Movements of Molecular Motors:

Random Walks and Traffic Phenomena

Theo Nieuwenhuizen Stefan KlumppReinhard Lipowsky

Motor traffic

Traffic problems: unbinding, diffusive excursions traffic jams coordination of traffic

Overview

• Molecular motors

• Single motors: random walks on pinning line, in fluid

• Cooperative traffic phenomena: traffic jams, phase transitions

1) Concentration profiles in closed systems2) Boundary-induced phase transitions3) Two species of motors

Molecular motors

Kinesin

Microtubule

• proteins which convert chemical energy into directed movements

• movements along filaments of cytoskeleton

• various functions in vivo: transport, internal organization of the cell, cell division, ...

• processive motors: large distances

Hirokawa 1998

microtubule +neurofilaments

cargo

In vitro-experiments

Measurements of transport properties of single motor molecules: velocity: ~ µm/sec = 0.1 m/month step size ~ 10 nm, step time ~ 10 ms ...

Janina Beeg

In vitro-experiments

Measurements of transport properties of single motor molecules: velocity: ~ µm/sec step size ~ 10 nm ...

Vale & Pollock in Alberts et al. (1999)

Modeling – separation of scales

Directed walk along filament~ 1 µm ~ 100 steps

Talk Imre Derenyi

Random walks: on filaments, in fluid: unbinding - binding many µm – mm

This talk

(I) (II) (III)

Molecular dynamics of single step ~ 10 nm

Vale & Milligan (2000) Visscher et al. (1999)

Talk Dean Astumian

Lattice models for the random walks of molecular motors

• biased random walk along a filament

• unbound motors: symmetric random walk

• detachment rate & sticking probability ad

simple and generic model

parameters can be adapted to specific motors

motor-motor interactions can be included (hard core)

Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, 108101 (2001)

Independent motors, d=2, full space

In bulk:

On line:

Above line:

Below line:

Full space: Exact solution via Fourier-Laplace transform Useful to test numerical routines

Initial condition: motors start at t=0 at origin on the line

speed on line of one motor: 1bv

Full space: Fourier-Laplace transform techniques apply

Integration over q yields = Fourier-Laplace transform on line:),( srPb

Nieuwenhuizen, Klumpp, Lipowsky, Europhys Lett 58 (2002) 468

Phys Rev E 69 (2004) 061911& June 15, 2004 issue of Virtual Journal of Biological Physics Research

Results for d=2 at large t

survival fraction

average spead

diffusion coefficient: enhanced

Spatio-temporal distribution on line: scaling form

Unbound motors in d=2

average spead

Diffusion coefficients: longitudinal enhanced

transversal normal

Random walks of single motors in open compartments

Half space Slab Open tube

Behavior on large scales:many cycles of binding/ unbinding

How fast do motors advance ?

Effective drift velocity

Tube:

Slab, 2d:

Half space, 3d:

const. ~v

t1/ ~v

t1/ ~v

Tube

Slab

Half space

Behavior on large scales

ad/ /1

Effective velocity: Scaling

Tube:

)/(

v

)/(1

vv v

ad

b

ad

b

ubb

bb

tt

t

Diffusive length scale: tDL ub~

Slab:tDh ub

adb

v ~v

tDhhL ub~~

Half space:tDub

adb

v ~v

tDL ub

2 ~~

Average position

Tube

Half space

Slab

Tube: (‚normal‘ drift)

Slab:

Half space:

tx ~

tx ~

tx ln ~

‚anomalous‘ drift

• Scaling arguments• analytical solutions (Fourier-Laplace transforms)

b

Nieuwenhuizen, Klumpp, Lipowsky, EPL 58,468 (2002)

Exclusion and traffic jams

Mutual exclusion of motors from binding sitesclearly demonstrated in decoration experiments

simple exclusion: no steps to occupied binding sitesmovement slowed down (molecular traffic jam)velocity:

1) Concentration profiles in closed compartments

Stationary state: Balance of directed current of bound motors and diffusive current of unbound motors

ububbbb ρx

)ρ1(ρv

D

Motor-filament binding/ unbinding:

bbubadbbb )1()1(v x

Local accumulation of motors Exclusion effects: reduced binding + reduced velocity

Concentration profiles and average current

„traffic jam“

# motors within tube

Average bound current

• # motors small: localization at filament end• # motors large: filament crowded

Density ofbound motors

Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, 108101 (2001)

• Intermediate # motors: coexistence of a jammed region and a low density region, maximal current

exponential growth

2) Boundary-induced phase transitions in open tube systems

• Tube coupled to reservoirs • Exclusion interactions

• Variation of the motor concentration in the reservoirs boundary-induced phase transitions • Dynamics along the filament: Asymmetric simple exclusion process (ASEP)

Periodic boundary conditions

exactly solvable in mean field: bound and unbound densities constant radial equilibrium:

current

)1()1( bubadubb

)1(v bbb J Current

Number of motors within the tube

Open tubes

2/1)0(b

4/vbJ

far from the boundaries: plateau with radial equilibrium

low density (LD): high density (HD): maximal current (MC):

2/1)0(b 2/1)0(

b

Transitions: LD-HD discontinuous LD/HD-MC continuous Klumpp & Lipowsky, J. Stat. Phys. 113, 233 (2003)

Phase diagrams

4

v

L

/ badub

2

DRCondition for the presence of the MC phase:

LD

HD

MC

Radial equilibrium at the boundaries

depending on the choice of boundary conditions

Motors diffuse in/out

HD

LD

3) Two species of motors

bound motor stimulates binding of further motorseffective interaction mediated via the filament

Experimental indications for cooperative binding of motors to a filament

Vilfan et al. 2001

50nmMotors with opposite directionality hinder each other

1q

Spontaneous symmetry breaking

• weak interaction: symmetric state

• strong interaction broken symmetry, only one motor species bound

cqq

0,0b Jm

0,0b Jm

Equal concentrations of both motor species

Total current JJJDensity difference b,b,bm

cq

Klumpp & Lipowsky, Europhys. Lett. 66, 90 (2004)

Spontaneous symmetry breaking

Total current

JJJDensity difference

b,b,bm

MC simulations

mean field equations

Hysteresisupon changing the relative motor concentrations

cqq

Total current JJJDensity difference b,b,bm

Fraction of ‚minus‘ motors

cqq

Phase transition induced by the binding/ unbinding dynamics along the filament robust against choice of the boundary conditions

Summary

• Lattice models for movements of molecular motors over large scales

• Interplay of directed walks along filaments and diffusion

Random walks of single motors: anomalous drift in slab and half space geometries active diffusion

Traffic phenomena: exclusion and traffic jams phase transitions: boundaries vs. bulk dynamics

Thanks to

Stefan KlumppReinhard Lipowsky

Janina Beeg