Abstract
Renewal-reward processes are used to provide a framework for the mathematical description of single-molecule bead-motor assays for processive motor proteins. The formulation provides a more powerful, general approach to the fluctuation analysis of bead-motor assays begun by Svoboda et al. (Proc. Natl. Acad. Sci. USA 91(25):11782, 1994). Fluctuation analysis allows one to gain insight into the mechanochemical cycle of motor proteins purely by measuring the statistics of the displacement of the cargo (e.g., bead) the protein transports. The statistical parameters of interest are shown to be the steady-state slopes (in time) of the cumulants of the bead (the cumulant rates). The first two cumulant rates are the steady-state velocity and slope of the variance. The cumulant rates are shown to be insensitive to experimental disturbances such as the initial state of the enzyme and from the viewpoint of modeling, unaffected by substeps. Two existing models—Elston (J. Math. Biol. 41(3):189–206, 2000) and Peskin and Oster (Biophys. J. 68(4):202S–211S, 1995)—are formulated as renewal-reward processes to demonstrate the insight that the formulation affords. A key contribution of the approach is the possibility of accounting for wasted hydrolyses and backward steps in the fluctuation analysis. For example, the randomness parameter defined in the first fluctuation analysis of optical trap based bead-motor assays (Svoboda et al. in Proc. Natl. Acad. Sci. USA 91(25):11782, 1994), loses its original purpose of estimating the number of rate-determining steps in the chemical cycle when backward steps and wasted hydrolyses are present. As a simple application of our formulation, we extend the randomness parameter’s scope by showing how it can be used to infer the presence of wasted hydrolyses and backward steps with certainty. A more powerful fluctuation analysis using higher cumulant rate measurements is proposed: the method allows one to estimates the number of intermediate reactions, the average chemical rate, and the probability of stepping backward or forward. The stability of the method in the presence of measurement errors is demonstrated numerically to encourage its use in experiments.
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References
Astumian, R. D., & Haenggi, P. (2002). Brownian motors. Phys. Today, 55(11), 33–39.
Atzberger, P. J., & Peskin, C. S. (2006). A Brownian dynamics model of kinesin in three dimensions incorporating the force-extension profile of the coiled-coil cargo tether. Bull. Math. Biol., 68(1), 131–160.
Bier, M. (1997). Brownian ratchets in physics and biology. Contemp. Phys., 38(6), 371–379.
Bier, M. (2005). Modelling processive motor proteins: moving on two legs in the microscopic realm. Contemp. Phys., 46(1), 41–51.
Block, S. M. (2003). Probing the kinesin reaction cycle with a 2 D optical force clamp. Proc. Natl. Acad. Sci. USA, 100(5), 2351–2356.
Block, S. M. (2007). Kinesin motor mechanics: binding, stepping, tracking, gating, and limping. Biophys. J., 92(9), 2986–2995.
Block, S. M., Goldstein, L. S. B., & Schnapp, B. J. (1990). Bead movement by single kinesin molecules studied with optical tweezers. Nature, 348, 348–352.
Boyer, P. D. (1997). The ATP synthase-a splendid molecular machine. Annu. Rev. Biochem., 66(1), 717–749.
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. Nature, 435(7040), 308–312.
Cinlar, E. (1975). Introduction to stochastic processes. Englewood Cliffs: Prentice-Hall.
Cox, D. R., & Miller, H. D. (1977). The theory of stochastic processes. London: Chapman & Hall/CRC.
Cross, R. A. (2004). The kinetic mechanism of kinesin. Trends Biochem. Sci., 29(6), 301–309.
Elston, T. C. (2000). A macroscopic description of biomolecular transport. J. Math. Biol., 41(3), 189–206.
Fisher, M. E., & Kolomeisky, A. B. (2001). Simple mechanochemistry describes the dynamics of kinesin molecules. Proc. Natl. Acad. Sci. USA, 98(14), 7748–7753.
Geeves, M. A., & Holmes, K. C. (1999). Structural mechanism of muscle contraction. Annu. Rev. Biochem., 68(1), 687–728.
Gilbert, S. P., Webb, M. R., Brune, M., & Johnson, K. A. (1995). Pathway of processive ATP hydrolysis by kinesin. Nature, 373(6516), 671–676.
Goodson, H. V., Kang, S. J., & Endow, S. A. (1994). Molecular phylogeny of the kinesin family of microtubule motor proteins. J. Cell. Sci., 107(7), 1875.
Grimmett, G., & Stirzaker, D. (2001). Probability and random processes. Cambridge: Oxford University Press.
Guydosh, N. R., & Block, S. M. (2006). Backsteps induced by nucleotide analogs suggest the front head of kinesin is gated by strain. Proc. Natl. Acad. Sci. USA, 103(21), 8054–8059.
Hill, T. L. (1989). Free energy transduction and biochemical cycle kinetics. Berlin: Springer.
Howard, J. (1996). The movement of kinesin along microtubules. Annu. Rev. Physiol., 58(1), 703–729.
Howard, J. (1997). Molecular motors: structural adaptations to cellular functions. Nature, 389(6651), 561–567.
Howard, J. (2001). Mechanics of motor proteins and the cytoskeleton. Sunderland: Sinauer.
Howard, J., Hudspeth, A. J., & Vale, R. D. (1989). Movement of microtubules by single kinesin molecules. Nature, 342(6246), 154.
Johnson, W. P. (2002). The curious history of Faà di Bruno’s formula. Am. Math. Mon., 109, 217–234.
Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes. New York: Academic Press.
Kramers, H. A. (1940). Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7(4), 284–304.
Krishnan, A. (2008). The random walker: stochastic mechano-chemical models for motor proteins. Master’s thesis, Mechanical Engineering, University of Michigan.
Krishnan, A., & Epureanu, B. I. (2008). A stochastic mechano-chemical model for cooperative motor protein dynamics. In Proceedings of SMASIS 2008. New York: ASME.
Lindén, M., & Wallin, M. (2007). Dwell time symmetry in random walks and molecular motors. Biophys. J., 92(11), 3804–3816.
Mehta, A. (2001). Myosin learns to walk. J. Cell. Sci., 114(11), 1981.
Mehta, A. D., Rief, M., Spudich, J. A., Smith, D. A., & Simmons, R. M. (1999a). Single-molecule biomechanics with optical methods. Science, 283(5408), 1689.
Mehta, A. D., Rock, R. S., Rief, M., Spudich, J. A., Mooseker, M. S., & Cheney, R. E. (1999b). Myosin-V is a processive actin-based motor. Nature, 400(6744), 590–596.
Mogilner, A., Fisher, A. J., & Baskin, R. J. (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.
Peskin, C. S., & Oster, G. (1995). Coordinated hydrolysis explains the mechanical behavior of kinesin. Biophys. J., 68(4), 202S–211S.
Prager, T., Schimansky-Geier, L., & Sokolov, I. M. (2005). Periodic driving controls random motion of Brownian steppers. J. Phys., Condens. Matter, 17(47), 3661–3672.
Purcell, E. M. (1977). Life at low Reynolds number. Am. J. Phys., 45(1), 3–11.
Qian, H., & Elson, E. L. (2002). Single-molecule enzymology: stochastic Michaelis–Menten kinetics. Biophys. Chem., 101, 565–576.
Reimann, P., & Hänggi, P. (2002). Introduction to the physics of Brownian motors. Appl. Phys. A, Mater. Sci. Process., 75(2), 169–178.
Rice, S., Lin, A. W., Safer, D., Hart, C. L., Naber, N., Carragher, B. O., Cain, S. M., Pechatnikova, E., Wilson-Kubalek, E. M., & Whittaker, M. (1999). A structural change in the kinesin motor protein that drives motility. Nature, 402, 778–784.
Rief, M., Rock, R. S., Mehta, A. D., Mooseker, M. S., Cheney, R. E., & Spudich, J. A. (2000). Myosin-V stepping kinetics: a molecular model for processivity. Proc. Natl. Acad. Sci. USA, 97(17), 9482.
Ross, S. M. (1983). Stochastic processes [M]. New York: Willey.
Santos, J. E., Franosch, T., Parmeggiani, A., & Frey, E. (2005). Renewal processes and fluctuation analysis of molecular motor stepping. Phys. Biol., 2, 207–222.
Schief, W. R., & Howard, J. (2001). Conformational changes during kinesin motility. Curr. Opin. Cell Biol., 13(1), 19–28.
Schnitzer, M. J., & Block, S. M. (1997). Kinesin hydrolyses one ATP per 8-nm step. Nature, 388(6640), 386–390.
Sheetz, M. P., & Spudich, J. A. (1983). Movement of myosin-coated fluorescent beads on actin cables in vitro. Nature, 303(5912), 31–35.
Smith, W. L. (1958). Renewal theory and its ramifications. J. R. Stat. Soc., Ser. B, Stat. Methodol., 20, 243–302.
Smith, W. L. (1959). On the cumulants of renewal processes. Biometrika, 46(1–2), 1–29.
Spudich, J. A. (1994). How molecular motors work. Nature, 372, 515.
Svoboda, K., & Block, S. M. (1994). Force and velocity measured for single kinesin molecules. Cell, 77(5), 773–784.
Svoboda, K., Schmidt, C. F., Schnapp, B. J., & Block, S. M. (1993). Direct observation of kinesin stepping by optical trapping interferometry. Nature, 365(6448), 721–727.
Svoboda, K., Mitra, P. P., & Block, S. M. (1994). Fluctuation analysis of motor protein movement and single enzyme kinetics. Proc. Natl. Acad. Sci. USA, 91(25), 11782.
Tsygankov, D., Lindén, M., & Fisher, M. E. (2007). Back-stepping, hidden substeps, and conditional dwell times in molecular motors. Phys. Rev. E, 75(2), 021909.
Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of the Brownian motion. Phys. Rev., 36(5), 823–841.
Van Kampen, N.G. (2007). Stochastic processes in physics and chemistry. Amsterdam: North-Holland.
Visscher, K., Schnitzer, M. J., & Block, S. M. (1999). Single kinesin molecules studied with a molecular force clamp. Nature, 400(6740), 184–189.
Wang, H. (2007). A new derivation of the randomness parameter. J. Math. Phys., 48(10), 103301.
Wang, H., & Qian, H. (2007). On detailed balance and reversibility of semi-Markov processes and single-molecule enzyme kinetics. J. Math. Phys., 48(1), 013303.
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Krishnan, A., Epureanu, B.I. Renewal-Reward Process Formulation of Motor Protein Dynamics. Bull Math Biol 73, 2452–2482 (2011). https://doi.org/10.1007/s11538-011-9632-x
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DOI: https://doi.org/10.1007/s11538-011-9632-x