P1254503540CaQPv

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DNA CODES BASED ON HAMMING STEM SIMILARITIES
A.G. Dyachkov1, A.N. Voronina1
1 Dept. of Probability Theory, MechMath., Moscow
State University, Russia
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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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DNA STRANDS
Single DNA strand

• DNA strands consist of nucleotides, composed of
  sugar and phosphate backbone and 1 base
• There are 4 types of bases

5 end
adenine
guanine
cytosine
thymine

• Base A is said to be complement to T and C to G
• DNA strands are oriented. Thus, for example,
  strand AATG is different from strand GTAA
• 2 oppositely directed strands containing
  complement bases at corresponding positions are
  called reverse-complement strands. For example,
  this 2 strands are reverse-complement

Bases
Nucleotide
The strands have different directions
Sugar phosphate backbone
3 end
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HYBRIDIZATION

• 2 oppositely directed DNA strands are capable of
  coalescing into duplex, or double helix
• The process of forming of duplex is referred to
  as hybridization
• The basis of this process is forming of the
  hydrogen bonds between complement bases
• Duplex, formed of reverse-complement strands is
  called a Watson-Crick duplex. Here is the example
  of it

Watson-Crick duplex
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CROSS-HYBRIDIZATION AND ENERGY OF HYBRIDIZATION

• Though, hybridization is not a perfect process
  and non-complementary strands can also hybridize
• This is one example of cross-hybridization

This bases are not complement
This bases are not complement

• The indicator of strength, or stability of
  formed duplex is its energy of hybridization. Its
  value depends on the total number of bonds formed
• Thus, the greatest hybridization energy is
  obtained when Watson-Crick duplex is formed
  rather than is case of cross-hybridization

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LONE BONDS AND PAIRWISE METRIC

• If a pair of bases is bonded but neither of its
  neighbor bases form a bond as well, then it is
  called a lone bond. Here it is

Lone bond does not contribute to hybr. energy
A triplet is counted as 2 adjacent pairs
A pair of bonds add 1 to total hybr. energy
Hybr. Energy 3

• The lone bond is too weak to form a strong
  connection, so it does not contribution much to
  the total energy of hybridization
• Moreover, in fact, the energy of hybridization
  depends not on the number of bonds formed, but on
  the number of pairs of adjacent bonds
• Thus, if we suppose, that hybridization energy is
  equal to the number of pairs, then in the example
  above it is equal to 3, not 5 or 6

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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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NOTATIONS

• General notations
• Let be an arbitrary
  even integer
• Denote by
  the standard alphabet of size
• Denote by the largest (smallest)
  integer
• Reverse-complementation
• For any letter , define
  the complement of the
  letter
• For any q-ary sequence
  , define its reverse
  complement
• Note, that if , then
  for any .

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STEM HAMMING SIMILARITY

• For 2 q-ary sequences of length n
• 
  and
• stem Hamming similarity is equal to
• 
  where
• is equal to the total number of
  common 2-blocks containing adjacent symbols in
  the longest common Hamming subsequence

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HAMMING VS. STEM HAMMING

• Hamming similarity is element-wise while stem
  Hamming similarity is pair-wise (though still
  additive)
• Re-ordering the elements in the sequence does not
  influence Hamming similarity, but may change stem
  Hamming similarity
• Example

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STEM HAMMING DISTANCE

• Note, that
  and if and
  only if
• Stem Hamming distance between
  is
• Example
• Let and
  
• The longest common Hamming subsequence is
• Stem Hamming similarity is equal to
• Stem Hamming distance is equal to

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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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MOTIVATION

• Study of DNA codes was motivated by the needs of
  DNA computing and biomolecular nanotechnology
• In these applications, one must form a collection
  of DNA strands, which will serve as markers,
  while the collection of reverse-complement (to
  that first strands) DNA strands will be utilized
  for reading, or recognition

Probing Complement Strands for Reading
Coding Strands for Ligation

1. Collection of mutually reverse-complement pairs
2. No self-reverse complement words
3. No cross-hybridization

TACGCGACTTTC ATCAAACGATGC TGTGTGCTCGTC ATTTTTGCGTT
A CACTAAATACAA GAAAAAGAAGAA
GAAAGTCGCGTA GCATCGTTTGAT GACGAGCACACA TAACGCAAAAA
T TTGTATTTAGTG TTCTTCTTTTTC
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DNA CODE

• 
  is a code of length and size
• , where
  are the codewords of code
• is called a DNA -code based
  on stem Hamming similarity if the following 2
  conditions are fulfilled
• For any , there exists
  , such that
• For any
• Let be the maximal size of
  DNA -codes.
• Is called a rate of DNA codes

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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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Q-ARY REED-MULLER CODES

• q-ary Reed-Muller codeLet
• Define mapping
  , with
• Reed-Muller code of order
  is the image
• Reed-Muller code of order 1
  satisfy the condition of reverse-complementarity
• It may contain self-reverse complement words,
  that should be excluded from the final
  construction

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EXAMPLE OF CODE
Let q4 and m1
Mutually-reverse complement
0 1 2 3
0 0 0 0
0 1 2 3
0 1 2 3
0 2 0 2
0 3 2 1
1 1 1 1
1 2 3 0
1 3 1 3
1 0 3 2
2 2 2 2
2 3 0 1
2 0 2 0
2 1 0 3
3 3 3 3
3 0 1 2
3 1 3 1
3 2 1 0
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OUTLINE

• DNA background
• Modeling the hybridization energy
• DNA codes
• Example of DNA codes
• Bounds on the rate on DNA codes
• Lower Gilbert-Varshamov bound
• Upper bounds
• Graphs
• On sphere sizes
• Possible generalizations
• Bibliography

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RANDOM CODING

• and are independent identically
  distributed random sequences with uniform
  distribution on
• Define
• Probability distribution of
• Sum of

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GILBERT-VARSHAMOV BOUND

• Let . Introduce
• We construct random code as a collection of
  independent variables and their
  reverse-complements. This fact leads to necessity
  of special random coding technique for DNA codes
• One can check, that
• Random coding bound (Gilbert-Varshamov bound)
  if then

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CALCULATION OF THE BOUND

• are dependent variables and
  both depend on and
• do not constitute a Markov chain
• 
  vs.
• are deterministic functions of Markov chain
  
• and
• We cannot apply standard technique as in case of
  Hamming similarity
• We have to use Large Deviations Principle for
  Markov chains for

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GILBERT-VARSHAMOV BOUND

• Introduce
• Gilbert-Varshamov lower bound on the rate
  If
  then , where
• and is a
  decreasing -convex function with

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OUTLINE

• DNA background
• Modeling the hybridization energy
• DNA codes
• Example of DNA codes
• Bounds on the rate on DNA codes
• Lower Gilbert-Varshamov bound
• Upper bounds
• Graphs
• On sphere sizes
• Possible generalizations
• Bibliography

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UPPER BOUNDS

• Plotkin upper bound
• If , then
  and
• 
  if
• Elias upper boundIf
  , then ,
  where is presented by parametric
  equation
• Elias bound improves Plotkin bound for small
  values of . We
  calculated and
  .

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OUTLINE

• DNA background
• Modeling the hybridization energy
• DNA codes
• Example of DNA codes
• Bounds on the rate on DNA codes
• Lower Gilbert-Varshamov bound
• Upper bounds
• Graphs
• On sphere sizes
• Possible generalizations
• Bibliography

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BOUNDS ON THE RATE (Q2)
Bound on the rate of DNA code, q2
0.75
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BOUNDS ON THE RATE (Q4)
Bound on the rate of DNA code, q4
0.9375
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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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FIBONACCI NUMBERS

• q-ary Fibonacci numbers are defined by recurrent
  equation
• with initial conditions
• q-ary Fibonacci numbers may also be calculated as
  sum
• q-ary Fibonacci number may be
  interpreted as the numberof q-ary sequences of
  length , which do not contain 2-stems of the
  form (0,0)

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COMBINATORIAL CALCULATION

• Space with metric is
  homogeneous, i.e., the volume of a sphere does
  not depend on its center
• Define
• for any
• Consider a sphere with center
  . Anysequence
  must have no
  common2-stems (pairs) with . In other
  words, is must have no 2-stems of type (0,0).
  Thus,
• Sphere sizes for other may be obtained using
  the same technique with some corresponding
  modifications

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GRAPH OF PROBABILITIES
Probability distribution
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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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B-STEM HAMMING SIMILARITY

• -stem Hamming similarity in spite of
  counting the number of 2-stems (pairs)
  calculate the number of -stems
• 
  where

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WEIGTHED STEM HAMMING SIMILARITY

• Weighted stem Hamming similarity assign weight
  to each type of q-ary pairs and take it into
  account while calculating the sum
• Let
  be a weight function such that
• Similarity is defined as follows
• , where

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INSERTION-DELETION STEM SIMILARITY

• Insertion-deletion stem similarityallow loops
  and shifts at the DNA duplex
• is a common block
  subsequence between and , if is an
  ordered collection of non-overlapping common (
  , )-blocks of length
• common ( , )-block of length ,
  is a subsequence of and ,
  consisting of consecutive elements of and
• is the set of all common block
  subsequences between and
• is the minimal number of
  blocks of consecutive elements of and in
  the given subsequence
• Similarity is defined as follows

Shift
Loop
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OUTLINE

1. DNA background
2. Modeling the hybridization energy
3. DNA codes
4. Example of code construction
5. Bounds on the rate on DNA codes
6. On sphere sizes
7. Further generalizations
8. Bibliography

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BIBLIOGRAPHY

• Probability theory and Large Deviation Principle
• V.N. Tutubalin, The Theory of Probability and
  Random Processes. Moscow Publishing House of
  Moscow State University, 1992 (in Russian).
• A. Dembo, O. Zeitouni, Large Deviations
  Techniques and Applications. Boston, MA Jones
  and Bartlett, 1993.
• DNA codes
• D'yachkov A.G., Macula A.J., Torney D.C.,
  Vilenkin P.A., White P.S., Ismagilov I.K.,
  Sarbayev R.S., On DNA Codes. Problemy Peredachi
  Informatsii, 2005, V. 41, N. 4, P. 57-77, (in
  Russian). English translation Problems of
  Information Transmission, V. 41, N. 4, 2005, P.
  349-367.
• Bishop M.A.,D'yachkov A.G., Macula A.J., Renz
  T.E., Rykov V.V., Free Energy Gap and Statistical
  Thermodynamic Fidelity of DNA Codes. Journal of
  Computational Biology, 2007, V. 14, N. 8, P.
  1088-1104.
• A. Dyachkov, A. Macula, T. Renz and V. Rykov,
  Random Coding Bounds for DNA Codes Based on
  Fibonacci Ensembles of DNA Sequences. Proc. of
  2008 IEEE International Symposium on Information
  Theory, Toronto, Canada, 2008, in print.