Math 8803/4803, Spring 2008: Discrete Mathematical Biology

Transcription

1 Math 8803/4803, Spring 2008: Discrete Mathematical Biology Prof. Christine Heitsch School of Mathematics Georgia Institute of Technology Lecture 11 February 1, 2008

2 and give one secondary structure for this sequence in Fig. l(a), where the base pairs are indicated by dashes. This structure is referred to in the biological literature as a cloverleaf and is the secondary structure assumed by transfer RNA molecules. In RNA secondary structures as nested arcs R = cagcaucacauccgcgggguaaacgcuaaacgcu 318 W.R. Schmitt, M.S. Waterman 1 Discrete Applied Mathematics 51 (1994) (4 C o-o G A C A I U I C o-0 U A C A C C (I-4 G G G (1-o c O-4 Fig. 1. Two representations of secondary structure How many possible S(R) for a sequence R of length n? structures determine the shape and hence the function of these important biological molecules. The structure of RNA is utilized in regulating the expression of genes, in the assembly of protein molecules, and in many other fundamental biological processes. For an excellent general reference to molecular biology, see Lewin [3]. As an example of secondary structure, we consider the sequence C. E. Heitsch, GA Tech 1 CAGCAUCACAUCCGCGGGGUAAACGCU

3 RNA foldings as plane trees Abstract folded sequence to its skeleton : stacked base pairs edges, single-stranded regions vertices. 1 loop leaf vertex 4 loop vertex of degree stacked base pairs edge of weight 6 6 external loop root vertex How many possible plane trees T with n edges? C. E. Heitsch, GA Tech 2

4 Some graph theory A graph G is a set of vertices V, a set of edges E, and a mapping which associates each edge e E to an unordered pair of vertices {x, y} for x, y V. A weighted graph G has a real number associated to each edge e. A simple closed path is a walk in a graph G which begins and ends at the same vertex but is otherwise a sequence of distinct vertices and distinct edges. A connected graph that contains no nontrivial simple closed paths is a tree. A rooted tree has a unique identified vertex (the root). A child of a (parent) vertex is a vertex one edge farther away from the root. A subtree of a vertex in a rooted tree is a child and all its descendents. C. E. Heitsch, GA Tech 3

5 Plane / ordered / linear trees From Stanley s Enumerative Combinatorics, Vol. 2 : Definition. A plane tree T is a rooted tree whose subtrees at any vertex are linearly ordered. A vertex with k children has degree k. T n = {T n edges (and n + 1 vertices)} T 3 = {,,,, } C. E. Heitsch, GA Tech 4

6 1 edge = 2 half-edges Left half right half Label the half-edges of T T n with the string s = n Let e(i, j) be the edge with the ith and jth half-edges (i < j). By parity, there is an even number of k between i and j in s. C. E. Heitsch, GA Tech 5

7 Counting plane trees T 3 = {,,,, } We know that T 3 = 5. What about when n = 4? T always has an edge e(1, j) for j = 2k, 1 k n. e(1, j) e(1, 2) e(1, 4) e(1, 6) e(1, 8) # of T T 3 T 1 T 2 T 2 T 1 T 3 C. E. Heitsch, GA Tech 6

8 Catalan numbers e(1, j) e(1, 2) e(1, 4)... e(1, 2n) # of T T 0 T n 1 T 1 T n 2... T n 1 T 0 Let T n = C n. Then C n = n 1 j=1 C j C n j 1. C n = 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,... C n = 1 n + 1 ( ) 2n n C. E. Heitsch, GA Tech 7

9 Catalan formula derivation G(x) = n=0 C n x n xg(x) 2 G(x) + 1 = 0 x G(x) = 1 1 4x 2 = x 0 (1 4t) 1/2 dt ( ) 1/2 n = ( 2n n ) ( 4) n (1 4x) 1/2 = n=0 ( 2n n ) x n C. E. Heitsch, GA Tech 8

10 and give one secondary structure for this sequence in Fig. l(a), where the base pairs are indicated by dashes. This structure is referred to in the biological literature as a cloverleaf and is the secondary structure assumed by transfer RNA molecules. In RNA secondary structures as nested arcs R = cagcaucacauccgcgggguaaacgcuaaacgcu 318 W.R. Schmitt, M.S. Waterman 1 Discrete Applied Mathematics 51 (1994) (4 C o-o G A C A I U I C o-0 U A C A C C (I-4 G G G (1-o c O-4 Fig. 1. Two representations of secondary structure How many possible S(R) for a sequence R of length n? structures determine the shape and hence the function of these important biological molecules. The structure of RNA is utilized in regulating the expression of genes, in the assembly of protein molecules, and in many other fundamental biological processes. For an excellent general reference to molecular biology, see Lewin [3]. As an example of secondary structure, we consider the sequence C. E. Heitsch, GA Tech 9 CAGCAUCACAUCCGCGGGGUAAACGCU

11 Counting sets of base pairs Let R = b 1 b 2... b n and suppose that any base pair b i b j is possible for i + 1 < j. Let S(n) be the number of secondary structures for R. For n = 6, S(n) = 17. Theorem. [10] We have S(0) = 0, S(1) = 1, and for n 2, S(n + 1) = S(n) + S(n 1) + n 2 k=1 S(k)S(n k 1) C. E. Heitsch, GA Tech 10

12 Generating functions Lemma. [10] If φ(x) = k 0 S(k)xk, φ 2 (x)x 2 φ(x)(1 x x 2 ) + x = 0. Proof. Let φ(x) = y. Recall S(n + 2) S(n + 1) S(n) = n 1 S(n k)s(k). y 2 = n 1 ( S(n k)s(k))x n k=1 n=1 k=1 = n 1 (S(n + 2) S(n + 1) S(n))x n = S(n + 2)x n S(n + 1)x n S(n)x n n 1 n 1 n 1 = y x2 x x 2 y x x y C. E. Heitsch, GA Tech 11

13 Asymptotics Theorem. [10] As n, S(n) π n 3/2 ( ) n. 2 Proof. From before, F (x, y) = x 2 y 2 (1 x x 2 )y + x = 0. There is a theorem that asserts when F (x, y) = 0, r > 0, s > S(0), is the unique solution of F (r, s) = 0, F y (r, s) = 0, the n-the coefficient of φ(x), S(n) satisfies r S(n) 2 F x (r, s) 2πF yy (r, s) n 1/2 r n. Solving for r and s yields s = and 1 r = C. E. Heitsch, GA Tech 12

14 Counting subtypes of secondary structures Let P n,k be the set of secondary structures on b 1... b n with exactly k basepairs and set P (n, k) = P n,k. Theorem. [10] Set P (n, 0) = 1 for all n and P (n, k) = 0 for k n/2. Then for n 2, P (n + 1, k + 1) = P (n, k + 1) + n 1 k ( j=1 i=0 P (j 1, i)p (n j, k i)) Corollary. [10] P (n) = n/2 k=0 P (n, k) C. E. Heitsch, GA Tech 13

15 s(n + 1) = s(n) + C s(j - l)s(n -j), (1) j=l for n 3 2, and s(o) = s(l) = s(2) = 1. Asymptotic formulae for s(n) and related sequences determined by a recurrence relation generalizing the above are given by Stein and Waterman [S]. Let 9& Some be the set of secondary examples structures on [n] which have exactly k pairs, and let s(n,k) = lyn,,j. For example, the set.jy ~,~ is shown in Fig. 2. The numbers s(n, k) are also studied in [2], where it is shown that they satisfy the recurrence s(n+l,k+l)=s(h,k+l)+ 1 xs(j-l,i)s(n-j,k-i), 1 (2) P 6,2 : I I Fig. 2. The set Y,,,. Let L n,k be the set of plane trees having n vertices, k of which are not leaves. 320 W.R. Schmitt, M.S. Waterman / Discretr Applied Mathematics 51 (1994) L 5,3 : Fig. 3. The set Y5,,, for II 3 2, where s(n, 0) = 1, for all n, and s(n, k) = 0 for k 3 $I. Using vector notation, this recurrence can be written in a form identical to that of Eq. (1). Hence, the sequences (s(y1, k): k > 0) can be viewed as vector generalizations of Catalan numbers. It turns out that the numbers s(n, k) are much easier to compute than the s(n). Our main result is a remarkably simple formula for s(n, k), which is obtained by constructing a bijection from Y,,k onto a certain set of trees. A linear tree is a rooted tree together with a linear ordering on the set of children of each vertex in the tree. An isomorphism of linear trees is a root-preserving tree C. E. Heitsch, GA Tech 14

16 Another bijection between structures and trees W.R. Schmitt, MS. Waterman / Discrete Applied Mathematics 51 (1994) Theorem. Example. [a) {FZ\. Z\. [10] For all n, k 1, there exists a bijection (b) φ : S n+k 2,k 1 L n,k Fig. 4. A secondary structure and its corresponding tree W.R. Schmitt, MS. Waterman / Discrete Applied Mathematics 51 (1994) [a) {FZ\. Z\ (b) Fig. 4. A secondary structure and its corresponding tree Fig. 5. Labelling of a linear tree, determining corresponding secondary structure. It is straightforward to verify that the maps c$ and CI are inverses of one another. C. E. Heitsch, GA Tech 15 Theorem 2.2. The number of secondary structures over a sequence of length n having exactly k pairs is given by

17 More enumerative results Theorem. [10] For n, k 0, S(n, k) = 1 k ( n k k + 1 )( n k + 1 k 1 ). Corollary. [10] n k=1 ( )( ) 1 n k n k + 1 k k + 1 k π n 3/2 ( ) n 2 Corollary. [10] S(n + k, k) = S(2n k 1, n k 1) C. E. Heitsch, GA Tech 16

18 Acknowledgments Predicted RNA foldings courtesy of Michael Zuker s mfold algorithm, available online through bioinfo.math.rpi.edu/ zukerm/. Neil Sloane s On-Line Encyclopedia of Integer Sequences: njas/sequences/seis.html. C. E. Heitsch, GA Tech 17

19 References [1] N. Dershowitz and S. Zaks. Enumerations of ordered trees. Discrete Math., 31(1):9 28, [2] W. Fontana, D. Konings, P. F. Stadler, and P. Schuster. Statistics of RNA secondary structures. Biopolymers, 33(9): , Sept [3] I. L. Hofacker, P. Schuster, and P. F. Stadler. Combinatorics of RNA secondary structures. Discrete Appl. Math., 88(1-3): , [4] M. Nebel. Combinatorial properties of RNA secondary structures, [5] A. Orlitsky and S. S. Venkatesh. On edge-colored interior planar graphs on a circle and the expected number of RNA secondary structures. Discrete Appl. Math., 64(2): , [6] J. Riordan. Combinatorial identities. John Wiley & Sons Inc., New York, [7] W. R. Schmitt and M. S. Waterman. Linear trees and RNA secondary structure. Discrete Appl. Math., 51(3): , [8] R. P. Stanley. Enumerative combinatorics. Vol. 2. Cambridge University Press, Cambridge, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. [9] P. R. Stein and M. S. Waterman. On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math., 26(3): , [10] M. S. Waterman. Introduction to Computational Biology. Chapman & Hall/CRC, C. E. Heitsch, GA Tech 18