1. Andreas Schadschneider Institute for Theoretical Physics University of Cologne www.thp.uni-koeln.de/~aswww.thp.uni-koeln.de/ant-traffic Cellular Automata Modelling of Traffic in Human and Biological Systems

2. Introduction Modelling of transport problems: space, time, states can be discrete or continuous various model classes

3. Overview 1.Highway traffic 2.Traffic on ant trails 3.Pedestrian dynamics 4.Intracellular transport Unified description!?!

4. Cellular Automata Cellular automata (CA) are discrete in space time state variable (e.g. occupancy, velocity) Advantage: very efficient implementation for large-scale computer simulations often: stochastic dynamics

5. Asymmetric Simple Exclusion Process

6. Asymmetric Simple Exclusion Process (ASEP): 1.directed motion 2.exclusion (1 particle per site) q q Caricature of traffic: For applications: different modifications necessary

7. Influence of Boundary Conditions open boundaries: Applications: Protein synthesis Surface growth Boundary induced phase transitions exactly solvable!

8. Phase Diagram Low-density phase J=J(p,  ) High-density phase J=J(p,  ) Maximal current phase J=J(p)

9. Highway Traffic

10. Cellular Automata Models Discrete in Space Time State variables (velocity) velocity

11. Update Rules Rules (Nagel-Schreckenberg 1992) 1)Acceleration: v j ! min (v j + 1, v max ) 2)Braking: v j ! min ( v j, d j ) 3)Randomization: v j ! v j – 1 (with probability p) 4)Motion: x j ! x j + v j (d j = # empty cells in front of car j)

12. Example Configuration at time t: Acceleration (v max = 2): Braking: Randomization (p = 1/3): Motion (state at time t+1):

13. Interpretation of the Rules 1)Acceleration: Drivers want to move as fast as possible (or allowed) 2)Braking: no accidents 3)Randomization: a) overreactions at braking b) delayed acceleration c) psychological effects (fluctuations in driving) d) road conditions 4) Driving: Motion of cars

14. Simulation of NaSch Model Reproduces structure of traffic on highways - Fundamental diagram - Spontaneous jam formation Minimal model: all 4 rules are needed Order of rules important Simple as traffic model, but rather complex as stochastic model

15. Fundamental Diagram Relation: current (flow) $ density

16. Metastable States Empirical results: Existence of metastable high-flow states hysteresis

17. VDR Model Modified NaSch model: VDR model (velocity-dependent randomization) Step 0: determine randomization p=p(v(t)) p 0 if v = 0 p(v) = with p 0 > p p if v > 0 Slow-to-start rule

18. NaSch model VDR-model: phase separation Jam stabilized by J out < J max VDR model Simulation of VDR Model

19. Dynamics on Ant Trails

20. Ant trails ants build “road” networks: trail system

21. Chemotaxis Ants can communicate on a chemical basis: chemotaxis Ants create a chemical trace of pheromones trace can be “smelled” by other ants follow trace to food source etc.

22. q q Q 1.motion of ants 2.pheromone update (creation + evaporation) Dynamics: f f f parameters: q < Q, f Ant trail model qqQ

23. Fundamental diagram of ant trails different from highway traffic: no egoism velocity vs. density Experiments: Burd et al. (2002, 2005) non-monotonicity at small evaporation rates!!

24. Spatio-temporal organization formation of “loose clusters” early timessteady state coarsening dynamics

25. Pedestrian Dynamics

26. Collective Effects jamming/clogging at exits lane formation flow oscillations at bottlenecks structures in intersecting flows ( D. Helbing)

27. Pedestrian Dynamics More complex than highway traffic motion is 2-dimensional counterflow interaction “longer-ranged” (not only nearest neighbours)

28. Pedestrian model Modifications of ant trail model necessary since motion 2-dimensional: diffusion of pheromones strength of trace idea: Virtual chemotaxis chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“

29. Floor field cellular automaton Floor field CA: stochastic model, defined by transition probabilities, only local interactions reproduces known collective effects (e.g. lane formation) Interaction: virtual chemotaxis (not measurable!) dynamic + static floor fields interaction with pedestrians and infrastructure

30. Transition Probabilities Stochastic motion, defined by transition probabilities 3 contributions: Desired direction of motion Reaction to motion of other pedestrians Reaction to geometry (walls, exits etc.) Unified description of these 3 components

31. Transition Probabilities Total transition probability p ij in direction (i,j): p ij = N¢ M ij exp(k D D ij ) exp(k S S ij )(1-n ij ) M ij = matrix of preferences (preferred direction) D ij = dynamic floor field (interaction between pedestrians) S ij = static floor field (interaction with geometry) k D, k S = coupling strength N = normalization (  p ij = 1)

32. Lane Formation velocity profile

33. Intracellular Transport

34. Intracellular Transport Transport in cells: microtubule = highway molecular motor (proteins) = trucks ATP = fuel

35. Several motors running on same track simultaneously Size of the cargo >> Size of the motor Collective spatio-temporal organization ? Fuel: ATP ATP ADP + PKinesin Dynein Kinesin and Dynein: Cytoskeletal motors

36. Practical importance in bio-medical research DiseaseMotor/TrackSymptom Charcot-Marie tooth disease KIF1B kinesinNeurological disease; sensory loss Retinitis pigmentosaKIF3A kinesinBlindness Usher’s syndromeMyosin VIIHearing loss Griscelli diseaseMyosin VPigmentation defect Primary ciliary diskenesia/ Kartageners’ syndrome DyneinSinus and Lung disease, male infertility Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….

37. ASEP-like Model of Molecular Motor-Traffic   q DA Parmeggiani, Franosch and Frey, Phys. Rev. Lett. 90, 086601 (2003) ASEP + Langmuir-like adsorption-desorption Also, Evans, Juhasz and Santen, Phys. Rev.E. 68, 026117 (2003)

38. position of domain wall can be measured as a function of controllable parameters. Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005) KIF1A (Red) MT (Green) 10 pM 100 pM 1000pM 2 mM of ATP 2  m Spatial organization of KIF1A motors: experiment

39. Summary Various very different transport and traffic problems can be described by similar models Variants of the Asymmetric Simple Exclusion Process Highway traffic: larger velocities Ant trails: state-dependent hopping rates Pedestrian dynamics: 2d motion, virtual chemotaxis Intracellular transport: adsorption + desorption

40. Collaborators Cologne: Ludger Santen Alireza Namazi Alexander John Philip Greulich Duisburg: Michael Schreckenberg Robert Barlovic Wolfgang Knospe Hubert Klüpfel Thanx to: Rest of the World: Debashish Chowdhury (Kanpur) Ambarish Kunwar (Kanpur) Katsuhiro Nishinari (Tokyo) T. Okada (Tokyo) + many others