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 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, (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:

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) & 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 spaceSlab 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: Tube Slab Half space Behavior on large scales

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

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

Effective diffusion Active diffusion: spreading of concentration profiles by  filaments of opposite polarity  switching between motors of opposite directionality  change between bound and unbound state Effective diffusion coefficient can be large compared to D ub Effective diffusion coefficient

Active diffusion in 2d arrays of filaments patterned surface selective binding of filaments (I) (II) (III) non-small 3 regimes:

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 Motor-filament binding/ unbinding: 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 of bound motors Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, (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 Current Number of motors within the tube

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

Phase diagrams Condition 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

Density profiles: LD/HD phase  Bound/ unbound densities:  Plateau value is approached exponentially:  Plateau densities fulfill radial equilibrium.

Density profiles: MC phase bound motor density: mean field: with scale-dependent diffusion coefficient: crossover to crossover length

3) Two species of motors bound motor stimulates binding of further motors effective interaction mediated via the filament Experimental indications for cooperative binding of motors to a filament Vilfan et al nm Motors with opposite directionality hinder each other

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

Spontaneous symmetry breaking Total currentDensity difference MC simulations mean field equations

Mean field equations motor currents: binding/ unbinding:

Mean field equations: stationary solutions symmetric solution: solutions with broken symmetry:

Hysteresis upon changing the relative motor concentrations Total current Density difference 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:  Stationary concentration profiles in closed compartments: balance of directed and diffusive currents  exclusion and traffic jams  boundary-induced phase transitions in open systems  2 species: phase transitions (hysteresis) driven by unbound motor concentration

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 Reinhard Lipowsky Theo Nieuwenhuizen NKL, Phys Rev E 69 (2004) June 15, 2004 issue of Virtual Journal of Biological Physics Research KL, Europhys Lett 66 (2004) 90 KL, J. Stat Phys 113 (2003) 233 NKL, Europhys lett 58 (2002) 468 Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, (2001)

Thanks to Stefan Klumpp Reinhard Lipowsky Janina Beeg

Density patterns 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

Density profiles and average current „traffic jam“ # motors within tube Average bound current # motors small: localization at filament end # motors large: filament crowded Density of bound motors Lipowsky, Klumpp, Nieuwenhuizen, PRL 87, (2001) Intermediate # motors:  coexistence of a jammed region and a low density region,  maximal current

far from the boundaries: plateau with radial equilibrium low density (LD): high density (HD):maximal current (MC): Boundary-induced phase transitions in open tube systems Tube coupled to reservoirs Changing the motor concentration in the reservoirs boundary-induced phase transitions

Phase diagrams LD HD MC Radial equilibrium at the boundaries Motors diffuse in/out HD LD Klumpp & Lipowsky, J. Stat. Phys. 113, 233 (2003) depend on the precise choice of boundary conditions

Two species of motors Cooperative binding of motors to a filament bound motor stimulates binding of further motors Motors with opposite directionality mutually repress binding Spontaneous symmetry breaking Hysteresis Klumpp & Lipowsky, Europhys. Lett. 66, 90 (2004)

Summary Lattice models for movements of molecular motors Interplay of directed walks along filaments and unbound diffusion Motor-motor interactions, traffic phenomena... Reinhard Lipowsky Theo Nieuwenhuizen Thanks to...