Dynamics of Molecular Motors on Heterogeneous Tracks

1
Applications of non-equilibriummodels in
biological systems
Yariv Kafri Technion, Israel
2
General plan

• Overview of molecular motors (the biological
  system we will consider)
  why study?
  physical conditions?
  experimental studies
• Theoretical models of single motors
  different approaches
  effects of disorder
• Many interacting motors
  different kinds of interactions
  help from driven
  diffusive systems

3
Is it helpful to use non-equilibrium models to
understand such systems? (for example, help
understand experiments)
4
D. Nelson D. Lubensky J. Lucks M. Prentiss C.
Danilowicz R. Conroy V. Coljee J. Weeks J.-F.
Joanny O. Campas K. Zeldovich J. Casademunt,
5
Why? The central dogma of biology
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The central dogma of biology
replication
DNA
transcription
RNA
translation
Protein
7
Molecular Motors complexes of proteins which
use chemical energy to perform mechanical work

• Move vesicles
• Replicate DNA

8

• Produce RNA
• Produce proteins
• Motion of cells
• And much much (much) more

MOVIE
MOVIE
9
What do motors need to function? (basics for
modeling)
1. Fuel (supplies a chemical potential gradient)
These vary! (examples before) But for the
systems we will discuss typically the following
holds (Kinesin)
ATP
ATP
ATP
ATP
discrete fuel
ATP
ATP
10
How much energy released?
created in cell or in experiment
for ATP gives about
Other sources GTP,UTP,CTP (no TTP) about the
same
11
What do motors need to function?
2. Track
Again these vary! (examples before)
DNA
microtubules
actin (myosin motors), circular tracks..one
dimensional
12
Scales
bacteria
kinesin
1 micro-meter (your cells 20 micro-meters)
fluid density
Reynolds number inertial forces/viscous forces
coefficient of viscosity
13
Another implication of scale
local thermal equilibrium motor time scales

equilibrium time scale

• No inertia (diffusive behavior)
• Can assume local thermal equilibrium (namely,
  transition rates obey a local version of
  detailed balance in a few slides)

Scale of nm
14
Experimental Technique(s)
15
Single molecule experiments
Study behavior of single motor under an external
perturbation (force)

• deduce characteristics (e.g. force exerted)
• understand chemical cycle better

tweezers exert forceopposing motion
K. Vissher, M. J. Schnitzer, S. M. Block Nature
400, 184 (1999)
MOVIE
16
8nmstep size
K. Vissher, M. J. Schnitzer, S. M. Block Nature
400, 184 (1999)
17
Velocity-Force Curve
K. Vissher, M. J. Schnitzer, S. M. Block Nature
400, 184 (1999)
stall force
The stall force is the force exerted by the motor
18
Kinesin

• Utilizes ATP energy
• Moves along microtubules, monomer size 8 nm
  (always in a certain direction)
• Processivity about 1 micron ( 100 steps)
• Exerts a force of about 6-7 pN

19
Forces pN Distances nM
Thermal fluctuations are important!
20
Theory How do the motors use chemical energy
to function?
two approaches
Brownian Ratchets Powerstroke
Both rely on the motor havinginternal states
Basic idea



ATP
ADP
M
P
M
21
Powerstroke models (Huxley, 1957)
Idea some internal spring is activated using
chemical energy
description in terms of a biased randomwalker
Can complicate by putting in many internal
state(Fisher and Kolomeisky on Kinesin)
22
Brownian ratchets
(Julicher, Ajdari, Prost,1994)
rectify Brownian motion

• Two channels for transition, chemical and
  thermal
• If have detailed balance, no
  motion
• Must have asymmetry
• Must have rates which depend on the location
  on the track

x
23
Treatment two coupled Fokker-Plank equations
with
or
Get conditions that under
24
Get conditions that under

• asymmetric potentials
• no detailed balance

effective potential for random walker described
by is tilted
diffusion with drift
25
Simple lattice version
Setup modeled
Lattice model
26

• Two channels for transition, chemical and
  thermal
• Included external force

describe coarse graineddynamics by
effectiveenergy landscape
27
force x size of monomer

• No chemical potential difference (have detailed
  balance)
• Symmetric potential
• Otherwise have an effective tilt

diffusion with drift
28
Simple enough that can calculate velocity and
diffusion constant
diffusion with drift
29
Back to ratchets vs. powerstroke
?
Personal opinion ratchet more generic and can
be made to behave as
powerstroke
30
Short Summary

• Molecular motors are complexes of proteins
  which use chemical energy to perform mechanical
  work.
• Single molecule experiments provide data on
  traces of motors giving information such as
  stall force
  velocity
  step size ..
  ..
• Models including internal states provide a
  justification for treating the motors as biased
  random walkers

31
So far motors which move on a periodic substrate
Not always the case!
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Example RNA polymerase

• Utilizes energy from NTPs
• Moves along DNA making RNA
• very high processivity
• Forces
• Step size 0.34 nm

15nm
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M. Wang et al, Science 282, 902 (1998)
30 bp/s
15 pN
34
Conventional explanation by model with jumps of
varying lengthinto off-pathway state
M.E. Fisher PNAS (2001)
kinesin moves along microtubuleswhich is a
periodic substrate
RNAp moves along DNAwhich is a disordered
substrate
35
Applications of non-equilibriummodels in
biological systems
Yariv Kafri Technion, Israel
36
Yesterday
Molecular motors on periodic tracks are described
by biased random walkers
in one hour
37
Many motors do not move on a periodic substrate
38
Example RNA polymerase

• Utilizes energy from NTPs
• Moves along DNA making RNA
• very high processivity
• Forces
• Step size 0.34 nm

15nm
39
M. Wang et al, Science 282, 902 (1998)
30 bp/s
15 pN
40
Conventional explanation by model with jumps of
varying lengthinto off-pathway state
M.E. Fisher PNAS (2001)
kinesin moves along microtubuleswhich is a
periodic substrate
RNAp moves along DNAwhich is a disordered
substrate
41
Recall
Randomness??
42
Randomness ???
functions of location along track
for this setup is not
43
Effective energy landscape is a random forcing
energylandscape
This results only from the use of chemical energy
coupled with the substrate
44
effective energy landscape
with chemical energy and disorder
pauses at specific sites
45
rough energy landscape

• anomalous dynamics
• shape of velocity-force curve
• pauses during motion

no chemical bias
diffusion with drift
(-)
heterogeneoustrack
with chemical bias
periodic track
Finite time convex curve
46
Random forcing energy landscapes
toy model
assume directed walk among traps (convection
by force vs. trapping)
with prob
prob of a barrier or size
rare but dominating events
time stuck at trap of this size
power law distribution
47
moves between traps
consider
can neglect trapping times larger than
48
Fluctuations in time
anomalous diffusion
49
exact solution of model with disorder
50
Motor model simple enough to solve exactly
51
Possible experimental test of predications
finite time effects ?
convexvelocity force curve !
windowdependent effectivevelocity
(MCS)
52
Single experimental traces
low force
higher force
53
Phase diagram for anomalous velocity
Important how large is this region in
experiments? (say RNA polymerase)
54
Before other sources of random forcing
RNA polymerase
produces RNAusing NTP energy
random chemical energy different energy for
each base in solution
55
Size of region for model
Assume effective energy difference has a Gaussian
distribution
variance
mean
For RNA polymerase gives a few pN
56
Another candidate system for anomalous dynamics
DNA polymerase / exonuclease system
Wuite et al Nature, 404, 103 (2000)
model not motor butdsDNA/ssDNA junction
57
Wuite et al Nature, 404, 103 (2000)
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Exoneclease
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DNA unzipping
(only explicit contribution)
Danilowicz et al PNAS 100, 1694 (2003), PRL 93,
078101 (2004).
3 different DNAs unzipped _at_ 15 pN
4 different DNAs unzipped _at_20pN
60
Using very naïve model can predict rather well
location of pause points
61
Summary of Infinite Processivity

• Using chemical energy leads to a rough energy
  landscape
• Anomalous dynamics near the stall force with a
  window dependent velocity
• Power law distribution of pause times
• It seems that the general role for biological
  systems is disorder implies
  random forcing

62
So far motors never fell from the
track (infinitely processive motors)What are
the implications of falling off?
63
(simple arguments, real results through analysis
of spectra of evolution operator and toy model)
Allow motor to leave track Influence on
dynamics?
Discuss in steps

• Homogeneous track and rates for leaving track
• Homogeneous track and heterogeneous rates for
  leaving track
• Heterogeneous track and rates for leaving track

64
Homogeneous track and rates for leaving track
diffusion with drift with homogeneous falling off
rates
probability to stay on track
motor moves until it falls off At long times
the probability to find motors on
specificlocation along it is equal.
(experiment put motors at random on track and
look at probability to find them
as a function of time averaging over
results from many motors)
65
Homogeneous track and heterogeneous rates for
leaving track
diffusion with drift with heterogeneous falling
off rates
66
Long times
small disorder in hopping off rates probability
profile
(decaying in time)
large disorder in hopping off rates probability
profile
(decaying in time stalled)
67
Possible to see transitions through the spectrum
of the evolution operator
using matrix for motor model with hopping off
included
For periodic boundary conditions and periodic
track no hopping off
eigenfunctions
biased motionsignature in imaginary component
68
eigenfunctions
spectrum
exponential decay of probability to beon track
only change is shift in energy
delocalized eigenfunction(have a contribution
from the velocity)
Can disorder modify this picture drastically ?
69
add hopping off rates
study the eigenvalue spectrum
imaginary component carries
current or delocalized
no imaginary component no
current or localized
Just look at spectrum
70
Possible to see transitions through the spectrum
of the evolution operator
diffusion and drift regime
no hopping off
71
Heterogeneous track and rates for leaving track
anomalous drift regime
always localized when disorder In hopping off
72
anomalous drift regime
always localized when disorder In hopping off
Can prove with toy model
73
Random forcing energy landscapes
(Bouchaud et al Ann. Phys. 201, 285 (1990))
toy model
assume directed walk among traps (convection
by force vs. trapping)
with prob
prob of a barrier or size
rare but dominating events
time stuck at trap of this size
power law distribution
74
dwell time distribution
In terms of rates
Hopping off
Master equation
Laplace transform
75
With periodic boundary conditions
assuming non of probabilitiesto be at one
site are zero!
76
diverges
and
diverge
77
For infinite processivity get (as numerics show)

system size
78
Falling off?
Simple model, two rates for falling off
with prob
with prob
need imaginary part of eigenvalue to solve (real
part from higher orders)
look at n0
decay can not be faster
no solution!!
79
Implies that at least one of sites has zero
probability
Can show that only purely real eigenvalue in this
case
and
exponentially localized at particular site
80
Heterogeneous track and rates for leaving track
Moving very slowly Analysis shows always
localized!!!
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Summary of Finite Processivity

• Disorder in hopping off rates leads to a
  localization transition
• When dynamics are anomalous always localized

82
Medium Summary

• Simple model for Brownian ratchets
• Exactly solvable with and without disorder
• Disorder induces a rough energy landscape
• Anomalous dynamics near the stall force, shape
  of velocity force curve pauses
• Hopping off of motors from tracks lead to
  localization of long lasting motors (always in
  anomalous dynamics region)

83
Applications of non-equilibriummodels in
biological systems
Yariv Kafri Technion, Israel
84
Past two lectures

• Molecular motors on periodic tracks are described
  by
• biased random walkers
• To study molecular motors on disordered
  substrateshave to know about random forcing
  energy landscapes

85
Next Systems with many motors
86
Work on Molecular Motors

• Experiments and models for single motors -
  single molecule experiments - general
  mechanisms for generating motion - attempts to
  understand details of a specific motor
• Studies of collective behavior of motors -
  experimental work (some discussion will follow)
  - simple models which capture general behavior
  - classification

87
Porters vs. Rowers (Leibler and Huse)
Processive Motors
Non-processive Motors
work best is small groups (e.g. kinesin)
work in large (but finite) groups (e.g. myosin II)
Porters
Rowers
88
rigid or elastic coupling between motors
(microtubule)
cant move since it is held back by other motors
protein friction
89
Much work under this classification (e.g.
Julicher and Prost, Vilfan and Frey .)
Sometimes the assumptions which underlie the
classification failsspecifically the rigid
coupling
Examples which will be discussed in this talk
motors pulling a liquid membranes tube
90
Motors carrying a vesicle
vesicle can be carried by different numbers of
motors
To leading approximation radius of vesicle so
large that essentially flat for motors
91
Outline of remaining part

• Discuss tube experiment
• Define simple model (consider only processive
  motors)
• Velocity force curves
• Effects of interactions (short ranged) between
  motors (possibility of detecting the
  interactions through such or similar
  experiments)
• Detachment effects?
• Origin of interactions between motors
  (generically expect interactions due to internal
  states)
• Summary

92
Experimental system Tube extraction by molecular
motors
microtubule
P.Bassereau group
93
Ignore the unbinding of motors (come back later)
How do motors work collectively to pull the tube?
! due to liquid membrane force acts only on
motors at the tip !
94
Can also think of single moleculeexperiment with
bead connectedonly to leading motoror vesicle
experiment
Typical scenario assumed (lipid vesicles) force
shared equally between motors(the presence of
other motors does not change anything)
stall force
95

• Relation used for
• Modeling of collective behavior
• Extracting the number of motors pulling a
  vesicle
• Extracting the force the motors exert
• Analyzing histogram of velocities (similar to
  above)

Is this reasonable?
96
Model as a driven diffusive system (particles
hopping on a lattice)

• - index labeling the particle
• total number of motors
• allow interactions between motors
• assume force acting only on front motor

97
force acting only on leading motor
rest of motors
98
Look at two motors
Solving master equation (as long as
have a bound state of particles)
99
stall force?
only when
(can show that this is general for any number of
motors)
100
stall force depends only on the ratio (u and v
could be very small (large) but with a much
larger (smaller) stall force)
stall force smaller than
stall force larger than
101
Velocity-Force Curve
black single motor
Possible indication for attractive interactions
between motors
102
Many motors
A specific limit can be solved exactly (following
M. R. Evans 96)
find
stall force
103
v
p1, q0.9
beyond curves the same
N 1 3 5 ..
f
v
p1, q0.1
real kinesin is in this limit! Functionally
already two behave like many!
(cant see the curves since so slow)
f
104
Why slow at large forces?
trying to moveforward
tries to move back
motion controlled by propagation of a hold from
one side to another
exp small in force
105
stall force

• In the limit discussed easy to show
• In general can show that when
  there is detailed balance at
  stall force. Always have

Corrections due to interactions
when ratios are not equal no current but no
detailed balanceinteractions break detailed
balance
106
Numerics with interactions
5,10
2
1
attractive v0.7 u0.5repulsive v1.54
u1.1 (same ratio 1.4)
p1, q0.1, v10, u1
repulsive v1.21 u1.1attractive v0.55
u0.5(same ratio 1.1)
p1 q0.833 (p/q1.2)
107
Falling off from the track?

• Expect uniform for motors behind leading one
• Leading one experiences a force which is not
  completely parallel to direction of motion
  detachment rate increases
  exponentially
  with f

108
Falling off from the track?

• Homogeneous density of detached
• Includes the effect that detachment of leading
  one grows exponentially with f

109
Mini Summary

• Simple driven diffusive system suggests that
  collective behavior of motors pulling a tube
  is different than simple picture
• Measurement of velocity force curves for many
  motors might (at least) indicate the nature of
  the interactions between the motors

Where can the interaction come from? (should
they be expected generically??)
110
Models of molecular motors (ratchets)
(Julicher, Ajdari, Prost,1994)
low Reynolds numbers
local thermal equilibrium
motor
time scales ,
equilibrium
time scale
rectify Brownian motion

• Two channels for transition, chemical and
  thermal
• If have detailed balance, no
  motion
• Must have asymmetry

x
111
simulate with only excluded volume interactions
between the particles
Internal states of the motor lead to repulsive
interactions between the motors
112
Attractive interactions ?

• Possibly by exploring more the phase space of
  parameters in the two state model?
• Or even simpler ATP binding site is obscured by
  near motor

113
Summary

• Simple driven diffusive system suggests that
  collective behavior of motors pulling a
  tube/vesicle is different than simple picture
• Measurement of velocity force curves for many
  motors might (at least) indicate the nature of
  the interactions between the motors
• Internal states of molecular motors induce
  effective repulsive or attractive interactions
  (on top of others that may be present)