applications of non-equilibrium models in biological systems
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Applications of non-equilibrium models in biological systems. Yariv Kafri Technion, Israel. General plan. - PowerPoint PPT PresentationTRANSCRIPT
Applications of non-equilibriummodels in biological systems
Yariv Kafri
Technion, Israel
• 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
General plan
Is it helpful to use non-equilibrium models to understand such systems?(for example, help understand experiments)
D. Nelson D. Lubensky J. LucksM. Prentiss C. Danilowicz R. ConroyV. Coljee J. WeeksJ.-F. Joanny O. Campas K. Zeldovich J. Casademunt,
Why? The central dogma of biology
hard disk
RAM
output device
The central dogma of biology
DNA
transcription
replication
translation
RNA
Protein
study in detailthe machines -the dogma in action
Molecular Motors: complexes of proteins which use chemical energy to perform mechanical work
• Move vesicles
• Replicate DNA
•Produce RNA
• Produce proteins
• Motion of cells
• And much much (much) more
MOVIE
MOVIE
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
ATPATP
ATPATP ``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
~1 micro-meter(your cells 20 micro-meters)
bacteria kinesin
Reynolds number =inertial forces/viscous forces
coefficient ofviscosity
fluid density
(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
swimming inpitch
local thermal equilibrium
motor time scales
equilibrium time scale
Another implication of scale
Scale of nm
• No inertia (diffusive behavior)
• 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 forceopposing motion
K. Vissher, M. J. Schnitzer, S. M. BlockNature 400, 184 (1999)
MOVIE
K. Vissher, M. J. Schnitzer, S. M. BlockNature 400, 184 (1999)
8nmstep size
Extract velocity for different forces velocity force curve
Velocity-Force Curve
stall forceK. Vissher, M. J. Schnitzer, S. M. BlockNature 400, 184 (1999)
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 ~ pNDistances ~ 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 havinginternal states
Basic idea
ATP ADP ++ M P + M
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)
Brownian ratchets
``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
(Julicher, Ajdari, Prost,…1994)
x
Treatment – two coupled Fokker-Plank equations
withor
Get conditions that under
Get conditions that under
• asymmetric potentials
• no detailed balance
effective potential for random walker described by is tilted
diffusion with driftmuch better ratchet
Simple lattice version
Setup modeled
Lattice model
• Two channels for transition, chemical and thermal
• Included external force
describe coarse graineddynamics by effectiveenergy landscape
• No chemical potential difference (have detailed balance)
• Symmetric potential
• Otherwise have an effective tilt diffusion with drift
force xsize of monomer
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 involvedmove alongdisordered substrates(DNA and RNA have given sequences)
Example: RNA polymerase
• Utilizes energy from NTPs
• Moves along DNA making RNA
• very high processivity
• Forces
• Step size 0.34 nm
~15nm
M. Wang et al, Science282, 902 (1998)
~30 bp/s
~15 pN
convex
M. Wang et al, Science282, 902 (1998)
Conventional explanation by model with jumps of varying lengthinto off-pathway state
small, simple
big, complicated
kinesin – moves along microtubuleswhich is a periodic substrate
RNAp – moves along DNAwhich is a disordered substrate
M.E. Fisher PNAS (2001)
Applications of non-equilibriummodels 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 involvedmove alongdisordered substrates(DNA and RNA have given sequences)
Example: RNA polymerase
• Utilizes energy from NTPs
• Moves along DNA making RNA
• very high processivity
• Forces
• Step size 0.34 nm
~15nm
M. Wang et al, Science282, 902 (1998)
~30 bp/s
~15 pN
convex
M. Wang et al, Science282, 902 (1998)
Conventional explanation by model with jumps of varying lengthinto off-pathway state
small, simple
big, complicated
kinesin – moves along microtubuleswhich is a periodic substrate
RNAp – moves along DNAwhich is a disordered substrate
M.E. Fisher PNAS (2001)
Recall
Randomness??
Randomness ???
functions of location along track
for this setup is not
sum over independent random variablesfluctuations which grow as
Effective energy landscape is a random forcing energylandscape
This results only from the use of chemicalenergy coupled with the substrate
effective energy landscape
with chemical energyand disorder
barriers which growas
(diffusion with drift)
no chemical energy(no ATP)
barriers of typical size
(diffusion)
pauses at specific sites
rough energy landscape
•anomalous dynamics•shape of velocity-force curve
•pauses during motion
no chemical bias
with chemical bias
periodic track
heterogeneoustrack
diffusion with drift(-)
Finite time convex curve
Random forcing energy landscapes
toy model
with prob
+ assume directed walk among traps (convection by force vs. trapping)
prob of a barrier or size
time stuck at trap of this size
power law distribution
rare but dominating events
moves between trapsconsider
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
finite time effects ?
convexvelocity force curve !
Possible experimental test of predications
windowdependent effectivevelocity
(MCS)
Single experimental traces
higher force
low 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 RNAusing NTP energy
effective energy landscape
explicit random forcing
random chemical energy + different energy for each base in solution
Size of region for model
Assume effective energy difference has a Gaussian distribution
variancemean
larger variance region of anomalous dynamics larger
For RNA polymerase gives a few pN
DNA polymerase / exonuclease system
Wuite et al Nature, 404, 103 (2000)
model not motor butdsDNA/ssDNA junction
Another candidate system for anomalous dynamics
Wuite et al Nature, 404, 103 (2000)
Exoneclease
Perkins et al, Science, 301, 1914 (2003)
DNA unzipping
3 different DNA’s unzipped @ 15 pN4 different DNA’s unzipped @20pN
Danilowicz et al PNAS 100, 1694 (2003), PRL 93, 078101 (2004).
(only explicit contribution)
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 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)
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 timeslocalization 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
biased motionsignature in imaginarycomponent
eigenfunctions
eigenfunctions
spectrum
delocalized eigenfunction(have a contribution from the velocity)
only change is shift in ``energy’’ exponential decay of probability to beon track
Can disorder modify this picture drastically ?
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
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 disorderIn hopping off
anomalous drift regime
always localized when disorderIn hopping off
Can prove with toy model
Random forcing energy landscapes (Bouchaud et al Ann. Phys. 201, 285 (1990))
toy model
with prob
+ assume directed walk among traps (convection by force vs. trapping)
prob of a barrier or size
time stuck at trap of this size
power law distribution
rare but dominating events
dwell time distribution
In terms of rates
Master equation
Laplace transform
Hopping off
With periodic boundary conditions
need
Interested in long-time limit
average only over W (denote )
assuming nonof probabilitiesto be at one siteare zero!
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-equilibriummodels 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 substrateshave 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 of collective behavior of motors
- experimental work (some discussion will follow) - simple models which capture general behavior - classification
Studies of collective behavior of motors carrying a load
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
Porters vs. Rowers (Leibler and Huse)
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 failsspecifically 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
P.Bassereau group microtubule
Need more than one motor to pullcollective behavior
Ignore the unbinding of motors (comeback later)
How do motors work collectively to pull the tube?
! due to liquid membrane force acts only on motors at the tip !
Typical scenario assumed (lipid vesicles): force shared equally between motors(the presence of other motors does not change anything)
stall force
Can also think of single moleculeexperiment with bead connected
only to leading motoror vesicle experiment
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
A specific limit can be solved exactly (following M. R. Evans 96)
find
stall force
Many motors:
v
f
p=1, q=0.9
N= 1 3 5 …..
v
f
p=1, q=0.1
real kinesin is in this limit!
Functionally already two behavelike many!
(can’t see the curves since so slow)
beyond curves the same
Why slow at large forces?
tries to move backtrying to moveforward
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
• 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
Numerics with interactions
p=1, q=0.1, v=10, u=1
1 25,10
attractive v=0.7 u=0.5repulsive v=1.54 u=1.1(same ratio 1.4)
repulsive v=1.21 u=1.1attractive v=0.55 u=0.5(same ratio 1.1)
p=1 q=0.833… (p/q=1.2)
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)
``rectify’’ Brownian motion
• Two channels for transition, chemical and thermal
• If have detailed balance, no motion
• Must have asymmetry
low Reynolds numbers + local thermal equilibrium motor time scales , equilibrium time scale
(Julicher, Ajdari, Prost,…1994)
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
• 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)
Summary