Movements of Molecular Motors:
Random Walks and Traffic Phenomena
Theo Nieuwenhuizen
Stefan Klumpp
Reinhard 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 systems
2) Boundary-induced phase transitions
3) Two species of motors
Molecular motors
cargo
microtubule
+
neurofilaments
• proteins which convert
chemical energy into directed movements
• movements along filaments of cytoskeleton
Hirokawa 1998
Kinesin
• various functions in vivo: transport, internal
organization of the cell, cell division, ...
• processive motors: large distances
Microtubule
In vitro-experiments
Janina Beeg
Measurements of transport properties of single motor molecules:
 velocity: ~ µm/sec = 0.1 m/month
 step size ~ 10 nm, step time ~ 10 ms
 ...
In vitro-experiments
Vale & Pollock in Alberts et al. (1999)
Measurements of transport properties of single motor molecules:
 velocity: ~ µm/sec
 step size ~ 10 nm
 ...
Modeling – separation of scales
(I)
(II)
Vale & Milligan (2000)
(III)
Visscher et al. (1999)
Molecular dynamics
of single step
~ 10 nm
Directed walk
along filament
~ 1 µm
~ 100 steps
Talk Dean Astumian
Talk Imre Derenyi
Random walks:
on filaments, in fluid:
unbinding - binding
many µm – mm
This talk
Lattice models for the random walks
of molecular motors
• biased random walk along a
filament
• unbound motors: symmetric
random walk
• detachment rate e & sticking
probability pad
Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, 108101 (2001)
 simple and generic model
 parameters can be adapted to specific motors
 motor-motor interactions can be included (hard core)
Independent motors, d=2, full space
In bulk:
On line:
Above line:
Below line:
speed on line of one motor:
vb  1  
   e
Initial condition: motors start at t=0 at origin on the line
Full space:
Exact solution via Fourier-Laplace transform
Useful to test numerical routines
Full space:
Fourier-Laplace transform techniques apply
Integration over q yields
Pb ( r , s )= Fourier-Laplace transform on line:
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
Behavior on large scales:
many cycles of binding/ unbinding
How fast do motors advance ?
Open tube
Effective drift velocity
Behavior on large scales
Tube:
v ~ const.
Tube
Slab, 2d:
v ~ 1/ t
Slab
Half space, 3d:
v ~ 1/t
Half space
Effective velocity: Scaling
Tube:
1/ e
v btb
vb
vb
v


t b  t ub 1  (e / pad ) (e / pad )
 / pad
Diffusive length scale:
Slab:
 ~ hL ~ h Dubt
Half space:
 ~ L ~ Dubt
2
L ~ Dubt
v b pad
v~
e h Dubt
v b pad
v~
e Dubt
Average position
Tube:
(‚normal‘ drift)
Tube
Slab
x~t
Half space
Slab:
x~ t
Half space:
x ~ ln t
‚anomalous‘ drift
b
• Scaling arguments
• analytical solutions (Fourier-Laplace transforms)
Nieuwenhuizen, Klumpp, Lipowsky, EPL 58,468 (2002)
Exclusion and traffic jams
Mutual exclusion of motors from binding sites
clearly demonstrated in decoration experiments
simple exclusion: no steps to occupied binding sites
movement 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

v bρ b (1  ρ b )  Dub ρ ub
x
Motor-filament binding/ unbinding:

v b  b (1   b )  pad  ub (1   b )  eb
x
Local accumulation of motors
Exclusion effects: reduced binding + reduced velocity
Concentration profiles and average current
Density of
bound
motors
„traffic jam“
• # motors small:
localization at filament end
• # motors large:
filament crowded
exponential growth
Average bound current
• Intermediate # motors:
 coexistence of a jammed region
and a low density region,
 maximal current
Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, 108101 (2001)
# motors within tube
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:
eb (1  ub )  pad ub (1  b )
 current
J  v bb (1  b )
Current
Number of motors within the tube
Open tubes
far from the boundaries: plateau with radial equilibrium
low density (LD):
(b0)  1 / 2
Transitions:
 LD-HD discontinuous
 LD/HD-MC continuous
high density (HD):
 (b0 )  1 / 2
maximal current (MC):
 (b0 )  1 / 2
J  vb / 4
Klumpp & Lipowsky, J. Stat. Phys. 113, 233 (2003)
Phase diagrams
depending on the choice of boundary conditions
Radial equilibrium at the boundaries
Motors diffuse in/out
HD
LD
HD
MC
LD
Condition for the presence
of the MC phase:
e / pad v b
pR Dub

L
4
2
3) Two species of motors
Experimental indications for cooperative
binding of motors to a filament
bound motor stimulates binding of
further motors
effective interaction mediated via
the filament
Motors with opposite directionality
hinder each other
50nm
Vilfan et al. 2001
q 1
Spontaneous symmetry breaking
Equal concentrations of both motor species
Density difference
mb   b,   b,
Total current
J  J  J
qc
• weak interaction:
symmetric state mb  0, J  0
• strong interaction q  qc
broken symmetry, only one motor species bound
mb  0, J  0
Klumpp & Lipowsky, Europhys. Lett. 66, 90 (2004)
Spontaneous symmetry breaking
MC simulations
mean field equations
Density difference
mb   b,   b,
Total current
J  J  J
Hysteresis
upon changing the relative motor concentrations
Density difference
mb   b,   b,
Total current
J  J  J
q  qc
q  qc
Fraction of ‚minus‘ motors
 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 Klumpp
Reinhard Lipowsky
Janina Beeg