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. Matrix Algorithms for Genome Evolution?
A computational approach for modeling intricacies of the human genome evolution
Alexei Fedorov,
Associate Professor, Department of Medicine
University of Toledo, HSC

23. “1000 Genomes” international project

24. The 1000 Genome Project ASW African Americans YRI Nigeria LWK Kenya
CEU Europeans in America
TSI Italy
FIN Finland
GBR Great Britain
IBS Spain
CHB Chinese in Beijing
CHS Chinese from South
JPT Japan
CLM Colombians
MXL Mexicans
PUR Puerto Rican
1000 Genome Project
LWK
FIN
JPT
GBR
TSI
PUR
MXL
IBS
CLM
CEU
ASW
YRI
CHS
CHB

25. BOTTOM LINE: Two individuals, even from the same population, differ from one another by millions of SNPs

26. Major haplotypes of human hemeoxygenase-1 gene (only of frequent SNPs are shown)
The bottom line: Mutations never exist alone but in groups linked with each other and forming haplotypes that slowly change due to meiotic recombination and selection/drift

27. Every human being has 50- 100 de novo mutations
Many of these mutations will drift away. Some of them
will be eventually fixed in the population. The rest will be present as polymorphic biomarkers for a considerable period of time.
The major question I would like to answer at this presentation:
What parameters are most important for maintaining the fitness of population under the intense influx of novel mutations when deleterious ones overwhelm beneficial?

28. Mathematical modeling versus Computer modeling in Population Genetics
Not trivial
Considerable simplificaton of reality
Several opposing theories
FROM: Weissman et al The rate of fitness-valley crossing in sexual populations

29. Mathematical modeling versus Computer modeling in Population Genetics
Cellular Automata
Game of life
BOTTOM LINE: Instead of utilizing mathematical modeling, prediction of the fate of mutations can be approached more fruitfully from a different dimension: taking advantage of the enormous power of contemporary supercomputers.
Computer experiments are able to reveal unexpected results and open a whole new way of looking at the operation of our universe

30. Advantages of computer modeling
No mathematics
Allows observation of cooperation of thousands elements (SNPs)
Behavior of the system with multiple parameters and inhomogeneity

31. Matrix Algorithms for Genome Evolution (MAGE)
Advanced version
MAGE.java
150 pages of scripts
Core version
MAGE.pl
15 pages

32. Genetically Identical Starting Population, size N
Mutations in each individual
Mating between N/2 mating pairs
Combine male and female gametes to make offspring
Calculate fitness of offspring
N most fit offspring to survive and reproduce
Observe population fitness & DNA sequences changes after a certain number of generations

33. A large portion of a real genomic sequence (even whole chromosomes of human or other species) can be assigned as a reference genome for a model population
A user specifies the number of individuals in the population (parameter N)
Each individual is constructed as a diploid genome that descended as two haploid gamete genomes from its parents.

34. Each individual accumulates a number of mutations in their germ cells
Mutations are created randomly taking a user-defined parameter μ (number of novel mutations per gamete)
Each individual accumulates a number of mutations in their germ cells
Effects fitness of next generation inheriting the mutations
Effect of mutation on fitness of offspring: “selective advantage” value (s)
Beneficial: improves fitness
Deleterious: decreases fitness

35. Upon generation of a mutation, MAGE assigns a selection coefficient (s) to the mutation using a user-defined s-distribution.
Since s is an experimentally immeasurable parameter, MAGE assigns non-normalized values of s for generated mutations.
When s > 0 the mutation is beneficial
When s< 0, the mutation is deleterious
BOTTOM LINE: Absolute values of s are not of primary importance, yet the distribution of these s-values among all mutations is crucial.
Distribution of s is a D-parameter

36. Our popular distributions (D parameters)
Deleterious mutations are 9 times more abundant than beneficial

37. r=1 R=15 Scheme of meiotic recombination
Number of recombination events per gamete – parameter r
Maternal 5’ 3’
Paternal 5’ 3’
r=1
Gamete 1 5’ 3’
R=15
Gamete 2 5’ 3’

38. Mating schemes and generation of offspring
Number of offspring per individual parameter α
α = 3
α = 6

39. Calculation of fitness for each generated offspring
Maternal allele: fitness Wm
Paternal allele: fitness Wp
MAGE calculates fitness for each gene for both parental chromosomes by summing all the s-values of mutations within that gene.
The fitness of the maternal allele for the given gene (wm) will be a sum of s-values for all SNPs within maternal haplotype of this gene, while the fitness of the paternal allele (wp) will be a sum of s-values for all the paternal SNPs.

40. w = min(wm, wp) + h×abs(wm-wp)
The fitness of a gene in is calculated from the wm and wp values and also another user-defined parameter, the dominance coefficient (h parameter).
In a co-dominance mode (h=0.5), the gene fitness is the average of the fitness of maternal and paternal alleles.
Under a recessive mode (h=1), which corresponds to recessive genes, the fitness is the maximum between wm and wp values (heterozygotes with one deleterious allele are healthy).
for a dominant mode (h=0), which corresponds to dominant genes, the gene fitness is the minimum between wm and wp values.
For a general case, the gene fitness is calculated by the formula:
w = min(wm, wp) + h×abs(wm-wp)
Fitness of an individual is the sum of fitnesses of all his/her genes

41. Darwinian selection – The fittest are survive only
N fittest offspring will survive and create new generations.
Fitness

42. Results of MAGE simulations
Population size N from 50 to 200
Mutations per gamete μ from 1 to 50
Recombinations per gamete r from 0 to 48
Dominance coefficient h 0, 0.5, or 1
Distribution of selection coefficients D: (deleterious mutations 8-10 overwhelm beneficial ones)
Offspring per individual α from 2 to 10
Does NOT
matter
Number of genes in the genome
Length of genes bp -10,000 bp

43. MAGE modeling number of SNPs in population

44. MAGE modeling number of SNPs in population

45. Change of population fitness in generations when deleterious mutations overwhelm beneficial ones

46. Change of population fitness in generations when deleterious mutations overwhelm beneficial ones

47. Major Conclusion Our formula for healthy population r ≥μ
Computer modeling with MAGE has demonstrated that the number of meiotic recombination events per gamete is among the most crucial factors influencing population fitness.
In humans, these recombinations create a gamete genome consisting on an average of 48 pieces of corresponding parental chromosomes. Such highly mosaic gamete structure allows preserving fitness of population under the intense influx of novel mutations even when the number of mutations with deleterious effects is up to ten times more abundant than those with beneficial effects.
Our formula for healthy population r ≥μ

48. How haplotypes look like in MAGE modeling?

49. Probability of fixation of a mutation with s = +1

50. MAGE outcome on the probability of fixation of a mutation with selection coefficient s
For a specific set of parameters, probability of fixation of neutral mutation significantly deviates from 1/2N. For example, for (α =10, h=0, r=1, D=expC, µ=1, N=50) it is 2.1 times higher.
A common view that the probability of ultimate fixation of a beneficial mutation (π) should be proportional to s (π ≅ 2s) and not depend on the population size (Patwa and Wahl 2008). Our MAGE results demonstrate that the probability of fixation of a beneficial mutation π also depends on the combination of the six aforementioned parameters (N, μ, r, h, α, D). This phenomenon can be explained by the linkage of deleterious, beneficial, and neutral mutations within haplotypes and selecting them as whole units.

51. Mating Schemes with random pairs
Scheme2: Promiscuity
Scheme1: True family values
Random assigned
Random assigned male and female
repeat
Random match
Random match
Each couple has the same number of offspring
The random mating continues until equal number of offspring are born.
Constant mating pairs
Temporary mating pairs

52. Mating Schemes sorted by fitness
Scheme 4: Artificial scheme
Best-to-Least fit
Scheme 3: Best-to-Best fit
Random assigned
Random assigned
Best fit matches the least fit
Best fit matches each other
Each couple has the same number of offspring
Each couple has the same number of offspring

53. Mating Schemes sorted by fitness & variable offspring numbers
Scheme 6: Couple fitness ->
Number of offspring
Scheme 5: Male fitness ->
Number of offspring
Random assigned
Random assigned male and female
Best fit matches each other
Random match
The number of offspring is based on the male fitness
The number of offspring is based on the couple fitness

54. Mating Schemes: Herd – Only the Best male is reproduced
Same number of the offspring
Scheme7
Random assigned
Only the best fit
male is left

56. 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?

57. Homework #2
READ THE PAPER: 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
DESRIBE THE DIFFERNCES between the CA for DNA modelling and Wolfram’s Elementary CA