , A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem . Numer. Math. 84 ( 2000 ) 375 - 393 . Zbl 0968.76069 · Zbl 0968.76069 · doi:10.1007/s002110050002
[4] M. Chipot , D. Kinderlehrer and M. Kowalczyk , A variational principle for molecular motors . Meccanica 38 ( 2003 ) 505 - 518 . Zbl 1032.92005 · Zbl 1032.92005 · doi:10.1023/A:1024719028273
[5] C. Doering , B. Ermentrout and G. Oster , Rotary DNA motors . Biophys. J. 69 ( 1995 ) 2256 - 2267 .
[6] J. Dolbeault , D. Kinderlehrer and M. Kowalczyk , Remarks about the flashing rachet , in Proc. PASI 2003 (to appear). MR 2091497 | Zbl 1064.35065 · Zbl 1064.35065
[7] D.D. Hackney , The kinetic cycles of myosin, kinesin, and dynein . Ann. Rev. Physiol. 58 ( 1996 ) 731 - 750 .
[8] S. Hastings and D. Kinderlehrer , Remarks about diffusion mediated transport: thinking about motion in small systems . (to appear). MR 2159989 | Zbl 1099.35005 · Zbl 1099.35005
[9] J. Howard , Mechanics of Motor Proteins and the Cytoskeleton . Sinauer Associates, Inc. ( 2001 ).
[10] A.F. Huxley , Muscle structure and theories of contraction . Prog. Biophys. Biophys. Chem. 7 ( 1957 ) 255 - 318 .
[11] R. Jordan , D. Kinderlehrer and F. Otto , The variational formulation of the Fokker-Planck equation . SIAM J. Math. Anal. 29 ( 1998 ) 1 - 17 . Zbl 0915.35120 · Zbl 0915.35120 · doi:10.1137/S0036141096303359
[12] D. Kinderlehrer and M. Kowalczyk , Diffusion-mediated transport and the flashing ratchet . Arch. Rat. Mech. Anal. 161 ( 2002 ) 149 - 179 . Zbl 1065.76183 · Zbl 1065.76183 · doi:10.1007/s002050100173
[13] D. Kinderlehrer and N. Walkington , Approximation of parabolic equations based upon Wasserstein’s variational principle . ESAIM: M2AN 33 ( 1999 ) 837 - 852 . Numdam | Zbl 0936.65121 · Zbl 0936.65121 · doi:10.1051/m2an:1999166
[14] J.S. Muldowney , Compound matrices and ordinary differential equations . Rocky Mountain J. Math. 20 ( 1990 ) 857 - 872 . Article | Zbl 0725.34049 · Zbl 0725.34049 · doi:10.1216/rmjm/1181073047
[15] Y. Okada and N. Hirokawa , A processive single-headed motor: kinesin superfamily protein KIF1A . Science 283 ( 1999 ) 19.
[16] Y. Okada and N. Hirokawa , Mechanism of the single headed processivity: diffusional anchoring between the K-loop of kinesin and the C terminus of tubulin , in Proc. Nat. Acad. Sciences 7 ( 2000 ) 640 - 645 .
[17] F. Otto , Dynamics of labyrinthine pattern formation: a mean field theory . Arch. Rat. Mech. Anal. 141 ( 1998 ) 63 - 103 . Zbl 0905.35068 · Zbl 0905.35068 · doi:10.1007/s002050050073
[18] F. Otto , The geometry of dissipative evolution equations: the porous medium equation . Comm. PDE 26 ( 2001 ) 101 - 174 . Zbl 0984.35089 · Zbl 0984.35089 · doi:10.1081/PDE-100002243
[19] P. Palffy-Muhoray , T. Kosa and E. Weinan , Dynamics of a light driven molecular motor . Mol. Cryst. Liq. Cryst. 375 ( 2002 ) 577 - 591 .
[20] A. Parmeggiani , F. Jülicher , A. Ajdari and J. Prost , Energy transduction of isothermal ratchets: generic aspects and specific examples close and far from equilibrium . Phys. Rev. E 60 ( 1999 ) 2127 - 2140 .
[21] C.S. Peskin , G.B. Ermentrout and G.F. Oster , 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 York ( 1995 ).
[22] M. Protter and H. Weinberger , Maximum principles in differential equations , Prentice Hall, Englewood Cliffs, N.J. ( 1967 ). MR 219861 | Zbl 0153.13602 · Zbl 0153.13602
[23] P. Reimann , Brownian motors: noisy transport far from equilibrium . Phys. Rep. 361 ( 2002 ) 57 - 265 . Zbl 1001.82097 · Zbl 1001.82097 · doi:10.1016/S0370-1573(01)00081-3
[24] M. Schliwa , Molecular Motors . Wiley-VCH Verlag, Wennheim ( 2003 ).
[25] B. Schwarz , Totally positive differential systems . Pacific J. Math. 32 ( 1970 ) 203 - 230 . Article | Zbl 0193.04501 · Zbl 0193.04501 · doi:10.2140/pjm.1970.32.203
[26] A. Tudorascu , A one phase Stefan problem via Monge-Kantorovich theory . CNA Report 03-CNA-007. · Zbl 1072.35079
[27] R.D. Vale and R.A. Milligan , The way things move: looking under the hood of motor proteins . Science 288 ( 2000 ) 88 - 95 .
[28] C. Villani , Topics in optimal transportation , Providence. AMS Graduate Studies in Mathematics 58 ( 2003 ). MR 1964483 | Zbl 1106.90001 · Zbl 1106.90001
[29] E. Zeidler , Nonlinear functional analysis and its applications . I Springer, New York ( 1986 ). MR 816732 | Zbl 0583.47050 · Zbl 0583.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.