Alexei Fedorov January, 2011 CELLULAR ATOMATA and its applications to pattern formation, self-organization, and evolution of genomic DNA Alexei Fedorov January, 2011

© Copyright 2011, Alexei Fedorov, University of Toledo Redistribution of this lecture is strictly prohibited. All materials are exclusively for the use of registered students. 

cellular automata Pattern formation and self organization in a variety of systems that are formed by networks of interacting units Cellular automaton was introduced by John von Neumann and Stanislaw Ulam as a possible idealization of biological systems with the particular purpose of modeling biological self-replication in 1950es

Homework: Watch Wolfram’s lecture 2002 and discussion on his work in 2009 and answer the following questions: http://videolectures.net/mitworld_wolfram_anko/ Lecture http://video.google.com/videoplay?docid=6394331726775613239#docid=6897792627313669933 discussion Describe applications of CA in a) Physics and Quantum Mechanics; b) Biology; c) Computer Science and Math. Define and exemplify a) Principle of Computation Irreducibility; b) Principle of Computation Undecidability Where is the place of CA in modern Science?

Game of Life by Conway (1970) Links http://www.bitstorm.org/gameoflife/ http://en.wikipedia.org/wiki/Conway's_Game_of_Life RULES: (every cell has 8 contacts with their neighbors) For a cell that is at an “alive” state: Each cell with one or no alive neighbors dies, as if by loneliness. Each cell with four or more alive neighbors dies, as if by overpopulation. Each cell with two or three alive neighbors survives. For a cell that is at “dead” state Each cell with three alive neighbors becomes alive.

Elementary cellular automaton 8 possible combination for a cell under consideration (in the middle)

256 rules (28) for the Elementary cellular automaton Cell under consideration Cell under consideration Neighbor on the left Neighbor on the right Neighbor on the left Neighbor on the right http://atlas.wolfram.com/01/01/views/3/TableView.html

Representation of time in the second dimension

Fractals from CA

Links to CA on the web http://math.hws.edu/xJava/CA/ (about CA) http://mathworld.wolfram.com/CellularAutomaton.html (see rules; rule 30) http://math.hws.edu/xJava/CA/ (about CA) http://en.wikipedia.org/wiki/Cellular_automaton (rule 30 and 110)

Four types of behavior of CA simple repetition (for the simplest rules e.g. rule 250) Nesting structures (fractals) (when the rules are slightly more complicated e.g. rule 90) Randomness (rules beyond some threshold of complexity, which is, however, very low, e.g. rule 30) Localized structures (Complex behavior partitioned into a mixture of regular and irregular parts) (e.g. rule 110) (see page 52)

Fractals in nature, math, and art

More complex CA Three colors (>7*1012 rules) Mobile automata Turing Machines Substitution systems Tag systems Register machines Symbolic systems http://math.hws.edu/xJava/CA/ (example)

conclusions Even for extremely simple rules certain CA can produce behavior of great complexity Phenomenon of complexity is universal and independent of the details of particular system There is no clear correlation between the complexity of rules and the complexity of behavior they produce (even with complex rules, very simple behavior still occur)

Main statement of S. Wolfram (not accepted by everybody) CA represent a NEW KIND OF SCIENCE for studying complexity applicable to many areas of physics, biology, chemistry, computer science, mathematics, and elsewhere. Fundamental issues in biology (p 403, 417)

Richard Dawkins http://www.rennard.org/alife/english/biomintrgb.html

Genomic DNA could have some features described by CA Fractals (Nested structures) Repetition motifs non-randomness in nucleotide sequences (pattern formation and self-organization)

Literature on the genomic DNA pattern structures Sirakoulis G, Karafyllidis I, Mizas C, Mardiris V, Thanailakis A, Tsalides P: A cellular automaton model for the study of DNA sequence evolution, Comput Biol Med 2003, 33:439-453 Mizas C, Sirakoulis G, Mardiris V, Karafyllidis I, Glykos N, Sandaltzopoulos R: Reconstruction of DNA sequences using genetic algorithms and cellular automata: towards mutation prediction?, Biosystems 2008, 92:61-68 Rigoutsos I, Huynh T, Miranda K, Tsirigos A, McHardy A, Platt D: Short blocks from the noncoding parts of the human genome have instances within nearly all known genes and relate to biological processes, Proc Natl Acad Sci U S A 2006, 103:6605-6610 Meynert A, Birney E: Picking pyknons out of the human genome, Cell 2006, 125:836-838 Fractals in DNA sequence analysis Yu Zu-Guo et al 2002 Chinese Phys. 11 1313-1318

Rigoutsos at al. 2006 Pyknons in the 3′ UTRs of the apoptosis inhibitor birc4 (shown above the horizontal line) and nine other genes.

Homework #1: Watch Wolfram’s lecture 2002 and answer the following questions: http://videolectures.net/mitworld_wolfram_anko/ Lecture http://www.youtube.com/watch?v=_eC14GonZnU (OPTIONAL: updated lecture of Wolfram for those who are really interested in CA ) Describe applications of CA in a) Physics and Quantum Mechanics; b) Biology; c) Computer Science and Math. Define and exemplify a) Principle of Computation Irreducibility; b) Principle of Computation Undecidability Where is the place of CA in modern Science?