Brownian Transport I: Brownian Transport I: Brownian Transport I: Brownian Transport I: Molecular MotorsMolecular Motors

fl t ti i ll bi i• fluctuations in small bio-engines

• … and the II Law of Thermodynamics

• noise rectification mechanisms

RD Astumian Sci Am July 2001 57RD Astumian, Sci. Am., July 2001, 57

P. Hanggi and F.M., Rev. Mod. Phys., 81 (2009) 387

Self propulsionSelf propulsionfrom macro to micro scales

scallops, 10-2mshell flaps, jets

1D

high Reynolds numbers

R=avρ/η~100

bacteria, 10-5m

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

low Reynolds numbers R~10-4

flagellum strokes

k 2D

corkscrew, v ∝ ω

flexible oar, v ∝ ω2

η Η2Ο = 10-2 g/cm s

Myosin motor:

100-1000 ATP molecules hydrolyzed per second

k f h l l i h ffi i f b20kT from each ATP molecule with an efficiency of about 50%

>>> power from fuel: 10-17-10-16W

Heat bath:

Energy scale: kT=0 025eV = 4 10-21JEnergy scale: kT 0.025eV 4 10 J

Time scale: τ=10-13

>>> power from bath: 10-8Wp

myosin, 10-8m

biological motor on a track: 10-17-10-16W from ATP vs. 10-8W from heat bathpower strokes: ATP hydrolysis, ATP→ADP+20kBT, efficiency ~50%; power strokes: ATP hydrolysis, ATP→ADP+20kBT, efficiency 50%; power from “fuel” 8-9 orders of magnitude smaller than from/to environment

Brownian motion: time to diffuse a particle length is a2/D, i.e. much Brownian motion: time to diffuse a particle length is a /D, i.e. much shorter than the drift time a/v — D=TkB/6πηa, v~3μm/s

not a deterministic

engine, rather a

directed random walker

and still

a very efficient motor!!(Yanagida, 1999)

Rectifying thermal fluctuations?Rectifying thermal fluctuations?

R. FeynmanL. da Vincipawl ratchetunbalanced wheel

SPRINGSPRING

VANEVANE

PAWLPAWL

VANEVANE

RATCHETRATCHET

Lippmann 1900, v. Smoluchowski 1912

noise harvesting,

noise-powered small devices

E. Coli ATP synthase enzymey y

reverse reaction

ADP + Pi→ATP

phosphor lationWang&Oster, Nature (1998)

phosphorylation

ATP (adenosine triphosphate) consists of adenosine — composed of an adenine ring and a ribose sugar — and three phosphate groups (triphosphate). The phosphoryl groups, starting with the group closest to the ribose, are referred to as the alpha (α), beta (β), and gamma (γ) phosphates. ATP is highly soluble in water and is quite stable in solutions between pH 6.8–7.4, but is rapidly hydrolysed at extreme pH. ATP is an unstable molecule in unbuffered water, in which it hydrolyses to ADP and phosphate.

ATP synthase is a general term for an enzyme that can synthesize ATP from ADP and inorganic phosphate by using a form of energy. This energy is often in the form

f i d l h i l diof protons moving down an electrochemical gradient, such as from the lumen into the stroma of chloroplasts or from the inter-membrane space into the matrix in mitochondria The overall reaction sequence is:mitochondria. The overall reaction sequence is:

ADP + Pi → ATPThese enzymes are of crucial importance in almost all organisms, because ATP is the common "energyorganisms, because ATP is the common energy currency" of cells.

impossible (at equilibrium)!

assign ratchet and vane

The Feynman Lectures on Physics, I-46

assign ratchet and vane

temperatures T1 and T2;

at equilibrium T1 = T2

ε

at equilibrium T1 = T2

τ

ratchet angular velocity

)(

)(12 //)( TT

BF ffετθεθ

νθ−+−

=−=Ωrectification

τ

)( 12 //)( TT ee ετθενθ + −= rectificationΩ T1 = T2

Maxwell daemonMaxwell daemon

If an automated devices doesn’t work,J C Maxwellwhat 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 p jto the same laws as those of universal Turing machines

P. Curie (1894): Rectification of statistical fl t ti i i lt fluctuations requires simultaneous breaking of spatial and time symmetry

Brownian motorsBrownian motors

assumptions:• overdamped particle on a periodic substrate V(x)=V(x+L)• overdamped particle on a periodic substrate V(x)=V(x+L)• zero-mean fluctuating ξ(t) and/or deterministic forces F(t)

V(x) = cos(x)ΔV

x

)()()( tFtxVx ++′−= ξ& Langevin equation

′

different non-equilibrium options → 0≠x&

)()()( tFtxVx ++′−= ξ&

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

1. ξ(t) Gaussian, stationary and white (equilibrium noise)

=&

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

F(t)=F1cos(Ω1t) sinusoidal signal, F(-t) -F(t)=&

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

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

0=x&no transport current,

harmonic mixingharmonic mixing

F(t) bi-harmonic signal, F(t) = F1cos(Ω1t+φ1) + F2cos(Ω2t+φ2);( ) g , ( ) 1 ( 1 φ1) 2 ( 2 φ2);commensurate frequencies, Ω1/Ω2 = m/nw/ or w/o the noise ξ(t)

)cos( 1221 φφ nmFFx mn −∝&

rectification due to the interplay of li it d d i t

2

F(t)nonlinearity and drive asymmetry

F(-t) -F(t) biased, we cheated! 0.5

1

1.5F(t)

≠&( ) ( ) b a d, a d2 4 6 8 10 12

-1

-0.5

≠

ratchet effect: rocked, pulsated, thermal

b. asymmetric substrate: V(-x) ≠ V(x)

1. rocked: F(t) additive sinusoidal signal, F(t)=F1cos(Ω1t), w/ or w/o noise;

2. pulsated: ξ(t) Gaussian and white, ‹ξ(t)ξ(0)›=2Dδ(t); F(t) multiplicative sinusoidal signal, i.e. modulates substrate amplitude F(t) V(x)cos(Ωt) modulates substrate amplitude, F(t)=εV(x)cos(Ωt)

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

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 periodicallyswitches on and off periodically pushed right/left periodically

F=0On

Off

-Fx

Fx

On

POSITIONPOSITION

thermal ratchets

LRFR=LLFL

qualitative argument (for ‘good’ potentials, only)

L=LL+LR

LRFR LLFL

barrier height dominates escape

FR-FLnew time scale τ

ξLL LR

& 0)(⎥⎤

⎢⎡

DLL

compare ‹ξ-Fi›τwith Li , i=L,R

x 0),( <⎥⎦

⎢⎣

−= DwLL LR

τ

0 0),( ,0⎯⎯ →⎯ ∞→ττ Dw

General General titi

J

thermal

propertiesproperties

t h i

Dτ

• resonant mechanismvs. D or F, τ or Ω

τ

• sensitive to parameters rockedJ

› substrate profile

› particle mass

› inter particle interactions› inter-particle interactions

• current inversionscurrent inversions

FFL FR

Applications:Applications: biology inspired nanobiology inspired nano--devicesdevices

P.Hanggi & FM, Rev Mod Phys 81, 387 (2009)P.Hanggi & FM, Rev Mod Phys 81, 387 (2009)

Optical tweezersOptical tweezers ArtificialArtificial μμ--porespores Cold atoms trapsCold atoms trapsOptical tweezersOptical tweezers Artificial Artificial μμ porespores Cold atoms trapsCold 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).

1 m mSuperconducting devicesSuperconducting devices

single vortex experiments

PRL 99 PRL 01Triangular traps PRL 04PRL 99 PRL 01Triangular traps PRL 04

… binary mixture experiments

H H

an example:an example: vortices in the vortices in the tiltedtilted magnetic fieldsmagnetic fields

k (PV) J h (JV) ti

HH

pancake (PV) vs. Josephson (JV) vortices

HHabab

HHcc

JJ((tt))FFJJ

abab

(Savelev & Nori, 2003; Bending et al., Bath, UK)

JV“active” vortices

thermal noise

repulsive interaction

Low T : Thermal noise is too weak to overcome barriers: red particles Low T : Thermal noise is too weak to overcome barriers: red particles move to the potential minima, green ones to the potential maxima

High T : Thermal noise shakes particles enough to jump over barriers

ConclusionsConclusionsConclusionsConclusions

¶ bi l i i d d i

G. G. CasatiCasati, H. , H. LinkeLinke, F. , F. MarchesoniMarchesoni

¶ biology inspired nano-devices powered by noise

¶ role of noise at the small scales reconsidered

¶ noise harvesting to power nano-devices for ICT

ConclusionsConclusionsConclusionsConclusions

¶ biology inspired nano-devices powered by noise

¶ role of noise at the small scales reconsidered

¶ noise harvesting to power nano-devices for ICT