Math 8803/4803, Spring 2008: Discrete Mathematical Biology

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1 Math 883/483, Spring 28: Discrete Mathematical Biology Prof. Christine Heitsch School of Mathematics Georgia Institute of Technology Lecture 5 January 5, 28

2 Nucleic acid hybridization DNA is an oriented (5 to 3 ) biochemical chain consisting of four nucleotides: a (adenine), c (cytosine), g (guanine), and t (thymine). DNA base pairing (hybridization) occurs in an antiparallel (5 to 3 with 3 to 5 ) and complementary (g with c and a with t) fashion. C. E. Heitsch, GA Tech

3 Approximating thermodynamic interactions Dissimilar melting temperatures bias physical behavior in vitro. Different estimation methods, related to oligonucleotide length: Very short ( 5 bases or fewer): gc content or the 2-4 rule Relatively short: nearest neighbor calculation of free energy and enthalpy based on experimental parameters. Longer (not less than 5 bases): Wetmur s equation for the stability of hybridized DNA Remember that these values are only approximations! C. E. Heitsch, GA Tech 2

4 DNA microarrays Short segments of DNA (codewords) are immbobilized on a surface. Other (possibly complementary) segements are introduced in solution. If hybridization occurs, it can be experimentally detected (by fluorescence). F 5 tacggcat 3 3 atgc cgta 5 C. E. Heitsch, GA Tech 3

5 A fundamental problem Hybridization occurs between nonexact complements! F F 5 tacggcat 3 3 atgc cgta 5 5 atcggcta 3 3 atgc cgta 5 How to design sets of DNA codewords W i which hybridize only with their correct complements C i? C. E. Heitsch, GA Tech 4

6 Biomedical motivations Gene expression analysis on DNA surface arrays SNP detection by rolling circle amplification (RCA) C. E. Heitsch, GA Tech 5

DNA double crossover (DX) array and truncated octahedron

7 Biotechnological applications Rational synthesis of periodic matter Self-assembly of a biochip computer? DNA double crossover (DX) array and truncated octahedron from the laboratory of Prof. Ned Seeman, NYU. C. E. Heitsch, GA Tech 6

8 Designing DNA codewords Problem: Minimize all possible mishybridizations for {W i }, {C i }. Also satisfy biochemical constraints on melting temperature and free energy. F F Complements W i and C j for i j Reverse-complements W i and W j (or C i and C j ) for i j Inverted repeats W i with itself Previously approaches were either top-down or bottom-up. C. E. Heitsch, GA Tech 7

9 Top-down: Hamming distance and coding theory Suppose A = {, } and x, y A k. Let x = x x 2... x k and y = y y 2... y k for x i, y i A. Definition. Define the Hamming distance of x and y as d(x, y) = {i : i n, x i y i } Example. Let x =, y =. Then d(x, y) = 4. Hamming distance has been used as the basis for coding theoretic approaches (such as [4, 5]) to the DNA word design problem but with only limited success. C. E. Heitsch, GA Tech 8

10 Bottom-up: massive computation Courtesy of Michael Shortreed, Laboratory of Prof. Lloyd Smith, UW Madison. C. E. Heitsch, GA Tech 9

11 A mixed solution strategy Top-down: Consider the set of (4!) distinct De Bruijn sequences with n = 4 and k = 3 having length 4 3. F F (, ) (, ) (,) (, ) Bottom-up: Select B(4, 3) (uniformly at random) which minimize other mishybrizations and satisfy required biochemical constraints. C. E. Heitsch, GA Tech

12 Problematic repetitions Complement mishybridization W i and C j for i j F F 5 tacggcat 3 3 atgc cgta 5 5 atcggcta 3 3 atgc cgta 5 Control complements by preventing repeated substrings. C. E. Heitsch, GA Tech

13 De Bruijn sequences Definition. A circular n-ary De Bruijn sequence of order k, B(n, k), contains every n-ary string of length k exactly once. Binary strings of length 3:,,,,,,,. The two distinct B(2, 3): and There are (n!) nk n k different B(n, k) with length n k. C. E. Heitsch, GA Tech 2

14 De Bruijn graph B(n, k) is an Euler circuit in G(n, k) = (V, E). (, ) V = {,,..., n } k v i = v i v i2... v ik E = {(v i, v j ) v i v jk = v i v j } (, B (2, 3) = and B 2 (2, 3) = ) (,) (, ) C. E. Heitsch, GA Tech 3

15 BEST theorem Theorem. Let G be a directed graph on n vertices having an Euler circuit. Let A i be the number of spanning arborescences rooted at vertex v i and denote by r j the number of out-going edges for vertex v j. Then the number of distinct Eulerian circuits in G is A i n j= (r j )!. Definition. An arborescence is a rooted directed tree with all the edges pointing in the direction of the root. Definition. A spanning tree of a connected graph G is a subgraph which contains all the vertices and is a tree. C. E. Heitsch, GA Tech 4

16 Illustrative example, part I Vertices v, v, v 2, v 3 where vertex v i is labeled with integer i in binary. An Euler circuit: v 3 v 2 v v v v 2 v v 3 v 3 (, A spanning arborescence rooted at v 3 : E = {(v, v ), (v 2, v ), (v, v 3 )} ) (,) (, ) (, ) C. E. Heitsch, GA Tech 5

17 Tutte matrix tree theorem Theorem. Let G be a directed graph with vertices x,..., x n. Define the matrix M, or M(G), as follows: M = (m ij ) where m ii is the number of edges pointing away from x i, not counting loops, and for i j, m ij is the negative of the number of edges directed from x i to x j. Then number of A l, or A l (G), of spanning arborescences of G with x l as root is the minor of position (l, l) in M, i.e. the determinant of the n n principal submatrix of the matrix M obtained by dropping row l and column l. C. E. Heitsch, GA Tech 6

18 Illustrative example, part II M = A 3 (G) = = 2 (, ) (,) (, ) (, ) m A i (r j )! = 2 j= (2 )! = 2 = 2! 23 /2 3 = n! nk n k 2 2 j= C. E. Heitsch, GA Tech 7

19 Combinatorial explosion The number of B(n, k): , , 38, , , 8, C. E. Heitsch, GA Tech 8

20 Random walks on the De Bruijn graph (, ) (, ) (,) (, ) C. E. Heitsch, GA Tech 9

21 Random walks on the De Bruijn graph (, ) (, ) (, ) (,) C. E. Heitsch, GA Tech 9

22 Uniformly random De Bruijn sequences Total of (4!) distinct B(4, 3) with length 4 3. Randomized Algorithm. For vertex v i, i =,..., 5, chose a random permutation p i of {,, 2, 3}. Beginning at v, build a sequence by walking around G(4, 3) according to p i. Stop when a permutation is exhausted. Accept if sequence has length 64. The sequence is De Bruijn with probability /64; about 64 trials on average will generate a B(4, 3). C. E. Heitsch, GA Tech 2

23 Surface plasmon resonance (SPR) imaging Apparatus Technique C. E. Heitsch, GA Tech 2

24 Verification of complementary hybridization W = ctaacaatacgcgatg W 3 = gtgtatccgacatgtg W 2 = agtgtcacgttggaag C 2 = cttccaacgtgacact W W 2 W 3 Proof of principle experimental results support using random De Bruijn sequences to design DNA codewords. C. E. Heitsch, GA Tech 22

25 Acknowledgments Ming Li for his SPR Imaging results and DNA9 PowerPoint slides. Prof. Robert M. Corn, UC Irvine Chemistry Department. Full Genome on a Chip, Laboratory of Dr. Patrick O. Brown, Stanford University. HYBRIDIZATION NOTES - Melting and Annealing Temperature frank/hybnotes.html C. E. Heitsch, GA Tech 23

26 References [] K. J. Breslauer, R. Frank, H. Blöcker, and L. A. Marky. Predicting DNA duplex stability from the base sequence. Proc Natl Acad Sci U S A, 83(): , June 986. [2] N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch., Proc., 49: = Indagationes Math. 8, (946), 946. [3] S. M. Freier, R. Kierzek, J. A. Jaeger, N. Sugimoto, M. H. Caruthers, T. Neilson, and D. H. Turner. Improved free-energy parameters for predictions of RNA duplex stability. Proc Natl Acad Sci U S A, 83(24): , December 986. [4] A. G. Frutos, Q. Liu, A. J. Thiel, A. M. W. Sanner, A. E. Condon, L. M. Smith, and R. M. Corn. Demonstartion of a word design strategy for DNA computing on sur faces. Nucleic Acids Res., 25(23): , 997. [5] M. Li, H. J. Lee, A. E. Condon, and R. M. Corn. DNA word design strategy for creating sets of non-interacting sets of oligonucleotides for DNA microarrays. Langmuir, 8(3):85 82, 22. [6] J. G. Propp and D. B. Wilson. How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms, 27(2):7 27, th Annual ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA, 996). [7] W. Tutte and C. A. B. Smith. On unicursal paths in a network of degree 4. Amer. Math. Monthly, 48(i4): , April 94. C. E. Heitsch, GA Tech 24

27 [8] W. T. Tutte. Graph theory, volume 2 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2. With a foreword by Crispin St. J. A. Nash-Williams, Reprint of the 984 original. [9] T. van Aardenne-Ehrenfest and N. G. de Bruijn. Circuits and trees in oriented linear graphs. Simon Stevin, 28:23 27, 95. [] J. H. van Lint. Combinatorial Theory Seminar, Eindhoven University of Technology. Springer-Verlag, Berlin, 974. Lint, K. A. Post, C. P. J. Schnabel, J. J. Seidel, H. C. H. Timmermans, J. A. P. M. van de Wiel and Mathematics, Vol [] J. H. van Lint and R. M. Wilson. A course in combinatorics. Cambridge University Press, Cambridge, second edition, 2. [2] R. B. Wallace, J. Shaffer, R. F. Murphy, J. Bonner, T. Hirose, and K. Itakura. Hybridization of synthetic oligodeoxyribonucleotides to phi chi 7 4 DNA: the effect of single base pair mismatch. Nucl. Acids Res, 6(): , Aug 979. [3] J. G. Wetmur. DNA probes: applications of the principles of nucleic acid hybri dization. Crit Rev Biochem Mol Biol, 26(3-4):227 59, 99. C. E. Heitsch, GA Tech 25

Math 8803/4803, Spring 2008: Discrete Mathematical Biology

Math 8803/4803, Spring 2008: Discrete Mathematical Biology Math 883/483, Spring 28: Discrete Mathematical Biology Prof. Christine Heitsch School of Mathematics Georgia Institute of Technology Lecture 6 January 8, 28 Designing DNA codewords Problem:Minimize all