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Cellular Automata Modeling of Physical, Chemical and Biological Systems

Cellular Automata Modeling of Physical, Chemical and Biological Systems. Peter HANTZ Sapientia University, Department of Natural and Technical Sciences. Marine Genomics Europe Summer Course, Naples, 3 July 2007. space. Game on a String:. Two States: 0 (black) and 1 (white). time.

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Cellular Automata Modeling of Physical, Chemical and Biological Systems

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  1. Cellular Automata Modeling of Physical, Chemical and Biological Systems Peter HANTZ Sapientia University, Department of Natural and Technical Sciences Marine Genomics Europe Summer Course, Naples, 3 July 2007

  2. space Game on a String: Two States: 0 (black) and 1 (white) time 3 Black = White 2 Black = Black 1 Black = Black 3 White = White Neighborhood: 3 cells A Transition Rule: 2 x 2 x 2=8 possibilities have to be specified Generic Rule:

  3. Denomination of the Rules: Wolfram Convention current pattern: 111 110 101 100 011 010 001 000 new 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

  4. A Game or Something More? http://www.ifi.unizh.ch/ailab/teaching/FG05/script/FM3-CA.pdf Sierpinski triangle: a fractal I. a trivial state II. simple or periodic structures III. chaotic structures IV. complex “migrating” structures

  5. Game on a 2D grid Neumann Neighborhood Moore Neighborhood NN - Possibilities to be specified: 25 = 32 => Number of Possible Rules: 232 = huge

  6. Simple “totalistic” rules: Conway’s Game of Life (Moore Nbh.) Any live cell with fewer than 2live neighbors dies, as if by loneliness. Any live cell with more than 3live neighbors dies, as if by overcrowding. Any live cell with 2 or 3live neighbors lives, unchanged, to the next generation. Any empty (dead) cell with exactly 3live neighbors comes to life. Features of the Game of Life Steady objects Oscillators Gliders and Spaceships

  7. CA in Biology/Chemistry: Excitable Media Greenberg-Hastings Model: cells with several states Q: one quiescent state E1…Ee: e excited states R1…Rr: r refractory states Rules: Q →E1 when a NN is excited excited cells: Ek→Ek+1, Ee→R1 refractory cells: Rk →Rk+1, Rr →Q Weimar, 1998 http://psoup.math.wisc.edu/java/jgh.html Bub et al., 1998 BZ Reaction

  8. 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 what fraction of sites will contain blue particles at the beginning. The shape of the branching structure (~fractal) depends on the initial density of blue particles. http://germain.umemat.maine.edu/faculty/hiebeler/java/CA/DLA/DLA.html Ben-Jacob et al., 1994

  9. CA in Physics: Traffic modeling Nagel-Schreckenberg model: a probabilistic cellular automaton The street is divided into cells, that may contain cars with velocity v http://www.thp.uni-koeln.de/~as/Mypage/traffic.html Cyclic boundary conditions

  10. Traffic modeling continued http://www.grad.hr/nastava/fizika/ole/simulation.html green: v=4, khaki: v=3, brown: v=2, yellow: v=1, red: v=0 Having p=0: second-order phase transition at a critical car density Poore, 2006

  11. Self-reproduction The Idea of János NEUMANN, in 1948, before the discovery of the DNA! => development of the Cellular Automata Science Langton’s loop http://necsi.org/postdocs/sayama/sdsr/ 8 states, 29 rules: Tempesti, 1998

  12. 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/

  13. 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)) 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

  14. Separating Length Scales: combining CA and Differential Equations Hantz, 2002

  15. Thank You!

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