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=avρ/η~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/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?
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
ffετθενθ
νθ−+− −=
=−=Ωrectification
τ
Ω
The Feynman Lectures on Physics, I-46
T1 = T2
ε
τ
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 ξ(t) and/or deterministic forces F(t)
V(x) = cos(x)
x
∆V
)()()( tFtxVx ++′−= ξ Langevin equation
)()()( tFtxVx ++′−= ξ
0=x
different non-equilibrium options →
no transport current,
0≠x
• symmetric substrate: V(-x) V(x)
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)]
=
=
harmonic mixing
F(t) bi-harmonic signal, F(t) = F1cos(Ω1t+φ1) + F2cos(Ω2t+φ2); commensurate frequencies, Ω1/Ω2 = m/n w/ or w/o the noise ξ(t)
)cos( 1221 φφ 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(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)=εV(x)cos(Ωt)
3. thermal: w/ or w/o drive; ξ(t) Gaussian and colored, ‹ξ(t)ξ(0)›=(D/τ)exp(-|t|/τ);
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
φ*x
y
F
(c)
actransverse
ac longitudinal
dc drive
pulsated
ratchet
General General propertiesproperties
D
J
τ
• resonant mechanism
vs. D or F, τ or Ω
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 µµ -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
3µm
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 0≤s≤t, 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
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