Movements of Molecular Motors: Random Walks and Traffic Phenomena

1. Movements of Molecular Motors:Random Walks and Traffic Phenomena Theo Nieuwenhuizen Stefan KlumppReinhard Lipowsky

2. Traffic problems: • unbinding, diffusive excursions • traffic jams • coordination of traffic Motor traffic

3. 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

4. Kinesin Microtubule Molecular motors cargo microtubule + neurofilaments • 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

5. 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 • ...

6. 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 • ...

7. Modeling – separation of scales (I) (II) (III) Vale & Milligan (2000) Visscher et al. (1999) Molecular dynamics of single step ~ 10 nm Directed walk along filament ~ 1 µm ~ 100 steps Talk Imre Derenyi Random walks: on filaments, in fluid: unbinding - binding many µm – mm This talk Talk Dean Astumian

8. 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)

9. Independent motors, d=2, full space In bulk: On line: Above line: Below line: speed on line of one motor: 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

10. Full space: Fourier-Laplace transform techniques apply Integration over q yields = 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

11. Results for d=2 at large t survival fraction average spead diffusion coefficient: enhanced Spatio-temporal distribution on line: scaling form

12. Unbound motors in d=2 average spead Diffusion coefficients: longitudinal enhanced transversal normal

13. Random walks of single motorsin open compartments Half space Slab Open tube Behavior on large scales: many cycles of binding/ unbinding How fast do motors advance ?

14. Tube Tube: Slab Slab, 2d: Half space, 3d: Half space Effective drift velocity Behavior on large scales

15. Diffusive length scale: Slab: Half space: Effective velocity: Scaling Tube:

16. b • Scaling arguments • analytical solutions (Fourier-Laplace transforms) Average position Tube: (‚normal‘ drift) Tube Slab Half space Slab: Half space: ‚anomalous‘ drift Nieuwenhuizen, Klumpp, Lipowsky, EPL 58,468 (2002)

17. 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:

18. 1) Concentration profiles in closed compartments Stationary state: Balance of directed current of bound motors and diffusive current of unbound motors Motor-filament binding/ unbinding: Local accumulation of motors Exclusion effects: reduced binding + reduced velocity

19. Concentration profiles and average current Density of bound motors • # motors small: • localization at filament end • # motors large: • filament crowded „traffic jam“ 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

20. 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)

21. Current Periodic boundary conditions • exactly solvable in mean field: • bound and unbound densities constant • radial equilibrium: • current Number of motors within the tube

22. Open tubes far from the boundaries: plateau with radial equilibrium high density (HD): maximal current (MC): low density (LD): Transitions: • LD-HD discontinuous • LD/HD-MC continuous Klumpp & Lipowsky, J. Stat. Phys. 113, 233 (2003)

23. Phase diagrams depending on the choice of boundary conditions Radial equilibrium at the boundaries Motors diffuse in/out HD HD LD MC LD Condition for the presence of the MC phase:

24. 50nm Vilfan et al. 2001 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

25. Spontaneous symmetry breaking Equal concentrations of both motor species Density difference Total current • weak interaction: • symmetric state • strong interaction • broken symmetry, only one motor species bound Klumpp & Lipowsky, Europhys. Lett. 66, 90 (2004)

26. Spontaneous symmetry breaking MC simulations mean field equations Density difference Total current

27. Hysteresis upon changing the relative motor concentrations Density difference Total current Fraction of ‚minus‘ motors • Phase transition induced by the binding/ unbinding dynamics • along the filament • robust against choice of the boundary conditions

28. 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

29. Thanks to Stefan Klumpp Reinhard Lipowsky Janina Beeg