Motor proteins
Part I
Motor proteins
Motor proteins perform active transport inside a cell
Why are they necessary?
Since diffusion is so slow over long distances, you need active transport to move more than a µm or so.
Many processes in the cell are out of equilibrium. A motor can transport against a concentration gradient,
maintaining out-of-equilibrium concentrations (e.g. ATP/ADP, ion concentrations, PMF, osmotic pressure).
Motors that are attached to the outside world can push, pull or rotate a cell, moving it to a better environment.
Even within a cell, motors can actively change a cell’s physical shape.
Any motor requires a source of energy
Many motors consume ATP, though some consume protons (PMF).
ANY source of free energy can power a motor: some polymerization motors (e.g. RNAP) use the free energy of the
polymerization reaction.
Most motors are cyclical
Motors generally iterate a cycle, each time consuming a unit of free energy and advancing one step.
This is similar to an enzyme, though the “reaction” is physical motion.
Motor proteins
Microscopic motors are different from macroscopic motors
Energy works differently.
Molecular motors work at energy scales close to kT.
Both their energy source and the work done per step are a few kT.
Thermal fluctuations are important.
It is impossible to store kinetic energy microscopically.
There is no molecular version of the flywheel in your car’s engine.
The energy required for a step must come from consuming a single unit of free energy. (Usually)
Fluctuations dissipate excess energy instantly.
There are microscopic springs
elastic deformations of proteins can be used to store energy.
Friction works differently.
Microscopic friction is not well understood, but for most purposes friction (or viscous drag) is negligible.
Microscopic motors are reversible.
Under the right conditions, most motors can change from forward (consuming free energy to produce work) to
reverse (transforming work into free energy)
This is most often seen in artificial situations (non-biological conditions in vitro), but some motors (e.g. F0F1-ATPase) are designed
to run in both directions.
Other motors
To make a motor, you need
Source of free energy
Spatial asymmetry (in motor, substrate or external conditions)
Spatial or temporal cycling
Anchor point
Brownian ratchets
Brownian ratchets
The simplest candidate for a microscopic motor is the Brownian ratchet
Symmetric Brownian ratchet
Imagine a linear object, studded with spring-loaded ratchets,
that moves randomly through a pore.
Elastic potential energy of !ky2 is necessary to compress the
spring, up to its maximum compression Y where != !kY2.
This energy is released when the ratchet pops through the pore, and
the cycle repeats every distance L.
Brownian ratchets
Though it can diffuse back and forth, this ratchet cannot move in a particular direction (on average).
1. There is no bulk free energy difference driving ratchet, so it cannot move purposefully in any direction.
That is, leftward and rightward motion are indistinguishable (energywise, over long distances), so there is no energy basis to prefer one
over the other.
This is easiest to see in the Arrhenius limit of deep minima separated by a high-energy transition state: the forward and reverse rates
should be identical, since they have identical "G‡.
2. Imagine using the ratchet to produce a force f
If the ratchet moved (even at a slow speed) against f, we could use it to
extract work fx. This would be work extracted directly from kBT: a violation
of the second law of thermodynamics.
Actually, coupling the ratchet to f shifts the free energy surface up to the
right: if anything, this would cause the ratchet to move leftward, so it
would be “producing” negative work. Not a good motor!
3. Imagine bending the ratchet into a circle. Wait a long time and look at the distribution of positions.
The final distribution is (a) steady-state and (b) in thermal equilibrium.
(a) only means that flux in = flux out. This can be accomplished by having uniform (constant) rotation rate
(b) must obey detailed balance as well: flux left = flux right at any point
no net rotation
Brownian ratchets
Asymmetric (nonuniform) ratchet
It is possible to make a Brownian ratchet that moves in a net direction.
This is called rectified Brownian motion.
Add latches to the springs, and start with the springs precompressed.
Every time a spring pops up, the free energy of the ratchet decreases.
Now leftward and rightward motion are different: G decreases if the ratchet moves to the right.
Even if it is pulling against a small force, the net progress must be rightward.
For small f, the free energy landscape slopes down to the right.
But if you make the force too large, you can switch the direction of motion
If f > !/L, the free energy landscape slopes down to the left.
This relies on springs being able to spontaneously contract due to thermal fluctuations.
This is a slightly artificial example because the ratchet is preloaded
We payed the energy cost up front when the ratchet was prepared and the springs were compressed. That energy is available to
produce work later.
This is a one-shot motor.
Brownian ratchets
Stall force
The limiting force for which G still slopes rightward is fstall = !/L
When f = fstall, there is no longer a preferred direction to G
Net motion must cease, so v = 0, though there is still Brownian diffusion (random thermal motions).
All the preloaded energy (! per spring spacing L) is being harvested as work when the motor diffuses back and forth.
No wasted energy
perfectly reversible motion
no net motion
The barrier to backwards motion comes from the “wasted” energy ! – fL.
Waste energy
heat
increased entropy
breaking of time reversal symmetry
net forward motion
Or, a 100% efficient ratchet cannot move!
Your macroscopic intuition can mislead you about microscopic phenomena.
For instance, on our (human) length scale, a barbed cactus glochid will enter your finger more easily than it comes
out.
This would not be the case if you made it molecular size (10,000x smaller )
836
CHAPTER 16. DYNAMICS OF MOLECULAR MO
Brownian ratchets
Translocation ratchet
The nonuniform ratchet is a directly
analogous to translocation through a pore
(A) THERMAL RATCHET
thermal motion
inner
mitochondrial
membrane
One of the models for protein import into the
mitochondrion ...
MATRIX
TIM23
complex
mitochondrial
hsp70
ATP
ADP + Pi
(B) CROSS-BRIDGE RATCHET
882
CHAPTER 16. DYNAMICS OF MOLECULAR MOTORS
binding proteins
MATRIX
... can be mapped directly onto a Brownian ratchet
model by mapping ratchets to binding proteins.
TIM23
complex
The spring spacing L corresponds to the distance between
protein binding sites.
The ratchet compression energy ! corresponds to the
binding energy.
F
mitochondrial
hsp70
translocating
polymer
ATP
ADP
+ Pi
v
energy-dependent
conformational
change in hsp70
De
d
Figure 16.16: Thermal ratchet model and cross-bridge ratchet model of p
import into the mitochondria. (A) Thermal ratchet model of protein t
membrane
cation. Thermal motion of the polymer in and out of the pore is biased
presence
of Translocation
binding proteins
on Motion
only one
of theacross
barrier.
(B) CrossFigure
16.50:
ratchet.
of aside
polymer
a membrane
model
translocation.
Binding
on one
ofmotion
the barri
isratchet
ratcheted
by theofpresence
of binding
proteinsproteins
which prevent
theside
back
use energy-dependent conformational changes to further ensure th
ofalso
the polymer.
cargo polymer moves in only one direction. (Adapted from B. Alberts
Molecular Biology of the Cell, 4th ed. New York: Garland Science, 2002
Ratchet rates
Ratchet rates
How fast does a Brownian ratchet move?
The rectified Brownian ratchet is a random walk with absorbing and reflecting boundary
conditions.
The ratchet diffuses back and forth until it happens to move a distance L to the right, when the next spring engages.
At this time,
The ratchet starts at position x = 0.
If the free energy drop ! is very high, the ratchet cannot move to the left.
That is, leftward flux is not allowed, so the LH boundary condition (at x = 0) is perfectly reflecting
If the free energy drop ! is finite, there is actually a probability of moving leftward, but it’s small if !/kBT is large
If the ratchet ever reaches position x = L, the next spring is triggered.
If the free energy drop ! is very high, this is a first-passage process like going bust in the gambler’s ruin: the position x = L acts like a
perfectly absorbing boundary.
If the free energy drop ! is finite, the walker simply falls over a large (but not infinite) cliff and is unlikely to ever return.
fL
The unloaded ratchet diffuses
on a piecewise flat free
energy landscape, with drops
! every distance L.
!
G(x)
x
L
with a load f, the
landscape tilts but the
drop height is unchanged
!
L
G(x)
x
Ratchet rates
The movement of the ratchet follows the Smoluchowski equation
∂p
∂
=D
∂t
∂x
!
∂p
f
+
p
∂x kB T
"
(Sign convention has positive f for a retarding force (a load)
In principle we could solve this (at least numerically) for a ratchet starting at x=0 at t=0, but
We haven’t put the energy drops in this equation in a natural way.
Solving the whole PDE is time-consuming and probably overkill anyway, because we’re only interested in the long-term (steadystate) behavior of the ratchet.
Ratchet rates
We’re interested in the steady-state solution
There is a transient that depends on exactly where the motor starts from, but we will only care about the long-term
speed of the motor.
This is not thermodynamic equilibrium because energy ! is released in every step.
If !p/!t=0, then the flux must be constant everywhere:
J = −D
!
"
f
dp
+
p
dx kB T
= const
This is only a first order ODE rather than a second order PDE. Big improvement.
Start with the special case ! ! #
The ratchet can never backstep, so x = L is a perfect absorber.
In mathematical terms, p = 0 at x = L.
The boundary condition at x = 0 is more subtle: molecules coming from the left are reflected, but those coming from the right are
transmitted
This is not the same as simple reflection, so we do not have !p/!x = 0 at x = 0.
In statistical mechanics, this phenomenon is known as Maxwell’s demon
The flux J and probability p will be proportional to the number
of molecules n contained in L.
!
L
We will normalize to a single motor by dividing by n, so that
The solution to the above ODE is not particularly hard,
p(x)dx = 1
0
p(x) = C0 e−f x/kB T + C1
but implementing the boundary conditions to get the proper values of C0 and C1 makes the algebra messy.
Ratchet rates
The solution is
!
"
f!
−f ! (x! −1)
1−e
p(x , f ) =
1 + f ! − ef !
!
x! ≡ x/L
!
f! ≡
fL
kB T
p !, f '
n L
Brownian ratchet probability distribution for different loads
4
f ' = 0.01
f ' = 0.1
f'=1
f ' = 10
f ' = 100
3
2
1
0.2
0.4
0.6
0.8
1.0
! xL
Ratchet rates
The average ratchet speed v satisfies (by definition) v / L = J. It is
v "D!L#
2.0
f !2
D
v=
L ef ! − 1 − f
1.5
f! ≡
fL
kB T
1.0
0.5
0
2
4
6
8
10
f L ! kB T
Notice that
These functions depend only on the ratio f L / kBT. As expected, energies are “large” or “small” compared to thermal energy kBT.
For vanishingly small loads, the ratchet speed maxes out at 2D / L. This is the diffusion limit for a Brownian ratchet.
Even without any load to speak of, it takes some (finite) time to diffuse from x = 0 to x = L.
Specifically, v = L / ", where " is the mean first passage time to L.
None of these expressions depends on !.
The energy drop is used only to rectify motion. As long as it’s large enough, the exact value of ! has no effect on v.
Ratchet rates
For a finite energy drop (renamed "G instead of !)
The basic setup is similar
we have the same ODE describing the steady state: J = −D
and the same normalization condition:
!
L
!
"
f
dp
+
p
dx kB T
p(x)dx = 1
0
p(L)
= e−∆G/kB T
p(0)
but the probability at L is only exponentially small in "G:
−f x/kB T
+ C1
The solution is still of the form p(x) = C0 e
but implementing the boundary conditions is even more algebraically messy.
Eventually,
!
!
!
!
!
1 − ef −g − (1 − e−g )e−f (x −1)
!
! !
!
p(x ; f , g ) = f !
(f + 1) + (1 − f ! )ef ! −g! − ef ! + e−g!
with nondimensional versions of length, load and energy drop:
x! ≡
x
L
f! ≡
Again, all energies are compared to thermal energy kBT
fL
kB T
g! ≡
∆G
kB T
Ratchet rates
probability
6
These solutions look like
g' = 0.01
g' = 0.1
g' = 1
g' = 10
g' = 100
5
4
Probability distribution:
(f ’ = 5)
3
2
1
–1
0
1
2
3
x/L
The limiting cases are simpler:
g’ ! 1 and g’ ! f ’:
p(x! ; f ! ) =
!
"
f!
−f ! (x! −1)
1
−
e
1 + f ! − ef !
This is the same as our previous expression: all dependence on "G has vanished!
With increasing f ’, the distribution gets pushed more and more against the left-hand barrier. Eventually p(0) increases to the point that it
overcomes the barrier and a substantial leftward flux occurs
g’ " 0
p(x! ; f ! ) = 1
With no drop, nothing distinguishes “steps” and we get a spatially uniform distribution.
Ratchet rates
Ratchet speed:
!2
!
−f !
−g !
"
e
f
−e
D
v(f , g ) =
L 1 + e−(f ! +g! ) − (1 − f ! )e−g! − (1 + f ! )e−f !
!
!
v (L/D)
g=0
g=1
g=2
g=4
g=8
2
2
–2
Again, some limits are simpler:
f ’ = g’
g’ ! 1
v=0
v=
f !2
D
!
L ef − 1 − f ! )
!
f’ # 1
v=2
D 1 − e−g
L 1 + e−g!
g’ ! 1 and f ’ # 1
v=2
D 2D !
−
f
L
3L
4
6
8
10
f ' = FL / kBT
Ratchet rates
There will be variability around this mean rate
Since there is a distribution of first passage times, there will be a distribution of distances moved per step
Even when the net velocity is forward, their may be a noticeable backward stepping rate
This variability gives rise to a macroscopic diffusion D* on top of the general drift.
D* (diffusion from step to step) " D (microscopic diffusion within a step)
More on this later.
Summary
"G only prevents backward motion.
Forward motion is by diffusion only
The baseline speed of a Brownian ratchet is step length / first passage time = 2D / L
Once "G is large enough to prevent backsliding, further increasing "G does not increase ratchet velocity.
A Brownian ratchet must have “wasted” free energy
As efficiency goes to 100%, speed goes to zero.
One-state ratchets
One-state ratchets
Many molecular motors move along regular, periodic tracks
A generic cartoon represents (among others) kinesin moving along a microtubule
Microtubules have a 4 µm periodic structure. (The step length is actually 2x4 µm)
This is a “one-state” model:
Motor moves forward (with a force-dependent rate constant k+) or backward (k–), but all bound motors are
equivalent.
In reality, there must be a (free) energy difference between different locations along the track in order to have k+ " k–.
Since all positions along the track are equivalent, the difference does not lie in the motor/track interaction.
All motors burn energy of some kind (often ATP), so the stepping reaction is really
motorn + ATP ! motorn+1 + ADP + Pi
The free energy that allows motion comes from progressive consumption of the cell’s ATP.
All motor / track binding energies are the same, but there is still a slope to the free energy surface of the motor / track system.
We can control the free energy difference between steps (in vitro, anyway) because
∆GATP = ∆G0 + kB T ln
[ADP ][Pi ]
[AT P ]
Figure 16.27: Randomness parameter measured for kinesin under a variety of
conditions. (A) At high (saturating) concentrations of ATP, the randomness is
generally low, indicating that the motor moves at a fairly constant speed. When
the load force is very high, approaching the stall force, randomness increases.
(B) Randomness also increases at very low levels of ATP (i.e. when the binding
rate of ATP may become rate-limiting), and there is a complex relationship
between force and randomness at low ATP concentrations, again with higher
forces causing higher randomness, or more variability in motor speed. Although
this data cannot be fit by a simple one-state model like that described in the
text, models invoking more internal states do a better job of accounting for the
(Adapted from K. Visscher et al., Nature, 400:184, 1999.)
flatmeasurements.
between steps.
One-state ratchets
A real ratchet free energy probably isn’t
It probably (maybe?) looks something like:
k–
Gmotor
Here the reaction coordinate is a real distance
(or maybe step number)
As before, an applied load tilts the landscape.
k+
DGhydrolysis
DG+barrier
DG–barrier
a
In principle we could solve the Smoluchowski equation
for this bumpy landscape
In practice we don’t usually know the landscape
n–1
n
n+1
16.28:
The free energy
a motor
processing
along a filament. n labels
If the rate-limiting step is jumping over a free energyFigure
barrier
(a transition
state),of we
can just
use Arrhenius:
‡
+f l)/kB T
‡
+∆G+f (l−L))/kB T
k+ = Ae−(∆G
k− = Ae−(∆G
the discrete positions of the motor on the filament. The overall tilt of the free
energy surface leads to motion of the motor with an average speed to the right.
linear-free-energy-like
termrates are determined by the free energy barriers
The
forward and backward
between the n and n + 1 state.
k+
= e−(∆G+f L)/kB T
k−
(my "G is "Ghydrolysis in the figure)
One-state ratchets
Recall that a random walk with rate constants k+ and k– maps to the diffusion equation:
k!
- t
k!
- t
i+1
!t
k+
v = L [k+ − k− ]
!t
k+
L2
[k+ + k− ]
D =
2
∗
i
1-(k-+k+)!t
∂p
∂p
∂ p
= D∗ 2 − v
∂t
∂x
∂x
i-1
n
1-(k-+k+)!t
k!
- t
2
n+1
i-1
i
D* is an effective site-to-site macroscopic diffusion coefficient.
Unless "G ! kBT and f L # kBT, there should be a non-negligible backward flux.
We should see occasional backsteps, with increasing probability with increasing load.
i+1
One-state ratchets
We do!
Vol 435|19 May 2005|NATURE
ARTICLES
Kinesin on a microtubule makes discrete steps, but those steps are not always forward:
Figure 1 | Example optical trapping records. Three superimposed records
Two successive experiments on the same molecule are shown. For automated
showing the movement of single kinesin
molecules
towards
stall
force,
over
dwell
time step”,
calculations
and step-averaging,
a t-test step finder was applied to
Carter and Ross, “Mechanics of the kinesin
Nature
435:308 (2005)
time. In record 1 (black), the trap and microtubule remain fixed throughout,
the bead position data. The inset shows the t-test profile for the first part of
and the kinesin walks with 8-nm steps away from the trap centre (dashed
record 1. Steps are defined where the t-value exceeds a preset threshold value
line) to stall force and finally detaches. In record 2 (red), on reaching 4 pN
(dotted line). The located steps are shown offset just above record 1. In all
the microtubule is moved rapidly (upward arrow), pulling the kinesin to
records, detachment events (steps larger than 12 nm recognized by the step
about 14 pN force. In some instances the kinesin responds with processive
finder) are marked with D. Conditions: single kinesin molecules on 560-nm
backward steps to stall force (7–8 pN). More commonly at forces above
polystyrene beads, 1 mM ATP. The trap stiffnesses for the beads in these
10 pN, the kinesin would detach after a few or no backward steps, the bead
records were 0.064, 0.067 and 0.064 pN nm21, respectively. The force scale
nanometry
probe tip
manipulate
glass
e
needl
streptavidin
evanescent field
S1
field of view
One-state ratchets
!"##"$% #& laser
'(#)$"
in
laser out
displacement
actin bundle
whisker
needle
single molecule imaging
Different myosins on actin also step predominantly
forward but occasionally backward
actin bundle
~90°
objective lens
muscle myosin (not very processive)
laser
myosin V (transport molecule: very processive)
(c)
10 nm
0.1 s
low load
35 nm
5.5 nm
5 ms
5.5 nm
10 ms !"#$%& ' !"#$%&&'(% &)%**'+, -. /.#&'+01 #-&%"(%2 3&'+, )4% 23560-%5/ #*)'$56
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7. Manipulation of single myosin molecules using a scanning probe and measurement of substeps of myosin.
#D@? 2;7>H 5>>;2 5 ?@4B>9 <
matic of the measurement. See reference Kitamura et al. (2005) for details. (b) The schematic of the tip of the scanning
5>;4B 5 ?@4B>9 R>5<946 2@6D
A single myosin head was Kitamura
capturedetat
tip of a of
whisker
al, the
“Mechanism
muscle attached
contractionat the glass microneedle. Motion of myosin head is
?=895H @4 6D9 ?69=O?@M9 <
d geometrically. (c) Stepwise
in the of
rising
of themotors
displacements. Some backward steps were observed
based onmovements
stochastic properties
singlephase
actomyosin
Mehta et al, “Myosin-V is a processive
!!+O4< <95?789<946 =89
ated by arrows.
observed in vitro. Biophysics 1:1–19. (2005)
actin-based motor” Nature 400:590 (1999!)
6D56 <K;?@4O( <5K 4;6 <;C
856D98 :54 A@4H 5>;4B 6D9
o a different actin protofilament, resulting in a
Taking advantage of the laser trap method in which
' !"#$%&&'(% &)%**'+, -. /.#&'+01 #-&%"(%2
3&'+,
)4% 23560-%5/
#*)'$56
=%5+ 2C%66
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5 ;3+$)'#+ the
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> "= @A!
6D9 =?97H
the
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motion
of a *9
trapped
bead
step than the!"#$%&
single
myosin step. This change
)"5*7 (8)9 :5/*6% )"5$%& #; &)%**'+, -%45('#3" 5) < /=9 >? "/ 5+2 > "= @A!9
G$'"$6%&H7 &9 K5$LC5"2 &)%**'+, -%45('#3" G5) >7M & 5+2 I &H 5) 4',4 6#52
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torsional motion of actin filaments is measured while
tated by rotating the actin filament (Kitamura
"%&*%$)'(%6.7 BC%66 *%"'#2& !' 5) > "= @A! D) &'+,6% %E*#+%+)'56 &)5)'&)'$&9 5&
@A!7
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the actin is manipulated. The actin filaments are held
005). Given that the properties of an isolated
F32,%2 -. G/%5+Z! ' "! ' # "# [ $ "# H "%/5'+'+, $#+&)5+) 5,5'+&) (5".'+, "# G"%;7 I?H7
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taut vertically by trapping a bead attached at the end by
motor are well described, the next step is to
#D9?9 H565 5>>;2 7? 6; 9?6
a laser while the other end is fixed on the glass surface
and the behaviour of motors in systems where
5.5 nm
10 ms
One-state ratchets
Unfortunately, kinesin is not a simple one-state ratchet
PBoC sets up a straw argument:
The one-state ratchet has only two parameters: forward and backward rate “constants”
From thermodynamics,
k+
= e−∆G/kB T ∝ [ATP]
k−
Either the forward or the backward rate could change with [ATP].
If we put the [ATP] dependence in k_,
1
k− ∝
[ATP]
!
v = a k+ −
0
k−
[ATP]
"
(speed saturates with [ATP])
If we put the [ATP] dependence in k+,
k+ ∝ [ATP]
What does v([ATP]) actually look like?
! 0
"
v = a k+
[ATP] − k−
(speed is linear in [ATP])
One-state ratchets
16.2.Neither
RECTIFIED
BROWNIAN MOTION AND MO
one!
For kinesin, neither pure k– modulation nor pure k+ modulation is consistent with data:
(B)
1.05 pN
3.59 pN
5.63 pN
velocity (nm s–1)
1,000
100
10
1
1
10
100
1,000
ATP concentration (mM)
10,000
velocity (nm s–1), 5 mM ATP
(A)
60
50
40
30
20
10
0
0
1
2
3
4
load (p
linear Speed
at low [ATP] of kinesin
saturates motors
at high [ATP] varying with
Figure 16.33:
applied force. (A) Kinesin speed increases with increa
One-state ratchets
How do we know that cleverer forms for k–([ATP]) and k+([ATP]) wouldn’t fit the data for v(f,
[ATP])?
If transitions between states are controlled by activation barriers, the waiting time for each transition will be
exponentially distributed (with mean time " = 1/k)
One-state and two-state waiting time distributions look fundamentally different:
For a one-state ratchet we expect a dwell time distribution
p(t) ∝ e−t/τ
A two-state ratchet moves when two sequential transitions (each a simple
exponential) have occurred. The total dwell time distribution should look like
p(t) ∝ e−t/τ1 − e−t/τ2
rel. number of event
1.0
0.8
0.6
0.4
0.2
0
2
4
6
8
10
time
0
0
0
2
1
2
0
1
Time (s)
One-state ratchets
Time (s)
e
d
100
Displacement (nm)
50
Displa
(
100
2
300
Kinesin has a bi-exponential
dwell time distribution:
16.2. RECTIFIED
BROWNIAN
MOTION AND MOLECULAR
MOTORS86
Myosin V also is at least two-state
.8 1.0 1.2
ncy
ng
en;
Count
probability
(A)
(21)
it is (at least) a two-state ratchet
50
0
40
20
0
0
200 400 600 800 1,000
Dwell time (ms)
200
(B)
100
0
0
number of events
(26)
140
120
100
80
60
5
10
15
40
Number
of spikes
20
transitions between
one-head-bound and0 0two-head-bound
states.
FRET
0.1
0.2
0.3
0.4
0.5
time (t)
time (s)dotted lines, w
changesFullgreater
than 0.3 are marked by vertical dwell
magenta
disclosure: at high [ATP] the fast step
may become
so fast that
the process
transitions
towards
high
and low FRET states indicated by arrows and
looks one-state.
16.35: Probability
distribution
for waiting
times times
for a two-state
moto
of Figure
arrowheads,
respectively.
d, Distributions
of dwell
in the high-FR
(A) Theoretical
result,
as stated
in eqn. 16.51.
(B)(arrows
Data onindwell
time distr
c) (bottom).
So
state (top) and
between
the high-FRET
spikes
7
fordotted
a single-headed
of myosin V, and
at 10
µM ATP. (Adapted
h bution
.
and
lines showversion
single-exponential
double-exponential
fitsfro
et al., 2005, between
and J. C.
Liao etofal.,
Proc. Nat.
Acad.
Sci., 104:317st
e, Relationship
number
transitions
towards
the high-FRET
d Purcell
Mori et al, “How kinesin waits between steps”,
Nature 450:750 (2007)
One-state ratchets
semble average for 13 complexes in 1 mM NTP, 1 (M PPi. Fit
parameter a ' 1.96 ) 0.99 *
104; +F1/2, ' 23.9 pN, corresponding to # in the range 7.0 to
8.7 bp.
delay
cal w
force,
this c
izes t
chang
tent s
integr
Eq
scalin
vidua
Downloaded from www.s
ships for RNAP. (A) Example of averaged
a
cal waiting times that are independent of
force-velocity relationsingle-molecule F-V curve, deforce, again leads to Eq. 1, with # ' d. For
ships for RNAP. (A) Example of a
rived from the data of Fig. 2, C
this class of models, the distance d charactersingle-molecule F-V curve, deand D. Notches produced by
izes the physical extent of the conformational
rived from the data of Fig. 2, C
transcriptional pausing appear in
change between the competent and incompeunprocessed data (dotted line),
and D. Notches produced by
tent states and need not correspond to an
but are absent in data from edtranscriptional pausing appear in
ited traces (solid line). Experiintegral multiple of base pairs.
unprocessed data (dotted line),
mental conditions: 1 mM NTP, 1
Equation 1 may be used to derive a useful
but are absent in data from ed(M PPi. (B and C) Averaged F-V
scaling formula, permitting data from indiited traces (solid line). Expericurves. Individual curves were
vidual RNAP molecules manifesting differnormalizedNot
as described.
(B) Enmental
1 mM NTP, 1
all motors
are soconditions:
complicated:
semble average for eight com(M PPi. (B and C) Averaged F-V
plexes in 1 mMDNA
NTP, transcription
1 mM PPi;
is driven
by the free
energy
of nucleotide polymerization.
curves.
Individual
curves
were
showing data with (thin solid
(As opposednormalized
to the search as
for the
RNAP binding
site, which is an
described.
(B) Enline) and without (thin dotted
energy-neutral
diffusional
process).
semble average for eight comline) pauses removed. Fit to the
f -v relationship (Appendix, Eq.
plexes in 1 mM NTP, 1 mM PPi;
11, thick lines). Fit parameter
showing data with (thin solid
a ' 5.08 ) 4.11 * 104; +F1/2, '
line) and without (thin dotted
15.8 pN, corresponding
to #linear
in
A one-state
energy model for the RNA polymerase surface ....
line)free
pauses
removed. Fit to the
the range 4.6 to 5.2 bp. (C) Ensemble average for 13 complexf -v relationship (Appendix, Eq.
es in 1 mM NTP, 1 (M PPi. Fit
11, thick lines). Fit parameter
parameter a ' 1.96 ) 0.99 *
a ' 5.08 ) 4.11 * 104; +F1/2, '
104; +F1/2, ' 23.9 pN, corre15.8 pN, corresponding to # in
sponding to # in the range 7.0 to
the range 4.6 to 5.2 bp. (C) En8.7 bp.
Fig. 5. Schematic energy diagram, depicting a
single barrier between two potential wells separated by distance d. The barrier is a distance !
from the initial site, where RNAP (sphere) begins its cycle. The application of an external
load, F, alters the potential by the amounts
indicated (dotted line, initial; solid line, final).
... fits the observed force-velocity curve well
www.sciencemag.org SCIENCE VOL 282 30 OCTOBER 1998
Wang et al, “Force and Velocity Measured for Single
Molecules of RNA Polymerase” Science 282:902-907 (1998)
Fig. 5
905 single
arated
from
gins i
load,
indica
www.sciencemag.org SCIENCE VOL 282 30 OCTOBER 199
Two-state ratchets
Two-state ratchets
Most motor energy landscapes cannot be reduced to a single (distance)
reaction coordinate.
We oversimplified by assuming there was a single transition state: WRONG.
transition state
one-state model
Work (motion against f) and
burning of ATP occur concurrently
one-state cartoon
Two-state ratchets
Make a more detailed model with intermediate
state(s)
856
CHAPTER 16.
DYNAMICS OF MOLECULAR MOTORS
This adds (at least) another reaction coordinate
This allows ATP burning (potentially 3 steps in itself) to be decoupled from physical movement.
If they are completely decoupled, the former will be insensitive to f and the latter will be insensitive to [ATP].
(A)
+
kB
–
kB
1
+
kA
–
kA
0
1
transition state(s)
n–1
n
n+1
+
(B)
+
kA
kB
–
–
kA
0
kB
1
0
transition state(s)
n–1
multi-state model
n
n+1
Figure 16.32: Two-state motor model. (A) The rates for the transitions that
can occur to change the occupancy of internal state 0. The dark icon indicates
the current state of the motor head and the two light icons indicate the two
possible states in the next time step. In the two-state model, the motor head is
constrained to convert from internal state 0 to internal state 1. This can occur
either while the motor remains stationary with respect to the filament (with
rate constant kA ) or while the motor takes a single step backwards (with rate
constant kB ) (B) The rates for the transitions that can occur to change the
occupancy of internal state 1. The motor head can convert to internal state 0,
two-state cartoon
Two-state ratchets
For myosin V, the two-state model with orthogonal energy release and motion looks good:
Fitting the dwell time distributions to biexponentials gives one rate constant (k1) that is completely independent of
[ATP] and one rate constant (k2) that is almost independent of load.
Unfortunately, PBoC chooses to display a dwell time distribution that looks monoexponential because k2 ! k1 at very high [ATP]
PBoC also switches k2 and k1 compared to the source paper.
Fig. 4. Force and ATP dependence for MV-1IQ-S1. Rates shown are derived
from a maximum-likelihood fit of the dwells to two sequential rates. Rates
from different ATP concentrations are plotted on a log-log scale. The fits are
estimates based on one ATP-independent rate (k1) and one saturating ATPdependent rate (k2).
Fig. 4. Force and ATP dependence for MV-1IQ-S1. Rates shown are derived
from a maximum-likelihood fit of the dwells to two sequential rates. Rates
Purcell et al, “A force-dependent from
state controls
the
coordination
of on a log-log scale. The fits are
concentrations
are plotted
myosin Vdifferent
to actin.ATP
This
load dependence
suggests that this
processive myosin V”, PNAS
102:13873–13878
(2005)
estimates
based
on one ATP-independent
and oneissaturating ATPtransition
involves
a mechanical
displacement.rate
This(k1)
transition
dependent
(k2).
hypothesized
to rate
occur
while the actin-bound motor is in its ADP
and actin-bound state, changing from its weakly actin-bound
(A!M!ADP) to its strongly actin-bound (A!M*!ADP) state. A
backward
loadVleads
an increase
the motor’s
population
of
myosin
to to
actin.
This in
load
dependence
suggests
that this
weakly
bound state,
resulting
in the motor’sdisplacement.
dissociation fromThis
the transition is
transition
involves
a
mechanical
!1. Recent studies on unloaded singleactinhypothesized
filament at "1.5
to soccur
while the actin-bound motor is in its ADP
and actin-bound state, changing from its weakly actin-bound
(A!M!ADP) to its strongly actin-bound (A!M*!ADP) state. A
backward load leads to an increase in the motor’s population of
filaments moved with the pointed (2) end leading on surfaces tha
were coated either with E698-2R or with rabbit skeletal musc
heavy meromyosin (HMM). This result agrees with myosin-I
derived motors moving towards the barbed (þ) end. By contras
E698-Q2R attached to an anti-His-tag antibody-coated surfac
moved filaments with the pointed (2) end trailing (Fig. 3a
indicating that E698-Q2R is a pointed (2)-end-directed moto
(see also Supplementary movies). The sliding velocities of labelle
actin filaments obtained following their specific attachment ar
summarized in Fig. 3b. Positive velocity indicates movemen
towards the barbed (þ) end, and negative velocity indicates move
ment towards the pointed (2) end of actin filaments. The wide
Two-state ratchets
Kinesin models generally invoke more than two states
There is still considerable disagreement about how kinesin works.
Two competing models:
A powerstoke model, where ATP binding and/or hydrolysis causes a conformational change, amplified through a lever arm, that
moves kinesin
A diffusion-to-capture model, where ATP binding / ADP release control head attachment to microtubule but motion occurs by
passive diffusion of one head past the other.
Actually, there are hybrid models where ADP release causes a conformation change (twisting of the linker) that biases diffusion .
A principle problem of both models is how to coordinate ATP processing in the two kinesin heads. Assumption is that strain
transferred from the leading head to the trailing (through the linker) is responsible.
In practice, it’s not so easy to distinguish between steps that involve motion of the kinesin heads (which should
couple to f) and steps that only involve conformational change of a lever arm
Since f is actually applied to the attached bead (in vitro) or the attached vesicle or actin filament (in vivo), both head and lever
arm couple to f
These could be distinguished by
applying force directly to the kinesin head (impractical), or
changing the length of the lever arm. This has a clear effect in myosin I;
ambiguous in kinesin.
In fact, you can switch the
direction of myosin I motion by
rotating its lever arm!
Tsiavaliaris et al, “Molecular engineering of a backwardsFiguremoving
3 Direction
of movement
of myosin
constructs. a, Movement of dual-labelled
myosin
motor” Nature
427:558-561
F-actin was visualized using a conventional in vitro motility assay24. The green tip mar
the pointed (2) end. HMM and E698-2R move filaments with their green tips leading
E698-Q2R moves them with their green tips trailing. Panels display the same spatial fie
at the relative times shown on the left. Bar, 10 mm. b, Histograms of the velocities of
F-actin moving on surface-adsorbed HMM or on His-tag antibody-bound E698-2R an
Two-state ratchets
Compliance (springiness) between the motor and load makes it hard to infer motor motion by looking at load
motion:
Volof435|19
2005|NATURE
The position
the beadMay
(as a function
of time) depends on the bead size: it tells you more about drag forces and linker
compliance than about the time course of motor head movement.
bead position (nm)
500 nm bead
800 nm bead
Carter and Ross, “Mechanics of the kinesin step”, Nature 435:308 (2005)
Two-state ratchets
Downloaded from rstb.royalsocietypublishing.org on 13 April 2009
Myosin VI is single-headed yet processive! (though
only with a load attached)
Single molecule measurements T. Yanagida et al. 2131
(b)
cargo
net movement
(a)
slow Brownian motion
ATP binding
displacement (nm)
(i)
400
ATP
myosin VI
(+)
(–)
200
0
–200
dissociation
0
0.3
0.6
0.9
P
AT
– 400
1.2 1.5
ATP
displacement (nm)
(ii)
rapid Brownian motion
300
ADP preferential landing
200
100
ADPPi
0
–100
0
0.5
1.0 1.5
time (s)
2.0
isomerization
or Pi release
conformational change
2.5
(Pi)
ADP(Pi)
power stroke and /
or hopping ?
non-diffusive process diffusive process
36 nm
Figure 9. Movements ofIwaki,
single-headed
myosin VI. (a) Time
courseaofwild-type
displacement
of single-headedmyosin-VI
myosin VI (i) without bead
et al, “Cargo-binding
makes
single-headed
and (ii) with bead. The traces in (i)move
were measured
by the fluorescence
of GFP fused
with myosin head and the trace in
processively.”
Biophys. J.position
90:3643–3652
(2006).
(ii) was measured by the bead position. (b) Stepping model for cargo-bound single-headed myosin VI. Huge cargo keeps the
myosin head near the actin filament and the head repeatedly attaches and detaches to several actin monomers during diffusion.
Finally, the head preferentially lands on the forward binding site.
unbound head and extends to bind to actin. The
wide distribution of the step size is consistent with
away from the actin filament. The preferential binding
to a site in one direction biases the diffusive motion to
Two-state ratchets
This can be modeled as a temporally coupled random walker
Flashing ratchet model:
asymmetric spatial potential (i.e. preferential binding at particular actin or tubulin locations)
free diffusion between episodic binding events
bind at n
release and diffuse freely
rebind at (n–1), n or (n+1)
Two-state ratchets
One-headed kinesins exist too, but they are more random than two-headed kinesin.
!
"
x(t)2 = v 2 t2
!
2
(∆x(t))
"
= 2Dt
Part II