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

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

3. 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

4. Homework: Watch Wolfram’s lecture 2002 and discussion on his work in 2009 and answer the following questions:
Lecture 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?

5. Game of Life by Conway (1970)
Links
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.

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

7. 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

8. Representation of time in the second dimension

9. Fractals from CA

10. Links to CA on the web http://math.hws.edu/xJava/CA/ (about CA)
(see rules; rule 30)
(about CA)
(rule 30 and 110)

11. 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)

12. Fractals in nature, math, and art

13. More complex CA Three colors (>7*1012 rules) Mobile automata
Turing Machines
Substitution systems
Tag systems
Register machines
Symbolic systems
(example)

16. 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)

17. 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)

18. Richard Dawkins

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

20. 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:
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:
Meynert A, Birney E: Picking pyknons out of the human genome, Cell 2006, 125:
Fractals in DNA sequence analysis Yu Zu-Guo et al 2002 Chinese Phys

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

22. Homework #1: Watch Wolfram’s lecture 2002 and answer the following questions:
Lecture (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?