A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether

Original Article


The kinesin family of motor proteins are involved in a variety of cellular processes that transport materials and generate force. With recent advances in experimental techniques, such as optical tweezers can probe individual molecules, there has been an increasing interest in understanding the mechanisms by which motor proteins convert chemical energy into mechanical work. Here we present a mathematical model for the chemistry and three dimensional mechanics of the kinesin motor protein which captures many of the force dependent features of the motor. For the elasticity of the tether that attaches cargo to the motor we develop a method for deriving the non-linear force-extension relationship from optical trap data. For the kinesin heads, cargo, and microscope stage we formulate a three dimensional Brownian Dynamics model that takes into account excluded volume interactions. To efficiently compute statistics from the model, an algorithm is proposed which uses a two step protocol that separates the simulation of the mechanical features of the model from the chemical kinetics of the model. Using this approach for a bead transported by the motor, the force dependent average velocity and randomness parameter are computed and compared with the experimental data.


Molecular Motor Protein Kinesin Brownian Dynamics Stochastic Processes Statistical Mechanics 


  1. Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walker, P., 2002. Molecular Biology of the Cell. Garland Publishing, New York.Google Scholar
  2. Amos, L.A., 2000. Focusing-in on microtubule. Curr. Opin. Struc. Biol. 10, 236–241.CrossRefGoogle Scholar
  3. Astumian, R.D., Derenyi, I., 1999. A chemically reversible brownian motor: Application to kinesin and ncd. Biophys. J. 77, 993–1002.Google Scholar
  4. Berliner, E., 1995. Failure of a single-headed kinesin to track parallel to microtubule protofilaments. Nature 373(23), 718–721.CrossRefGoogle Scholar
  5. Block, S., Asbury, C., Shaevitz, J., Lang, M., 2003. Probing the kinesin reaction cycle with a 2d optical force clamp. Proc. Natl. Acad. Sci. U.S.A. 100, 2351–2356.CrossRefGoogle Scholar
  6. Bustamante, C., Keller, D., Oster, G., 2001. The physics of molecular motors. Acc. Chem. Res. 34(6), 412–420.CrossRefGoogle Scholar
  7. Case, R.B., Rice, S., Hart, C.L., Ly, B., Vale, R., 2000. Role of the kinesin neck linker and catalytic core in microtubule-based motility. Curr. Biol. 10, 157–160.CrossRefGoogle Scholar
  8. Chen, Y., Yan, B., Rubin, R.J., 2002. Fluctuations and randomness of movement of bead powered by a single kinesin molecule in a force-clamped motility array: Monte-carlo simulations. Biophys. J. 83, 2360–2369.Google Scholar
  9. Coppin, C., Finer, J.T., Spudich, J.A., Vale, R.D., 1996. Detection of sub-8-nm movements of kinesin by high-resolution optica-trap microscopy. Proc. Natl. Acad. Sci. U.S.A. 93, 1913–1817.CrossRefGoogle Scholar
  10. Coppin, C., Pierce, D., Hsu, L., Vale, R., 1997. The load dependence of kinesin's mechanical cycle. Proc. Natl. Acad. Sci. U.S.A. 94, 8539–8544.CrossRefGoogle Scholar
  11. Coy, D.L., Wagenbach, M., Howard, J., 1999. Kinesin takes one 8-nm step for each atp that it hydrolyzes. J. Biol. Chem. 274, 3667–3671.CrossRefGoogle Scholar
  12. Cross, R.A., 2004. The kinetic mechanism of kinesin. Trends Biochem. Sci. 29(6), 301–307.CrossRefGoogle Scholar
  13. Cross, R.A., Crevel, I., Carter, N.J., Alonso, M.C., Hirose, K., Amos, L.A., 2000. The conformational cycle of kinesin. Philos. Trans. R. Soc. Lond. Ser. B 355, 459–464.CrossRefGoogle Scholar
  14. Downing, K., Nogales, E., 1998. Tubulin and microtubule structure. Curr. Opin. Cell Biol. 10, 16–22.CrossRefGoogle Scholar
  15. Elston, T.C., Peskin, C.S., 2000. The role of protein flexibility in molecular motor function: Coupling diffusion in a tilted periodic potential. SIAM J. Appl. Math. 60(3), 842–867.MATHCrossRefMathSciNetGoogle Scholar
  16. Elston, T.C., You, D., Peskin, C.S., 2000. Protein flexibility and correlation ratchet. SIAM J. Appl. Math. 61(3), 776–791.MATHCrossRefMathSciNetGoogle Scholar
  17. Fisher, M.E., Kolomeisky, A.B., 2001. Simple mechanochemistry describes the dynamics of kinesin molecules. PNAS 98(14), 7748–7753.CrossRefGoogle Scholar
  18. Fox, R.F., 1998. Rectified brownian movement in molecular and cell biology. Phys. Rev. E 57(2), 2177–2203.CrossRefGoogle Scholar
  19. Gilbert, S., Johnson, K., 1995. Pathway of processive atp hydrolysis by kinesin. Nature 373, 671–676.CrossRefGoogle Scholar
  20. Goldstein, L. S.B., 2001. Molecular motors: From one motor many tails to one motor many tales. Trends Cell Biol. 11(12), 477–482.CrossRefGoogle Scholar
  21. Hoenger, A., Thormahlen, M., Diaz-Avalos, R., Doerhoefer, M., Goldie, K.N., Muller, J., Mandelkow, E., 2000. A new look at the microtubule binding patterns of dimeric kinesins. J. Mol. Biol. 297, 1087–1103.CrossRefGoogle Scholar
  22. Howard, J., 2001. Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates, Sunderland.Google Scholar
  23. Julicher, F., Ajdari, A., Prost, J., 1997. Modeling molecular motors. Rev. Modern Phys.s 69(4), 1269–1281.CrossRefGoogle Scholar
  24. Kamal, A., Goldstein, L.S., 2002. Principles of cargo attachments to cytoplasmic motor proteins. Curr. Opin. Cell Biol. 14, 63–68.CrossRefGoogle Scholar
  25. Karsenti, E., Vernos, I., October 2001. The mitotic spindle: A self-made machine. Science 294(5542), 543–547.CrossRefGoogle Scholar
  26. Kikkawa, M., Sablin, E.P., Okada, Y., Yajima, H., Fletterick, R.J., Hirokawa, N., 2001. Switch-based mechanisms of kinesin motors. Nature 411, 439.CrossRefGoogle Scholar
  27. Kloeden, P.E., Platen, E., 1992. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin.MATHGoogle Scholar
  28. Knight, A.E., Molloy, J.E., 1999. Coupling atp hydrolysis to mechanical work. Nat. Cell Biol. 1(4), E87–E89.CrossRefGoogle Scholar
  29. Kozielski, F., Sack, S., Marx, A., Thormahlen, M., Schonbrum, E., Biou, V., Thompson, A., Mandelkow, E.M., Mandelkow, E., 1997. The crystal structure of dimeric kinesin and implications for microtubule-dependent motility. Cell 91, 985.CrossRefGoogle Scholar
  30. Kull, F.J., Sablin, E.P., Lau, R., Fletterick, R.J., Vale, R.D., 1996. Crystal structure of the kinesin motor domain reveals a structural similarity to myosin. Nature 380, 550–555.CrossRefGoogle Scholar
  31. Landau, D., Binder, K., 2000. A Guide to Monte-Carlo Simulations in Statistical Physics. Cambridge University Press, Camridge.MATHGoogle Scholar
  32. Li, J., Pfister, K., Brady, S., Dahlstrom, A., 1999. Axonal transport and distribution of immunologically distinct kinesin heavy chains in rat neurons. J. Neurosci. Res. 58, 226–241.CrossRefGoogle Scholar
  33. Maes, C., van Wieren, M.H., 2003. A markov model for kinesin. J. Stat. Phys.s 112(112), 329–355.MATHMathSciNetGoogle Scholar
  34. Mandelkow, E., Hoenger, A., 1999. Structures of kinesin and kinesin-microtubule interactions. Curr. Opin. Cell Biol. 11, 34–44.CrossRefGoogle Scholar
  35. Mogilner, A., Fisher, A., Baskin, R., 2001. Structural changes in the neck linker of kinesin explain the load dependence of the motor's mechanical cycle. J. Theor. Biol. 211(2), 143–157.CrossRefGoogle Scholar
  36. Nishiyama, M., Muto, E., Inoue, Y., Yanagida, T., Higuchi, H., 2001. Substeps within the 8 nm step of the atpase cycle of single kinesin molecules. Nat. Cell Biol. 3.Google Scholar
  37. Nogales, E., Whittaker, M., Milligan, R., Downing, K., 1999. High-resolution model of the microtubule. Cell 96, 79–88.CrossRefGoogle Scholar
  38. Oksendal, B., 2000. Stochastic Differential Equations: An Introduction with Applications. Springer-Verlag, Berlin.Google Scholar
  39. Peskin, C.S., Odell, G.M., Oster, G.F., 1993. Cellular motions and thermal fluctuations: The brownian ratchet. Biophys. J. 65, 316–324.CrossRefGoogle Scholar
  40. Peskin, C., Oster, G., 1995. Coordinated hydrolysis explains the mechanical behavior of kinesin. Biophys. J. 68, 202–211.Google Scholar
  41. Ray, S., Meyhöfer, E., Milligan, R., Howard, J., 1993. Kinesin follows the microtubule's protofilament axis. J. Cell Biol. 121, 1083–1093.CrossRefGoogle Scholar
  42. Reichl, L.E., 1998. A Modern Course in Statistical Physics. Wiley, New York.MATHGoogle Scholar
  43. Rice, S., Cui, Y., Sindelar, S., Naber, N., Matuska, M., Vale, R., Cooke, R., 2003. Thermodynamic properties of the kinesin neck-region docking to the catalytic core. Biophys. J. 84, 1844–1854.Google Scholar
  44. Rice, S., Lin, A.W., Safer, D., Hart, C., Naber, N., Carragher, B., Cain, S., Pechatnikova, E., Wilson-Kubalek, E.W., Whittaker, M., Pate, E., Cooke, R., Taylor, E.W., Milligan, R., Vale, R., 1999. A structural change in the kinesin motor protein that drives motility. Nature 402(6763), 778–784.CrossRefGoogle Scholar
  45. Ross, S., 1995. Stochastic Processes. Wiley, New York.Google Scholar
  46. Sablin, E.P., Fletterick, R.J., 2001. Nucleotide switchets in molecular motors: Structural analysis of kinesins and myosins. Current Opin. Struc. Biol. 11, 716–724.CrossRefGoogle Scholar
  47. Sack, S., Muller, J., Marx, A., Thormahlen, M., Mandelkow, E.M., Brady, S.T., Mandelkow, E., 1997. X-ray structure of motor and neck domains from rat brain kinesin. Biochemistry 36, 16155.CrossRefGoogle Scholar
  48. Schliwa, M., Woehlke, G., 2001. Switching on kinesin. Nature 411, 424–425.CrossRefGoogle Scholar
  49. Sharp, D., Rogers, G., Scholey, J., 2000. Microtubule motors in mitosis. Nature 407, 41–45.CrossRefGoogle Scholar
  50. Sindelar, C., Budny, M., Rice, S., Naber, N., Fletterick, R., Cooke, R., 2002. Two conformations in the human kinesin power stroke defined by X-ray crystallography and epr spectroscopy. Nat. Struc. Biol. 9(11), 844–848.Google Scholar
  51. Song, Y.H., Marx, A., Muller, J., Woehlke, G., Schliwa, M., Krebs, A., Hoenger, A., Mandelkow, E., 2001. Structure of fast kinesin: Implications for atpase mechanisms and interactions with microtubules. EMBO J. 20, 6213.CrossRefGoogle Scholar
  52. Svoboda, K., Block, S., 1994. Force and velocity measured for single kinesin molecules. Cell 77, 773–784.CrossRefGoogle Scholar
  53. Svoboda, K., Mitra, P., Block, S., 1994b. Fluctuation analysis of motor protein movement and single enzyme kinetics. Proc. Natl. Acad. Sci. U.S.A. Vol. 91.Google Scholar
  54. Svoboda, K., Schmidt, C.F., Schnapp, B.J., Block, S.M., 1993. Direct observation of kinesin stepping by optical trapping interferometry. Nature 365, 721–727.CrossRefGoogle Scholar
  55. Tuma, C., Zill, A., Bot, N.L., Vernos, I., Gelfand, V., December 1998. Heterotrimeric kinesin ii is the microtubule motor protein responsible for pigment dispersion in xenopus melanophores. J. Cell Biol. 143(6), 1547–1558.CrossRefGoogle Scholar
  56. Vale, R., Fletterick, R., 1997. The design plan of kinesin motors. Annu. Rev. Cell Dev. Biol. 13, 745–777.CrossRefGoogle Scholar
  57. Visscher, K., Schnitzer, M., Block, S.M., July 1999. Single kinesin molecules studies with a molecular force clamp. Nature 400, 184–189.Google Scholar
  58. Woehlke, G., Ruby, A., Hart, C., Ly, B., Hom-Booher, N., Vale, R., 1997. Microtubule interaction site of the kinesin motor. Cell 90(2), 207–216.CrossRefGoogle Scholar
  59. Yun, M., Bronner, C.E., Park, C.G., Cha, S.S., Park, H.W., Endow, S.A., 2003. Rotation of the stalk/neck and one head in a new crystal structure of the kinesin motor protein. EMBO J. 22, 1.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations