DNA C ODES B ASED ON H AMMING S TEM S IMILARITIES

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

2. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

3. A A A C C C G G G T T T 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 guanine adenine 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

4. A A A C C G G T T T 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

5. A A A A A A C C C C C C C G G G G G T T T T 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

6. A A A A A C C C C C G G T T T T 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

7. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

8. 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 .

9. 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

10. 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

11. 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

12. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

13. 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 Collection of mutually reverse-complement pairs No self-reverse complement words No cross-hybridization TACGCGACTTTC ATCAAACGATGC TGTGTGCTCGTC ATTTTTGCGTTA CACTAAATACAA GAAAAAGAAGAA GAAAGTCGCGTA GCATCGTTTGAT GACGAGCACACA TAACGCAAAAAT TTGTATTTAGTG TTCTTCTTTTTC

14. 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

15. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

16. Q-ARY REED-MULLER CODES • q-ary Reed-Muller code:Let 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

17. Self-reverse complement EXAMPLE OF CODE Let q=4 and m=1 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

18. 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

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

20. 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

21. 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 forMarkov chainsfor

22. GILBERT-VARSHAMOV BOUND • Introduce • Gilbert-Varshamov lower bound on the rate : If then , where and is a decreasing -convex function with

23. 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

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

25. 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

26. Gilbert-Varshamov bound Plotkin bound Hamming bound Elias bound BOUNDS ON THE RATE (Q=2) Bound on the rate of DNA code, q=2 0.75

27. Gilbert-Varshamov bound Plotkin bound Hamming bound Elias bound BOUNDS ON THE RATE (Q=4) Bound on the rate of DNA code, q=4 0.9375

28. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

29. 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)

30. COMBINATORIAL CALCULATION • Space with metric is homogeneous, i.e., the volume of a sphere does not depend on it’s 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

31. n = 5 n = 10 n = 20 n = 30 n = 40 GRAPH OF PROBABILITIES Probability distribution

32. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

33. B-STEM HAMMING SIMILARITY • -stem Hamming similarity:in spite of counting the number of 2-stems (pairs) – calculate the number of -stems where

34. 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

35. INSERTION-DELETION STEM SIMILARITY • Insertion-deletion stem similarity:allow loops and shifts at the DNA duplex • is a common block subsequencebetween 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

36. OUTLINE • DNA background • Modeling the hybridization energy • DNA codes • Example of code construction • Bounds on the rate on DNA codes • On sphere sizes • Further generalizations • Bibliography

37. 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. D’yachkov, 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.