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Thermodynamics and Kinetics of aBrownian Motor
R. Dean Astumian
Nonequilibrium fluctuations, whether generated externally or by
a chemical reaction far
from equilibrium, can bias the Brownian motion of a particle in
an anisotropic mediumwithout thermal gradients, a net force such as
gravity, or a macroscopic electric field.
Fluctuation-driven transport is one mechanism by which chemical
energy can directlydrive the motion of particles and macromolecules
and may find application in a wide
variety of fields, including particle separation and the design
of molecular motors andpumps.
A small particle in a liquid is subject torandom collisions with
solvent molecules.The resulting erratic movement, or Brown-ian
motion, has been described theoretical-ly by Einstein (1) and
independently by
Langevin (2). Langevin hypothesized thatthe forces on the
particle due to the solventcan be split into two components: (i)
afluctuating force that changes direction andmagnitude frequently
compared to any oth-er time scale of the system and averages tozero
over time, and (ii) a viscous drag forcethat always slows the
motions induced bythe fluctuation term. These two forces arenot
independent: The amplitude of thefluctuating force is governed by
the viscos-ity of the solution and by temperature, sothe
fluctuation is often termed thermalnoise.
At equilibrium, the effect of thermal
noise is symmetric, even in an anisotropicmedium. The second law
of thermodynamicsrequires this: Structural features alone, nomatter
how cleverly designed, cannot biasBrownian motion (3, 4). To
illustrate thispoint, Feynman discussed the possibility ofusing
thermal noise in conjunction with an-isotropy to drive a motor in
the context of aratchet and pawl device shrunk to micro-scopic size
(4). He showed that when allcomponents of such a device are
treatedconsistently, net motion is not achieved inan isothermal
system, despite the anisotropyof the ratchets teeth. However, a
thermal
gradient in synergy with Brownian motioncan cause directed
motion of a ratchet andcan be used to do work. As a practical
mat-ter, large thermal gradients are essentiallyimpossible to
maintain over small distances.Particularly in biology and
chemistry, thethermal gradients necessary to drive signifi-cant
motion are not realistic.
It might seem then that, despite its per-
vasive nature, Brownian motion cannot beused to any advantage in
separating or mov-ing particles, either in natural systems (suchas
biological ion pumps and biomolecularmotors) or by artificial
devices. Recent work
has focused, however, on the possibility of anenergy source
other than a thermal gradientto power a microscopic motor. If
energy issupplied by external fluctuations (58) or anonequilibrium
chemical reaction (9, 10),Brownian motion can be biased if the
medi-um is anisotropic, even in an isothermalsystem. Thus, directed
motion is possiblewithout gravitational force, macroscopicelectric
fields, or long-range spatial gradientsof chemicals.
In devices based on biased Brownianmotion, net transport occurs
by a combi-nation of diffusion and deterministic mo-tion induced by
externally applied time-
dependent electric fields. Because parti-
cles of different sizes experience differentlevels of friction
and Brownian motion, an
appropriately designed external modula-
tion can be exploited to cause particles ofslightly different
sizes to move in oppositedirections (1116). One can imagine
anapparatus where a mixture is fed into thesystem from the middle
and purified frac-tions are continuously removed from ei-ther side.
Because part of the energy re-quired for transport over energy
barriers isprovided by thermal noise and because theexternal forces
are exerted on a smalllength scale, such devices may be able
tooperate with small external voltages. Incontrast, many
conventional techniquessuch as electrophoresis, centrifugation
and chromatography, must be turned offand on each time a new
batch of particlesis added. These methods rely on motioncaused by
long-range gradients, where themajor influence of thermal noise is
todegrade the quality of separation by diffu-sive broadening of the
bands.
Biased Brownian Motion
As a specific example of biased Brownianmotion (Fig. 1A),
picture a charged parti-cle moving over an array of
interdigitatedelectrodes with a spatial period of 10 m.When a
voltage is applied, the potential
energy of the particle is approximately ananisotropic sawtooth
function (Fig. 1C)Although the electric generator is certain-
The author is in the Departments of Surgery and of Bio-chemistry
and Molecular Biology, University of Chicago,MC6035, Chicago, IL
60637, USA. E-mail: [email protected]
( 1)L0
LLL
U(x)
x
C
A
B
-
V
x
L ~ 10-8m
+ ++
U
HS H+ + S
+10-5 m
-
Fig. 1.Two specific models by which an anisotro-
picperiodicpotential canarise. (A) When a voltage
difference Vis applied to interdigitated electrodes
deposited on a glass slide with anisotropically po-
sitioned teeth, the resulting potential is spatially
periodic but anisotropic. With photolithography it
is possible to achieve a spatial period L for such a
structure as small as 105 m or even somewhat
less. (B) A linear array of dipoles aligned head to
tail on which a charged Brownian particle (possi-
bly a protein) moves. The individual dipoles couldbe
macromolecular monomers that aggregate to
form an extended linear polymer. The length of
the individual monomer is L 108 m, a reason-able value for many
aggregating proteins. If the
particle catalyzes the reaction HS 3H S,the charge on the
particle, and hence its electro-
static potential energy of interaction with the di-
pole array U(x, t), fluctuates depending on its
chemical state. (C) The potential U(x) for both (A) and (B) is
approximately an anisotropic sawtooth
function Usaw, drawn for simplicity as a piece-wise linear
function. The amplitude is U, and the wells(potential minima) are
spaced periodically at positions iL. The anisotropy is
parameterized by . Theamplitude Uof the sawtooth potential can be
modulated in (A) by using an external switching device tocontrol
the applied voltage.
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ly a macroscopic device, the electric fieldin the x direction
averaged over a spatialperiod is zero no matter what the
voltage,and so there is no net macroscopic force.A nonequilibrium
fluctuation can be pro-duced by using a switching device
thatimposes an externally defined but possiblyrandom modulation of
the voltage. Recentexperiments have shown that unidirec-
tional motion of microscopic particles canbe induced by
modulating the amplitudeof such an anisotropic sawtooth
potential(17, 18). Theory shows that the directionof flow is
governed by a combination ofthe local spatial anisotropy of the
appliedpotential, the diffusion coefficient of theparticle, and the
specific details of howthe external modulation is carried
out(1116). Under some conditions, particlesof slightly different
sizes can move in op-posite directions, possibly providing thebasis
for a continuous separation process.Several technological hurdles
remain,
however, before a practical device can beconstructed.In a second
example (Fig. 1B), a
Brownian particle moves along a lattice ofelectric dipoles
arranged head to tail (7).The individual dipoles could be, for
exam-ple, macromolecular monomers that ag-gregate to form an
extended linear poly-mer. Imagine that the particle has an ac-tive
site to catalyze a hydrolysis reactionHS 3 H S, where S denotes
somesubstrate molecule. The charge on an in-
dividual particle will then fluctuate, de-
pending on whether H or S is bound.The potential energy of the
particle alongthe axis of the dipole lattice is approxi-mately
described by a sawtooth function(Fig. 1C), but the amplitude
depends onthe charge and hence on the chemicalstate. When the
chemical reaction HS 3H S is far from equilibrium, thefluctuation
of the charge on the particledue to the reaction can cause
unidirec-tional transport. This effect may be impor-tant in the
chemomechanical energy con-version of biomolecular motors and
pumps, proteins that convert energy froma chemical reaction such
as adenosinetriphosphate (ATP) hydrolysis to drive
unidirectional transport. Although it isunlikely that any
biological motor oper-ates exactly as a simple Brownian
ratchetusing chemical energy to bias thermalnoise, some of the
basic principles of op-eration may be the same.
Recent experiments have shown that,consistent with a ratchet
mechanism (5,19, 20), external oscillating (21) or fluctu-
ating (22) electric fields can drive transportby a molecular ion
pump, the sodium-po-tassium adenosine triphosphatase. Energyfrom
the field substitutes for the energynormally provided by ATP
hydrolysis eventhough the average value of the field is
zero.Fluctuation-driven transport provides anexplicit mechanism for
coupling energyfrom a nonequilibrium chemical reaction tocause
local fluctuations similar to those im-posed externally to drive
unidirectionaltransport (20). Ultimately, it may be possi-ble to
construct microscopic motors andpumps that use nonequilibrium
chemical
reactions as their fuel.The examples in Fig. 1 are provided
tomake our discussion more concrete. Thegeneral idea of diffusion
on a periodic po-tential arises in many other contexts
(23),including enzyme catalysis, where an en-zyme cycles through
several intermediatestates in carrying out its function (19);
thesolid-state physics of transistors (24), wherean electron moves
through several p-n junc-tions; models of Josephson junctions
(25);and generalized models of computation(26). The details of the
interactions givingrise to the potential are not as critical as
themechanisms by which spatial anisotropy
conspires with thermal noise to allow ener-
gy from external fluctuations or a nonequi-librium chemical
reaction to drive unidirec-tional flow. Anisotropic potentials such
asthat shown in Fig. 1C are known as ratchetpotentials, in analogy
to macroscopic ratch-ets such as a car jack that allow motion
inonly one direction by a series of asymmetricgears.
A Fluctuating Potential, orFlashing Ratchet
To see how thermal noise combined with
an externally modulated anisotropic poten-
tial can lead to unidirectional transport,consider the model (7,
27) shown in Fig.2A. A particle subject to viscous dampingmoves
along a track where an anisotropicpotential with periodically
spaced wells atpositions iL cyclically turns, or flashes, onand
off. To show that work can be doneagainst an external force, the
potential pro-files are illustrated with a net force Fextacting
from right to left.
If the external force is not too large, theparticle is pinned
near the bottom of one of
0 L(1)L LL
U
x
tF/
(4Dt)
A
B
C
5 10 15 20
0.1
0.1
0.2
0.2
0.3
toff (s)
U
Uon = Usaw xFext
Uon = xFext
V
Fgrav sin
Smallparticles
Largeparticles
+-
(m/s)
Fig. 2. How a fluctuating potential causes uphill
transport. (A) Schematic representation of a fluc-
tuating potential energy profile. Whenthe potential
is on, the potential energy Uon of a particle is a
sawtooth function Usaw xFext with periodicallyspaced wells at
positions iL. The anisotropy of
Usaw results in two legs of the potential, one of
length L on which the force is U/(L) Fext,and the other of
length(1 )L on which theforceis U/[(1 )L] Fext. When the potential
is off,theenergy profile is flat with force Fext everywhere.
The curve below the two potential profiles shows aGaussian
probability function resulting from turn-
ing the potential off at t 0, with the particlestarting at x 0.
The result is resolved into twocomponents: the downhill drift and
the diffusive
spreading of the probability distribution. For inter-
mediate times, it is more likely for a particle to be
between L and (1 )L, where it would betrapped in the well at L
if the potential were turned
back on, than between ( 1)L and ( 2)L,where it would be trapped
in the well at L if thepotential were turned back on. Thus, turning
the
potential on and off cyclically can cause motion to
the right despite the net force to the left. (B) Aver-
age velocity of a spherical particle with radiusrp 2 m (solid
line) and rp 5 m (dashed line)
induced by cyclically turning an anisotropic saw-tooth potential
( 0.1) onand off versusthe timespent in the off state toff (with
ton toff). Theperiod was L 10 m with an external force dueto
gravity Fext Vp(dp dw)gsin(); Vp is the vol-ume of theparticle, dp
1.05 g/cm
3 is thedensity
of the particle, dw is the density of water, g 9.8 m/s2 is the
acceleration due to gravity, and 45.
The coefficient of viscous drag was 6r 4 108 Ns/m, where 1
centipoise is the viscosityof water. The Einstein relation yields D
kBT/ 1 10
13 m2/s for the diffusion coefficient. The
amplitude Uwas taken to be greater than 2 1016 J (U/L 20 pN) so
that the particle has time toreach the bottom of the well while the
potential is on. Such a large U is necessary for the
approxima-tions inherent in this simple picture given to be
fulfilled. An exact calculation using the diffusion equation
results in qualitatively the same behavior for smaller values of
U. (C) Schematic illustration of anapparatus for separating
particles with the use of biased Brownian motion combined with
gravity.
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the wells (for example, at x 0) when thepotential is on. Without
noise, the particlewould move to the left with a velocityFext/ when
the potential is off, where isthe coefficient of viscous drag.
Thermalnoise changes the situation dramatically.Because of Brownian
motion, a randomwalk is superimposed on the deterministicdrift, and
the position of the particle can be
determined only in terms of a probabilitydistribution. While the
potential is off, theprobability distribution drifts downhill witha
velocity Fext/ and spreads out like aGaussian function. After a
time toff, thedistribution (Fig. 2A) is (28)
P 0x;toff
exp x toffFext /
2
4Dtoff
4Dtoff(1)
where D is the diffusion coefficient. Whenthe potential is
turned on again, the particleis trapped in the well at iL if it is
between
L(i 1 ) and L(i ). The probabilityof finding the particle in
each well, PiL
, iscalculated by integrating the probabilitydensity (Eq. 1)
between these limits. Be-cause of the anisotropy of the potential,
aparticle starting at iL is more likely to betrapped in the well at
(i 1)L than in thewell at (i 1)L for small enough Fext
andintermediate values of toff. The averagenumber of steps R in a
cycle of turning thepotential on for a time ton long enough thatthe
particle reaches the bottom of a well(ton L
2/U), off for a time toff, and thenback on again, is R
i
iPiL
, and theaverage velocity is v RL/(toff ton) (Fig.
2B). Motion to the right occurs despitethe fact that the
macroscopic force pushesthe particle to the left. For large U/L,
theexternal force necessary to cause the flowto be identically
zero, Fstop (the stoppingforce), can be evaluated exactly (21) to
beFstop (L/toff)( 1/2). For toff (L/F)( 1/2), only the terms due
tointegration of Eq. 1 with i 1 contrib-ute to the velocity. In
this regime, veloc-ities scale with D/L and times with L2/D.
The mechanism illustrated in Fig. 2works not despite diffusion
and thermalnoise but because of diffusion. When the
potential is turned off, the particle diffusessymmetrically.
Then, when the anisotro-pic potential is turned on, it is more
likelyfor the particle to feel a force to the rightthan to the
left. Thus, net motion to theright occurs, even in the presence of
asmall homogeneous force to the left. Theenergy to drive the
transport, however,does not come from thermal noise but isprovided
when the potential is turned on.This energy can be calculated by
integrat-ing over space the product of Uon(x) andthe probability
density P(0x; toff). Be-
cause the potential remains on for longenough to allow the
particle to reach thebottom of a well, no energy is returnedwhen
the potential is turned off. The out-put energy per cycle is the
number of stepsR multiplied by the energy gain per stepFextL. The
difference in these energies isdissipated as heat resulting from
viscousdrag of the particle with the solvent. For
the simple model shown in Fig. 2, thethermodynamic efficiency is
less than 5%(29). Other features of biased Brownianmotion make it
more promising for theconstruction of microscopic motors. Witha
spatial period of 10 m, velocities ofaround 0.3 m/s can be achieved
for a2-m particle with toff 2 s in aqueoussolution (Fig. 2B, solid
line). This motorcan do work against a force as large as 0.05pN,
easily overcoming gravity, which for a2-m particle with a density
of 1.05 g/cm3
in water is 0.004 pN. Larger particles,however, cannot overcome
gravity and
drift to the left when toff is large enough(Fig. 2B, dashed
line).The velocity depends nonmonotoni-
cally on the frequency of modulation (Fig.2B). This can be
understood in terms ofthe competing requirements that a parti-cle
be able to diffuse at least the shortdistance L but not the longer
distance(1 )L while the potential is off. Thenonmonotonic behavior
allows for thepossibility of tuning a device to a frequen-cy
characteristic to a particular particle.This tuning may be useful
in applicationsfor separating particles on the basis of size,but
the bandwidth of the response is rela-
tively broad. One can exploit this depen-
dence, however, by imposing an externalforce so that some
particles will move tothe right and others to the left. The
sim-plest example (Fig. 2C) is to tilt an appa-ratus such as that
shown in Fig. 1A so thatthe particles experience a net
gravitation-al force Fgravsin(), where is the anglerelative to a
horizontal surface. Small par-ticles will feel only a small force
due togravity and a larger amount of Brownianmotion, which can be
biased by the ratch-et to cause motion uphill to the right.
Thelarger particles have less Brownian motion
and feel a greater force due to gravity andso move downhill to
the left. In such anapparatus, a mixture of particles of differ-ent
sizes can be introduced in the middle,and purified fractions can be
collectedcontinuously on either end. Because of theclose spacing of
the electrodes, only asmall voltage is necessary to achieve
suf-ficiently large energy barriers.
The approach shown in Fig. 2C is notappropriate for smaller
particles becausegravity is not sufficiently strong. However,other
similar strategies can be adopted, in-
cluding the use of a constant electric field asthe homogeneous
force (16). It has evenbeen shown that with a more complicatedpulse
protocol than a simple square wave,particles of different sizes can
be induced tomove in opposite directions without anyhomogeneous
long-range force at all (12,13, 15). By exploiting the different
scalingfor biased Brownian motion and motion
induced by deterministic gravitational andelectric forces,
approaches for continuousseparation of particles ranging from
macro-molecular (108 m) to mesoscopic (105
m) size are possible.The basic principles of motion induced
by a fluctuating potential have recentlybeen tested in simple
model systems. Rous-selet et al. (17) applied a
time-dependentratchet potential to colloidal particles usinga
series of interdigitated Christmas treeelectrodes deposited on a
glass slide byphotolithography. Cyclically turning thepotential on
and off resulted in net flow
Despite the complexity of dealing with asuspension of many
interacting particlesthe data agreed at least
semiquantitativelywith the predictions of the simple theory.A
similar effect was demonstrated by Fau-cheux et al. (18), who used
a single col-loidal particle in an optical trap modulat-ed to
generate a sawtooth potential. Thissetup has the advantage that
there is onlyone particle, thus eliminating
interparticlehydrodynamic interactions.
Modulating the potential certainly re-quires work, and there is
no question ofthese devices being perpetual motion ma-chines. The
surprising aspect is that f low is
induced without a macroscopic forceallof the forces involved are
local and act on alength scale of the order of a single period
ofthe potentialyet the motion persists in-definitely, for many
periods. In contrastconventional techniques for particle
sepa-ration such as centrifugation, chromatogra-phy, or
electrophoresis all rely on macro-scopic gradients and can only
operate asbatch separators.
A Fluctuating Force, orRocking Ratchet
In the mechanism shown in Fig. 2, thespatial average of the
force is independentof time, and the temporal modulation
onlychanges the shape of the potential locallyAnother well-studied
approach to drivingflow on a ratchet potential involves apply-ing a
fluctuating net force (6, 11, 14, 23,30, 31). This approach can be
visualized asrocking the sawtooth potential shown inFig. 1C back
and forth between the limitsFmax (Fig. 3). IfU/[(1 )L] FmaxU/(L),
the potential energy decreasesmonotonically to the left when the
force is
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Fmax, but when the force is Fmax, thereremain minima that trap a
particle as itmoves to the right in response to the ap-plied force
(Fig. 3A). Thus, even withoutthermal noise, a slow oscillation of
the forcebetween Fmax causes net flow to the left.Fast oscillation
does not result in net flowbecause the particle does not have time
tomove a period L before the force reverses
sign. Applying a fluctuating or oscillatingforce is analogous to
the standard way ofdriving an electrical rectifier or a
macro-scopic mechanical ratchet such as a car jackor ratchetting
screwdriver.
The presence of thermal noise allows asubthreshold fluctuating
force {Fmax U/[(1 )L]} to cause flow (6). For aslow, zero-average
square wave modula-tion of the force between Fmax, the av-erage
rate can be calculated using theanalytic formula given by Magnasco
(6).In Fig. 3B, the average velocity is plottedas a function of
Fmax where the dashed
lines indicate the location of the thresholdforces U/[(1 )L] and
U/(L). InFig. 3C, the average velocity is plotted asa function of
the thermal noise strengthkBT(where kB is Boltzmanns constant andT
is temperature) for a subthreshold ap-plied force (Fmax 0.4 pN). We
see that,to a point, increasing the noise can actu-ally increase
the flow induced by the fluc-tuating force, which suggests that in
sometechnological applications, it may be use-ful to electronically
add noise to the sys-tem. For forces near the optimum (Fmax
1.5 pN in Fig. 3B), however, the velocitydecreases almost
monotonically with in-creasing noise.
A More General Approach
A more general approach to investigatingfluctuation-driven
transport involves solv-ing the diffusion equation
P x,t
t
x
U
x P x,t D
2P x,t
x2
(2)
At equilibrium, the probability density isgiven by a Boltzmann
distribution P(x) exp(U/kBT). Any external temporal mod-ulation
ofU, whether accomplished period-ically or randomly in time,
requires energyinput and drives the system away from equi-librium,
disturbing the Boltzmann distribu-tion. The time dependence ofU for
a fluc-tuating potential ratchet is U(x, t) f(t)
Usaw(x), and for a fluctuating force ratchet,is U(x, t) Usaw(x)
xf(t). Equation 2 canbe solved for P(x, t) for any explicit
timedependence of f(t) with appropriate (typi-cally periodic)
boundary conditions, andthe result can be used to calculate the
av-erage velocity. It turns out, however, that inmany cases, it is
simpler to solve Eq. 2 when
f(t) is modeled as a position-independentstochastic variable
(31). This situation mustnot be confused with an equilibrium
fluctu-ation. At equilibrium, the probability toundergo a
transition from one state to an-
other depends on the energy difference be-tween the states
according to a Boltzmannrelation (also known as detailed
balance),and the energy difference for a ratchet po-tential clearly
depends on the position. Anexternal modulation, whether random
ornot, does not conform to detailed balanceand cannot model
equilibrium fluctuations.
Unlike motion driven by forces such as
gravity or macroscopic electric fields, wherethe direction of
motion is determined bythe sign of the force, the direction of
flowinduced by zero-average modulation of apotential or force
depends on the details ofhow the modulation is carried out.
Recent-ly, several schemes have been proposed thatallow for flow
reversal to occur as a functionof the modulation frequency, with
differentreversal frequencies for particles with differ-ent
diffusion coefficients (1318).
Chemically Driven Transport
Consider the one-
dimensional motion of acharged particle along a periodic lattice
ofdipoles arranged head to tail (Fig. 1B), andlet the interaction
between the particle andlattice be purely electrostatic (7). The
po-tential energy profile then consists of aperiodic series of
minima (wells) and max-ima (barriers). For simplicity, this
profilecan be represented as a sawtooth function(Fig. 1C). Because
of thermal noise, theparticle undergoes Brownian motion.
Occa-sionally, it will have enough energy to passover one of the
two barriers surrounding thewell in which it starts and move to the
wellon the right or on the left. Despite the
anisotropy, without an energy supply, thetwo probabilities are
exactly equal.
Now, imagine that this particle catalyzesa chemical reaction HS
^ H S. Be-cause the product molecules H and S arecharged, the
amplitude of the interactionpotential between the particle and the
di-pole lattice depends on whether H or S
are bound to the particle, resulting in acoupling between the
chemical reactionand the diffusion of the particle along thedipole
lattice. Because of the charge on H
and S , their local concentrations vary asa function of position
along the axis of the
dipole lattice (the hydrogen ion concentra-
tion [H] is larger near the negative end ofthe dipole than near
the positive end, andthe opposite is true for [S]). The radii ofthe
ions H and S are very small com-pared to the radius of the
particle, so theirlocal concentrations equilibrate quickly,leading
to [S] [S]bulkexp[U(x)/kBT]and [H] [H]bulkexp[U(x)/kBT],where the
subscript bulk denotes the con-centration far from the surface of
the dipolelattice. For the specific mechanism shownin Fig. 4A, k21,
k23, and k31 are first-order
0.2
0.02
0.01
0.1
1.0 2.0 3.00
4 8 16 20 2400
0
U= UsawxIFmaxI
U= Usaw + xIFmaxI
U= Usaw
A
B
C
Fmax(pN)
kBT(10-21 J)
(m/s)
(m/s)
Fig. 3. The basic mechanism by which a fluctuating oroscillating
force cancausedirectedmotion along a ratch-
et potential Usaw. The parameters U 4 1018 J, L
10 m, and 0.2 were used. (A) Because of theanisotropy of the
potential Usaw, starting at the bottom of
any well, the force required for the particle to move to the
right, U/(L) 2 pN, is greater than the force neces-sary to move
to the left, U/[(1 )L] 0.5 pN. Whenan external force between these
values (such as 1.5pN)is applied, thenet potential U(x) U
sawx(1.5pN)
decreases monotonically to the left, and a particle acted
on by this potential drifts to the left with no impediment.
When an external force of 1.5 pN is applied, the netpotential
U(x) U
saw x(1.5 pN) decreases on aver-
age to the right, but there remain wells (local potential
minima, indicated by the vertical arrows) that tend to trap
theparticle. Thus a square wave modulation of theexter-nal force
Fmax
between 1.5 pN and 1.5 pN results innet flow to the left. (B)
For a low-frequency square wave
modulation, the average velocity can be calculated as a
function of Fmax
with the analytic formula given by Mag-
nasco (6). The average velocity v depends nonmono-tonically on
the amplitude F
maxand is maximum between
Fmax
U/[(1 )L] 0.5 pN and Fmax
U/(L) 2pN (indicated by the dashed lines). Because of
thermal
noise, a subthreshold force Fmax
U/[(1 )L] caninduce some flow. (C) The dependence of the flow on
the
thermal noise amplitude kBT induced by a 0.4-pN force.
The curve is nonmonotonic, reminiscent of a phenome-
non known as stochastic resonance (32, 33).
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rate constants and do not depend on theposition along the
dipole, and k12 k12[SH], k13 k13[H
], and k32 k32[S]
are pseudofirst-order rate coefficients intowhich the
concentrations of the reactantshave been incorporated. Because [H]
and[S] depend on the position x along thedipole lattice, the rate
coefficients k13 andk32 also depend on position.
If the chemical reaction is at equilibri-
um, a direct transition from the unchargedstate i 3 to the
charged state i 2 is morelikely near the positive end of the
dipole,where [S] is relatively high, than near thenegative end of
the dipole, where [S] isrelatively low. Similarly, direct
transitionfrom the charged state i 1 to the un-charged state i 3 is
more likely near thenegative end of the dipole, where [H]
isrelatively high, than near the positive endof the dipole, where
[H] is relatively low.The net effect is that transitions from
thecharged to the uncharged state are more
likely near the negative end of the dipole,and transitions from
the uncharged tocharged state are more likely near the pos-itive
end of the dipole. In this case (and inthe absence of an external
force), the prob-ability density of the particle is
distributedeverywhere according to a Boltzmann dis-tribution (Fig.
4B, dashed line), and theaverage velocity of the particle is
zero.
If the chemical reaction is away fromequilibrium, this relation
between the posi-tion of the particle and transition probabil-ities
does not necessarily hold (Fig. 4B, solidline). In the extreme case
where [H] [S] 0, direct transitions from state 1 to
state 3 and from state 3 to state 2 cannotoccur at all. None of
the rate coefficientsfor the remaining transitions depend on
theposition of the particle, and so the proba-bility of a
transition from the charged to theuncharged state is independent of
the posi-tion along the axis of the dipole lattice, as isthe
probability for a transition from theuncharged to the charged
state. This situa-tion is essentially identical to that describedas
a fluctuating potential ratchet (Fig. 2),and as before, directed
motion can result.
Thus, we have a rudimentary motordriven by a chemical reaction.
The mech-
anism shown in Fig. 4A is a tremendousoversimplification as a
description of anyactual molecular motor. The interaction be-tween
the motor and dipole lattice and theeffect of the catalyzed
reaction were as-sumed to be purely electrostatic, and theresulting
periodic potential was grossly sim-plified. Nevertheless, with
reasonable val-ues of the rate constants and other param-eters, the
motor moves pretty well (Fig.4C), with a maximal velocity of
severalmicrometers per second. The stoichiometryis poor: It takes
more than 3 HS molecules
on average to cause a single step (displace-ment by a period L),
and less than 1 pN ofapplied force stops the motion. What
thissimplified motor lacks is what a real biomo-lecular motor can
certainly provide: morecomplex interaction between the motorand
track, provided by conformational flex-ibility, and better
regulated coupling be-tween chemical and mechanical events,
possibly provided by allosteric interactionbetween the motor and
track.
Perspective and Outlook
Noise is unavoidable for any system in ther-mal contact with its
surroundings. In tech-nological devices, it is typical to
incorporatemechanisms for reducing noise to an abso-lute minimum.
An alternative approach isemerging, however, in which attempts
aremade to harness noise for useful purposes.
One example involves signal detection bythreshold detectors
(32). Typically, suchdevices have been designed so that thesystem
is isolated from noise, and thethreshold is as small as possible.
Thesegoals, however, are often difficult and ex-pensive to attain.
It may instead be possibleto operate the system with a threshold
larg-er than the signal to be measured and to use
added noise to provide a boost that willallow the signal to be
detected above thenoise. The phenomenon of
noise-enhancedsignal-to-noise ratio is known as stochasticresonance
and holds for a wide variety ofnonlinear systems (32, 33).
Similarly, fluc-tuation-driven transport uses noise to ac-complish
mass transport without macro-scopic forces or gradients. Three
ingredientsare essential: (i) thermal noise to causeBrownian
motion; (ii) anisotropy arisingfrom the structure of the medium in
which
L/2 L/2
0.5
0.6
0.7
L L
G/RT1 1 2 3 4
1
2
3
0
-
-
-
-
SH
i= 1
i= 2
i= 3
i= 1
k13k31
k32k23
k21k12
Position
Chemicalstate
P3(x)/Pi(x)
A
B
C
S
H+
H+
SH
(m/s)
Fig. 4.Transport driven by a chemical reaction. (A)
Schematic illustration of how the potential energyprofile of a
Brownian particle that catalyzes a reac-
tion SH ^ S H changes depending on itschemical state, even in a
constant electric potential
due toa dipolearrayas shown inFig. 1B. (B) Prob-
ability of the particle being uncharged P3(x)/Pi(x)as a function
of position when the chemical reac-
tion SH^S H is at equilibrium (dashed curve)and far
fromequilibrium (solid curve). For simplicity,
we treat the case thatthe particle bears a charge of1 and that
the interaction potential between theparticle and dipole track
(Fig. 1B) is purely electro-
static. Because S and H are charged, their local
concentrations depend on the local potential ac-
cording to [S] [S]bulkexp[U(x)/kBT] and [H]
[H]bulkexp[U(x)/kBT]. The rate constants used
were k12 10[SH], k21 106, k23 106, k32 10[S], k13 100[H
], and k31 100, all in inverse
milliseconds, and the concentrations were [SH] 4,
[H]bulk 2, and [S]bulk 2 for the equilibrium
case, and [SH] 20, [H]bulk 0.05, and [S]bulk
0.05for the far fromequilibrium case, all in millimolar.
The amplitude of the potential U was 1020 J. (C)
Average velocity as a function of the G of the chem-
ical reaction using the rateconstants and field ampli-
tude as in (B). This result can be calculated in general
by using reaction-diffusion equations to describe the
combined physical motion and chemical reaction of
a catalytic Brownian particle as described elsewhere
(9, 10). Here, k12 and k23 were taken to be large so
that state 2 could be treated as a steady-state inter-
mediate. Then, there are effectively only two states,
off and on, as in Fig. 2, with the effective timeconstants off
{5[S
]bulkexp[U(x)/kBT] 100}1
and on {5[SH] 100[H]bulkexp[U(x)/kBT]}
1,
as described in (9). The value ofG RTln{[SH]/
([S]bulk[H]bulk)} (R is the universal gas constant)
was varied by varying [SH] with constant [S]bulk
[H]bulk 1. The period L was taken to be 108 m,
and the viscosity at the interface was taken to be
greater than in bulk water (int 100 cP), resulting in 6r 2 108
Ns/m for a particle with a
Stokes radius of r 108 m and a diffusion coeffi-
cient D 2 1013 m2/s. The inset shows that there is a linear
relation between flow and G close to
equilibrium. The nonmonotonic dependence on G arises because G
is changed by varying the chemica
concentrations, which also varies the time scales.
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the particle diffuses; and (iii) energy sup-plied either by an
external variation of theconstraints on the system or by a
chemicalreaction far from equilibrium.
It is often overlooked that noise-assist-ed processes
incorporate features analo-gous to chemical kinetics (34).
Transi-tions along a chemical pathway are de-scribed in terms of
rate constants that
reflect the probability that thermal noisewill provide
sufficient energy to surmountan energy barrier separating
chemicalstates. Experiments on ion pumps haveshown that energy from
externally im-posed electric oscillations (21) and fluctu-ations
(22) can substitute for energy fromhydrolysis of ATP to power
uphill trans-port of ions. The likely mechanism in-volves biasing
thermally activated steps inthe reaction pathwaya Brownian ratch-et
mechanism (5, 19). As Lauger summa-rized, Ion pumps do not function
by apower-stroke mechanism; instead, pump
operation involves transitions betweenmolecular states, each one
of which is veryclose to thermal equilibrium with respectto its
internal degrees of freedom, even atlarge overall driving force
(35).
It is not yet clear whether molecular mo-tors such as muscle
(myosin) or kinesinmove by using an ATP-driven powerstrokea
viscoelastic relaxation process inwhich the protein starts from a
nonequilib-rium conformation after phosphate releaseand which does
not require thermal activa-tionor whether energy from hydrolysis
ofATP is used to bias thermally activatedsteps. The recent work on
Brownian ratchets
shows that a thermally activated mechanismis not inconsistent
with a process in which aprotein moves a distance of 10 nm in a
singlestep and generates a force of several pico-newtons. This
question may be resolved soonwith the use of recently developed
tech-niques for studying molecular motors at thelevel of a single
molecule (36).
Incorporation of thermal motion as anessential element in the
design of micro-scopic machines is essentially the equiva-lent of
adopting design principles bor-rowed from chemistry rather than
fromclassical macroscopic physics. A remark-
able feature is that the fundamentally sto-
chastic nature of noise-assisted activated
transitions can give rise to mechanismsthat rival purely
deterministic processes inthe predictability of specific
outcomes.This possibility has been discussed by Ben-net in a
thermodynamic comparison ofcomputers based on Brownian versus
bal-listic principles (37). Recent work showsthat coupling many
particles in a modu-lated ratchet potential gives rise to ex-
traordinarily rich behavior and thermody-
namic efficiencies up to 50% (38).The use of noise in
technological appli-
cations is still in its infancy, and it is farfrom clear what
the future holds. Neverthe-less, there is already much active
research inthe area of noise-enhanced magnetic sens-ing (39) and
electromagnetic communica-tion (40). The recent work on
fluctuation-driven transport leads to optimism that sim-ilar
principles can be used to design micro-scopic pumps and
motorsmachines thathave typically relied on deterministicmechanisms
involving springs, cogs, and
leversfrom stochastic elements modeledon the principles of
chemical reactions andnoise-assisted processes. Such devices
wouldbe consistent with the behavior of mole-cules, including
enzymes, and could pavethe way to construction of true
molecularmotors and pumps.
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