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