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Mathematical and Computational Methods for Studying Energy Transduction in Protein Motors

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Abstract

Protein motors play a central role in many cellular functions. Due to the small size of these molecular motors, their motion is dominated by high viscous friction and large thermal fluctuations. There are many levels of modeling molecular motors: from simple chemical kinetic models with a small number of discrete states to all atom molecular dynamics simulations. Here we describe a mathematical framework for an intermediate level of description. In this approach the major conformational changes of the motor protein are treated as continuous motions and changes in the chemical state of the motor are modeled as discrete Markov transitions. We discuss a numerical method for solving the Fokker-Planck equations that result from this mathematical framework and describe its extension to solving motor-cargo systems. We show that when the potential is discontinuous, detailed balance is a necessary condition for numerical convergence. We study the behavior of a motor-cargo system where the motor is driven by a tilted periodic potential. In particular, we derive a formula for the effective diffusion coefficient in the weak spring limit and analytically show that if the ratio of the motor size to that of the cargo is sufficiently large, then the velocity does not obtain its maximum value in the weak spring limit.

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References

  1. H. C. Berg, Random walks in biology (Princeton University Press, Princeton, N.J., 1993).

    Google Scholar 

  2. H. Wang and G. Oster, Energy transduction in the F1 motor of ATP synthase. Nature 396:279–282 (1998).

    Article  ADS  Google Scholar 

  3. G. Oster and H. Wang, Reverse engineering a protein: the mechanochemistry of ATP synthase. Biochimica et Biophysica Acta (Bioenergetics) 1458:482–510 (2000).

    Article  Google Scholar 

  4. C. S. Peskin, G. M. Odell and G. Oster, Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. J. 65:316–324 (1993).

    Google Scholar 

  5. T. Elston, H. Wang and G. Oster, Energy transduction in ATP synthase. Nature 391:510–514 (1998).

    Article  ADS  Google Scholar 

  6. A. Mogilner and G. Oster, The polymerization ratchet model explains the force-velocity relation for growing microtubules. Eur. J. Biophys. 28:235–242 (1999).

    Article  Google Scholar 

  7. R. Astumian, Thermodynamics and Kinetics of a Brownian Motor. Science 276:917–922 (1997).

    Article  Google Scholar 

  8. P. Reimann, Brownian motors: noisy transport far from equilibrium. Phy. Rep. 361:57–265 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. K. Visscher, M. Schnitzer and S. Block, Single kinesin molecules studied with a molecular force clamp. Nature 400:184–189 (1999).

    Article  ADS  Google Scholar 

  10. T. Elston and C. Peskin, The role of protein exibility in molecular motor function: Coupled diffusion in a tilted periodic potential. SIAM J. Appl. Math. 60:842–867 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Xing, F. Bai, R. Berry and G. Oster, The torque-speed relationship of the bacterial flagellar motor. Proc. Natl. Acad. Sci. 103:1260–1265 (2006).

    Article  ADS  Google Scholar 

  12. A. Raj and C. S. Peskin, The influence of chromosome flexibility on chromosome transport during anaphase A. Proc. Natl. Acad. Sci. 103:5349–5354 (2006).

    Article  ADS  Google Scholar 

  13. S. M. Block and H. C. Berg, Successive incorporation of force-generating units in the bacterial rotary motor. Nature 309:470–473 (1984).

    Article  ADS  Google Scholar 

  14. H. C. Berg, The rotary motor of bacterial flagella. Annu. Rev. Biochem. 72:19–54 (2003).

    Article  Google Scholar 

  15. R. Vale, Kinesin: possible biological roles for a new microtubule motor. TIBS 11:464–468 (1986).

    Google Scholar 

  16. J. Howard, A. J. Hudspeth and R. D. Vale, Movement of microtubules by single kinesin molecules. Nature 342:154–158 (1989).

    Article  ADS  Google Scholar 

  17. C. Coppin, D. Pierce, L. Hsu and R. Vale, The Load Dependence of Kinesin's Mechanical Cycle. Proc. Natl. Acad. Sci. USA 94:8539–8544 (1997).

    Article  ADS  Google Scholar 

  18. J. Abrahams, A. Leslie, R. Lutter, and J. Walker, Structure at 2.8Å resolution of F1-ATPase from bovine heart mitochondria. Nature 370:621–628 (1994).

    Article  ADS  Google Scholar 

  19. D. Sabbert, S. Engelbrecht and W. Junge, Intersubunit Rotation in Active F-ATPase. Nature 381:623–625 (1996).

    Article  ADS  Google Scholar 

  20. H. Noji, R. Yasuda, M. Yoshida and K. Kinosita, Direct observation of the rotation of F1-ATPase. Nature 386:299–302 (1997).

    Article  ADS  Google Scholar 

  21. R. I. Menz, J. E. Walker and A. G. Leslie, Structure of Bovine Mitochondrial F1-ATPase with Nucleotide Bound to All Three Catalytic Sites: Implications for the Mechanism of Rotary Catalysis. Cell 106:331–341 (2001).

    Article  Google Scholar 

  22. J. Prost, J. Chauwin, L. Peliti and A. Ajdari, Asymmetric Pumping of Particles. Phys. Rev. Lett. 72:2652–2655 (1994).

    Article  ADS  Google Scholar 

  23. F. Julicher, A. Ajdari and J. Prost, Modeling molecular motors. Rev. Mod. Phy. 69:1269–1281 (1997).

    Article  ADS  Google Scholar 

  24. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw Hill, New York, 1965).

    Google Scholar 

  25. H. Risken, The Fokker-Planck Equation, 2 nd ed. (Springer-Verlag, New York, 1989).

    MATH  Google Scholar 

  26. N. P. Money, More g's than the Space Shuttle: ballistospore discharge. Mycologia 90:547–558 (1998).

    Article  Google Scholar 

  27. M. Fischer, J. Cox, D. J. Davis, A. Wagner, R. Taylor, A. M. Huerta and N. P. Money, New information on the mechanism of forcible ascospore discharge from Ascobolus immersus. Fungal Genet. Biol. 41:698–707 (2004).

    Article  Google Scholar 

  28. P. Dimroth, H. Wang, M. Grabe and G. Oster, Energy transduction in the sodium F-ATPase of Propionigenium modestum. Proc. Natl. Acad. Sci. USA 96:4924–4929 (1999).

    Article  ADS  Google Scholar 

  29. P. Boyer, The binding change mechanism for ATP synthase–some probabilities and possibilities. Biochim. Biophys. Acta 1140:215–250 (1993).

    Article  ADS  Google Scholar 

  30. J. Weber and A. E. Senior, Catalytic mechanism of F1-ATPase. Biochim. Biophys. Acta 1319:19–58 (1997).

    Article  Google Scholar 

  31. P. Boyer, The ATP synthase–-a splendid molecular machine. Annu. Rev. Biochem. 66:717–749 (1997).

    Article  Google Scholar 

  32. C. Gardiner, Handbook of Stochastic Methods, 2 nd ed. (Springer-Verlag, New York, 1985).

    Google Scholar 

  33. C. S. Peskin, G. B. Ermentrout and G. F. Oster, in Mechanochemical coupling in ATPase motors (Springer-Verlag, Les Houches, 1994).

    Google Scholar 

  34. C. Doering, W. Horsthemke and J. Riordan, Nonequilibrium fluctuation-induced transport. Phys. Rev. Lett. 72:2984–2987 (1994).

    Article  ADS  Google Scholar 

  35. H. Wang and G. Oster, The Stokes efficiency for molecular motors and its applications. Europhy. Lett. 57:134–140 (2002).

    Article  ADS  Google Scholar 

  36. G. Oster and H. Wang, Rotary protein motors. Trends Cell Biol. 13:114–121 (2003).

    Article  Google Scholar 

  37. M. O. Magnasco, Forced thermal ratchets. Phys. Rev. Lett. 71:1477–1481 (1993).

    Article  ADS  Google Scholar 

  38. M. O. Magnasco, Correlations in cellular patterns. Philosophical Magazine B 69:397–429 (1994).

    Google Scholar 

  39. D. Keller and C. Bustamante, The mechanochemistry of molecular motors. Biophys. J. 78:541–556 (2000).

    Article  Google Scholar 

  40. H. Wang, C. Peskin and T. Elston, A Robust numerical algorithm for studying biomolecular transport processes. J. Theo. Biol. 221:491–511 (2003).

    Article  MathSciNet  Google Scholar 

  41. T. Elston and C. Doering, Numerical and analytical studies of nonequilibrium fuctuation induced transport processes. J. Stat. Phys. 83:359–383 (1996).

    Article  ADS  Google Scholar 

  42. J. Xing, H. Wang and G. Oster, From Continuum Fokker-Planck Models to Discrete Kinetic Models. Biophys. J. 89:1551–1563 (2005).

    Article  Google Scholar 

  43. L. Stryer, Biochemistry, 4th ed. (W. H. Freeman, New York, 1995).

    Google Scholar 

  44. A. Katchalsky and P. Curran, Nonequilibrium Thermodynamics in Biophysics (Harvard University Press, Cambridge, MA, 1965).

    Google Scholar 

  45. D. K. Kondepudi and I. Prigogine, Modern thermodynamics (John Wiley, New York, 1998).

    MATH  Google Scholar 

  46. T. Elston, D. You and C. Peskin, Protein exibility and the correlation ratchet. SIAM J. Appl. Math. 61:776–791 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  47. J. Fricks, H. Wang and T. Elston, A numerical algorithm for investigating the role of the motor-cargo linkage in molecular motor driven transport. J. Theo. Biol. 239:33–48 (2006).

    Article  MathSciNet  Google Scholar 

  48. S. M. Block, C. L. Asbury, J. W. Shaevitz and M. J. Lang, Probing the kinesin reaction cycle with a 2D optical force clamp. Proc. Natl. Acad. Sci. USA 100:2351–2356 (2003).

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA, 95064, USA

    Hongyun Wang

  2. Department of Pharmacology, University of North Carolina, Chapel Hill, NC, 27599, USA

    Timothy C. Elston

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  1. Hongyun Wang
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  2. Timothy C. Elston
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Correspondence to Hongyun Wang.

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Wang, H., Elston, T.C. Mathematical and Computational Methods for Studying Energy Transduction in Protein Motors. J Stat Phys 128, 35–76 (2007). https://doi.org/10.1007/s10955-006-9169-9

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  • DOI: https://doi.org/10.1007/s10955-006-9169-9

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