Mathematical principles underlying genetic structures

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Mathematical principles underlying genetic
structures

• Matthew Berryman

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Overview

• Mathematics of DNA
• The DNA code
• Number of bases, number of amino acids
• Coding properties
• Evolution of sequences
• Evolution of introns, alternative splicing
• Non-coding DNA what is it good for?
• Mathematics for analysing DNA
• Mutual information
• Correlations

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DNA primer

• Bases with complimentary pairing
• Cytosine with Guanine
• Adenosine with Thymine
• Triplets code for 20 amino acids (includes start
  codon) stop codon, some degeneracy (43gt21).
• Other sequences binding sites, introns,
  telomeres, junk DNA.

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Eukaryotic gene
http//web.indstate.edu/thcme/mwking/gene-regulati
on.html
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Types of mutations (1)

• Insertions
• attgcctgggtgc -gt attcgcctgggtgcc
• I A W V -gt I R L G A
• Point mutations
• attgcctgggtgc -gt attacctgggtgc
• I A W V -gt I T W V
• Deletions
• attgcctgggtgc -gt attcctgggtgc
• I A W V -gt I P G C

M. Spanò, F. Lillo, S. Miccichè and R. N.
Mantegna, Inverted repeats in viral genomes,
Fluctuations and Noise Letters, 5(2)L193-L200
(2005).
http//www.ebi.ac.uk/2can/disease/genes5.html
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Types of mutations (2)

• Gene duplication
• attgcctgggtac -gt
• attgcctgggttgcctgggtac
• Repeating elements
• attgcctgggtac -gt
• attgtcttcttcttctcctgggtac
• Flips
• attgcctgggtac -gt
• attgggtccgtac

http//www.ebi.ac.uk/2can/disease/genes5.html
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Types of mutations (3)

• Recombination
• Find sites based on short target sequences
  insert different genes / parts of genes.
• Gene/operon shuffles
• b0357 b0362
• b0358 b0363
• b0359 -gt b0357
• b0362 b0358
• b0363 b0359

http//www.bio.davidson.edu/courses/movies.html
M.D. Ermolaeva, O. White and S.L. Salzberg,
Prediction of operons in microbial genomes,
Nucleic Acids Res. 29(5)1216-1211 (2001).
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Optimal number of amino acids
M.A. Soto C.J. Tohá, A hardware interpretation
of the evolution of the genetic code,
Biosystems, 18209-215 (1985).
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Optimal number of amino acids
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Optimal no. of bases and amino acids
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Doublet ancestor code
A. Patel, The Triplet Genetic code had a Doublet
Predecessor, arXivq-bio.GN/0403036
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Gray code

• A Gray code is an encoding of numbers so that
  adjacent numbers have a single digit differing by
  1.
• List all strings of bits of a given length in a
  sequence such that each string differs from its
  successor in only a single bit position.
• Thus the essential property is one of minimal
  change between the neighbouring bit strings.
• In DNA terms, if we change a codon by a single
  mutation we expect to end up mapping to the same,
  or similar amino acid.

R. Swanson, A unifying concept for the amino
acid code, Bulletin of Mathematical Biology,
46(2)187-203 (1984).
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DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited,
The 2003 International Conference on Mathematics
and Engineering Techniques in Medicine and
Biological Sciences
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DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
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DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
18
DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
19
DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
20
DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
21
DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
22
DNA Tower of Hanoi
A Gray coding-based assignment of codons to amino
acids can be formulated as the traveling salesman
problem1, which is analogous to solving the Tower
of Hanoi game.
1 D. Bonacki, H.M.M. ten Eikelder, P.A.J.
Hilbers, Genetic code as a Gray code revisited.
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Gene for infidelity
0 fidelity, 1 infidelity
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Gene for infidelity
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Information theory

• Entropy
• Mutual information
• Chaitin-Kolmogorov entropy the algorithmic
  complexity, ie what is the shortest program
  which will reproduce the original information.
  Depends on software libraries / set of genes
  (routines) and proteins (running operating system)

X
Y
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Evolution of introns

• Alternative splicing as a form of compression
• subfunction1subfunction2subfunction3
  subfunction1subfunction3
• subfunction1subfunction2subfunction3
• Advantages higher information content gt more
  complexity gt more adaptable.
• Trouble how to control this in an efficient,
  error free manner, with the required flexibility.
• actggggcttaa
• Introns allow us/cell machinery to efficiently
  mark the start and end of subfunctions/exons.
• actgtagggggctgtagtaa
• Efficient because we can use the same splicing
  machinery throughout.
• We can now insert extra information into the
  genome, eg if nerve cell, include the following
  exon and generate novelty, eg. include exons
  which can vary with mutation of intron length, or
  include coded sites for recombination.

M.V. Bell, A.E. Cowper, M.-P. Lefranc, J.I. Bell
G.R. Screaton, Influence of intron length on
alternative splicing of CD44, Molecular and
Cellular Biology 18(10)5930-594 (1998).
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Non-coding DNA

• We know that, ignoring binding sites, non-coding
  DNA can, fairly easily, through point mutations,
  turn into coding DNA and vice versa.
• There are also other things--the splicing that
  goes on through sexual reproduction, and we know
  from somatic mutations in genes "for" MHC
  complexes that mutation rates can be
  controlled--that suggest that non-coding DNA may
  actually be a recycling ground for de novo
  creation of new genes.
• If this were the case, then in some species which
  "had to" undergo rapid mutation (ie, non-coding
  DNA was selected for, by virtue of its ability to
  generate new genes that better equipped a
  species, thus gaining a higher game theory score
  than others) then these species would be expected
  to carry around more non-coding DNA, "just in
  case" they needed to generate new genes again.

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Correlation and mutual information
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Mutual information - real DNA vs. random sequence
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Genomic Signal Processing
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Conclusions

• Mathematics shows us which solutions to
  evolutionary problems are optimal, and therefore
  what to expect
• 4 bases, 20 amino acids.
• Properties of the genetic code.
• Frequencies of genes in a population.
• Genes that allow us to increase the complexity
  and make systems that are more adaptable
  (alternative splicing, introns).
• Mathematics allows us to analyse the patterns
  structures that form.
• Correlation detection
• Mutual information
• Signal processing

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Evolution directing evolution

• Not only has life evolved but life has evolved
  to evolve. That is, correlations within protein
  structure have evolved, and mechanisms to
  manipulate these correlations have evolved in
  tandem. The rates at which the various events
  within the hierarchy of evolutionary moves occur
  are not random or arbitrary but are selected by
  Darwinian evolution
• D.J. Earl and M.W. Dean, PNAS 101(32)11531-11536
  (2004)

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Examples

• Vertebrate immune system.
• Bacteria under nutrient starvation conditions.
• Centromere proteins in animals and humans.
• Cytochrome P450 genes.
• CYP19 (cancer)
• CYP2C9 (blood coagulation)