Cellular Automata Modeling ofPhysical, Chemical and Biological Systems

Peter HANTZSapientia University, Department of Natural and Technical Sciences

Marine Genomics Europe Summer Course, Naples, 3 July 2007

Game on a String:

Neighborhood: 3 cells

A Transition Rule: 2 x 2 x 2=8 possibilities have to be specified

Two States: 0 (black) and 1 (white)

3 Black = White

2 Black = Black

1 Black = Black

3 White = White

Generic Rule:

time

space

Denomination of the Rules: Wolfram Convention

current pattern: 111 110 101 100 011 010 001 000new state for center cell 0 1 1 1 1 1 1 0

In binary notation: 0 + 26 + 25 + 24 + 23 + 22 + 21 + 0 = 126

Possible patterns: Outputs: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

=>256 possible general rules

A Game or Something More?

I. a trivial state II. simple or periodic structures

III. chaotic structures IV. complex “migrating” structures

Sierpinski triangle:a fractal

http://www.ifi.unizh.ch/ailab/teaching/FG05/script/FM3-CA.pdf

Game on a 2D grid

Neumann Neighborhood Moore Neighborhood

NN - Possibilities to be specified: 25 = 32 => Number of Possible Rules: 232 = huge

Simple “totalistic” rules: Conway’s Game of Life (Moore Nbh.)

Any live cell with fewer than 2 live neighbors dies, as if by loneliness.

Any live cell with more than 3 live neighbors dies, as if by overcrowding.

Any live cell with 2 or 3 live neighbors lives, unchanged, to the next generation.

Any empty (dead) cell with exactly 3 live neighbors comes to life.

               

Features of the Game of Life

Steady objects Oscillators Gliders and Spaceships

CA in Biology/Chemistry: Excitable Media

Bub et al., 1998

Q: one quiescent stateE1…Ee: e excited states

R1…Rr: r refractory states

Rules: Q →E1 when a NN is excitedexcited cells: Ek→Ek+1, Ee →R1

refractory cells: Rk →Rk+1, Rr →Q

Greenberg-Hastings Model: cells with several states

Weimar, 1998

http://psoup.math.wisc.edu/java/jgh.htmlBZ Reaction

CA in Physics/Biology: Growth Phenomena: diffusion-limited aggregation

Particles (blue) move randomly in the grid. When a blue particles touches a green solid particle, it also turns green and "sticks“. Particle density (parameter): a percentage, saying whatfraction of sites will contain blue particles at the beginning.

The shape of the branching structure (~fractal) depends on the initial density of blue particles.

Ben-Jacob et al., 1994http://germain.umemat.maine.edu/faculty/hiebeler/java/CA/DLA/DLA.html

CA in Physics: Traffic modelingNagel-Schreckenberg model: a probabilistic cellular automaton

Step 1: accelerationAll cars that have not already reached the maximalvelocity vmax, accelerate by one unit: v -> v+1

Step 2: safety distanceIf a car has d empty cells in front of it, and its velocity isv larger then d, it reduces the velocity to d: v ->min{d,v}

Step 3: randomizationWith probability p, the velocity is reduced by one unit: v -> v-1

Step 4: drivingCar n moves forward vn cells: xn -> xn+vn.

The street is divided into cells, that may contain cars with velocity v

Cyclic boundary conditionshttp://www.thp.uni-koeln.de/~as/Mypage/traffic.html

Traffic modeling continued

Having p=0: second-order phase transition at a critical car density

green: v=4, khaki: v=3, brown: v=2, yellow: v=1, red: v=0

http://www.grad.hr/nastava/fizika/ole/simulation.html

Poore, 2006

Self-reproduction

The Idea of János NEUMANN, in 1948, before the discovery of the DNA!=> development of the Cellular Automata Science

Langton’s loop

8 states, 29 rules:

Tempesti, 1998

http://necsi.org/postdocs/sayama/sdsr/

Self-reproduction with Evolution? Artificial Life, Evoloop

“Collision” of two SR loops may lead to third, different SR loop “Hurted” SR loops may disappear

Smaller individuals were naturally selected? Or just fragmentation?

http://necsi.org/postdocs/sayama/sdsr/

So, what are cellular automata?

Dynamical System: a rule that describes the time evolution of the state (x) of an arbitrary system

The system state at time t is a description of the system which is sufficient to predict the future states of the system without recourse to states prior to t.

Continuous in state and time: differential equations

Discrete in state and time: cellular automata

x(t+1)=f(x(t))

0)(

dt

tdx

Some Special Behavior of Dynamical Systems:

Fixed point x(t+1)=x(t); for all t>t0

Limit cycle x(t+k)=x(t) for all t>t0

))(()(

txfdt

tdx

Separating Length Scales: combining CA and Differential Equations

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Hantz, 2002

Thank You!