Brownian Transport I:
Molecular Motors
• fluctuations
fl t ti
iin small
ll bi
bio-engines
i
• … and the II Law of Thermodynamics
• noise rectification mechanisms
RD Astumian,
Astumian Sci.
Sci Am.,
Am July 2001,
2001 57
P. Hanggi and F.M., Rev. Mod. Phys., 81 (2009) 387
Self propulsion
from macro to micro scales
scallops, 10-2m
1D
shell flaps, jets
high Reynolds numbers
R=avρ/η~100
Purcell’s (scallop) theorem
bacteria, 10-5m
low Reynolds numbers R~10-4
flagellum strokes
corkscrew,
k
v∝ω
flexible oar, v ∝ ω2
2D
η Η2Ο = 10-2 g/cm s
Myosin motor:
100-1000 ATP molecules hydrolyzed per second
20kT
k ffrom each
h ATP molecule
l l with
i h an efficiency
ffi i
off about
b
50%
>>> power from fuel: 10-17-10-16W
Heat bath:
Energy scale: kT
kT=00.025eV
025eV = 4 10-21J
Time scale: τ=10-13
>>> ppower from bath: 10-8W
myosin, 10-8m
biological motor on a track: 10-17-10-16W from ATP vs. 10-8W
from heat bath
power strokes: ATP hydrolysis, ATP→ADP+20kBT, efficiency ~50%;
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
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?
unbalanced wheel
L. da Vinci
SPRING
pawl ratchet
R. Feynman
PAWL
VANE
RATCHET
Lippmann 1900, v. Smoluchowski 1912
noise harvesting,
noise-powered small devices
E. Coli ATP synthase
y
enzyme
y
reverse reaction
ADP + Pi→ATP
Wang&Oster, Nature (1998)
phosphor lation
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
off protons moving
i down
d
an electrochemical
l
h i l gradient,
di
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.
ADP + Pi → ATP
These enzymes are of crucial importance in almost all
organisms, because ATP is the common "energy
energy
currency" of cells.
impossible (at equilibrium)!
The Feynman
Lectures on Physics, I-46
assign ratchet and vane
ε
temperatures T1 and T2;
at equilibrium T1 = T2
τ
ratchet angular velocity
Ω = νθ ( f F − f B ) =
= νθ
θ (e
− ( ε +τθ ) / T2
−e
−ε / T1
τ
)
Ω
rectification
T1 = T2
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).
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 p
process information are subject
j
to the same laws as those of universal
Turing machines
P. Curie (1894): Rectification of statistical
fl t ti
fluctuations
requires
i
simultaneous
i
lt
breaking of spatial and time symmetry
Brownian motors
assumptions:
• overdamped particle on a periodic substrate V(x)=V(x+L)
• zero-mean fluctuating ξ(t) and/or deterministic forces F(t)
ΔV
V(x) = cos(x)
x
x& = − V ′( x) + ξ (t ) + F (t )
Langevin equation
different non-equilibrium options →
x& ≠ 0
x& = − V ′( x) + ξ (t ) + F (t )
a.
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)]
no transport current,
x& = 0
harmonic mixing
F(t)
( ) bi-harmonic signal,
g
, F(t)
( ) = F1cos(Ω
( 1t+φ1) + F2cos(Ω
( 2t+φ2);
commensurate frequencies, Ω1/Ω2 = m/n
w/ or w/o the noise ξ(t)
x&
∝ F1n F2m cos(mφ2 − nφ1 )
rectification due to the interplay
of
nonlinearity
li
it and
dd
drive
i
asymmetry
t
2
F(t)
1.5
1
F(-t)
( )
≠&
-F(t)
() b
biased,
a d, we cheated!
a d
0.5
2
-0.5
-1
4
6
8
10
12
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
amplitude, F(t)=
F(t) εV(x)cos(Ωt)
3. thermal: w/ or w/o drive; ξ(t) Gaussian and colored,
‹ξ(t)ξ(0)›=(D/
(0)› (D/τ)exp(-|t|/
)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 periodically
F=0
On
Off
-Fx
On
Fx
POSITION
POSITION
thermal ratchets
qualitative argument (for ‘good’ potentials, only)
LRFR=L
LLFL
L=LL+LR
-FL
barrier height
dominates
escape
FR
LL
new time scale τ
LR
⎡L
L ⎤
x& = ⎢ −
⎥ w(τ , D) < 0
⎣ LR LL ⎦
→0 ,∞
w(τ , D) ⎯τ⎯
⎯→ 0
compare ‹ξ-Fi›τ
with Li , i=L,R
thermal
General
properties
ti
J
• resonantt mechanism
h i
vs. D or F, τ or Ω
τ
rocked
J
• sensitive to parameters
› substrate profile
› particle mass
› inter-particle
inter particle interactions
• current inversions
FL FR
F
D
Applications:
biology inspired nano
nano--devices
P.Hanggi & FM, Rev Mod Phys 81, 387 (2009)
Optical tweezers
D. G. Grier et al, Appl. Phys.
Lett., 82, 3985 (2003).
Artificial μ-pores
Z. Siwy and A. Fulinski, Phys.
Rev. Lett. 89, 198103 (2002).
Cold atoms traps
F Renzoni et al, Phys. Rev. Lett.
95, 073003 (2005).
Superconducting devices
1m m
single vortex experiments
PRL 99
Triangular traps PRL 04
… binary mixture experiments
PRL 01
H
H
an example: vortices in the tilted magnetic fields
pancake
k (PV) vs. JJosephson
h
(JV) vortices
ti
Hc
Hab
J(t)
FJ
JV
(Savelev & Nori, 2003; Bending et al., Bath, UK)
“active” vortices
thermal
noise
repulsive interaction
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
Conclusions
G. Casati,
Casati, H. Linke,
Linke, F. Marchesoni
¶
biology
bi
l
iinspired
i d nano-devices
d i
powered by noise
¶
role of noise at the small
scales reconsidered
¶
noise harvesting to power
nano-devices for ICT
Conclusions
¶
biology inspired nano-devices powered by noise
¶
role of noise at the small scales reconsidered
¶
noise harvesting to power nano-devices for ICT