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PPT – Dynamics of Molecular Motors on Heterogeneous Tracks PowerPoint presentation | free to download - id: 80d44a-ZTNlO

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Applications of non-equilibriummodels 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

The central dogma of biology

replication

DNA

transcription

RNA

translation

Protein

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

ATP

ATP

discrete fuel

ATP

ATP

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)

DNA

microtubules

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

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

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. Block Nature

400, 184 (1999)

MOVIE

8nmstep size

K. Vissher, M. J. Schnitzer, S. M. Block Nature

400, 184 (1999)

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!

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

(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

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 is tilted

diffusion with drift

Simple lattice version

Setup modeled

Lattice model

- Two channels for transition, chemical and

thermal - Included external force

describe coarse graineddynamics by

effectiveenergy landscape

force x size of monomer

- No chemical potential difference (have detailed

balance) - 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

- 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

So far motors which move on a periodic substrate

Not always the case!

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, Science 282, 902 (1998)

30 bp/s

15 pN

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

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

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, Science 282, 902 (1998)

30 bp/s

15 pN

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

Recall

Randomness??

Randomness ???

functions of location along track

for this setup is not

Effective energy landscape is a random forcing

energylandscape

This results only from the use of chemical energy

coupled with the substrate

effective energy landscape

with chemical energy and disorder

pauses at specific sites

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

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

moves between traps

consider

can neglect trapping times larger than

Fluctuations in time

anomalous diffusion

exact solution of model with disorder

Motor model simple enough to solve exactly

Possible experimental test of predications

finite time effects ?

convexvelocity force curve !

windowdependent effectivevelocity

(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 RNAusing NTP energy

random chemical energy different energy for

each base in solution

Size of region for model

Assume effective energy difference has a Gaussian

distribution

variance

mean

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 butdsDNA/ssDNA junction

Wuite et al Nature, 404, 103 (2000)

Exoneclease

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

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

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 motionsignature in imaginary component

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 ?

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 disorder In hopping off

anomalous drift regime

always localized when disorder In hopping off

Can prove with toy model

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

dwell time distribution

In terms of rates

Hopping off

Master equation

Laplace transform

With periodic boundary conditions

assuming non of probabilitiesto be at one

site are zero!

diverges

and

diverge

For infinite processivity get (as numerics show)

system size

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!!

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

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

rigid or elastic coupling between motors

(microtubule)

cant move since it is held back by other motors

protein friction

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

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

microtubule

P.Bassereau group

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 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

- 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

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

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

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

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)

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)

(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

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)