Transcript Dynamics of Molecular Motors on Heterogeneous Tracks

Applications of non-equilibrium
models in biological systems
Yariv Kafri
Technion, Israel
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
Is it helpful to use non-equilibrium models to understand such systems?
(for example, help understand experiments)
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,
Why? The central dogma of biology
hard disk
RAM
output device
The central dogma of biology
replication
DNA
transcription
RNA
translation
Protein
study in detail
the machines the dogma in
action
Molecular Motors:
complexes of proteins which use chemical energy to perform mechanical work
• Move vesicles
• Replicate DNA
•Produce RNA
MOVIE
• Produce proteins
MOVIE
• Motion of cells
• And much much (much) more
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
ATP
ATP
``discrete’’ fuel
How much energy released?
created in cell or in experiment
for ATP gives about ~
Other sources GTP,UTP,CTP (no TTP)
about the same
What do motors need to function?
2. Track
Again these vary! (examples before)
microtubules
DNA
actin (myosin motors), circular tracks……..one dimensional
Scales
bacteria
kinesin
~1 micro-meter
(your cells 20 micro-meters)
fluid density
Reynolds number =inertial forces/viscous forces
coefficient of
viscosity
swimming in
pitch
(started in 1927 drop no. 9)
The pitch drop experiment (Ig Noble 2005)
R. Edgeworth, B.J. Dalton and T. Parnell
Eur. J. Phys (1984) 198-200
Another implication of scale
local thermal equilibrium
motor time scales
equilibrium time scale
• No inertia (diffusive behavior)
Scale of nm
• Can assume local thermal equilibrium
(namely, transition rates obey a local
version of detailed balance – in a few
slides)
Experimental Technique(s)
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 force
opposing motion
K. Vissher, M. J. Schnitzer, S. M. Block
Nature 400, 184 (1999)
MOVIE
8nm
step size
K. Vissher, M. J. Schnitzer, S. M. Block
Nature 400, 184 (1999)
Extract velocity for different forces
velocity force curve
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
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
Forces ~ pN
Distances ~ nM
Thermal fluctuations are important!
Ingredients for modeling:
• No inertia (some sort of biased brownian motion)
• Noisy (both temperature and discrete fuel)
• Safe to assume local thermal equilibrium
Theory: How do the motors use chemical energy
to function?
``two approaches’’
Brownian Ratchets
Powerstroke
Both rely on the motor having
internal states
Basic idea
ATP
+ M
ADP + P
+
M
Powerstroke models (Huxley, 1957)
Idea: some internal ``spring’’ is activated using chemical energy
description in terms of a biased random
walker
Can complicate by putting in many internal state
(Fisher and Kolomeisky on Kinesin)
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
x
• Must have rates which depend
on the location on the track
Treatment – two coupled Fokker-Plank equations
with
or
Get conditions that under
Get conditions that under
• asymmetric potentials
• no detailed balance
effective potential for random walker described by
diffusion with drift
much better ratchet
is tilted
Simple lattice version
Setup modeled
Lattice model
• Two channels for transition,
chemical and thermal
• Included external force
describe coarse grained
dynamics by effective
energy landscape
• No chemical potential difference (have detailed balance)
force x
size of
monomer
• Symmetric potential
• Otherwise have an effective tilt
diffusion with drift
Simple enough that can calculate velocity and diffusion constant
diffusion with drift
Back to ratchets vs. powerstroke
?
Personal opinion: ratchet more generic and can be
made to behave as powerstroke
Short Summary:
1. Molecular motors are complexes of proteins which
use chemical energy to perform mechanical work.
2. Single molecule experiments provide data on traces
of motors giving information such as:
stall force
velocity
step size
…..
…..
3. Models including internal states provide a justification
for treating the motors as biased random walkers
So far motors which move on a periodic substrate
Not always the case!
Motors involved
move along
disordered
substrates
(DNA and RNA
have given
sequences)
Example: RNA polymerase
• Utilizes energy from NTPs
~15nm
• Moves along DNA making RNA
• very high processivity
• Forces
• Step size 0.34 nm
M. Wang et al, Science
282, 902 (1998)
M. Wang et al, Science
282, 902 (1998)
~30 bp/s
convex
~15 pN
Conventional explanation by model with jumps of varying length
into off-pathway state
M.E. Fisher PNAS (2001)
kinesin – moves along microtubules
which is a periodic substrate
small, simple
RNAp – moves along DNA
which is a disordered substrate
big, complicated
Applications of non-equilibrium
models in biological systems
Yariv Kafri
Technion, Israel
Yesterday:
Molecular motors on periodic tracks are described by
biased random walkers
in one hour
Many motors do not move on a periodic substrate
Motors involved
move along
disordered
substrates
(DNA and RNA
have given
sequences)
Example: RNA polymerase
• Utilizes energy from NTPs
~15nm
• Moves along DNA making RNA
• very high processivity
• Forces
• Step size 0.34 nm
M. Wang et al, Science
282, 902 (1998)
M. Wang et al, Science
282, 902 (1998)
~30 bp/s
convex
~15 pN
Conventional explanation by model with jumps of varying length
into off-pathway state
M.E. Fisher PNAS (2001)
kinesin – moves along microtubules
which is a periodic substrate
small, simple
RNAp – moves along DNA
which is a disordered substrate
big, complicated
Recall
Randomness??
Randomness ???
functions of location along track
for this setup is not
sum over independent random variables
fluctuations which grow as
Effective energy landscape is a random forcing energy
landscape
This results only from the use of chemical
energy coupled with the substrate
effective energy landscape
no chemical energy
(no ATP)
barriers of typical
size
(diffusion)
with chemical energy
and disorder
barriers which grow
as
(diffusion with drift)
pauses at specific sites
rough energy landscape
•anomalous dynamics
•shape of velocity-force curve
•pauses during motion
diffusion with drift
no chemical bias
(-)
heterogeneous
track
with chemical bias
periodic track
Finite time
convex curve
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
consider
moves between traps
can neglect trapping times larger than
Total time
Subballistic
Fluctuations in time
anomalous diffusion
exact solution of model with disorder
Subballistic
Motor model simple enough to solve exactly
Possible experimental test of predications
finite time effects ?
convex
velocity force
curve !
window
dependent effective
velocity
(MCS)
Single experimental traces
low force
higher force
``Phase diagram’’ for anomalous velocity
Important: how large is this region in experiments?
(say RNA polymerase)
Before: other sources of random forcing
RNA polymerase
produces RNA
using NTP energy
random chemical energy + different energy for each base in solution
effective energy landscape
explicit random forcing
Size of region for model
Assume effective energy difference has a Gaussian distribution
mean
larger variance
variance
region of anomalous dynamics larger
For RNA polymerase gives a few pN
Another candidate system for anomalous dynamics
DNA polymerase / exonuclease system
Wuite et al Nature, 404, 103 (2000)
model not motor but
dsDNA/ssDNA junction
Wuite et al Nature, 404, 103 (2000)
Exoneclease
Perkins et al, Science, 301, 1914 (2003)
DNA unzipping
(only explicit contribution)
Danilowicz et al PNAS 100, 1694 (2003), PRL 93, 078101 (2004).
3 different DNA’s unzipped @ 15 pN
4 different DNA’s unzipped @20pN
Using very naïve model can predict rather well location of pause points
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
So far: motors never fell from the track
(infinitely processive motors)
What are the implications of falling off?
(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
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 specific
location 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)
Homogeneous track and heterogeneous rates for leaving track
diffusion with drift with heterogeneous
falling off rates
change
have a transition between two behaviors at large times
localization transition
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)
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 motion
signature in imaginary
component
eigenfunctions
spectrum
only change is shift in ``energy’’
exponential decay of probability to be
on track
delocalized eigenfunction
(have a contribution from the velocity)
Can disorder modify this picture drastically ?
add hopping off rates
study the eigenvalue spectrum
imaginary component
no imaginary component
carries current or delocalized
no current or localized
Just look at spectrum
Possible to see transitions through the spectrum of the evolution operator
diffusion and drift regime
no hopping off
Heterogeneous track and rates for leaving track
anomalous drift regime
always localized when disorder
In hopping off
anomalous drift regime
always localized when disorder
In hopping off
Can prove with toy model
Random forcing energy landscapes
toy model
(Bouchaud et al Ann. Phys. 201, 285 (1990))
+ 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
dwell time distribution
In terms of rates
Master equation
Laplace transform
Hopping off
With periodic boundary conditions
assuming non
of probabilities
to be at one site
are zero!
need
Interested in long-time limit
average only over W (denote
)
diverges
and
diverge
For infinite processivity get (as numerics show)
+
system size
using previous results:
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 n=0 :
decay can not be faster
no solution!!
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
Heterogeneous track and rates for leaving track
Moving very slowly
Analysis shows always localized!!!
Summary of Finite Processivity
• Disorder in hopping off rates leads to a localization transition
• When dynamics are anomalous – always localized
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)
Applications of non-equilibrium
models in biological systems
Yariv Kafri
Technion, Israel
Past two lectures:
•Molecular motors on periodic tracks are described by
biased random walkers
• To study molecular motors on disordered substrates
have to know about random forcing energy landscapes
Next: Systems with many motors
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
ofof
collective
behavior
of motors
Studies
collective
behavior
of motors carrying a load
- experimental work (some discussion will follow)
- simple models which capture general behavior
- classification
Porters vs. Rowers (Leibler and Huse)
Processive Motors
work best is small groups
(e.g. kinesin)
Porters
Non-processive Motors
work in large (but finite) groups
(e.g. myosin II)
Rowers
rigid or elastic coupling between motors (microtubule)
can’t move since it is held back by other motors = protein friction
can only move if most of the other motors are unbound
Much work under this classification
(e.g. Julicher and Prost, Vilfan and Frey ….)
Sometimes the assumptions which underlie the classification fails
specifically the rigid coupling
Examples which will be discussed in this talk:
motors pulling a liquid membranes tube
weakly coupled processive motors
very different behavior
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
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
Experimental system: Tube extraction by molecular motors
Need more than one motor to pull
collective behavior
P.Bassereau group
microtubule
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 !
Can also think of single molecule
experiment with bead connected
only to leading motor
or vesicle experiment
Typical scenario assumed (lipid vesicles): force shared equally between motors
(the presence of other motors does not change anything)
stall force
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?
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
force acting only on leading motor
rest of motors
Look at two motors
Solving master equation (as long as
have a bound state of particles)
stall force?
only when
(can show that this is general for any number of motors)
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
Velocity-Force Curve
black single motor
Possible indication for attractive interactions between motors
Many motors:
A specific limit can be solved exactly (following M. R. Evans 96)
find
stall force
v
p=1, q=0.9
beyond curves the same
N= 1
3
5 …..
f
v
p=1, q=0.1
real kinesin is in this limit!
Functionally already two behave
like many!
(can’t see the curves since so slow)
f
Why slow at large forces?
trying to move
forward
tries to move back
motion controlled by propagation of a hold from one side to another
exp small in force
stall force
• In the limit discussed easy to show
Corrections due to interactions
• In general can show that when
there is detailed
balance at stall force. Always have
when ratios are not equal no current but no detailed balance
interactions break detailed balance
Numerics with interactions
5,10
1
attractive v=0.7 u=0.5
repulsive v=1.54 u=1.1
(same ratio 1.4)
repulsive v=1.21 u=1.1
attractive v=0.55 u=0.5
(same ratio 1.1)
p=1 q=0.833… (p/q=1.2)
2
p=1, q=0.1, v=10, u=1
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
Falling off from the track?
• Homogeneous density of detached
• Includes the effect that detachment of leading one grows exponentially
with f
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??)
Models of molecular motors (ratchets)
low Reynolds numbers
+
(Julicher, Ajdari, Prost,…1994)
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
simulate with only excluded volume interactions between the particles
Internal states of the motor lead to ``repulsive interactions between the motors’’
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
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)