The Physics The Physics of Molecular Motorsof 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

Self propulsionSelf propulsion

from macro to micro scales

scallops, 10-2mshell flaps, jets

high Reynolds numbers

R=av~100

bacteria, 10-5mlow Reynolds numbers R~10-4

flagellum strokes

corkscrew, v ω

flexible oar, v ω2

1D

2D

Purcell’s (scallop) theoremPurcell’s (scallop) theorem

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/6a, v~3m/s

not a deterministic

engine, rather a

directed random walker

and still

a very efficient motor!!

(Yanagida, 1999)

Rectifying thermal fluctuations?Rectifying thermal fluctuations?

VANEVANE

RATCHETRATCHET

noise harvesting,

noise-powered small devices

R. FeynmanL. da Vincipawl ratchetunbalanced wheel

SPRINGSPRING

PAWLPAWL

E. Coli ATP synthase enzyme

Wang&Oster, Nature (1998)

reverse reaction

ADP + Pi→ATP

impossible (at equilibrium)!

assign ratchet and vane

temperatures T1 and T2;

at equilibrium T1 = T2

angular velocity of ratchet

)(

)(

12 //)( TT

BF

ee

ff

rectification

The Feynman Lectures on Physics, I-46

T1 = T2

Maxwell daemonMaxwell daemon

J C MaxwellIf an automated devices doesn’t work,

what about an intelligent one?

... 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).

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

Brownian motorsBrownian motors

assumptions:• overdamped particle on a periodic substrate V(x)=V(x+L)

• zero-mean fluctuating (t) and/or deterministic forces F(t)

V(x) = cos(x)

x

V

)()()( tFtxVx Langevin equation

)()()( tFtxVx

0x

different non-equilibrium options →

no transport current,

0x

a. symmetric substrate: V(-x) V(x)

1. (t) Gaussian, stationary and white (equilibrium noise) ‹(t)(0)›=2D(t);

F(t)=F1cos(1t) sinusoidal signal, F(-t) -F(t)

2. (t) Gaussian, stationary and colored, (non-equilibrium noise) ‹(t)(0)›=(D/)exp(-|t|/)

[w/ or w/o a sinusoidal signal F(t)]

harmonic mixing

F(t) bi-harmonic signal, F(t) = F1cos(1t+1) + F2cos(2t+2); commensurate frequencies, 1/2 =

m/nw/ or w/o the noise (t)

)cos( 1221 nmFFx mn

rectification due to the interplay of nonlinearity and drive asymmetry

F(-t) -F(t) biased, we cheated!2 4 6 8 10 12

-1

-0.5

0.5

1

1.5

2

F(t)

b. asymmetric substrate: V(-x) ≠ V(x)

1. rocked: F(t) additive sinusoidal signal, F(t)=F1cos(1t), w/ or w/o noise;

2. pulsated: (t) Gaussian and white, ‹(t)(0)›=2D(t); F(t) multiplicative sinusoidal signal, i.e. modulates substrate amplitude,

F(t)=V(x)cos(t)

3. thermal: w/ or w/o drive; (t) Gaussian and colored, ‹(t)(0)›=(D/)exp(-|t|/);

ratchet effect: rocked, pulsated, thermal

net transport current is the rule!

physical principles of ratchet operation

flashing ratchet: substrate switches on and off periodically

rocked ratchet: particle pushed right/left periodically

-Fx

Fx

POSITIONPOSITION

F=0

On

On

Off

counterintuitive effects: noise induced

anomalous negative mobility: transport can be directed against dc drive

*x

y

F

(c)

actransverse

ac longitudinal

dc drive

pulsated

ratchet

General General propertiesproperties

D

J

• resonant mechanism

vs. D or F, or

thermal

• sensitive to parameters › substrate profile

› particle mass

› inter-particle interactions

• current inversions

• optimization

rocked

FFL FR

J

D. G. Grier et al, Appl. Phys. Lett., 82, 3985 (2003).

Optical tweezersOptical tweezers

Z. Siwy and A. Fulinski, Phys. Rev. Lett. 89, 198103 (2002).

Artificial Artificial -pores-pores

ApplicationsApplications biology inspired nano-devices - biology inspired nano-devices - Rev Mod Phys 81, 387 (2009)

Cold atoms trapsCold atoms traps

F Renzoni et al, Phys. Rev. Lett. 95, 073003 (2005).

1 m m

PRL 99 PRL 01Triangular traps PRL 04

Superconducting devicesSuperconducting devices

single vortex experiments

… binary mixture experiments

• 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

3m

Toward a new thermodynamicsToward a new thermodynamics

folding - unfolding

RN

A/D

NA

han

dles

u-force

f-force

• n.e. forward process XF(t): XA XB ; T constant; XF(0) = XA initial equilibrium state; XF(t) = XB n.e. final state

• n.e. reverse process XR(t): XB XA

XR(0) = XB initial equilibrium 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 0st, with corresponding work p.d. PF(W) and PR(W)

(Crooks, 1999)

Tk

W

Tk

G

BB

expexp

Fluctuation theoremsFluctuation theorems

(Jarzynski, 1997)

ConclusionsConclusions

¶ biology inspired nano-devices powered by noise

¶ role of noise at the small scales reconsidered

¶ noise harvesting to power nano-devices for ICT