Andreas Schadschneider

Institute for Theoretical Physics

University of Cologne

www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic

Cellular Automata Modelling of Traffic in Human and

Biological Systems

Introduction

Modelling of transport problems:

space, time, states can be discrete or continuous

various model classes

Overview

1. Highway traffic

3. Traffic on ant trails

5. Pedestrian dynamics

7. Intracellular transport

Unified description!?!

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

Asymmetric Simple

Exclusion Process

Asymmetric Simple Exclusion Process

Asymmetric Simple Exclusion Process (ASEP):

• directed motion• exclusion (1 particle per site)

qq

Caricature of traffic:

For applications: different modifications necessary

Influence of Boundary Conditions

open boundaries:

Applications: Protein synthesis

Surface growth

Boundary induced phase transitions

exactly solvable!

Phase Diagram

Low-density phase

J=J(p,α)

High-density phase

J=J(p,β)

Maximal current phase

J=J(p)

Highway

Traffic

Cellular Automata Models

Discrete in • Space • Time• State variables (velocity)

velocity ),...,1,0( maxvv =

Update Rules

Rules (Nagel-Schreckenberg 1992)

• Acceleration: vj ! min (vj + 1, vmax)

• Braking: vj ! min ( vj , dj)

• Randomization: vj ! vj – 1 (with probability p)

• Motion: xj ! xj + vj

(dj = # empty cells in front of car j)

Example

Configuration at time t:

Acceleration (vmax = 2):

Braking:

Randomization (p = 1/3):

Motion (state at time t+1):

Interpretation of the Rules

• Acceleration: Drivers want to move as fast as possible (or allowed)

• Braking: no accidents

• Randomization: a) overreactions at braking b) delayed acceleration c) psychological effects (fluctuations in driving) d) road conditions

4) Driving: Motion of cars

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

Fundamental Diagram

Relation: current (flow) $ density

Metastable States

Empirical results: Existence of

• metastable high-flow states

• hysteresis

VDR Model

Modified NaSch model: VDR model (velocity-dependent randomization)

Step 0: determine randomization p=p(v(t))

p0 if v = 0

p(v) = with p0 > p

p if v > 0

Slow-to-start rule

NaSch model

VDR-model: phase separation

Jam stabilized by Jout < Jmax

VDR model

Simulation of VDR Model

Dynamics on

Ant Trails

Ant trails

ants build “road” networks: trail system

Chemotaxis

Ants can communicate on a chemical basis:

chemotaxis

Ants create a chemical trace of pheromones

trace can be “smelled” by otherants follow trace to food source etc.

q q Q

1. motion of ants

2. pheromone update (creation + evaporation)Dynamics:

f f f

parameters: q < Q, f

Ant trail model

q q Q

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!!

Spatio-temporal organization

formation of “loose clusters”

early times steady state

coarsening dynamics

Pedestrian

Dynamics

Collective Effects

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

Pedestrian Dynamics

More complex than highway traffic

• motion is 2-dimensional• counterflow • interaction “longer-ranged” (not only nearest neighbours)

Pedestrian model

Modifications of ant trail model necessary sincemotion 2-dimensional:• diffusion of pheromones• strength of trace

idea: Virtual chemotaxis

chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“

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

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

Transition Probabilities

Total transition probability pij in direction (i,j):

pij = N¢ Mij exp(kDDij) exp(kSSij)(1-nij)

Mij = matrix of preferences (preferred direction)

Dij = dynamic floor field (interaction between pedestrians)

Sij = static floor field (interaction with geometry)

kD, kS = coupling strength

N = normalization (∑ pij = 1)

Lane Formation

velocity profile

Intracellular

Transport

Intracellular Transport

Transport in cells:

• microtubule = highway• molecular motor (proteins) = trucks• ATP = fuel

• Several motors running on same track simultaneously

• Size of the cargo >> Size of the motor

• Collective spatio-temporal organization ?

Fuel: ATP

ATP ADP + P Kinesin

Dynein

Kinesin and Dynein: Cytoskeletal motors

Practical importance in bio-medical research

BlindnessKIF3A kinesinRetinitis pigmentosa

Sinus and Lung disease, male infertility

DyneinPrimary ciliary diskenesia/

Kartageners’ syndrome

Pigmentation defectMyosin VGriscelli disease

Hearing lossMyosin VIIUsher’s syndrome

Neurological disease; sensory loss

KIF1B kinesinCharcot-Marie tooth disease

SymptomMotor/TrackDisease

Goldstein, Aridor, Hannan, Hirokawa, Takemura,…………….

ASEP-like Model of Molecular Motor-Traffic

α βq

D A

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)

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 ATP2 µm

Spatial organization of KIF1A motors: experiment

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

Collaborators

Cologne:

Ludger SantenAlireza NamaziAlexander JohnPhilip Greulich

Duisburg:

Michael Schreckenberg Robert BarlovicWolfgang KnospeHubert Klüpfel

Thanx to:

Rest of the World:

Debashish Chowdhury (Kanpur)

Ambarish Kunwar (Kanpur)

Katsuhiro Nishinari (Tokyo)

T. Okada (Tokyo)

+ many others