Dynamics of Molecular Motors Molecular Motors

Home , Kinesin, Molecular motor, Myosin

MAE 545: Lecture 8 (10/13) Dynamics of molecular motors Molecular motors

Actin Typical motor speeds 2 v 10 -1 µm/s ⇠ Actin

Microtubule

A.B. Kolomeisky, J. Phys.: Condens. Matter 25, 463101 (2013) 2 Movement of molecular motors is powered by ATP molecules

Myosin motor walking Kinesin motor walking on actin in muscles on microtubule

Graham Johnson

3 Japanese Journal of Applied Physics 54, 06FG02 (2015) REGULAR PAPER http://dx.doi.org/10.7567/JJAP.54.06FG02

Structural parameter dependence of directed current generation in GaAs nanowire-based electron Brownian ratchet devices

Yushi Abe, Ryota Kuroda, Xiang Ying, Masaki Sato, Takayuki Tanaka, and Seiya Kasai*

Research Center for Integrated Quantum Electronics, and Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-8628, Japan E-mail: [email protected] Received December 1, 2014; accepted January 16, 2015; published online April 15, 2015

We investigated the structural parameter dependence of the directed current in GaAs-nanowire-based Brownian ratchet devices. The directed current was generated by flashing a ratchet potential array repeatedlyMolecular using multiple motors asymmetric gates vs with a periodic signal. The amount of current in the fabricated device increased as the nanowire width W decreased, which contradicted the theoretical model. The current also depended on the number of the gates N, when N was smaller thanBrownian 6. We discussed theratchets obtained results in terms of the structural parameter dependence of carrier transfer efficiency and the effect of electron reservoirs on current generation in flashing ratchet operation. © 2015 The Japan Society of Applied PhysicsMyosin motor Brownian ratchet ATP driven process net movement of particles is drives molecular motors achieved by periodic modulation of along the filaments asymmetric external potential 1. Introduction Asymmetric potential Biological systems are highly energy-epotentialfficient energy as comparedATP motor with artificial electronic systems.1)alongElectronic the filament implementation of the energy efficiency of biological systems has been an attractive issue toward ultralow-power electronics.cell whole From The 432 this point of view, a molecular motor is one of the important examples, which drives muscular contractionbound ATP exploiting no ATP fl 2–5) ffi uctuation. Its energy conversionactin e ciency is considered 6) to be as high as 50%. The mechanismfilament of the molecular motor is known as the Brownian ratchet,7,8) in which the motion of Brownian particles is rectified using a periodic array of asymmetric ratchet potentials.9,10) The Brownian (a) fi ratchet has been arti cially demonstrated using various 4 If the monomers of a biofi lament have a dipolar charge distribution, the head of a motor molecule 11molecule – 17)motor a of head the distribution, 18)charge dipolar a have lament biofi a of monomers the If 11.19 Fig. GaAs nanowire systems such as microparticles, This (left). form livingsawtooth soft a cells,with energy potential anda experiences charge positive Asymmetriclocal a carrying 19–23) electrons. So far, several electronfrom ATP like Brownianmolecule charged oppositely an ratchet captures head the if eliminated Schottkylargely is potential gates devices that generated directed current have been re- (right). solution INET ported;19–23) however, the main interest has been in the experimental demonstration and physical aspect. The next step is to achieve a high energy efficiency comparable to biological systems for application. Recently, we have fabricated an electron Brownian ratchet device using an etched GaAs nanowire with multiple asymmetric Schottky AlGaAs Square wave for wrap gates and demonstrated the generation of directed In a simplifi ed model for a thermal ratchet, the potential energy has a sharply peaked sawtooth form form sawtooth peaked sharply a has energy potential the ratchet, thermal a for model ed simplifi GaAs a In 11.20 Fig. flashing potential current at room temperature.24) The advantages of this device . We replace the Gaussian probability probability Gaussian the replace We . b α separation peak-to-trough minimum and b period with (b) in terms of energy efficiency are the collimation motion. of probability ofthe electroncalculate to w 2 base with one triangular a by distribution motion in the nanowire direction, and formation and flashing of ratchet potentials by an asymmetricnamely distributions, wraptwo the gateof thatdispersions the Equating 1. = x d ) x ( P ∫ LG LD 2 2 2 σ 2 provides a tight potential control owing to its three-dimen- relation the gives /6, w = 〉 x 〈 and = 〉 x 〈 , (11.13) (11.13) , σ 2.5 ≅ σ 6 √ = w sional gate configuration.25–28) In this study, we investigate W as one might intuitively expect from the shapes of the distributions. The e h T . s n o i t u b i r t s i d e h t f o s e p a h s e h t m o r f t c e p x e y l e v i t i u t n i t h g i m e n o s a the structural parameter dependence set. problem ofthe thein directedfurther tested is current approximation triangle the of accuracy in the GaAs-based nanowire to Brownian refer we ratchetmodel, ratchet device.the in motion We of calculation our begin To fabricate and characterize the devices the to having hopping particle various a of nano- probability the determine and 11.20 Fig. right. If the potential is turned off for too short a time, a particle dif- particle a time, a short too for off turned is potential the If right. wire widths and numbers of gates that in such terms narrow, of very is the amount distribution of probability the and locally fuses (c) current and carrier transfer effi - i r ciency. o s t i o t s n r u t e r y l p m i s e l c i t r a p a , s n o i t a u t i s h c u s n I . b α n a h t s s e l s i w ginal position once the potential is turned on again. For somewhat longer longer somewhat For again. on turned is potential Fig.the 1.once (Colorposition online)ginal (a) Mechanism of Brownian ratchet in the case of α , the particle may have moved so far that it it that far so moved have may particle the , b > w fl with times diffusion 2. Concept and experiment Now, restored. is potential the once minimum ashingneighboring operation,a out seeks (b) schematic illustration of a GaAs-nanowire-based we will assume that the potential is turned on so fast that any particle particle any that fast so on turned is potential Brownianthe that ratchetassume will devicewe with measurement circuit, and (c) a scanning The mechanism of the Brownian ratchet is schematically electron microscopy image of a fabricated device. shown in Fig. 1(a). In the case of the biological molecular motor, the actin filament is a one-dimensional rail for moving myosin. The chemical interaction between actin and myosin formed with an asymmetric electrostatic potential. Consider achieves the situation as shown in Fig. 1(a).3,9) In the case that the electrons with Brownian motion are accumulated in of the electron Brownian ratchet, the ratchet is electrically the potential valleys, as shown in the top illustration in 06FG02-1 © 2015 The Japan Society of Applied Physics Typical rates (timescales)

1 2000s 1 Actin- 100s Myosin-ATP Myosin-ADP-Pi Myosin-ATP

1 1 1 1µM s 30s

Actin- Actin- Actin-

Myosin 1 Myosin-ADP 1 Myosin-ADP-Pi 1000s & 10000s

1 Microtubule- 5000s Microtubule- Kinesin-ADP-Pi Kinesin-ADP

1 1 100s 50s

Microtubule- Microtubule- 1 1 Kinesin-ATP 2µM s Kinesin timescale=1/rate stepping motion for kinesin: ~100 µs

F. Fulga et al., Integr. Biol. 1, 150-169 (2009) 5 (A) As a result, Equation 16.47 can be written explicitly as

t τ/τ (t τ)/τ 1 p(t) e− A e− − B dτ . (16.50) = τ τ !0 A B This integral can be evaluated (this is assigned as a problem at the Internal states are revealedprobability by dwell times end of the chapter) and results in

Distribution of time needed for one cycle Let’s assume that for walking 1 t/τ t/τ of myosin V walkingtime on tactin at 10 µM ATP p(t) (e B e A ). (16.51) of molecular motors there are − − = τB τA − (B) − two subprocesses A and B 140 with time distributions 120 Figure 16.36 shows the functional form of this class of waiting time 1 t/t p (t)= e A distribution. In practice, observed dwell times for various molecular A t 100 A motors fit well to this kind of scheme. 1 t/tB 80 pB(t)= e tB 60 Because the rates of internal conformational changes for motors in position–state models depend on nucleotide hydrolysis state and Distribution of time to 40 complete both subprocesses 20 on applied force, we expect that dwell time distributions should also number events of depend on these factors. Figure 16.37 shows average dwell times for t 0 p(t)= d⌧ p (⌧)p (t ⌧) 00.10.20.30.40.5 myosin V as a function of force for two different concentrations of A B Z0 dwell time (s) t/t t/t ATP. At high ATP, dwell times are initially very short, but increase with e B e A p(t)= R. Phillips et al., Physical load as the motor is forced to do more work. At very low concentra- tB tA Figure 16.36: Probability distribution Biology of the Cell tions of ATP, the behavior of the motor is dominated by the state while for waiting6 times for a two-state motor. (A) Theoretical result, as stated in it is waiting for a nucleotide to bind, and is less sharply dependent on Equation 16.51. (B) Data on dwell time applied force. The conclusion of this analysis is that the overall distri- distribution for a single-headed version bution of dwell times provides a window on the composite processes of myosin V, at 10 µMATP.(Badapted that make up the overall motor dynamics and a filter on the different from T. J. Purcell et al., Proc. Natl Acad. Sci. USA 102:13873, 2005 and J. C. Liao classes of models set forth to greet data on dwell times. et al., Proc. Natl Acad. Sci. USA 104:3171, 2007.) 16.2.5 More General Motor Models

In general, the internal dynamics of molecular motors can be consid- erably more complex than the one- and two-state motors considered thus far. The P-state, N-position class of models introduced in Figure 16.20 is probably still a useful framework for thinking about these motors. For example, in order to ﬁt all the data available for kinesin, it appears that a model based upon four internal states is most realistic. The complexity of the mathematics and the prolifer- ation of parameters make these models largely beyond the scope of this book. However, before turning to other classes of motors besides the translational motors considered here, we ﬁrst make some general observations.

[ATP] = 1 mM

1 Figure 16.37: Average dwell times for myosin V as a function of load, for high

ATP (solid circles) and low ATP (open dwell time (s) [ATP] = 2 mM circles). The lines represent a ﬁt of the data to a two-state model. (Adapted 0.1 from A. D. Mehta et al., Nature 400:590, 1999 and A. B. Kolomeisky and M. E. Fisher, Biophys. J.84:1642, 0 0.5 1 1.5 2 2.5 3 2003.) force (pN)

656 Chapter 16 DYNAMICS OF MOLECULAR MOTORS

“chap16.tex” — page 656[#34] 29/9/2012 11:03 Two state model for molecular motor

ATP bound to motor head x 2a x a x x + a x +2a B u1 u2 u1 u2 u1 u2 u1 u2 u1

w1 w2 w1 w2 w1 w2 w1 w2 w1 A x 2a x a x x + a x +2a [ATP] motor

Forward rate u1 depends on concentration of ATP

u1 = kon[ATP]

Forward rate u2 corresponds to the slow rate limiting process.

Backward rates w1 and w2 are assumed to be small but non zero. What is the speed of molecular motor in such model?

7 Two state model for molecular motor

x 2a x a x x + a x +2a B u1 u2 u1 u2 u1 u2 u1 u2 u1

w1 w2 w1 w2 w1 w2 w1 w2 w1 A x 2a x a x x + a x +2a Master equation @p (x, t) A = u p (x a, t)+w p (x, t) (u + w )p (x, t) @t 2 B 1 B 1 2 A @p (x, t) B = u p (x, t)+w p (x + a, t) (u + w )p (x, t) @t 1 A 2 A 2 1 B Continuum limit @p (x, t) @p (x, t) a2u @2p (x, t) A = (u + w )p (x, t)+(u + w )p (x, t) au B + 2 B @t 1 2 A 2 1 B 2 @x 2 @x2 @p (x, t) @p (x, t) a2w @2p (x, t) B = (u + w )p (x, t)+(u + w )p (x, t)+aw A + 2 A @t 2 1 B 1 2 A 2 @x 2 @x2

8 Two state model for molecular motor

x 2a x a x x + a x +2a B u1 u2 u1 u2 u1 u2 u1 u2 u1

w1 w2 w1 w2 w1 w2 w1 w2 w1 A x 2a x a x x + a x +2a i!t ikx Fourier transform pA,B(x, t)= dkd!e p˜A,B(k, !) Z ! iu iw ,iw+ iu (1 + ika k2a2/2) p˜ (!,k) i 1 2 1 2 A =0 iu + iw (1 ika k2a2/2), ! iu iw p˜ (!,k) ✓ 1 2 2 1 ◆✓ B ◆ Only those Fourier modes are nonzero, that correspond to the matrix with zero determinant! (! iu iw )(! iu iw ) 1 2 2 1 + w + u (1 + ika k2a2/2) u + w (1 ika k2a2/2) =0 1 2 1 2 9 Two state model for molecular motor

x 2a x a x x + a x +2a B u1 u2 u1 u2 u1 u2 u1 u2 u1

w1 w2 w1 w2 w1 w2 w1 w2 w1 A x 2a x a x x + a x +2a Dispersion relation i !(k)= (u + u + w + w ) 2 1 2 1 2 i (u + u + w + w )2 +4ika(u u w w ) 2k2a2(u u + w w )+k4a4u w2 ±2 1 2 1 2 1 2 1 2 1 2 1 2 2 p Obtain motor velocity and diffusion constant by Taylor expansion !(k)=vk + iDk2 + ···

10 Two state model for molecular motor

x 2a x a x x + a x +2a B u1 u2 u1 u2 u1 u2 u1 u2 u1

w1 w2 w1 w2 w1 w2 w1 w2 w1 A x 2a x a x x + a x +2a Speed of molecular motor (u u w w ) v = a 1 2 1 2 (u1 + u2 + w1 + w2) Diffusion constant for molecular motor (u u + w w ) (u u w w )2 D = a2 1 2 1 2 1 2 1 2 2(u + u + w + w ) (u + u + w + w )3 ✓ 1 2 1 2 1 2 1 2 ◆ “Einstein force” v Fictitious force v = µF FE = kBT E D = µkBT D acting on the motor 11 Figure 16.53: Single-molecule s = 23.08 nm s = 3.44 nm N = 6 measurements of steps taken by 1 mol 900 s2 = 51.65 nm s = 4.11 nm Nstep = 92 19.2 nm myosin V along actin ﬁlaments. The 45.5 nm histogram in the inset is of all the 4 27.2 nm measured step sizes, which are 800 48.8 nm measured with 1.5 nm precision. 2 74.3 nm (Adapted from A. Yildiz et al., Science 700 18.7 nm 300:2061, 2003.) 58.6 nm number of steps 0 20 40 60 80 600 15.4 nm step size (nm) 48.3 nm 66.1 nm 500 82.1 nm 400 72.5 nm position (nm) 23.7 nm 67.2 nm 300 56.6 nm 26.0 nm 25.8 nm 51.5 nm 200 68.2 nm 23.5 nm 68.4 nm 52.0 nm 100 26.5 nm 73.5 nm 68.9 nm 0 0 10 20 30 40 50 60 time (s)

(b) Assume that the stepping rate is k.Thisisthe that is of the form probability per unit time that the motor will make a step. Calculate the waiting time distribution between the two p0(n, t) i(Kn ωt) A e − . (16.91) steps observed in the experiment if the fluorescent marker p1(n, t) = B is placed at position x found in (a). What is the expected ! " ! " distribution, assuming a hand-over-hand stepping mechanism if the marker is placed very close to one of the Find a relation between K and ω that guarantees the heads (that is, x 18 nm)? Use your calculated distributions existence of a solution of this form. This is the so-called to rationalize and≈ fit the data (from Yildiz et al., 2003) dispersion relation. provided on the book’s website. Does your analysis support (b) By substituting the trial solution ei[(K/a) ωt] into the the hand-over-hand mechanism? What value of k do you − differential equation for diffusion with drift (Equation 13.54, obtain? p. 530),ATP show concentration that the dispersion relation independent this case is speed of motors

Ignoring backwardK K2 16.3 Kinesin as an ATP-hydrolyzing enzyme ω v iD . (16.92) • = a − a2 u1u2 [ATP] As described in the chapter, careful measurements have rates and assuming v a = vmax ⇡ (u1 + u2) ([ATP] + Kd) been performed that examine the dependence of motor u1 = kon[ATP] velocity on ATP concentration. Under certain conditions, the where v is the drift velocity and D is the diffusion constant. hydrolysis reaction performed by a molecular motor can be described using the Michaelis–Menten model introduced in Kinesin motor on microtubules Section 15.2.7 (p. 596). In the particular case of kinesin, its Maximal speed 103 stepping is strongly coupled to its ATPase activity, which vmax = au2 0.6µm/s translates into relatively constant step sizes. Finally, its high Vmax ⇡ processivity allows for a clear definition of a speed, since Vmax/2 ATP concentration at kinesin takes many steps before falling off the microtubule.

Relate the reaction speed (the rate of ATP hydrolysis) to 102 half the maximal speed the maximum stepping speed of kinesin and determine its u dependence on ATP concentration. Fit your model to the 2 Kd = 50µM data by Schnitzer and Block (1997) shown in Figure 16.54 kon ⇡ and provided on the book’s website. Then work out what

change in substrate concentration is needed to increase the speed (nm/s) 101 Measured constants reaction rate from 0.1vmax to 0.9vmax. Schnitzer and Block data Michaelis–Menten fit a 8nm ⇡ 1 u2 70s 16.4 Kinetics of two-state motors ⇡ 1 1 • 0 kon 1µM s In the chapter, in order to obtain the velocity of a two-state 10 10 1 100 101 K 102 103 ⇡ motor, we made use of a trick to circumvent solving the d master equation directly. Here we take up this task and in R. Phillips et al., Physical [ATP] (mM) the process also derive an expression for the diﬀusion FigureBiology 16.54: ofATP the hydrolysis Cell by kinesin. Speed of12 a kinesin constant. motor as a function of ATP concentration and ﬁt of the data to (a) Consider a trial solution of the system of equations for aMichaelis–Mentenmodel.(AdaptedfromM.J.Schnitzerand p0(n, t) and p1(n, t),givenbyEquations16.33and16.34, S. M. Block, Nature 388:386, 1997.)

678 Chapter 16 DYNAMICS OF MOLECULAR MOTORS

“chap16.tex” — page 678[#56] 29/9/2012 11:03 letters to nature

Received 24 December 1998; accepted 17 May 1999. 1. Miyata, H. & Miyata, M. Mode of conjugation in homothallic cells of Schizosaccharomyces pombe. a J. Gen. Appl. Microbiol. 27, 365±371 (1981). 100 2. Egel, R. & Eie, B. Cell lineage asymmetry in Schizosaccharomyces pombe: unilateral transmission of a high-frequency state of mating-type switching in diploid pedigrees. Curr. Genet. 12, 429±433 (1987). 3. Klar, A. J. S. Differentiated parental DNA strands confer developmental asymmetry on daughter cells bead in ®ssion yeast. Nature 326, 466±470 (1987). 0 4. Klar, A. J. S. The developmental fate of ®ssion yeast cells is determined by the pattern of inheritance of parental and grandparental DNA strands. EMBO J. 9, 1407±1415 (1990). 5. Klar, A. J. S. & Bonaduce, M. J. The mechanism of ®ssion yeast mating-type interconversion: evidence for two types of epigenetically inherited chromosomal imprinted events. Cold Spring Harb. Symp. Motors carrying the load Quant. Biol. 58, 457±465 (1993). –100 6. Beach, D. H. Cell type switching by DNA transposition in ®ssion yeast. Nature 305, 682±687 (1983). Force exerted on kinesin motors carrying plastic 7. Nielsen, O. & Egel, R. Mapping the double-strand breaks at the mating-type locus in ®ssion yeast by beads can be controlled with optical tweezers genomic sequencing. EMBO J. 8, 269±276 (1989). 8. Klar, A. J. S. & Miglio, L. M. Initiation of meiotic recombination by double-strand DNA breaks in ∆x trap F kx

S. pombe. Cell 46, 725±731 (1986). (nm) Position –200 Effective spring⇡ constant 9. Moreno, S. A., Klar, A. J. S. & Nurse, P. Molecular genetic analysis of ®ssion yeast Schizosaccharomyces F bead pombe. Methods Enzymol. 194, 795±823 (1991). k depends on the bead 10. Egel, R., Beach, D. H. & Klar, A. J. S. Genes required for initiation and resolution steps of mating-type v size, refractive indices of switching in ®ssion yeast. Proc. Natl Acad. Sci. USA 81, 3481±3485 (1984). microtubule kinesin the bead and surrounding 11. Singh, J. & Klar, A. J. S. DNA polymerase-a is essential for mating-type switching in ®ssion yeast. –300 medium, and the gradient Nature 361, 271±273 (1993). – + of laser intensity 12. Caddle, M. S. & Calos, M. P. Speci®c initiation at an origin of replication from Schizosaccharomyces pombe. Mol. Cell. Biol. 14, 1796±1805 (1994). 13. Clyne, R. K. & Kelly, T. J. Genetic analysis of an ARS element from the ®ssion yeast –400 Schizosaccharomyces pombe. EMBO J. 14, 6348±6357 (1995). 0.0 0.5 1.0 1.5How motor 2.0 speed depends 14. Brewer, B. J., Lockshon, D. & Fangman, W. L. The arrest of replication forks in the rDNA of yeast focusedTime (s) on the loading force? occurs independently of transcription. Cell 71, 267±276 (1992). laser beam Note: viscous drag is negligible 15. Arcangioli, B. A site- and strand-speci®c DNA break confers asymmetric switching potential in ®ssion 200 7.4 Fdrag =6⇡⌘Rv yeast. EMBO J. 17, 4503±4510 (1998). b c 3 1 1 2 K. Visscher et al., Nature 400, 184-189 (1999)6.5 F 0.1drag pN6⇡10 kgm s 1µm 1µm/s 10 pN 16. Klar, A. J. S. Lineage-dependent mating-type transposition in ®ssion and budding yeast. Curr. Opin. 13 ± ⇠ · · ⇠ Genet. Dev. 3, 745±751 (1993). 17. Sambrook, J., Fritsch, E. F. & Maniatis, T. Molecular Cloning: A Laboratory Manual 2nd ed (Cold (nm) 175 6.5 x

Spring Harbor Laboratory Press, New York, 1989). ∆ 18. MacDonell, M. W., Simon, M. W. & Studier, F. W. Analysis of restriction fragments of T7 DNA and Load (pN) determination of molecular weights by electrophoresis in neutral and alkaline gels. J. Mol. Biol. 110, 119±146 (1977). 150 5.6 19. Egel, R. in Molecular Biology of the Fission Yeast (eds Nasim, A., Young, P. & Johnson, B. F.) 31±74 0.0 0.5 1.0 1.5 2.0 0 120 240 360 (Academic, San Diego, 1989). Time (s) Counts 20. Klar, A. J. S., Ivanova, A. V., Dalgaard, J. Z., Bonaduce, M. J. & Grewal, S. I. S. in Epigenetics (ed. Cardew, G.) 87±103 (Novartis Foundation Symposium 214, Wiley, Chichester, 1998). 21. Arcangioli, B. & Klar, A. J. S. A novel switch-activating site (SAS1) and its cognate binding factor Figure 1 Operation of the force clamp. a, Sample record from the force clamp, (SAP1) required for ef®cient mat1 switching in Schizosaccharomyces pombe. EMBO J. 10, 3025±3032 showing kinesin-driven bead movement and corresponding optical trap (1991). displacement (2 mM ATP). Discrete steps of 8 nm are readily apparent. Inset, 22. Styrkarsdottir, U., Egel, R. & Nielsen, O. The smt-0 mutation which abolishes mating-type switching in ®ssion yeast is a deletion. Curr. Genet. 23, 184±186 (1993). schematic representation of the motility assay used, showing the experimental 23. Gerring, S. L., Connelly, C. & Hieter, P.in Guide to Yeast Genetics and Molecular Biology (eds Fink, G. & geometry (not to scale). The separation between bead and trap was nominally Guthrie, C.) 57±76 (Academic, San Diego, 1991). 24. Bresch, C., Muller, G. & Egel, R. Genes involved in meiosis and sporulation of a yeast. Mol. Gen. Genet. ®xed at Dx 175 nm. Bead position was sampled at 20 kHz, ®ltered with a 12-ms 102, 301±306 (1968). boxcar window for the feedback on the trap de¯ection, and saved un®ltered at

Acknowledgements. We thank our colleagues at ABL for suggestions and helpful discussions; B. Brewer, 2.0 kHz. b, The measured bead±trap separation, Dx, for the record in a. c, M. Hoant and S. Hunt for help with the analysis of replication intermediates; A. Arthur for editorial Histogram of the displacements in b, converted to force by multiplying by the suggestions; and R. Frederickson for the art work. This work was sponsored by the Danish Natural Science -1 Research Council (J.Z.D.) and the National Cancer Institute, Department of Health and Human Services, trap stiffness (0.037 pN nm ). Solid red line is gaussian ®t to these data, yielding a under contract with ABL. The contents of this article do not necessarily re¯ect the views or politics of the load of 6:5 6 0:1 pN (mean 6 s:d:). DHHS, nor does mention of trade names, commercial products, or organizations imply endorsement by the US Government.

Correspondence and requests for materials should be addressed to A.K. (e-mail: [email protected]). mechanochemical studies under controlled external loads. Analysis of records of kinesin motion under variable ATP concentrations and loads revealed several new features. First, kinesin stepping appears to be tightly coupled to ATP hydrolysis over a Single kinesin molecules wide range of forces, with a single hydrolysis per 8-nm mechanical advance. Second, the kinesin stall force depends on the ATP studied with a molecular concentration. Third, increased loads reduce the maximum velocity as expected, but also raise the apparent Michaelis±Menten force clamp constant. The kinesin cycle therefore contains at least one load- dependent transition affecting the rate at which ATP molecules Koen Visscher*²³, Mark J. Schnitzer*§³ & Steven M. Block*²§ bind and subsequently commit to hydrolysis. It is likely that at * Department of Molecular Biology, ² Princeton Materials Institute, and least one other load-dependent rate exists, affecting turnover § Department of Physics, Princeton University, Princeton, New Jersey 08544, USA number. Together, these ®ndings will necessitate revisions to ³ These authors contributed equally to this work. our understanding of how kinesin motors function...... The mechanochemical coupling ratio relates the number of ATP Kinesin is a two-headed, ATP-driven motor protein that moves molecules hydrolysed to the distance travelled during a single processively along microtubules in discrete steps of 8 nm, prob- enzymatic cycle. Determination of this ratio over a range of loads ably by advancing each of its heads alternately in sequence1±4. sheds light on the mechanism for movement, and has been Molecular details of how the chemical energy stored in ATP is considered one of the most pressing challenges in the motility coupled to mechanical displacement remain obscure. To shed ®eld1. However, for the microtubule-based molecular motor kine- light on this question, a force clamp was constructed, based on a sin, the task is by no means straightforward. Direct comparisons feedback-driven optical trap capable of maintaining constant between solution ATPase measurements and motility in vitro are loads on single kinesin motors5. The instrument provides only feasible in the absence of external loads, as these cannot be unprecedented resolution of molecular motion and permits applied in bulk solution. Single-molecule ¯uorescence methods,

Free energy Work against u motor velocity 2 w proﬁle in the 2 the load force (u u (F ) w w (F )) v(F )=a 1 2 1 2 absence of load W = Fa (u1 + u2(F )+w1 + w2(F ))

Assuming forward rate Assuming backward rate Assuming both rates affected by the load affected by the load affected by the load

w2(F ) w2(F ) u2(F ) u2 u2(F ) w2

W ↵W W (1 ↵)W Fa/kB T +Fa/kB T ↵Fa/kB T u2(F )=u2e w2(F )=w2e u2(F )=u2e +(1 ↵)Fa/kB T w2(F )=w2e v v v ↵ =1 ↵ =0 0 < ↵ < 1

F F F 14 Motor velocity dependence on the load

↵Fa/kB T +(1 ↵)Fa/kB T kon[ATP]u2e w1w2e v(F )=a Assuming both rates ↵Fa/k T +(1 ↵)Fa/k T kon [ATP] + u2e B + w1 + w2e B affected by the load letters to nature kinesin walking on microtubules u2(F ) w2(F ) a halt movement almost entirely. These results are consistent with a 60 role for load in lowering the effective rate of ATP binding. At ↵W 2 mM ATP 800 limiting ATP, kb becomes much lower than kcat, and therefore the (1 ↵)W 50 5 µM ATP stall force becomes independent of the latter. However, at 2 mM ↵Fa/kB T M ATP ATP, the stall force probably depends on both k and k , as K rises u2(F )=u2e µ 600 b cat m +(1 ↵)Fa/k T 40 w (F )=w e B beyond ,300 mM at loads of 7 pN or more. Although the stall force

2 2 ), 5 ), 2 mM ATP

v –1 30 –1 may depend on additional factors at all ATP levels, stall forces are 400 expected to rise continuously from ,5.5 pN to ,7.0 pN as the ATP 0 < ↵ < 1 20 concentration is raised. 200 To further test this supposition, we measured the ATP depen- 10 dence of kinesin stall force by two additional methods, using either 17 Velocity (nm s Velocity (nm s Velocity an optical position clamp or a ®xed trap. The position clamp , 0 0 01234567 based on an optical trapping interferometer, is well suited to F Load (pN) isometric measurements of stall force16, and was used previously b to study force generation by individual RNA polymerase molecules18. Position clamp measurements on single kinesin 8 K. Visscher et al., Nature 400, 184-189 (1999) 15 motors gave stall forces that rose with ATP concentration, as anticipated (Fig. 3b). An inherent problem encountered in these 7 experiments was the dramatic decline in kinesin processivity at loads approaching stall12. Rapid detachment of beads from the microtubule made the collection of a large data set prohibitive, 6 and accounted for much of the experimental scatter (Fig. 3b). To

Stall force (pN) Stall force con®rm the position clamp measurements, we also measured stall Position clamp forces using a ®xed optical trap. Kinesin-coated beads moving 5 Fixed trap outwards in a stationary trap experience increasing loads until they arrive at a position where forward progress ceases: the ®nal 10 100 1,000 displacement, multiplied by the trap stiffness, supplies a measure of ATP concentration (µM) the stall force12. Kinesin stall forces determined by this method exhibit a similar increase with ATP concentration (Fig. 3b). Figure 3 Load dependence of motility. a, Average bead velocity, v How is the kinesin ATPase pathway coupled to mechanical (mean 6 s:e:m:), versus applied load for ®xed ATP concentrations (red triangles, movement under load? One possibility is that kinesin coupling is left axis, 5 mM ATP, N 19±57; blue circles, right axis, 2 mM ATP, N 37±87). The tight, involving no futile hydrolysis events that fail to generate velocity point at (5.6 pN, 5 mM ATP) is likely to represent an overestimate because movement, even under load (that is, e 1). In this case, any changes beads which stalled completely (v 0) were indistinguishable from beads lack- in the rates of load-dependent transitions must account entirely for ing active motors, and so were not included in the analysis. b, Stall force the observed decreases in velocity under load. However, more (mean 6 s:e:m:) versus ATP concentration, measured either with the position complicated scenarios may be entertained. Although the rise in 17,18 clamp (red triangles) or with a ®xed optical trap (blue circles). Stalls had to last a Km with load eliminates the simpler versions of loose coupling, minimum of 2 s to be included in the analysis. Data points represent an average of mechanochemical coupling in kinesin might display an admixture either 12±29 (position clamp) or 6±70 (®xed trap) stalls. of both tight and loose elements, with load-dependent transitions and unproductive hydrolyses conspiring jointly to lower the velocity. Another candidate mechanism for loose coupling involves an traps10±12 or microneedles13, all of which reported nearly linear increase in the frequency of 8-nm backwards steps10±12 with load. decreases in velocity with increased load, for both limiting and However, the latter mechanism may be discounted because the saturating ATP concentrations. In fact, such linear force±velocity fraction of backwards steps found in this study (data not shown) relations formed the basis of an argument for a load-independent and by others remains only ,5±10% at both negligible8,11 and 10 11,12 Km (refs 10, 19) which supported a simple, loosely coupled picture . substantial loads. However, the earlier studies lacked the improved resolution To study the coupling mechanism under load, we performed a afforded by the force clamp, and subtle differences in shape were ¯uctuation analysis23,24 of kinesin movement subjected to the force not distinguished within experimental error. Data from all previous clamp. Fluctuation analysis, which relies on the processive property studies consistently showed a lower stall force at limiting ATP than exhibited by some molecular motors, has been used previously to at saturating ATP,although the difference in force was insuf®cient in study ATP coupling in kinesin near zero load8, as well as the duty any given experiment to have been considered signi®cant. More- ratio of the bacterial rotary motor under torque25. Fluctuations in over, a greater discrepancy (,1 pN) between the stall force at displacement are quanti®ed by a randomness parameter, r, which limiting and saturating ATP was observed when the limiting measures the temporal irregularity of mechanical advances. For an concentration of ATP was ,35-fold lower than the unloaded Km ensemble of displacement records, x(t), r is de®ned by (ref. 13), as compared to when limiting ATP values were only ,6- x2 t 2 x t 2 fold lower (,0.5 pN, refs 10±12). We therefore measured velocity as r lim h i h i t ` d x t a function of load in the force clamp at both 2 mM and 5 mM ATP, ! h i choosing the lower ATP level to be .17-fold below the unloaded Km where d is the motor step size, and the angle brackets denote an to highlight any potential differences. ensemble average23,24. A perfectly clocklike motor displays no step- Force±velocity curves obtained with the force clamp displayed ping irregularity, and has r 0, whereas a `Poisson' motor with one different shapes at limiting and saturating ATP levels (Fig. 3a), and biochemical transition moves at exponentially distributed intervals, they also showed different stall forces. The 5-mM curve is nearly and has r 1. In general, r-1 provides a continuous measure of the linear, with a stall force of ,5.5 pN. In contrast, the 2-mM curve is number of rate-limiting transitions in the overall mechanochemical concave down, and supports a greater stall force of ,7.0 pN. cycle. Determinations of randomness are insensitive to thermal or Strikingly, a 5-pN load leads to a ,40% decline in speed for other stationary noise and can therefore be performed even when 2 mM ATP, whereas at 5 mM ATP an identical load is suf®cient to individual steps are not resolved. When steps can be discerned amid

a halt movement almost entirely. These results are consistent with a 60 role for load in lowering the effective rate of ATP binding. At 2 mM ATP 800 limiting ATP, kb becomes much lower than kcat, and therefore the 50 5 µM ATP stall force becomes independent of the latter. However, at 2 mM

M ATP ATP, the stall force probably depends on both k and k , as K rises µ 40 600 b cat m beyond ,300 mM at loads of 7 pN or more. Although the stall force ), 5 ), 2 mM ATP

–1 30 Stall force –1 may depend on additional factors at all ATP levels, stall forces are 400 maximal expectedpossible to force rise continuously from ,5.5 pN to ,7.0 pN as the ATP 20 v(Fs)=0 from energyconcentration conservation is raised. 200 To further test this supposition, we measured the ATP depen- 10 letterskBT to naturekon[ATP]u2 GATP 20kBT Fs = ln Fmax = dence of kinesin stall10pN force by two additional methods, using either a ⇡ 8nm ⇠ 17 Velocity (nm s Velocity Received 24 Decembera 1998; accepted 17 May 1999. w w (nm s Velocity 1 2 a an optical position clamp or a ®xed trap. The position clamp , 0 1. Miyata, H. & Miyata, M. Mode✓ of conjugation in homothallic cells of Schizosaccharomyces◆ pombe. 0 J. Gen. Appl. Microbiol. 27, 365±371 (1981). 100 012345672. Egel, R. & Eie, B. Cell lineage asymmetry in Schizosaccharomyces pombe: unilateral transmission of a based on an optical trapping interferometer, is well suited to high-frequency state of mating-type switching in diploid pedigrees. Curr. Genet. 12, 429±433 (1987). 16 3. Klar, A. J. S. Differentiated parental DNALoad strands confer (pN) developmental asymmetry on daughter cells bead isometric measurements of stall force , and was used previously in ®ssion yeast. Nature 326, 466±470 (1987). kinesin walking on microtubules 0 4. Klar, A. J. S. The developmental fate of ®ssion yeast cells is determined by the pattern of inheritance of Position clamp b parental and grandparental DNA strands. EMBO J. 9, 1407±1415 (1990). to study force generation by individual RNA polymerase 5. Klar, A. J. S. & Bonaduce, M. J. The mechanism of ®ssion yeast mating-type interconversion: evidence 18 for two types of epigenetically inherited chromosomal imprinted events. Cold Spring Harb. Symp. laser followsmolecules the .bead Position clamp measurements on single kinesin Quant. Biol. 58, 457±465 (1993). –100 8 6. Beach, D. H. Cell type switching by DNA transposition in ®ssion yeast. Nature 305, 682±687 (1983). letters7. Nielsen, O. & Egel, R. to Mapping nature the double-strand breaks at the mating-type locus in ®ssion yeast by and keepsmotors ﬁxed gave force stall forces that rose with ATP concentration, as genomic sequencing. EMBO J. 8, 269±276 (1989). ∆x 8. Klar, A. J. S. & Miglio, L. M. Initiation of meiotic recombination by double-strand DNA breaks in trap

S. pombe. Cell 46, 725±731 (1986). (nm) Position –200 anticipated (Fig. 3b). An inherent problem encountered in these Received9. Moreno, 24 December S. A., Klar, 1998; A. J. S. accepted & Nurse, 17 P. May Molecular 1999. genetic analysis of ®ssion yeast Schizosaccharomyces a bead 1. Miyata,pombe. Methods H. & Miyata, Enzymol. M. Mode194, 795±823 of conjugation (1991). in homothallic cells of Schizosaccharomyces pombe. experiments was the dramatic decline in kinesin processivity at 7 10.J. Egel, Gen. R., Appl. Beach, Microbiol. D. H. & Klar,27, 365±371 A. J. S. Genes (1981). required for initiation and resolution steps of mating-type 100 2. Egel,switching R. & in Eie, ®ssion B. Cell yeast. lineageProc. asymmetry Natl Acad. in Sci.Schizosaccharomyces USA 81, 3481±3485 pombe (1984).: unilateral transmission of a microtubule kinesin 12 11. Singh, J. & Klar, A. J. S. DNA polymerase-a is essential for mating-type switching in ®ssion yeast. –300 high-frequency state of mating-type switching in diploid pedigrees. Curr. Genet. 12, 429±433 (1987). – loads approaching+ stall . Rapid detachment of beads from the 3. Klar,Nature A.361, J. S. Differentiated271±273 (1993). parental DNA strands confer developmental asymmetry on daughter cells bead 12.in Caddle, ®ssion M. yeast. S. &Nature Calos, M.326, P.466±470 Speci®c initiation(1987). at an origin of replication from Schizosaccharomyces 0 4.pombe Klar, A.. Mol. J. S. The Cell. developmental Biol. 14, 1796±1805 fate of ®ssion (1994). yeast cells is determined by the pattern of inheritance of microtubule made the collection of a large data set prohibitive, 13.parental Clyne, R. and K. grandparental & Kelly, T. DNA J. Genetic strands. EMBO analysis J. of9, 1407±1415 an ARS element (1990). from the ®ssion yeast –400 5.Schizosaccharomyces Klar, A. J. S. & Bonaduce, pombe M.. EMBO J. The mechanism J. 14, 6348±6357 of ®ssion (1995). yeast mating-type interconversion: evidence 0.0 0.5 1.0and accounted 1.5 2.0 for much of the experimental scatter (Fig. 3b). To 6 14.for Brewer, two B.types J., Lockshon, of epigenetically D. & Fangman, inherited W. chromosomal L. The arrest imprinted of replication events. forksCold in Spring the rDNA Harb. of Symp. yeast TimeFixed (s) trap Quant.occurs independently Biol. 58, 457±465 of transcription. (1993). Cell 71, 267±276 (1992). –100 15. Arcangioli, B. A site- and strand-speci®c DNA break confers asymmetric switching potential in ®ssion 200 7.4 Stall force (pN) Stall force 6. Beach, D. H. Cell type switching by DNA transposition in ®ssion yeast. Nature 305, 682±687 (1983). con®rm the position clamp measurements, we also measured stall yeast. EMBO J. 17, 4503±4510 (1998). b laser positionc is is ﬁxed 7. Nielsen, O. & Egel, R. Mapping the double-strand breaks at the mating-type locus in ®ssion yeast by 6.5 ± 0.1 pN 16.genomic Klar, A. J. sequencing. S. Lineage-dependentEMBO J. 8, mating-type269±276 (1989). transposition in ®ssion and budding yeast. Curr. Opin. 8. Klar,Genet. A. Dev. J. S.3, &745±751 Miglio, L. (1993). M. Initiation of meiotic recombinationPosition by double-strand DNA clamp breaks in forces∆x using a ®xed optical trap. Kinesin-coated beads moving 17. Sambrook, J., Fritsch, E. F. & Maniatis, T. Molecular Cloning: A Laboratory Manual 2nd ed (Cold (nm) 175 trap 6.5

S. pombe. Cell 46, 725±731 (1986). (nm) Position x –200

9. Moreno,Spring Harbor S. A., Klar, Laboratory A. J. S. &Press, Nurse, New P. MolecularYork, 1989). genetic analysis of ®ssion yeast Schizosaccharomyces ∆ 5 Fixed trap outwardsLoad (pN) bead in a stationary trap experience increasing loads until 18.pombe MacDonell,. Methods M. W., Enzymol. Simon,194, M. W.795±823 & Studier, (1991). F. W. Analysis of restriction fragments of T7 DNA and 10. Egel,determination R., Beach, of D. molecular H. & Klar, weights A. J. S. Genes by electrophoresis required for initiation in neutral and and resolution alkaline gels. stepsJ. of Mol. mating-type Biol. 110, switching119±146 (1977). in ®ssion yeast. Proc. Natl Acad. Sci. USA 81, 3481±3485 (1984). 150 microtubulethey5.6 arrivekinesin at a position where forward progress ceases: the ®nal 19.11. Egel,Singh, R. J. in &Molecular Klar, A. J. Biology S. DNA of polymerase- the Fission Yeasta is essential(eds Nasim, for mating-type A., Young, P. switching & Johnson, in ®ssion B. F.) 31±74 yeast. –3000.0 0.5 1.0 1.5 2.0 0 120 240 360 Nature(Academic,10361, San271±273 Diego, (1993). 1989). 100 1,000 Time (s)– Counts + 12.20. Caddle,Klar, A. M. J. S., S. Ivanova, & Calos, A.M. V., P. Speci®c Dalgaard, initiation J. Z., Bonaduce, at an origin M. of J. replication & Grewal, S.from I. S.Schizosaccharomyces in Epigenetics (ed. displacement, multiplied by the trap stiffness, supplies a measure of pombeCardew,. Mol. G.) 87±103Cell. Biol. (Novartis14, 1796±1805 Foundation (1994). Symposium 214, Wiley, Chichester, 1998). Figure 1 Operation of the force clamp. a, Sample record from the force clamp,12 13.21. Clyne,Arcangioli, R. B.K. & & Klar,ATP Kelly, A. J. T. S. J. Aconcentration novelGenetic switch-activating analysis of an site ARS (SAS1) element and ( its from cognateM) the binding ®ssion factor yeast µ –400 the stall force . Kinesin stall forces determined by this method Schizosaccharomyces(SAP1) required for ef®cient pombe. EMBOmat1 switching J. 14, 6348±6357 in Schizosaccharomyces (1995). pombe. EMBO J. 10, 3025±3032 showing0.0 kinesin-driven 0.5bead movement 1.0 and corresponding 1.5 optical 2.0 trap (1991). 14. Brewer, B. J., Lockshon, D. & Fangman, W. L. The arrest of replication forks in the rDNA of yeast displacement (2 mM ATP). Discrete stepsTime of 8(s) nm are readily apparent. Inset, 22.occurs Styrkarsdottir, independently U., Egel, of R. transcription. & Nielsen, O.Cell The71, smt-0267±276 mutation (1992). which abolishes mating-type switching exhibit a similar increase with ATP concentration (Fig. 3b). 15.in Arcangioli, ®ssion yeast B. A is site- a deletion. and strand-speci®cCurr. Genet. DNA23, 184±186 break confers (1993). asymmetric switching potential in ®ssion K.schematic Visscher200 representation et of theal., motility Nature assay7.4 used, 400, showing the184-189 experimental (1999) Figure 3 Load23.yeast. Gerring, dependenceEMBO S. L., J. Connelly,17, 4503±4510 C. & Hieter, (1998). P.in ofGuide to motility. Yeast Genetics and Moleculara, Biology Average(eds Fink, G. & geometry bead (notb velocity, to scale). The separationv betweenHow beadc and trap is was the nominally kinesin ATPase pathway coupled to mechanical 16.Guthrie, Klar, A. J. C.) S. Lineage-dependent 57±76 (Academic, San mating-type Diego, 1991). transposition in ®ssion and budding yeast. Curr.16 Opin. 6.5 ± 0.1 pN ®xed at Dx 175 nm. Bead position was sampled at 20 kHz, ®ltered with a 12-ms 24.Genet. Bresch, Dev. C., Muller,3, 745±751 G. & Egel, (1993). R. Genes involved in meiosis and sporulation of a yeast. Mol. Gen. Genet. (mean 6 s:e:m:), versus17.102, Sambrook,301±306 J., applied (1968).Fritsch, E. F. & Maniatis, load T. Molecular for ®xed Cloning: A Laboratory ATP Manual concentrations2nd ed (Cold boxcar(nm) 175 window (red for triangles, the feedback on the trap6.5 de¯ection, and saved un®ltered at x movement under load? One possibility is that kinesin coupling is Spring Harbor Laboratory Press, New York, 1989). 2.0∆ kHz. b, The measured bead±trap separation, Dx, for the record in a. c, 18.Acknowledgements. MacDonell, M. W.,We Simon, thank M.our W. colleagues & Studier, at ABL F. W. for Analysis suggestions of restriction and helpful fragments discussions; of T7 B. DNA Brewer, and Load (pN) M. Hoant and S. Hunt for help with the analysis of replication intermediates; A. Arthur for editorial left axis, 5 mM ATP,determinationN 19±57; of molecular weights blue by electrophoresis circles, in neutral right and alkaline axis, gels. J. 2 Mol. mM Biol. 110, ATP,HistogramN of37±87). the displacements The in b, convertedtight, to force involving by multiplying by the no futile hydrolysis events that fail to generate suggestions; and R. Frederickson for the art work. This work was sponsored by the Danish Natural Science 119±146 (1977). 150 -1 5.6 Research Council (J.Z.D.) and the National Cancer Institute, Department of Health and Human Services, trap stiffness (0.037 pN nm ). Solid red line is gaussian ®t to these data, yielding a 19. Egel, R. in Molecular Biology of the Fission Yeast (eds Nasim, A., Young, P. & Johnson, B. F.) 31±74 0.0 0.5 1.0 1.5 2.0 0 120 240 360 velocity point at (5.6under contractpN, with 5 m ABL.M The ATP) contents of this is article likely do not necessarily to represent re¯ect the views or politics an of overestimate the load of 6:5 6 0:1 pN because (mean 6 s:d:). movement, even under load (that is, e 1). In this case, any changes DHHS,(Academic, nor does San mention Diego, of 1989). trade names, commercial products, or organizations imply endorsement by Time (s) Counts 20.the Klar,US Government. A. J. S., Ivanova, A. V., Dalgaard, J. Z., Bonaduce, M. J. & Grewal, S. I. S. in Epigenetics (ed. beads which stalledCardew, completely G.) 87±103 (Novartis Foundation (v Symposium0) were 214, Wiley, indistinguishable Chichester, 1998). from beads lack- in the rates of load-dependent transitions must account entirely for 21.Correspondence Arcangioli, B. and & Klar, requests A. J. for S. materials A novel switch-activating should be addressed site to (SAS1) A.K. (e-mail: and its [email protected]). binding factor Figure 1 Operation of the force clamp. a, Sample record from the force clamp, (SAP1) required for ef®cient mat1 switching in Schizosaccharomyces pombe. EMBO J. 10, 3025±3032 showingmechanochemical kinesin-driven studies bead movement under controlledand corresponding external optical loads. trap ing active motors,(1991). and so were not included in the analysis. b, Stall force the observed decreases in velocity under load. However, more displacementAnalysis of (2 records mM ATP). of Discrete kinesin steps motion of 8 nm under are readily variable apparent. ATP Inset,con- 22. Styrkarsdottir, U., Egel, R. & Nielsen, O. The smt-0 mutation which abolishes mating-type switching centrations and loads revealed several new features. First, kinesin (mean 6 s:e:m:) versusin ®ssion yeast ATP is a deletion. concentration,Curr. Genet. 23, 184±186 (1993). measured eitherschematic with representation the position of the motility assaycomplicated used, showing the experimental scenarios may be entertained. Although the rise in 23. Gerring, S. L., Connelly, C. & Hieter, P.in Guide to Yeast Genetics and Molecular Biology (eds Fink, G. & geometrystepping (not appears to scale). to The be tightlyseparation coupled between to bead ATP and hydrolysis trap was nominally over a 17,18 SingleGuthrie, C.) 57±76 kinesin (Academic, San Diego, 1991). molecules wide range of forces, with a single hydrolysis per 8-nm mechanical clamp (red triangles)24. Bresch, C., Muller,or with G. & Egel, a R. Genes ®xed involved optical in meiosis and sporulation trap of(blue a yeast. Mol. circles). Gen. Genet. ®xed Stalls at Dx had175 nm. to Bead last position a was sampledKm atwith 20 kHz, ®ltered load with a 12-ms eliminates the simpler versions of loose coupling, 102, 301±306 (1968). boxcaradvance. window Second, for the the feedback kinesin on the stall trap de¯ection, force depends and saved on un®ltered the ATP at studied with a molecular concentration. Third, increased loads reduce the maximum velo- minimum of 2 s toAcknowledgements. be includedWe thank in our colleagues the analysis. at ABL for suggestions Data and helpful points discussions; B. Brewer, represent2.0 kHz. b an, The average measured bead±trap of separation,mechanochemicalDx, for the record in a. c, coupling in kinesin might display an admixture M. Hoant and S. Hunt for help with the analysis of replication intermediates; A. Arthur for editorial Histogramcity as expected, of the displacements but also raise in b, converted the apparent to force Michaelis±Menten by multiplying by the suggestions; and R. Frederickson for the art work. This work was sponsored by the Danish Natural Science either 12±29 (positionforce clamp) clamp or 6±70 (®xed trap) stalls. constant. The kinesin-1 cycle therefore contains at least one load- Research Council (J.Z.D.) and the National Cancer Institute, Department of Health and Human Services, trap stiffness (0.037 pN nm ). Solid red line isof gaussian both ®t to these tight data, yielding and a loose elements, with load-dependent transitions under contract with ABL. The contents of this article do not necessarily re¯ect the views or politics of the loaddependent of 6:5 6 0 transition:1 pN (mean 6 affectings:d:). the rate at which ATP molecules KoenDHHS, nor Visscher does mention*²³ of trade, Mark names, J. commercial Schnitzer products,*§ or³ & organizations Steven imply M. endorsement Block*² by§ bind and subsequently commit toand hydrolysis. unproductive It is likely that at hydrolyses conspiring jointly to lower the velo- the US Government. * Department of Molecular Biology, ² Princeton Materials Institute, and least one other load-dependent rate exists, affecting turnover §CorrespondenceDepartment and of requestsPhysics, for Princeton materials should University, be addressed Princeton, to A.K. (e-mail: New Jersey [email protected]). 08544, USA mechanochemicalnumber. Together, studies these ®ndings undercity. will controlled necessitate Another external revisions loads. candidate to mechanism for loose coupling involves an 10±12 ³ These authors contributed13 equally to this work. Analysisour understanding of records of of how kinesin kinesin motion motors under function. variable ATP con- 10±12 ...... traps or microneedles , all of which reportedcentrationsThe nearly mechanochemical and loads linear revealed coupling several ratioincrease new relates features. the number in First, the kinesin of ATP frequency of 8-nm backwards steps with load. Kinesin is a two-headed, ATP-driven motor protein that moves moleculesstepping appears hydrolysed to be to tightly the distance coupled travelled to ATP hydrolysis during a over single a decreases in velocityprocessively along with microtubules increased in discrete load, steps of 8 nm, for prob- bothenzymatic limiting cycle. Determination and of thisHowever, ratio over a range the of loads latter mechanism may be discounted because the Single kinesin molecules 1±4 wide range of forces, with a single hydrolysis per 8-nm mechanical saturating ATPably concentrations. by advancing each of its heads In alternately fact, such in sequence linear. shedsadvance. force±velocity light Second, on the the mechanism kinesin stall forfraction force movement, depends andof on hasbackwards the been ATP steps found in this study (data not shown) Molecularstudied details with of how the a molecular chemical energy stored in ATP is consideredconcentration. one Third, of the increased most pressing loads reduce challenges the maximum in the motility velo- 1 8,11 coupled to mechanical displacement remain obscure. To shed ®eldcity as. However, expected, for but the also microtubule-based raise the apparent molecular Michaelis±Menten motor kine- relations formedlight on the this question,basis a of force an clamp argument was constructed, for based a on load-independenta sin, the task is by no means straightforward.and by Direct others comparisons remains only ,5±10% at both negligible and force clamp constant. The kinesin cycle10 therefore contains at least11,12 one load- feedback-driven optical trap capable of maintaining constant betweendependent solution transition ATPase affecting measurements the rate at and which motility ATPin molecules vitro are Km (refs 10, 19) which supported a5 simple, loosely coupled picture . substantial loads. loadsKoen Visscher on single*²³, kinesin Mark J. motors Schnitzer. The*§³ & instrument Steven M. Block provides*²§ onlybind feasible and subsequently in the absence commit of external to hydrolysis. loads, as It these is likely cannot that be at However, theunprecedented* Department earlier of Molecular resolution studies Biology, of² Princeton molecular lacked Materials motion Institute, the and and improved permits appliedleast one in resolution other bulk solution. load-dependent Single-molecule rate exists,To ¯uorescence study affecting methods, turnover the coupling mechanism under load, we performed a § Department of Physics, Princeton University, Princeton, New Jersey 08544, USA number. Together, these ®ndings will necessitate revisions to © 1999 Macmillan Magazines Ltd 23,24 afforded by the³184These force authors contributed clamp, equally and to this work. subtle differencesour in understanding shape of were how kinesinNATURE¯uctuation| motorsVOL 400 | 8 function. JULY 1999 | www.nature.com analysis of kinesin movement subjected to the force ...... The mechanochemical coupling ratio relates the number of ATP not distinguishedKinesin within is a two-headed, experimental ATP-driven motor error. protein that Data moves frommolecules all hydrolysed previous to the distanceclamp. travelled Fluctuation during a single analysis, which relies on the processive property processively along microtubules in discrete steps of 8 nm, prob- enzymatic cycle. Determination of this ratio over a range of loads studies consistentlyably by advancing showed each of a its lower heads alternately stall force in sequence at1±4 limiting. sheds light ATP on the than mechanism forexhibited movement, and by has some been molecular motors, has been used previously to Molecular details of how the chemical energy stored in ATP is considered one of the most pressing challenges in the motility 8 at saturating ATP,althoughcoupled to mechanical the displacement difference remain obscure. in force To shed was®eld insuf®cient1. However, for the microtubule-based in study molecular ATP motor coupling kine- in kinesin near zero load , as well as the duty any given experimentlight on this question, to have a force clamp been was constructed, considered based on signi®cant. a sin, the task is by More- no means straightforward.ratio of Direct the comparisons bacterial rotary motor under torque25. Fluctuations in feedback-driven optical trap capable of maintaining constant between solution ATPase measurements and motility in vitro are over, a greaterloads discrepancy on single kinesin motors (,15. pN) The instrument between provides theonly stall feasible in force the absence at of externaldisplacement loads, as these cannot are be quanti®ed by a randomness parameter, r, which unprecedented resolution of molecular motion and permits applied in bulk solution. Single-molecule ¯uorescence methods, limiting and saturating ATP was observed when the limiting measures the temporal irregularity of mechanical advances. For an © 1999 Macmillan Magazines Ltd 184 NATURE | VOL 400 | 8 JULY 1999 | www.nature.com concentration of ATP was ,35-fold lower than the unloaded Km ensemble of displacement records, x(t), r is de®ned by (ref. 13), as compared to when limiting ATP values were only ,6- x2 t 2 x t 2 fold lower (,0.5 pN, refs 10±12). We therefore measured velocity as r lim h i h i t ` d x t a function of load in the force clamp at both 2 mM and 5 mM ATP, ! h i choosing the lower ATP level to be .17-fold below the unloaded Km where d is the motor step size, and the angle brackets denote an to highlight any potential differences. ensemble average23,24. A perfectly clocklike motor displays no step- Force±velocity curves obtained with the force clamp displayed ping irregularity, and has r 0, whereas a `Poisson' motor with one different shapes at limiting and saturating ATP levels (Fig. 3a), and biochemical transition moves at exponentially distributed intervals, they also showed different stall forces. The 5-mM curve is nearly and has r 1. In general, r-1 provides a continuous measure of the linear, with a stall force of ,5.5 pN. In contrast, the 2-mM curve is number of rate-limiting transitions in the overall mechanochemical concave down, and supports a greater stall force of ,7.0 pN. cycle. Determinations of randomness are insensitive to thermal or Strikingly, a 5-pN load leads to a ,40% decline in speed for other stationary noise and can therefore be performed even when 2 mM ATP, whereas at 5 mM ATP an identical load is suf®cient to individual steps are not resolved. When steps can be discerned amid

17 Skeletal muscle contraction by myosin motors

30µm

2µm

(A) Z disc dark band light band (B) Z disc titin Z disc M line myosin (thick filament) Cap Z tropomodulin 60nm myofibril

minus end nebulin actin (thin filament) one sarcomere plus end of actin filament thick filament (myosin) (C) actin (thin filament) myosin (thick filament) thin filament (actin) 2.2µm light band dark band light band 18

M line Z disc Z disc 60 nm

myosin heads

Figure 16.7: The structure of muscle. (A) Thin-section electron micrograph showing the organization of a single sarcomere when a muscle is stretched. The dark band in the middle represents the location of the aligned myosin thick filaments and the light bands on the sides show the position of actin. The diagrams below show the change in sarcomere length during muscle contraction. (B) The regular structure of the sarcomere depends on the precise arrangement and alignment of many structural proteins. The long proteins titin and nebulin help to set the length of the actin thin filaments and determine the overall length of the sarcomere. The Z disc serves as an anchor point for the actin thin filaments, and ensures that they are all oriented in the same direction, so that the myosin heads walking toward the actin filament plus ends will cause the sarcomere to shorten. (C) A quick-freeze deep-etch electron micrograph shows the extremely regular spacing of thick myosin filaments alternating with thin actin filaments and the myosin heads bridging the gap between them. (A, courtesy of Roger Craig; B, adapted from B. Alberts et al., Molecular Biology of the Cell, 5th ed. Garland Science, 2008; C, courtesy of J. Heuser.)

properties of individual motors, how much force might we estimate can be applied by an array of motors such as are found in amuscle?Wecandevelopanestimateofthisforcebyexamin- ing the structure and function of muscles. Figure 16.7(B) shows acartoonofamusclecellmadeupofmusclefibersormyofib- rils. The myofibrils are themselves composed of contractile units called sarcomeres. Myosin molecules are arranged in a cylindrically symmetric structure called the thick filament, and exert forces on the outer actin filaments. We can estimate the net force per myosin molecule by appealing to a simple picture in which muscles are thought of as arrays of springs in series and parallel as shown in Figure 16.8. The number of myosins in a cross-section of muscle is roughly

cross-sectional area of muscle N myosin ≈ cross-sectional area of thick ﬁlament π(3cm)2 N 300 1014. × myosin/thick ﬁlament ≈ π(60 nm)2 × ≈

If we assume that the force scale associated with a muscle like the biceps in the upper arm is 100 N (corresponding to

THE DYNAMICS OF MOLECULAR MOTORS 631

“chap16.tex” — page 631[#9] 29/9/2012 11:03 Skeletal muscle contraction by myosin motors sarcomere 2.2µm

(A) Z disc dark band light band (B) Z disc titin Z disc M line myosin (thick filament) Cap Z tropomodulin 60nm myofibril

M line Z disc Z disc Muscles contract at twice Estimated60 force generated nm the speed of myosin motors by myosin motors 0.1-1µm/s 2pN ⇠ 300 20N/cm2 myosin heads 2 Muscles may contract by Figure 16.7: The structure⇥ of muscle.⇡(30nm) (A) Thin-section⇠ electron micrograph showing the organization of a single sarcomere when a muscle is stretched. The dark band in the middle represents the location of the aligned myosin thick filaments and the light bands on the sides show the position of actin. The diagrams below show the change in sarcomere length5%-45% during muscle per second! contraction. (B) The regular structure of the sarcomere depends on the precise arrangement and alignment of many structural proteins. The long proteins titin and nebulin help to set the length of the actin thin filaments and determine the overall length of the sarcomere. The Z disc serves as an anchor point for the actin thin filaments, and ensures that they are all oriented in the same direction, so that the myosin heads walking toward the actin filament plus ends will cause the sarcomere to shorten. (C) A quick-freeze deep-etch electron micrograph shows the extremely regular spacing of19 thick myosin filaments alternating with thin actin filaments and the myosin heads bridging the gap between them. (A, courtesy of Roger Craig; B, adapted from B. Alberts et al., Molecular Biology of the Cell, 5th ed. Garland Science, 2008; C, courtesy of J. Heuser.)

cross-sectional area of muscle N myosin ≈ cross-sectional area of thick ﬁlament π(3cm)2 N 300 1014. × myosin/thick ﬁlament ≈ π(60 nm)2 × ≈

If we assume that the force scale associated with a muscle like the biceps in the upper arm is 100 N (corresponding to

THE DYNAMICS OF MOLECULAR MOTORS 631

“chap16.tex” — page 631[#9] 29/9/2012 11:03 Skeletal muscle contraction is controlled by nerve cells muscle nerve cells(A)cell (B) muscle cells nerve cell nerve cell synaptic vesicles Electric signal from nerve cells releases Ca2+ from sarcoplasmic reticulum

synapses muscle cell myosin-binding site exposed (C) synapses tropomyosin blocking by Ca2+-mediated tropomyosin actin tropomyosin myosin-binding site movement troponin complex actin + Ca2+

2+ _ 2+ 2+ Low Ca , muscles are relaxed Ca High Ca , muscles are contracted 10 nm

Figure 16.39: Schematic of coordination in muscle contraction. (A) A bundle of muscle fibers is innervated by a single motor neuron that forms communication synapses with all the fibers in the bundle. (B) At each of these synapses, a differentiated portion of the nerve terminal is stuffed with acetylcholine-containing vesicles lined up across the intercellular gap from their target muscle cell. After the neuron releases its neurotransmitters, acetylcholine binding to receptors on the muscle cell triggers a nearly instantaneous influx of calcium ions throughout the entire cytoplasm of the giant muscle cell. (C) In the resting muscle cell, myosin heads are prevented from binding to the actin filaments because the actin is coated with a series of copies of a long, skinny protein called tropomyosin, which is also associated with several small calcium-binding proteins called troponins. When calcium ions bind to troponin, a conformational change causes tropomyosin to shift its position on the actin filament, revealing the myosin-binding sites, which can then be simultaneously attacked by the myosin motor heads poised nearby. (A, courtesy of T. Caceci; B, courtesy of J. Heuser; C, adapted from B. Alberts et al., Molecular Biology of the Cell, 5th ed. Garland Science, 2008.)

the rear head not release from the filament until the front head is firmly bound. Otherwise, the motor and its attached cargo would be at risk of drifting away due to thermal motion, which is significant at molecular distance scales. Thus, the two heads must somehow communicate through their linked domains to influence one another’s cycles of ATP hydrolysis and filament binding. The mechanisms of this form of molecular communication are currently under investigation. One interesting approach to this problem is illustrated in Figure 16.40. In this experiment, protein engineering was used to create a series of artifically oligomerized kinesin motor heads, linked by a series of rigid rod-shaped protein domains and elastic spring-like protein domains. While individual kinesin heads were able to generate microtubule glid- 20 ing at reasonable rates, linked pairs and triplets were able to cooperate to generate progressively faster motion. Cooperation was manifested even by these artificially engineered protein constructs, where the molecular structure of the links between the motor heads bears no resemblance to the links found in any real molecular motors. This result provides hope that the mechanisms of cooperation between linked heads can perhaps be understood in purely mechanical terms. Frequently, it is necessary for multiple copies of different kinds of motors to work together, for example in the intracellular transport of

RECTIFIED BROWNIAN MOTION AND MOLECULAR MOTORS 659

“chap16.tex” — page 659[#37] 29/9/2012 11:03 How muscles get ATP energy?

Aerobic respiration O2 C H O + 6O = 6CO + 6H O + 36ATP 6 12 6 2 ) 2 2 Anaerobic respiration

no O2 C H O = 2 Lactic Acids + 2ATP 6 12 6 ) (muscle fatigue) glucose pyruvate

Aerobic respiration

Note: Citric acid cycle = Krebs cycle

21 Electron transport chain

Mitochondrion

NADH products of the Cytric acid cycle are ATP synthase used to pump H+ to the space between outer and inner mitochondrial membrane.

Gradient of H+ concentration drives the ATP synthase motor that converts ADP to ATP.

Note: ATP synthase can run in reverse and use ATP to pump H+ at low concentrations.

22 Crawling of cells

Immune system: neutrophils chasing bacteria

migration of skin cells during wound healing spread of cancer cells during metastasis of tumors amoeba searching for food David Rogers, 1950s v 0.1µm/s ⇠

23 Crawling of cells 952 Chapter 16: The Cytoskeleton ﬁsh skin cell v =0.2µm/s

actin cortex lamellipodium substratum Figure 16–75 A model of how forces generated in the actin-rich cortex move a cell forward. The actin-polymerization- dependent protrusion and firm attachment of a lamellipodium at the leading edge of the cell move the edge forward (green arrows at front) and stretch the actin actin polymerization at cortex. Contraction at the rear of the cortex under tension plus end protrudes lamellipodium cell propels the body of the cell forward (green arrow at back) to relax some of PROTRUSION the tension (traction). New focal contacts are made at the front, and old ones are disassembled at the back as the cell crawls movement of unpolymerized actin forward. The same cycle can be repeated, moving the cell forward in a stepwise Figure 14.2: Organization of actin myosin II fashion. Alternatively, all steps can be filaments at the leading edge of a crawling cell. This fish skin keratocyte ATTACHMENT AND tightly coordinated, moving the cell forward was moving upward atcontraction the time that it TRACTION was fixed. Its membrane was stripped smoothly. The newly polymerized cortical off and the filamentous structures imaged after being coated with a very actin is shown in red. thin layer of platinum. The large white blob towards the bottom of the cell shows the location of the nucleus and the membranous organelles. The area focal adhesions within the white box is blown up actinbelow. Essentially all of the filaments (contain integrins) are actin. Two regions are shown at still higher magnification on the right. At the top, near the front edge of the cell, the actin filaments are short and frequently branched. Further back, individual actin filaments are longer and overlaid at nearly random angles. (Adapted from T. M. Svitkina et al., J. Cell Biol. 139:397, 1997.)

1 mm R. Phillips et al., Physical Filopodia, formed by migrating Albertsgrowth cones et al., of Molecular neurons and some types of fbroblasts, are essentially one-dimensional. Tey contain a core of long, bun- proteins in mitochondriaBiology can be manipulatedof the byCell the application of an 24 electric ﬁeld. The result of the application of a ﬁeld is to segregate Biology of the Cell the membrane proteins to one side of the membrane, as shown in dled actin flaments, which are reminiscent of those in microvilli but longer and the electron micrograph of freeze-fractured membranes. This image thinner, as well as more dynamic. Lamellipodia, formed by epithelial cells and and others demonstrate the large fraction of membrane devoted to membrane proteins. Indeed, in the case of the mitochondria, more fbroblasts, as well as by some neurons, are two-dimensional, sheetlike structures. than 50% of the membrane mass is donated by proteins. Interesting measurements of the relative mass of phospholipids and membrane Tey contain a cross-linked mesh of actin flaments, most of which lie in a plane proteins are reported in Mitra et al. (2004). Just as simple estimates on protein concentrations in the cytoplasm parallel to the solid substratum.MBoC6 Invadopodia m16.86/16.77 and related structures known as reveal a mean spacing comparable to protein size, similar estimates can be carried out for the cell membrane as well. To see this, we podosomes represent a third type of actin-rich protrusion. Tese extend in three recall that the membrane area for a bacterium like E. coli is roughly 6 µm2 and that both the inner and outer membranes are home to dimensions and are important for cells to cross tissue barriers, as when a meta- roughly 500,000 proteins. This tells us that the area per protein is roughly 10 nm2, or that the typical distance between two proteins is static cancer cell invades the surrounding tissue. Invadopodia contain many of of the order of 3 nm. The cell membrane is tightly packed indeed, with a crude estimate being that roughly half of the membrane area is the same actin-regulatory components as flopodia and lamellipodia, and they occupied by proteins. also degrade the extracellular matrix, which requires the delivery of vesicles con-

14.1.4 Consequences of Crowding taining matrix-degrading proteases.

So far, we have examined the structural features of the crowded envi- A distinct form of membrane protrusion called blebbing is often observed in ronment of cells. We have seen that this crowded structure that serves as the backdrop for the bustling biochemical metropolis of the cell vivo or when cells are cultured on a pliable extracellular matrix substratum. Blebs is quite different from the dilute and homogeneous environments of solution biochemistry. What are the consequences of this crowding for form when the plasma membrane detaches locally from the underlying actin cells? We explore two broad classes of consequence arising from this crowding, namely, how equilibrium binding is modified by crowding cortex, thereby allowing cytoplasmic fow to push the membrane outward (Fig- and how diffusive processes are altered as a result of the tight packing of molecules within cells. ure 16–76). Bleb formation also depends on hydrostatic pressure within the cell, which is generated by the contraction of actin and myosin assemblies. Once blebs CROWDING,have LINKAGE, extended, AND ENTANGLEMENT actin547 flaments reassemble on the bleb membrane to form a new

“chap14.tex” — page 547[#5] 29/9/2012 10:16 light micrograph GFP-actin merge

Figure 16–76 Membrane bleb induced by disruption of the actin cortex. On the left is a light micrograph showing a spherical membrane protrusion or bleb induced by laser ablation of a small region of the actin cortex. The cortex is labeled green in the middle image by expression of 3 µm GFP-actin. (Courtesy of Ewa Paluch.)

MBoC6 n16.208/16.78 Movement of bacteria Listeria monocytogenes moving in PtK2 cells

Actin polymerization is pushing bacteria Caps prevent further polymerization in comet tails

Actin is randomly (speeded up 150x) depolymerized Julie Theriot in comet tails v 0.1 0.3µm/s ⇠ L. A. Cameron et al., Nat. Rev. Mol. Cell Biol. 1, 110 (2000) 25 outer Swimming of sperm cells dynein arm radial spoke

inner sheath

central singlet nexin Sperm ﬂagellum is constructedmicrotubule from microtubules

outer dynein arm radial spoke

inner sheath plasma membrane inner central singlet nexin dynein arm microtubule (A) (B) (C) 100 nm A microtubule B microtubule outer doublet microtubule

plasma membrane inner dynein arm Figure 16.4: Structure of the axoneme. (A) Thin-section electron micrograph of a cross-section of the flagellum of the (A) microscopic algae Chlamydomonas(B) . (B) Diagram of the flagellar parts. The inner dynein(C) arms and outer dynein arms are motor proteins that100 nm make the flagellum beat by slidingA microtubule the microtubules B microtubule relative to one another. (C) A side view of the flagellum. (A, courtesy of Lewis Tilney.) outer doublet microtubule

Figure 16.4: Structure of the axoneme. (A) Thin-section+ electron micrograph of a cross-section of the flagellum ofFigure the 16.5: Bending of flagella and microscopic algae Chlamydomonas. (B) Diagram of the flagellar parts. The inner dynein arms and outer dynein armscilia are motordue to translational motors. (A) proteins that make the flagellum beat by sliding the microtubules relative to one another. (C) A side view of the flagellum.When adjacent filaments are not (A, courtesy of Lewis Tilney.)+ + ++ ++ tethered together, stepping of the linking motors will result in sliding of the proteins + adjacent filaments. (B) When the adjacent filaments are tethered Figure 16.5: Bending of flagella and + together, stepping of the motors will cilia due to translational motors. (A) result in deformation of the filaments. When adjacent filaments are not + + ++ (Adapted from B. Alberts et al., ++ tethered together, stepping of the +ATP linking bend motors will result in slidingMolecular of the Biology of the Cell, 5th ed. proteins + adjacent filaments. (B) WhenGarland the Science, 2008.) adjacent filaments are tethered Jeff Guasto together, stepping of the motors will result in deformation of the filaments. – (Adapted from B. Alberts et al., +ATP bend Molecular Biology of the Cell, 5th ed. v 50µm/s –– ⇠ –– –– Garland Science, 2008.) isolated doublet – normal microtubules– flagellum

(A) DYNEIN PRODUCES (B) ––DYNEIN CAUSES –– –– MICROTUBULE SLIDING MICROTUBULE TO BEND isolated R. Phillips et al., Physical doublet – normal microtubules flagellum 26 Biology of the Cell (A) DYNEIN compressivePRODUCES force that induces(B) DYNEIN bending. CAUSES In Figure 16.5(A), we see that MICROTUBULE SLIDING MICROTUBULE TO BEND if two adjacent filaments are not linked, then the motion of the molecular motors will induce sliding. On the other hand, if the filaments are tied together, these motors will generate bending, as is shown compressive forcein Figure that induces 16.5(B). bending. The energetics In Figure of 16.5(A), filament we seebending that can be modeled if two adjacentusing filaments beam are theory, not linked, as discussed then the motion in Chapter of the 10. molec- Spatial and temporal ular motors willcoordination induce sliding. of dynein On the motor other stepping hand, if thealong filaments the length of the cilium are tied together, these motors will generate bending, as is shown or flagellum gives rise to regular beat patterns that propagate down in Figure 16.5(B). The energetics of filament bending can be modeled the structure. For example, a snapshot of the flagellum of a swimming using beam theory, as discussed in Chapter 10. Spatial and temporal coordination of dynein motor stepping along the length of the cilium or flagellum gives rise to regular beat patterns that propagate down THE DYNAMICS OF MOLECULAR MOTORS 629 the structure. For example, a snapshot of the flagellum of a swimming

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