Biological motors
18.S995 - L06
lds Numbers in Biology
Reynolds numbers
number is dimensionless group that characterizes the ratio o
fined as
⇥U L
UL
Re =
=
µ
density of the medium the organism is moving through; µ is t
; is the kinematic viscosity; U is a characteristic velocity of
stic length scale. When we discuss swimming biological organ
eatures that are moving through water (or through a fluid with
hose of water). This means that the material properties µ and
ber is roughly determined by the size of the organism.
e characteristic size of the organism and the characteristic sw
rule-of-thumb, the characteristic locomotion velocity, U , in bi
y U L/second e.g. for people L 1 m and we move
at U 1
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E.coli (non-tumbling HCB 437)
Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS
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Bacterial motors
movie: V. Kantsler
~20 parts
20 nm
Berg (1999) Physics Today
source: wiki
Chen et al (2011) EMBO Journal
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Torque-speed relation
200 nm fluorescent bead attached to a flagellar motor
26 steps per revolution
30x slower than real time
2400 frames per second
position resolution ~5 nm
Berry group, Oxford
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Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
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Chlamydomonas alga
10 ㎛
~ 50 beats / sec
Goldstein et al (2011) PRL
10 ㎛
speed ~100 μm/s
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Chlamy
9+2
Merchant et al (2007) Science
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dunkel@math.mit.edu
Eukaryotic motors
Sketch: dynein molecule carrying cargo down a microtubule
http://www.plantphysiol.org/content/127/4/1500/F4.expansion.html
Yildiz lab, Berkeley
dunkel@math.mit.edu
Microtubule filament “tracks”
Dogic Lab, Brandeis
Drosophila oocyte
Physical parameters
(e.g. bending rigidity) from fluctuation
analysis
Goldstein lab, PNAS 2012
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unlike dyneins
(most) kinesins walk towards plus end of
microtubule
25nm
dunkel@math.mit.edu
0). Based
showed
oncluded
for kinen asym-
chnique,
Accuraking the
accuracy
NA, the
step is
rescence
a totalope. The
is a dif280 nm,
ponds to
ted with
plied the
lks in a
alternatcements,
(11).
s experiwith a
ch head
Fig. 1B)
d as the
e immoent conglutamic
cond homer with
ines and
43C and
B). Subthe hoof fluo-
(13). The dye’s position was monitored as the
kinesin moved on microtubules that were immobilized on a coverslip (13). Three different conposition
time. aHowever,
if the
observed
structs versus
were used:
homodimer
with
glutamic
17-nm
steps
arise
from
the
convolution
of
twohoacid mutated to cysteine (E215C), a second
sequential
17 nm,
nm. . .), thenwith
a
modimersteps
with (i.e.,
T324C,
and 0a heterodimer
dwell-time
of the number
of steps
one head histogram
lacking solvent-exposed
cysteines
and
versus
step-time
duration
will
be
the
convolution
the other head containing cysteines at S43C and
of T324C,
two exponential
(11).(Fig.
This1B).
yields
which areprocesses
2 nm apart
Sub2
exp(–kt),
thestoichiometric
dwell time probability,
P(t
)
$
tk
labeling was used for the howhich
is
zero
at
t
$ 0, quantal
rises initially,
andofthen
modimers, and single
bleaching
fluofalls,
when
k
is
the
stepping
rate
constant.
rescence confirmed that only a single dye In
was
contrast,
if
the
17-nm
steps
arise
from
a
single
present on each kinesin analyzed (fig. S1B). The
process,
then the
dwell-time
histogram
would
heterodimer
was
labeled with
an excess
ofbe
dye
and both single- and double-quantal bleaching
was observed (13).
In the absence of ATP, kinesins were stationary. In the presence of 340 nM ATP, discrete
steps were observed for the three different kinesin constructs (Fig. 2). A total of 354 steps from
35 kinesins were observed. We typically collected 4000 photons per 0.33-s image. Traces from
relatively bright kinesins ("5000 photons per
image) are shown in Fig. 2; a histogram of 143
steps from 26 molecules is shown in Fig. 3A.
The precision of step-size determination was 1.5
to 3 nm, based on measurement of the distance
between the average positions of the PSF centers
before and after a step (11, 14). The average step
size derived from the step-size histogram (Fig.
3A) is 17.3 # 3.3 nm. We did not observe
8.3-nm steps or odd multiples of 8.3 nm. These
data therefore strongly support a hand-over-hand
mechanism and not an inchworm mechanism.
The hand-over-hand mechanism predicts that
these 17-nm steps alternate with 0-nm steps,
which are not directly observable in a graph of
REPORTS
tional human kinesin, were mutated to cysteines for fluorescent
dye labeling as described in the
text. The bound nucleotide
(adenosine
diphosphate)
is
shown as a space-filling model in
cyan. This figure was made with
MolMol (22).
Kinesin walks hand-over-hand
expected to yield an exponential decay (the
Poisson-distributed rate). The dwell-time histogram of 347 steps for E215C and T324C (Fig.
3B) is well fit by the above convolution function
(with k $ 1.14 # 0.03 steps per s), and not by
the single-step decaying function. The rise near
t $ 0 is not due to instrument artifacts: An
exponential process for myosin V stepping (with
dyes located to show every step) at very similar
rates yields the expected monotonic decay with
the same instrument (11). We also have immo-
Fig. 1. (A) Examples of two alternative classes of mechanisms
for processive movement by kinesin. The hand-over-hand model (left) predicts that a dye on
the head of kinesin will move
alternately 16.6 nm, 0 nm, 16.6
nm, whereas the inchworm
mechanism (right) predicts uniform 8.3-nm steps. The inchworm model was adapted with
slight modification from (9). (B)
The positions of S43 (red), E215
(green), and T324 (blue) on the
rat kinesin crystal structure
[from (6), Protein Data Base
2KIN]. These residues, whose
numbers correspond to conventional human kinesin, were mutated to cysteines for fluorescent
dye labeling
as described
in the
Fig. 2. Position
versus time
for kinesin
motility. The blue and green traces are from E215C
nucleotide
homodimer text.
kinesin;The
the bound
red trace,
from the heterodimer S43C-T324C kinesin. The numbers
(adenosine
is
correspond to
the step sizediphosphate)
# %&. The uncertainties
were calculated as described (11). Red lines
shownpositions
as a space-filling
in between steps (plateau) and when the step occurs
represent average
of each model
duration
cyan.
This figure
was made with
(jumps) based
on data
analysis.
MolMol (22).
Yildiz et al (2005) Science
www.sciencemag.org SCIENCE VOL 303 30 JANUARY 2004
dunkel@math.mit.edu
67
0). Based
showed
oncluded
for kinen asym-
chnique,
Accuraking the
accuracy
NA, the
step is
rescence
a totalope. The
is a dif280 nm,
ponds to
ted with
plied the
lks in a
alternatcements,
(11).
s experiwith a
ch head
Fig. 1B)
d as the
e immoent conglutamic
cond homer with
ines and
43C and
B). Subthe hoof fluo-
nucleotides (15, 16), and a two-headed bound
not Rrotating
E P O R T Sthe stalk (20), implying it too is
species
inferred
to exist
during
thean exponential
likely decay
asymmetric.
Such a mechanism has
expected
to yield
(the
position
versushas
time.been
However,
if the observed
Poisson-distributed
dwell-time
histo17-nm
steps arise
from based
the convolution
two
catalytic
cycle
on a ofkinetic
analysis (17)rate). The
rather
stringent
biophysical constraints (9),
gram of 347 steps for E215C and T324C (Fig.
sequential steps (i.e., 17 nm, 0 nm. . .), then a
and
on
fluorescence
polarization
measurements
implications for how the rear head
3B)
is well fit by the above including
convolution function
dwell-time histogram of the number of steps
(with kHowever,
$ 1.14 # 0.03 steps
per s), and
by front head. Hoenger et al. (10)
versus
step-time duration
will concentration
be the convolution (18).
at saturating
ATP
passes
bynotthe
the single-step decaying function. The rise near
of two exponential processes (11). This yields
2 bound with one or two
whether
or
not
kinesin
is
have artifacts:
postulated
a model where the rear head
t $ 0 is not due to instrument
An
the dwell time probability, P(t ) $ tk exp(–kt),
which
is zero
at t $
0, rises initially,
and then
exponential
process
V stepping
heads
while
waiting
for ATP
during
motility
hasfor myosin
passes
the(with
front head in such a manner that
falls, when k is the stepping rate constant. In
dyes located to show every step) at very similar
been
unclear.
If
only
one
head
is
bound,
then
the monotonic
the neck-linker
wraps and unwraps around
contrast, if the 17-nm steps arise from a single
rates yields
the expected
decay with
process,
dwell-timealternate
histogram would
be
the16.6x
same instrument
have immostep then
sizethewould
between
and x,(11). We
thealsostalk
with alternating steps to minimize
Kinesin walks hand-over-hand
Fig. 3. The step sizes of an individual
head of a kinesin dimer and dwell-time
analysis support a hand-over-hand
mechanism. (A) The kinesin step-size
histogram from 124 steps of 22 molecules of E215C, 12 steps of 3 molecules
of T324C, and 7 steps of one S43CT324C heterodimer. The average step
size is 17.3 ! 3.3 nm (n $ 143, "# $
0.27 nm). The black solid line is
a Gaussian fit. (B) The dwell-time histogram of 347 steps from 33 kinesin
molecules, including 317 steps from 29
molecules of E215C and 30 steps from
4 molecules of T324C, at 340 nM ATP.
The black line is a best-fit curve to the
convolution function tk2exp(–kt), with
k $ 1.14 ! 0.03 s–1 and coefficient of
determination r 2 $ 0.984.
Fig. 1. (A) Examples of two alternative classes of mechanisms
for processive movement by kinesin. The hand-over-hand model (left) predicts that a dye on
the head of kinesin will move
alternately 16.6 nm, 0 nm, 16.6
nm, whereas the inchworm
mechanism (right) predicts uniform 8.3-nm steps. The inchworm model was adapted with
slight modification from (9). (B)
The positions of S43 (red), E215
(green), and T324 (blue) on the
rat kinesin crystal structure
[from (6), Protein Data Base
2KIN]. These residues, whose
numbers correspond to conventional human kinesin, were mutated to cysteines for fluorescent
dye labeling as described in the
text. The bound nucleotide
(adenosine
diphosphate)
is
shown as a space-filling model in
cyan. This figure was made with
MolMol (22).
Yildiz et al (2005) Science
dunkel@math.mit.edu
Intracellular transport
Chara corralina
http://damtp.cam.ac.uk/user/gold/movies.html
dunkel@math.mit.edu
wiki
dunkel@math.mit.edu
Muscular contractions:
Actin + Myosin
G-Actin
(globular)
F-Actin
helical filament
dunkel@math.mit.edu
Actin-Myosin
F-Actin
helical filament
Myosin
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Actin-Myosin
F-Actin
Myosin
helical filament
myosin-II
myosin-V
dunkel@math.mit.edu
step size of the stalk while
domain does not move. For
attached to the light chain
the inchworm model presize of 37 nm, whereas the
el predicts alternating steps
x, where x is the in-plane
from the midpoint of the
and 30 ms, respectively.
FIONA. A single fluorescent molecule
forms a diffraction-limited image of width %
&/2 N.A., or % 250 nm for visible light,
spots displayed
single quantal
bleaching, inwhere N.A. is the of
numerical
aperture
of the
dicative of a single molecule. Step sizes were
collection lens. The
center
thelabeled
image,
analyzed
only forofsingly
myosins.
which, under appropriate
conditions,
In the absence
of ATP, thecorrefluorescent spots
were immobile.
addition
of !300
sponds to the position
of theThedye,
can
be nM ATP
led to discernable steps, and the average steplocated to arbitrarily
precision
by col-ATP conpinghigh
rate increased
with increasing
total,
we observed
49 different
lecting a sufficientcentration.
numberIn of
photons.
Our
BR-labeled myosin V molecules and detected
method for determining
the center relies on
552 total steps. We observed three different
RESEARCH ARTICLES
Myosin walks hand-over-hand
dels, we have developed a
escence imaging technique
populations of myosin V molecules, exhibiting
either uniform 74-nm steps, alternating 52- and
and
Hand over hand
Inchworm
23-nm
(52-23) steps, or alternating 42- and 33del
nm
(42-33)
steps. Uniform 37-nm steps were
Catalytic
Cargo binding domain
. A
domain
not observed.
Light chain domain
n is
Specifically we detected 365 steps from 38
74 nm
x
myosin V’s, each of which stepped ! 74 nm
flu37(Fig.
nm 3; Movie
37 nm
S1). Thirty-two of these moleexcules were bright enough to yield a
osin
signal-to-noise ratio (SNR) " 10 for a total of
ain,
231 steps. A histogram of these steps showed that
37 nm — 2x
37size
nmis 73.8 # 5.3 nm (mean # SD), with
of
the step
74 nm
ons
an excellent fit to a normal distribution (r 2 $
37 nm
37 nm
0.994, %2r$ 1.67) (Fig. 3). We also detected six
atic
molecules that took a total of 92 alternating 52-23
posteps (Fig. 4), and six other molecules that took a
vertotal of 69 alternating 42-33 steps (Fig. 5). The
ear
37 nm + 2x
37 size
nm data shows three very
histogram of the step
fordistinct peaks for both the 52-23 and 42-33 data
sets. For the 52-23 data, the averages of these
ead
peaks37are
4.1nm
nm, 23.1 # 3.4 nm, and
talk
nm51.7 # 37
37 nm
37 nm
73.6 # 5.3 nm (mean # SD). For the 42-33 data,
the
the averages are 42.4 # 2.9 nm, 32.8 # 2.1 nm,
ing
and 74.1 # 2.2 nm. The peak centered around 74
the dye is a different distance from the stalk in the forward
versuswith
rear
lightVchain
nm is consistent
myosin
molecules taking
mmetry in the myosin V structure, then x is the average
distance
dye
two steps
(e.g., 52ofnmthe
& 23
nm from
$ 75 nm) within
0.5 or 1 and
s, which
be fully resolved
worm model, all parts of the myosin move 37-nm forward,
onecould
headnotalways
because
of
the
0.5-s
time
resolution.
The percentermission from (32).
age of such missed steps is consistent with a
probability distribution corresponding to exponentially distributed dwell times with an
average
NCE VOL 300 27 JUNE 2003
2061
'1
step rate of 0.3 s (Fig. 6).
When the myosin V was labeled with a Cy3calmodulin, the observed step sizes were consistent with those measured with BR (Figs. 3 and 5).
This acts as a control to ensure that the stepping
characteristics we see are not specific to a particular dye. In particular, BR, which is attached by
two points to the light chain, is highly polarized
on the 0.5-s time scale (6), whereas Cy3, a
monofunctional dye, is expected to have signifi-
These results strongly support a hand-overhand model and are not consistent with an inchworm model. The hand-over-hand model predicts the dye will take alternating steps equal to
the stalk-step-size # 2x, and we interpret the
uniform 74-nm steps arising from a dye attached
to a light chain near the catalytic domain (stalkstep-size $ 37 nm; x $ 18.5 nm), perhaps on the
first light chain. Using the same model, we interpret the 52-23 steps arising from a dye 6.5 to
7 nm from the midpoint in the direction of
motion, probably corresponding to a dye on the
fifth light chain, and the 42-33 steps arising from
a dye 2 to 2.5 nm from the midpoint, probably
corresponding to a dye on the sixth light chain
(Fig. 1).
The hand-over-hand model predicts that
for a dye very close to, or on, one catalytic
domain, the steps will alternate between 74
nm and 0 nm (74-0) (Fig. 1). The 0-nm steps
Fig. 3. Stepping traces of three different myosin V molecules displaying 74-nm steps and histogram
(inset) of a total of 32 myosin V’s taking 231 steps. Calculation of the standard deviation of step sizes
can be found (14). Traces are for BR-labeled myosin V unless noted as Cy3 Myosin V. Lower right trace,
see Movie S1.
Yildiz et al (2003) Science
dunkel@math.mit.edu
Bacteria-driven motor
Di Leonardo (2010) PNAS
dunkel@math.mit.edu
Feynman-Smoluchowski
ratchet
dunkel@math.mit.edu
generic model of
a micro-motor
dunkel@math.mit.edu
Basic ingredients for
rectification
•
•
•
•
some form of noise (not necessarily thermal)
some form of nonlinear interaction potential
spatial symmetry breaking
non-equilibrium (broken detailed balance) due to
presence of external bias, periodic forcing,
memory, etc.
dunkel@math.mit.edu
Eukaryotic motors
Sketch: dynein molecule carrying cargo down a microtubule
http://www.plantphysiol.org/content/127/4/1500/F4.expansion.html
Yildiz lab, Berkeley
dunkel@math.mit.edu
thermal equilibrium.
Generally speaking, the combination of broken spatial symmetry and non-equilibrium driving is sufficient for generating stationary currents by means of a ratchet e↵ect.
Most biological micro-motors operate in the low Reynolds number regime, where inertia
is negligible. A minimal model can therefore be formulated in terms of an over-damped
Ito-SDE
p
0
dX(t) = U (X) dt + F (t)dt + 2D(t) ⇤ dB(t).
(1.116)
19
For further reading, we refer to the review articles [HM09, Rei02].
25
thermal equilibrium.
Generally speaking, the combination of broken spatial symmetry and non-equilibrium driving is sufficient for generating stationary currents by means of a ratchet e↵ect.
Most biological micro-motors operate in the low Reynolds number regime, where inertia
is negligible. A minimal model can therefore be formulated in terms of an over-damped
Ito-SDE
p
0
dX(t) = U (X) dt + F (t)dt + 2D(t) ⇤ dB(t).
(1.116)
19
For further reading, we refer to the review articles [HM09, Rei02].
Here, U is a periodic potential
25
U (x) = U (x + L)
(1.117a)
with broken reflection symmetry, i.e., there is no x such that
U ( x) = U (x + x).
(1.117b)
1
U = U0 [sin(2⇡x/L) + sin(4⇡x/L)].
4
(1.117c)
A typical example is
The function F (t) is a deterministic driving force, and the noise amplitude D(t) can be
time-dependent as well.
The corresponding FPE for the associated PDF p(t, x) reads
@t p =
@x j ,
j(t, x) =
{[U 0
F (t)]p + D(t)@x p},
and we assume that p is normalized to the total number of particles, i.e.
(1.118)
66
P. Reimann / Physics Reports 361 (2002) 57 – 265
2
V(x)/V
0
1
0
-1
-2
-1
-0.5
0
0.5
1
x /L
Fig. 2.2. Typical example of a ratchet-potential V (x), periodic in space with period L and with broken spatial symmetry.
Plotted is the example from (2.3) in dimensionless units.
microscopic degrees of freedom of the environment. As discussed in detail in Sections A.1 and A.2
of Appendix A, our assumption that the environment is an equilibrium heat bath with temperature T
and that its e!ect on the system can be modeled by means of the phenomenological ansatz appearing
dunkel@math.mit.edu
on the right-hand side of (2.1) completely "xes [66,77–97] all statistical properties of the #uctuations
A typical example is
U = U0 [sin(2⇡x/L) +
1
sin(4⇡x/L)].
4
(1.117c)
The function F (t) is a deterministic driving force, and the noise amplitude D(t) can be
time-dependent as well.
The corresponding FPE for the associated PDF p(t, x) reads
@t p =
@x j ,
{[U 0
j(t, x) =
F (t)]p + D(t)@x p},
and we assume that p is normalized to the total number of particles, i.e.
Z L
NL (t) =
dx p(t, x)
(1.118)
(1.119)
0
gives the number of particles in [0, L]. The quantity of interest is the mean particle velocity
vL per period defined by
Z L
1
vL (t) :=
dx j(t, x).
(1.120)
NL (t) 0
Inserting the expression for j, we find for spatially periodic solutions with p(t, x) = p(t, x + L)
that
Z L
1
vL =
dx [F (t) U 0 (x)] p(t, x).
(1.121)
NL (t) 0
1.6.1
Tilted Smoluchowski-Feynman ratchet
As a first example, assume that F = const. and D = const. This case can be considered
as a (very) simple model for kinesin or dynein walking along a polar microtubule, with the
constant force F
0 accounting for the polarity. We would like to determine the mean
transport velocity v for this model.
vL =
1.6.1
NL (t)
dx [F (t)
U 0 (x)] p(t, x).
(1.121)
0
Tilted Smoluchowski-Feynman ratchet
As a first example, assume that F = const. and D = const. This case can be considered
as a (very) simple model for kinesin or dynein walking along a polar microtubule, with the
constant force F
0 accounting for the polarity. We would like to determine the mean
transport velocity vL for this model.
To evaluate Eq. (1.121), we focus on the long-time limit, noting that a stationary
solution p1 (x) of the corresponding FPE (1.118) must yield a constant current-density j1 ,
i.e.,
[(@x )p1 + D@x p1 ]
(1.122)
26
P. Reimann / Physics Reports 361 (2002) 5
4
2
3
2
1
<x>
eff
(x)
1
.
0
V
j1 =
0
-1
-2
-1
-3
-2
-1
-0.5
0
0.5
1
-4
-6
-4
x
Fig. 2.3. Typical example of an e!ective potential from (2.35) “tilted to
ample from (2.3) in dimensionless units (see Section A.4 in Appendix A
P. Reimann / Physics Reports 361 (2002) 57 – 265
73
Ve! (x) = sin(2!x)
+ 0:25 sin(4!x) + x.
2
4
vL =
1.6.1
NL (t)
dx [F (t)
U 0 (x)] p(t, x).
(1.121)
0
Tilted Smoluchowski-Feynman ratchet
As a first example, assume that F = const. and D = const. This case can be considered
as a (very) simple model for kinesin or dynein walking along a polar microtubule, with the
constant force F
0 accounting for the polarity. We would like to determine the mean
transport velocity vL for this model.
To evaluate Eq. (1.121), we focus on the long-time limit, noting that a stationary
solution p1 (x) of the corresponding FPE (1.118) must yield a constant current-density j1 ,
i.e.,
j1 =
where
[(@x )p1 + D@x p1 ]
(1.122)
26
(x) = U (x)
xF
(1.123)
is the full e↵ective potential acting on the walker. By comparing with (1.85), one finds
that the desired constant-current solution is given by
Z x+L
1
p1 (x) = e (x)/D
dy e (y)/D .
(1.124)
Z
x
This solution is spatially periodic, as can be seen from
Z x+2L
1 [U (x+L) (x+L)F ]/D
p1 (x + L) =
e
dy e[U (y) yF ]/D
Z
x+L
Z x+L
1 [U (x) (x+L)F ]/D
=
e
dz e[U (z+L) (z+L)F ]/D
Z
x
Z x+L
1
gives the number of particles in [0, L]. The quantity of interest is the mean particle velocity
as a (very) simple model for kinesin or dynein walking along a polar
e that p is normalized
to period
the total
number
vL per
defined
by of particles, i.e.
constant force F
0 accounting for the polarity. We would like to
Z L
Z L
1transport velocity vL for this model.
NL (t) =
dx p(t, x)
(1.119)
vL (t) :=
dx j(t,
x).
(1.120)
N
(t)
0
L
To
evaluate Eq. (1.121), we focus on the long-time limit, noti
0
Constant current solution
solution
p1 (x)solutions
of the corresponding
(1.118) must yield a constan
ber of particles in [0,
L]. Thethe
quantity
of interest
is the
particleperiodic
velocity
Inserting
expression
for j, we
findmean
for spatially
with p(t, x) = p(t,FPE
x + L)
defined by
i.e.,
that
where
Z L
Z L
1
1
0
j1 = (1.121)
[(@x )p
vL =
dx [F
(t)
U(1.120)
(x)]
vL (t) :=
dx j(t, x).
1 + D@x p1 ]
(x)
=
U
(x)
xFp(t, x).
(1.123)
N
(t)
N
(t)
L
where
L
0
0
isfind
thefor
full
e↵ective
potential
acting
By comparing with (1.85), one 26
finds
xpression for j, we
spatially
periodic
solutions
with on
p(t, the
x) =walker.
p(t, x + L)
1.6.1
Tilted Smoluchowski-Feynman
ratchet
(x) = U (x) xF
(1.123)
that the desired constant-current solution is given by
As Za Lfirst example, assume that F = const. and D = const. This case can be considered
Z x+L By comparing with (1.85), one finds
1
is the
e↵ective
potential
actingor1on
the walking
walker.
0
as afull
(very)
for x).
kinesin
dynein
along a(y)/D
polar microtubule, with the
vL =
dx [Fsimple
(t) Umodel
(x)] p(t,
(1.121)
(x)/D
p1 (x) =
e
dy e
.
(1.124)
NL (t)
that
the0 desired
solution
is given by
constant
force Fconstant-current
0 accounting
for
Zthe polarity. xWe would like to determine the mean
transport velocity vL for this model.
Z x+L
ted Smoluchowski-Feynman
ratchet
1
evaluateis Eq.
(1.121),
we focusason
thebelong-time
limit,
noting that a stationary
(x)/D
(y)/D
ThisTo
solution
spatially
periodic,
can
seen from
p
(x)
=
e
dy
e
. current-density j , (1.124)
1
solution
p
(x)
of
the
corresponding
FPE
(1.118)
must
yield
a
constant
1
1
Z
mple, assume that F = const. and D = const. This case can be considered
x
Z
x+2L
i.e., or dynein walking along a polar
1 microtubule,
mple model for kinesin
with]/D
the
[U (x+L) (x+L)F
p
(x
+
L)
=
e
dy e[U (y) yF ]/D
1
This
solution
is
spatially
periodic,
as
can
be
seen
from
F
0 accounting for the polarity. We wouldj like
determine
the mean
Z to [(@
(1.122)
1 =
x )p1 + D@x p1 ] x+L
Z
city vL for this model.
Zx+L
x+2L [U (z+L) (z+L)F ]/D
1
[U
(x)
(x+L)F
]/D
1noting
te Eq. (1.121), we focus on the long-time limit,
that a(x+L)F
stationary
[U (x+L)
]/D
=
e
dz edy e[U (y) yF ]/D
p1 (x + L) =
e
26
Z
) of the corresponding FPE (1.118) must yield a constant
current-density Zjx1 ,x+L
Z
x+L
1 [U (x) (x+L)F ]/DZ x+L
[U (z) (z+L)F ]/D
= 1 e[U (x) (x+L)F ]/D
dz e[U
(z+L) (z+L)F ]/D
=
dz
e
Ze
x
j1 = [(@x )p1 + D@x p1 ]
(1.122)
Z
x
= p1 (x),
(1.125)
Z x+L
1 [U (x) (x+L)F ]/D
=
e
dz e[U (z) (z+L)F ]/D
26
Z
where we have used the coordinate
transformation
L 2 [x, x + L] after the first
x z = y
line. Inserting p1 (x) into=Eq.p1
(1.121)
(x), gives
(1.125)
Z L
1
where we have used vthe =coordinate transformation
dx (@x ) p1 z = y L 2 [x, x + L] after the first
L
line. Inserting p1 (x) into Eq. N
(1.121)
gives
L 0
Z L
Z x+L
1Z L
(x)/D
(y)/D
=
dx
(@
)
e
dy
e
x
1ZN
L dx
0 (@x ) p1
x
vL =
Z x+L
NL Z L
[Usolution
(x) (x+L)F
]/D
[U (z+L) (z+L)F
]/D(1.118) must yield a constan
p
(x)
of
the
corresponding
FPE
1
=
e
dz
e
ber of particles in [0,
L].
The
quantity
of
interest
is
the
mean
particle
velocity
Inserting the expression for j, we find for spatially periodic solutions with p(t, x) = p(t, x + L)
Z
x
defined by
i.e.,
that
Z x+L
Z L
1Z L [U (x) (x+L)F ]/D
[U (z) (z+L)F ]/D
1
1
=
e
dz
e
0
j1 = (1.121)
[(@x )p1 + D@x p1 ]
vL =
dx [F (t) U(1.120)
(x)] p(t, x).
vL (t) :=
dx j(t, x).
Z
NL (t) 0
NL (t) 0
x
= p (x),
xpression for j, we find for spatially periodic solutions1
with p(t, x) = p(t, x + L)
1.6.1
Tilted Smoluchowski-Feynman ratchet
26
(1.125)
where
have
usedassume
the coordinate
transformation
= ycase L
x + L] after the first
As we
example,
that F = const.
and D = const.zThis
can2be[x,
considered
Za Lfirst
0 into
a (very)
forEq.
kinesin
or dynein
walking
along a polar microtubule, with the
line.1asInserting
p1 (x)
(1.121)
gives
vL =
dx [Fsimple
(t)
Umodel
(x)] p(t,
x).
(1.121)
NL (t)
0
constant
force F
0 accounting for the polarity. We would like to determine the mean
transport velocity vL for this model.Z L
1
ted Smoluchowski-Feynman
ratchet
To evaluate Eq.
(1.121),
we
focus on
vL =
dxthe
(@xlong-time
) p1 limit, noting that a stationary
NL FPE
p1 (x)
of D
the=corresponding
(1.118)
must yield a constant current-density j1 ,
0 can
mple, assume thatsolution
F = const.
and
const. This
case
be considered
Z
Z x+L
L
i.e.,
1 microtubule, with the
mple model for kinesin or dynein walking along a polar
(x)/D
(y)/D
dx
(@
)
e
dy
e
x
F
0 accounting for the polarity.
like
determine the mean
j1
= to
(1.122)
ZN
L [(@0x )p1 + D@x p1 ]
x
city vL for this model.
Z L
Z x+L
⇥
⇤
D noting that a stationary
te Eq. (1.121), we focus on the long-time limit,
(y)/D
26@x e (x)/D
=
dx
dy
e
.
) of the corresponding FPE (1.118) must yield aZN
constant current-density j1 ,
L 0
x
We =
would
Integrating
can be simplified to(1.122)
j1 = [(@x by
)p1parts,
+ D@x pthis
1]
Z L
Z x+L
D
26
vL =
dx e (x)/D @x
dy e (y)/D
ZNL 0
x
Z L
⇥ (x+L)/D
⇤
D
(x)/D
(x)/D
=
dx e
e
e
ZNL 0
Z L
D
=
dx 1 e[ (x+L) (x)]/D
ZNL 0
Z L
D
=
dx 1 e F [(x+L) x]/D
ZNL 0
DL
=
1 e F L/D ,
ZNL
(1.126)
(1.127)
Z
[Usolution
(x) (x+L)F
]/D
[U (z+L) (z+L)F
]/D(1.118) must yield a constan
x+L
p
(x)
of
the
corresponding
FPE
1
=
e
dz
e
ber of particles in [0,
L].
The
quantity
of
interest
is
the
mean
particle
velocity
Inserting the expression for j, we find for spatially periodic solutions
Z x+L with p(t, x) = p(t, x + L)
Z
x
1 i.e.,
defined by
that
=
e [U (x) (x+L)F ]/DZ x+L dz e[U (z+L) (z+L)F ]/D
Z L
1ZZL [U (x) (x+L)F ]/D x
[U (z) (z+L)F ]/D
1
1
=
e
dz
e
0
Z
j1 = (1.121)
[(@x )p1 + D@x p1 ]
vL =
dx [F (t) U(1.120)
(x)] p(t, x).
vL (t) :=
dx j(t, x).
x+L
Z
1
NL (t) 0
NL (t) 0
[U (x) (x+L)F ]/D x
[U (z) (z+L)F ]/D
=
e
dz e
= p1Z(x),
xpression for j, we find for spatially periodic solutions with p(t, x) = p(t, x + L) x
1.6.1
26
Tilted Smoluchowski-Feynman
ratchet
= p1 (x),
(1.125)
(1.125)
where
have
usedassume
the coordinate
transformation
= ycase L
x + L] after the first
As we
example,
that F = const.
and D = const.zThis
can2be[x,
considered
Za Lfirst
0 into
asInserting
a (very)
model
forEq.
kinesin
or dynein
walking
along azpolar
with
the after the first
we[Fsimple
have
the
coordinate
transformation
= ymicrotubule,
L 2 [x, x
+ L]
line.1where
p1 (x)
(1.121)
gives
vL =
dx
(t)
Uused
(x)] p(t,
x).
(1.121)
NL (t)
0
constant
force F p10(x)
accounting
the polarity.
line.
Inserting
into Eq.for(1.121)
gives We would like to determine the mean
transport velocity vL for this model.Z L
1
Z on
ted Smoluchowski-Feynman
ratchet
L the
To evaluate Eq.
(1.121),
we
focus
vL =
dx
(@xlong-time
) p1 limit, noting that a stationary
1
NL FPE
p1 (x)
of D
the=corresponding
(1.118)
a constant current-density j1 ,
vconst.
dxbe(@considered
) pyield
0 can
L = This
xmust
1
mple, assume thatsolution
F = const.
and
case
Z
Z x+L
N
L 0L
i.e.,
1 microtubule, with the
mple model for kinesin or dynein walking along a polar
Z
Z
(x)/D
(y)/D
L
x+L
dx
(@
)
e
dy
e
1
x
F
0 accounting for the polarity.
like
determine
the
mean (x)/D
= to
D@
= j1
dx+(@
dy e (y)/D(1.122)
ZN
x x p)1e]
L [(@0x )p1
x
city vL for this model.
ZN
Z LL 0
Z x+Lx
⇥
⇤
Z
Z x+L
D noting that
te Eq. (1.121), we focus on the long-time limit,
a⇥ stationary
L
(x)/D
(y)/D
⇤
26@x e
D
=
dx
dy
e
.
(x)/D
(y)/D
) of the corresponding FPE (1.118) must yield=aZN
constant current-density
j1 ,
dx
@
e
dy
e
.
x
L 0
x
ZN
We =
would
L
0
(1.126)
(1.126)
x
Integrating
this can be simplified toto
by
parts,
j1Integrating
= [(@x by
)p1parts,
+ D@
x p1 ] this can be simplified (1.122)
Z ZL L
ZZ x+L
x+L
DD
26
(x)/D
(y)/D
(x)/D
vL vL= =
dxdxe e
@@xx
dy
dy e (y)/D
ZN
ZN
L L0 0
xx
Z ZL L
⇥⇥ (x+L)/D
⇤⇤
DD
(x)/D
(x+L)/D
(x)/D
(x)/D
(x)/D
==
dxdxe e
ee
e
ZN
ZN
L L0 0
Z ZL L
D
D
(x+L) (x)]/D
(x)]/D
=
dx 1 1 e[e[ (x+L)
=
dx
ZN
ZN
L L0Z0
Z LL
D
F [(x+L) x]/D
D
F [(x+L) x]/D
=
dx
1
e
=
dx 1 e
ZN
L
0
ZNL 0
DL
F L/D
= DL
1 eF L/D
,
=
1
e
,
ZN
ZNL L
(1.127)
(1.127)
=solution
dx corresponding
ewith p(t, x) = p(t,
e FPE
e yield a constan
(x)/D
p1 (x)solutions
of the
(1.118)
must
ber of particles in [0,
L]. Thethe
quantity
of interest
is the
particleperiodic
velocity
Inserting
expression
for j, we
findmean
for spatially
x
+
L)
x
0
defined by
i.e., ZNL
that
Z L
L
0
x
Z L
Z L
D
1
[ j (x+L)
1
0
[(@x(x)]/D
)p1 + D@x p1 ]
=
1
e
vL =
dx [F (t) U(1.120)
(x)] p(t,dx
x).
vL (t) :=
dx j(t, x).
1 = (1.121)
NL (t) 0
NL (t) 0
ZNL 0
Z L)
xpression for j, we find for spatially periodic solutions with p(t, x) = p(t, x + L
26
1.6.1 Tilted Smoluchowski-Feynman
D ratchet
F [(x+L) x]/D
=
dx
1
e
As Za Lfirst example, assume that F = const. ZN
and D = const. This case can be considered
L 0
1as a (very)
L
x+L
0
simple
model
for
kinesin
or
dynein
walking
along a polar microtubule,
with the
vL =
dx [F (t) U (x)] p(t, x).
(1.121)
NL (t)
DL We would likeF to
0
constant
force F
0 accounting for the polarity.
determine the mean
(x)/D
L/D
1 e
,
transport velocity vL for this model.
L =
x
ZN
L
ted Smoluchowski-Feynman
ratchetwe focus on the
To evaluate Eq. (1.121),
long-time
limit,
L noting
0 that a stationary
x
p1 (x)
of D
the=corresponding
FPEcan
(1.118)
must yield a constant current-density j1 ,
mple, assume thatsolution
F = const.
and
const. This case
be considered
L
i.e.,
can be
expressed
as microtubule, with the
mple model forwhere
kinesin N
or dynein
walking
along a polar
=
dx @ e
ZN
dy
Integrating by parts, this can be simplified to
Z
Z
D
v =
dx e
@
dy
ZN
Z
⇥
D
(x)/D
(x+L)/D
27
F
0 accounting for the polarity. We wouldj like
to
determine
the
mean
dx e (1.122)
e
= Z
[(@ =
)p Z+ D@ p ]
city v for this model.
1
ZN
= noting that
dx26a stationary
dy
e L 0
.
(1.128)
te Eq. (1.121), we focus on the long-time N
limit,
Z current-density j , Z
) of the corresponding FPE (1.118) must yield a constant
L
D
[ (x+L)
(x)]/D
We thus obtain the final result
=
dx
1
e
j = [(@ )p + D@ p ]
(1.122)
ZN
1 Le 0
v = DL R
,
(1.129)
R
26
Z
dx
dy e L
D
F [(x+L) x]/D
=
dx 1 e
which holds for arbitrary periodic potentials U (x). Note that there is no net-current at
ZNL 0
equilibrium F = 0.
DL
F L/D
=
1 e
,
1.6.2 Temperature ratchet
ZNL
L
1
L
x
L
1
x+Lx
1
L
0
1
x
1
x
[ (x)
(y)]/D
1
x 1
F L/D
L
L
0
x+L
x
[ (x)
(y)]/D
As we have seen in the preceding sections, the combination of noise and nonlinear dynamics can yield surprising transport e↵ects. Another example is the so-called temperatureratchet, which can be captured by the minimal SDE model
p
27
Tilted Feynman-Smoluchowski
ratchet
P. Reimann / Physics Reports 361 (2002) 57 – 265
73
4
2
3
2
<x>
1
.
0
V
eff
(x)
1
0
-1
-2
-1
-3
-2
-1
-0.5
0
0.5
1
-4
-6
-4
-2
0
x
2
4
6
F
Fig. 2.3. Typical example of an e!ective potential from (2.35) “tilted to the left”, i.e. F¡0. Plotted is the example from (2.3) in dimensionless units (see Section A.4 in Appendix A) with L = V0 = 1 and F = −1, i.e.
Ve! (x) = sin(2!x) + 0:25 sin(4!x) + x.
Fig. 2.4. Steady state current ⟨ẋ⟩ from (2.37) versus force F for the tilted Smoluchowski–Feynman ratchet dynamics (2.5),
(2.34) with the potential (2.3) in dimensionless units (see Section A.4 in Appendix A) with " = L = V0 = kB = 1 and
T = 0:5. Note the broken point-symmetry.
st
dunkel@math.mit.edu
0
x
which holds for arbitrary periodic potentials U (x). Note that there is no net-current at
equilibrium F = 0.
1.6.2
Temperature ratchet
As we have seen in the preceding sections, the combination of noise and nonlinear dynamics can yield surprising transport e↵ects. Another example is the so-called temperatureratchet, which can be captured by the minimal SDE model
p
0
dX(t) = [F U (X)] dt + 2D(t) dB(t),
(1.130a)
where D(t) = D(t + T ) is now a time-dependent noise amplitude, such as for instance
D(t) = D̄ {1 + A sign[sin(2⇡t/T )]} ,
(1.130b)
where |A| < 1. Such a temporally varying noise strength can be realized by heating
and cooling the ratchet system periodically. Transport can be quantified in terms of the
combined spatio-temporal average
Z
Z L
1 t+T
hẊi :=
ds
dx j(t, x)
T t
0
Z t+T Z L
1
=
ds
dx [F U 0 (x)] p(t, x).
(1.131)
T t
0
This choice is motivated by the fact that the equations of motions are periodic in space
and time, which suggests an asymptotically oscillating solution p(t, x) = p(t, x + L) =
p(t + T, L) = p(t + T, x + L) for the probability density. Equation (1.130) has been
studied numerically (see slide and Sec. 2.6 in Ref. [Rei02]), and was found to predict
an counterintuitive e↵ect: In the presence of a small load force, optimally tuned periodic
thermal pumping allows particles to climb up-hill (see slides for an illustration).
can be solved numerically
Time-dependent temperature
P. Reimann / Physics Reports 361 (2002) 57 – 265
77
0.04
<x>
0.02
.
0
-0.02
-0.04
-0.02
0
0.02
F
Fig. 2.5. Average particle current ⟨ẋ⟩ versus force F for the temperature ratchet dynamics (2.3), (2.34), (2.47), (2.50)
in dimensionless units (see Section A.4 in Appendix A). Parameter values are ! = L = T = kB = 1, V0 = 1=2", T! = 0:5,
A = 0:8. The time- and ensemble-averaged current (2.53) has been obtained by numerically evolving the Fokker–Planck
equation (2.52) until transients have died out.
Fig. 2.6. The basic working mechanism of the temperature ratchet (2.34), (2.47), (2.50). The "gure illustrates how
Brownian particles, initially concentrated at x0 (lower panel), spread out when the temperature is switched to a very high
value (upper panel). When the temperature jumps back to its initial low value, most particles get captured again in the
basin of attraction of x0 , but also substantially in that of x0 + L (hatched area). A net current of particles to the right, i.e.
⟨ẋ⟩¿0 results. Note that practically the same mechanism is at work when the temperature is kept "xed and instead the
potential is turned “on” and “o# ” (on–o# ratchet, see Section 4.2).
dunkel@math.mit.edu
A conversion (recti"cation) of random $uctuations into useful work as exempli"ed above is called