The Physics of Molecular Motors

1. The Physics of Molecular Motors • fluctuations in small engines • … and the II Law of Thermodynamics • noise rectification mechanisms RD Astumian, Sci. Am., July 2001, 57 P. Hanggi and F.M., Rev. Mod. Phys., 81 (2009) 387

2. Self propulsion from macro to micro scales scallops, 10-2m shell flaps, jets high Reynolds numbers R=avr/h~100 1D Purcell’s (scallop) theorem bacteria, 10-5m low Reynolds numbers R~10-4 flagellum strokes corkscrew, v  ω flexible oar, v  ω2 2D

3. myosin, 10-8 biological motor on a track: 10-16-10-17W from ATP vs. 10-8W from heat bath power strokes:ATP hydrolysis, ATP→ADP+20kBT, efficiency ~50%; power from “fuel” comparable with power from/to environment Brownian motion: time to diffuse a particle length is a2/D, i.e. much shorter than the drift time a/v — D=kT/6pha, v~3mm/s not a deterministic engine, rather a directed random walker and still a very efficient motor!! (Yanagida, 1999)

4. Rectifying thermal fluctuations? unbalanced wheel pawl ratchet L. da Vinci R. Feynman SPRING PAWL VANE RATCHET noise harvesting, noise-powered small devices

5. E. Coli ATP synthase enzyme reverse reaction ADP + Pi→ATP Wang&Oster, Nature (1998)

6. e t impossible (at equilibrium)! The Feynman Lectures on Physics, I-46 assign ratchet and vane temperatures T1 and T2; at equilibrium T1 = T2 angular velocity of ratchet t rectification W T1 = T2

7. Maxwell daemon If an automated devices doesn’t work, what about an intelligent one? J C Maxwell ... if we conceive of a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, …. will raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics (1871).

8. also impossible, but … M. Smoluchowski (1914): No automatic, permanently effective perpetual motion machine can violate the II Law by taking advantage of statistical fluctuations. Such device might perhaps function if operated by intelligent beings. W. H. Zurek (1989): The II Law is safe from intelligent beings as long as their abilities to process information are subject to the same laws as those of universal Turing machines P. Curie (1894): Rectification of statistical fluctuations requires simultaneous breaking of spatial and time symmetry

9. V(x) = cos(x) DV x Brownian motors assumptions: • overdamped particle on a periodic substrate V(x)=V(x+L) • zero-mean fluctuating x(t) and/or deterministic forces F(t) Langevin equation

10. symmetric substrate: V(-x) V(x) • 1.x(t) Gaussian, stationary and white (equilibrium noise) ‹x(t)x(0)›=2Dd(t); • F(t)=F1cos(W1t) sinusoidal signal, F(-t) -F(t) • 2.x(t) Gaussian, stationary and colored, (non-equilibrium noise) ‹x(t)x(0)›=(D/t)exp(-|t|/t) • [w/ or w/o a sinusoidal signal F(t)] different non-equilibrium options → no transport current,

11. harmonic mixing F(t) bi-harmonic signal, F(t) = F1cos(W1t+f1) + F2cos(W2t+f2); commensurate frequencies, W1/W2 = m/nw/ or w/o the noise x(t) rectification due to the interplay of nonlinearity and drive asymmetry F(-t) -F(t) biased, we cheated! F(t)

12. ratchet effect: rocked, pulsated, thermal • b. asymmetric substrate: V(-x) ≠ V(x) • 1. rocked: F(t) additivesinusoidal signal, F(t)=F1cos(W1t), w/ or w/o noise; • 2. pulsated:x(t) Gaussian and white, ‹x(t)x(0)›=2Dd(t); F(t) multiplicative sinusoidal signal, i.e. modulates substrate amplitude, F(t)=eV(x)cos(Wt) • 3. thermal: w/ or w/o drive; x(t) Gaussian and colored,‹x(t)x(0)›=(D/t)exp(-|t|/t); net transport current is the rule!

13. physical principles of ratchet operation flashing ratchet: substrate switches on and off periodically rocked ratchet: particle pushed right/left periodically F=0 On Off -Fx On Fx POSITION POSITION

14. x f* (c) F y counterintuitive effects: noise induced pulsated ratchet ac transverse dc drive ac longitudinal anomalous negative mobility: transport can be directed against dc drive

15. thermal J J rocked D t F FL FR General properties • resonant mechanism • vs. D or F, t or W • sensitive to parameters • › substrate profile • › particle mass • › inter-particle interactions • current inversions • optimization

16. Applications biology inspired nano-devices - Rev Mod Phys 81, 387 (2009) Optical tweezers Artificial m-pores Cold atoms traps D. G. Grier et al, Appl. Phys. Lett., 82, 3985 (2003). Z. Siwy and A. Fulinski, Phys. Rev. Lett. 89, 198103 (2002). F Renzoni et al, Phys. Rev. Lett. 95, 073003 (2005).

17. 1 m m PRL 99 Triangular traps PRL 04 PRL 01 Superconducting devices single vortex experiments …binary mixture experiments

18. folding - unfolding u-force f-force RNA/DNA handles Toward a new thermodynamics • mechanical stretching of a single RNA molecule (20nm long), • at constant loading rate r (below: r = 7.5pN/s) • irreversible folding-unfolding cycles: hysteretic work is dissipated 3mm

19. Fluctuation theorems • n.e. forward process XF(t): XA XB ; T constant; • XF(0) = XAinitialequilibrium state; XF(t) = XB n.e. final state • n.e. reverse process XR(t): XB XA • XR(0) = XBinitialequilibrium state; XR(t)= XA n.e. final state • XR(t) is time reversed with respect to XF(t), i.e. XR(s) = XF(t-s)for • 0st, with corresponding work p.d. PF(W) and PR(W) (Jarzynski, 1997) (Crooks, 1999)

20. Conclusions • biology inspired nano-devices powered by noise • role of noise at the small scales reconsidered • noise harvesting to power nano-devices for ICT