movements of molecular motors: random walks and traffic phenomena
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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
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