Dwell time symmetry in random walks and molecular motors.

The statistics of steps and dwell times in reversible molecular motors differ from those of cycle completion in enzyme kinetics. The reason is that a step is only one of several transitions in the mechanochemical cycle. As a result, theoretical results for cycle completion in enzyme kinetics do not apply to stepping data. To allow correct parameter estimation, and to guide data analysis and experiment design, a theoretical treatment is needed that takes this observation into account. In this article, we model the distribution of dwell times and number of forward and backward steps using first passage processes, based on the assumption that forward and backward steps correspond to different directions of the same transition. We extend recent results for systems with a single cycle and consider the full dwell time distributions as well as models with multiple pathways, detectable substeps, and detachments. Our main results are a symmetry relation for the dwell time distributions in reversible motors, and a relation between certain relative step frequencies and the free energy per cycle. We demonstrate our results by analyzing recent stepping data for a bacterial flagellar motor, and discuss the implications for the efficiency and reversibility of the force-generating subunits.

[1]  Anatoly B. Kolomeisky,et al.  Molecular motors and the forces they exert , 1999 .

[2]  W. Greenleaf,et al.  Direct observation of base-pair stepping by RNA polymerase , 2005, Nature.

[3]  Masasuke Yoshida,et al.  Mechanically driven ATP synthesis by F1-ATPase. , 2004, Nature.

[4]  Masasuke Yoshida,et al.  An Alternative Reaction Pathway of F1-ATPase Suggested by Rotation without 80°/40° Substeps of a Sluggish Mutant at Low ATP , 2006 .

[5]  Hiroyasu Itoh,et al.  Resolution of distinct rotational substeps by submillisecond kinetic analysis of F1-ATPase , 2001, Nature.

[6]  A. Kolomeisky.  Exact results for parallel-chain kinetic models of biological transport , 2001 .

[7]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[8]  Frederick Sachs,et al.  Maximum likelihood estimation of molecular motor kinetics from staircase dwell-time sequences. , 2006, Biophysical journal.

[9]  Jaime E. Santos,et al.  Renewal processes and fluctuation analysis of molecular motor stepping , 2005, Physical biology.

[10]  William H. Press,et al.  Numerical recipes in C (2nd ed.): the art of scientific computing , 1992 .

[11]  Masasuke Yoshida,et al.  Mechanically driven ATP synthesis by F1-ATPase , 2004, Nature.

[12]  Z. Koza.  Maximal force exerted by a molecular motor. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  F. Jülicher,et al.  Modeling molecular motors , 1997 .

[14]  R. Simmons,et al.  Hidden-Markov methods for the analysis of single-molecule actomyosin displacement data: the variance-Hidden-Markov method. , 2001, Biophysical journal.

[15]  I. Derényi,et al.  Intrawell relaxation of overdamped Brownian particles. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  S. McKinney,et al.  Analysis of single-molecule FRET trajectories using hidden Markov modeling. , 2006, Biophysical journal.

[17]  Sean X. Sun,et al.  Dynamics of myosin-V processivity. , 2005, Biophysical journal.

[18]  A B Kolomeisky,et al.  The force exerted by a molecular motor. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Joshua W Shaevitz,et al.  Statistical kinetics of macromolecular dynamics. , 2005, Biophysical journal.

[20]  G. Oster,et al.  The physics of molecular motors. , 2001, Accounts of chemical research.

[21]  Hideo Higuchi,et al.  Overlapping hand-over-hand mechanism of single molecular motility of cytoplasmic dynein. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[22]  A. Vilfan.  Elastic lever-arm model for myosin V. , 2005, Biophysical journal.

[23]  H. Qian.  A simple theory of motor protein kinetics and energetics. , 1997, Biophysical chemistry.

[24]  Kazuhiko Kinosita,et al.  Catalysis and rotation of F1 motor: Cleavage of ATP at the catalytic site occurs in 1 ms before 40° substep rotation , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[25]  T. L. Hill,et al.  Free Energy Transduction and Biochemical Cycle Kinetics , 1988, Springer New York.

[26]  Hong Qian,et al.  On detailed balance and reversibility of semi-Markov processes and single-molecule enzyme kinetics , 2007 .

[27]  Shin'ichi Ishiwata,et al.  Mechanochemical coupling of two substeps in a single myosin V motor , 2004, Nature Structural &Molecular Biology.

[28]  Michio Homma,et al.  Direct observation of steps in rotation of the bacterial flagellar motor , 2005, Nature.

[29]  Denis Tsygankov,et al.  Back-stepping, hidden substeps, and conditional dwell times in molecular motors. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  William H. Press,et al.  Numerical Recipes in C, 2nd Edition , 1992 .

[31]  Frederick Sachs,et al.  Extracting dwell time sequences from processive molecular motor data. , 2006, Biophysical journal.

[32]  Jianhua Xing,et al.  Making ATP. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[33]  J. Armitage,et al.  The maximum number of torque-generating units in the flagellar motor of Escherichia coli is at least 11. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Matthias Rief,et al.  Force-dependent stepping kinetics of myosin-V. , 2005, Biophysical journal.

[35]  T. Elston,et al.  A robust numerical algorithm for studying biomolecular transport processes. , 2003, Journal of theoretical biology.

[36]  A. Kolomeisky,et al.  Simple mechanochemistry describes the dynamics of kinesin molecules , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[37]  A. Mehta,et al.  Myosin-V stepping kinetics: a molecular model for processivity. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[38]  Hong Qian,et al.  Generalized Haldane equation and fluctuation theorem in the steady-state cycle kinetics of single enzymes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Kazuhiko Kinosita,et al.  Chemomechanical coupling in F1-ATPase revealed by simultaneous observation of nucleotide kinetics and rotation , 2004, Nature Structural &Molecular Biology.

[40]  K. Svoboda,et al.  Fluctuation analysis of motor protein movement and single enzyme kinetics. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[41]  R. Cross,et al.  Mechanics of the kinesin step , 2005, Nature.

[42]  Michael E Fisher,et al.  Kinesin crouches to sprint but resists pushing. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[43]  Steven M Block,et al.  Backsteps induced by nucleotide analogs suggest the front head of kinesin is gated by strain. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[44]  H. Berg.  The rotary motor of bacterial flagella. , 2003, Annual review of biochemistry.

[45]  H. Berg,et al.  Constraints on models for the flagellar rotary motor. , 2000, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[46]  J. Baker,et al.  Myosin V processivity: multiple kinetic pathways for head-to-head coordination. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[47]  Anatoly B Kolomeisky,et al.  A simple kinetic model describes the processivity of myosin-v. , 2002, Biophysical journal.

[48]  R. A. Kennedy,et al.  Forward-backward non-linear filtering technique for extracting small biological signals from noise , 1991, Journal of Neuroscience Methods.

[49]  William H. Press,et al.  Numerical recipes in C , 2002 .

[50]  H. Noji,et al.  F1-ATPase is a highly efficient molecular motor that rotates with discrete 120 degree steps. , 1998, Cell.

[51]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[52]  Anatoly B. Kolomeisky,et al.  Periodic sequential kinetic models with jumping, branching and deaths , 2000 .

[53]  T. Yanagida,et al.  Entropy rectifies the Brownian steps of kinesin , 2005, Nature chemical biology.

[54]  Matthias Rief,et al.  Myosin-V is a mechanical ratchet. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[55]  P. Reimann.  Brownian motors: noisy transport far from equilibrium , 2000, cond-mat/0010237.

[56]  Kazuhiko Kinosita,et al.  ATP-driven stepwise rotation of FoF1-ATP synthase. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[57]  Anatoly B Kolomeisky,et al.  Understanding mechanochemical coupling in kinesins using first-passage-time processes. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  Kazuhiko Kinosita,et al.  F1-ATPase Is a Highly Efficient Molecular Motor that Rotates with Discrete 120° Steps , 1998, Cell.

[59]  Michael Börsch,et al.  Proton-powered subunit rotation in single membrane-bound F0F1-ATP synthase , 2004, Nature Structural &Molecular Biology.

[60]  Wei Yang,et al.  A Structure-Based Model for the Synthesis and Hydrolysis of ATP by F1-ATPase , 2005, Cell.

[61]  Toshio Yanagida,et al.  Chemomechanical coupling of the forward and backward steps of single kinesin molecules , 2002, Nature Cell Biology.

[62]  J. Howard,et al.  Mechanics of Motor Proteins and the Cytoskeleton , 2001 .