Math 8803/4803, Spring 2008: Discrete Mathematical Biology

Transcription

1 Math 8803/4803, Spring 2008: Discrete Mathematical Biology Prof. hristine Heitsch School of Mathematics eorgia Institute of Technology Lecture 12 February 4, 2008

2 Levels of RN structure Selective base pair hybridization structure and function UUU UUU U UUU UUU UU U Primary sequence secondary structure 3D molecule. E. Heitsch, Tech 1

3 Important biomathematical questions Prediction? UUU UUU U UUU UUU UU U nalysis? Design? How do RN sequences encode secondary structures?. E. Heitsch, Tech 2

4 Sequence to structure: a one-to-many mapping R = gcgga uuuagcuc aguuggga gagc g ccaga cugaa gaucugg agguc cugug uucgauc cacag a auucgc acca S 1 (R) = S 2 (R) = Hypothesis: RN sequences fold with minimal free energy.. E. Heitsch, Tech 3

5 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 RN molecules. In RN secondary structures as nested arcs R = cagcaucacauccgcgggguaaacgcuaaacgcu 318 W.R. Schmitt, M.S. Waterman 1 Discrete pplied Mathematics 51 (1994) (4 o-o I U I o-0 U (I-4 (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 RN 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]. s an example of secondary structure, we consider the sequence. E. Heitsch, Tech 4 UUUU

6 RN foldings as plane trees bstract 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?. E. Heitsch, Tech 5

7 How to compare RN secondary structures? d 30= [initially -43.4] 05Dec d = [initially -42.4] Dec d = [initially -42.3] Dec by D. Stewart and M. Zuker d = [initially -41.7] 05Dec d = [initially -42.4] 05Dec Secondary structures for the combinatorial RN sequence R = aaaa gggggg aaaa cccccc aaaa gggggg aaaa cccccc aaaa gggggg aaaa cccccc aaaa.. E. Heitsch, Tech 6

8 What is the space of possible RN configurations? d = [initially -43.4] Dec d = [initially -42.4] Dec d = [initially -42.3] Dec d = [initially -42.4] 05Dec d = [initially -41.7] 50 05Dec E. Heitsch, Tech 7

9 Metric: generalized distance Definition. Let X be a set. metric d on X is a function d : X X R where for all x, y, z X 1. d(x, y) 0, (nonnegativity) 2. d(x, y) = 0 if and only if x = y, (with nonnegativity gives positive definiteness) 3. d(x, y) = d(y, x), and (symmetry) 4. d(x, z) d(x, y) + d(y, z). (triangle inequality). E. Heitsch, Tech 8

10 Two metrics on strings Let be a finite set of symbols, and x, y +. Definition. If x = y, then the Hamming distance of x and y is d H (x, y) = {i : 1 i n, x i y i } Definition. The Levenshtein distance d L of x and y is the minimum number of insertions, deletions, or substitutions to transform x into y. Example. Let = {0, 1}, x = , and y = Then d H (x, y) = 6 and d L (x, y) = 2.. E. Heitsch, Tech 9

11 Norm: generalized size Definition. Let V be an n dimensional vector space over a field F. norm on V is a function : V R where for all v, u V and a F 1. v > 0 when v 0, (nonnegativity) 2. v = 0 if and only if v = 0, (with nonnegativity gives positive definiteness) 3. av = a v, and (positive homogeneity) 4. v + u v + u. (triangle inequality). E. Heitsch, Tech 10

12 Metrics and norms Let V be an n dimensional vector space over a field F with v, u, w V and a F. If V has a norm, define a metric d on V by d(v, u) = v u for all v, u V. onversely, if V has a metric d which satisfies 1. d(v, u) = d(v + w, u + w), and (translation invariance) 2. d(av, au) = a d(v, u), (homogeneity) then define a norm on V by v = d(v, 0).. E. Heitsch, Tech 11

13 Norms on spaces Let V be an n dimensional vector space over a field F with v = (v 1, v 2,..., v n ) V and v i F. For p = 1, 2,..., the L p -norm is defined as v p = ( n i=1 v i p ) 1/p with v = max i v i Example. Let F = R, n = 3, and v = (1, 2, 3). Then v 1 = 6, v 2 = 14, v 3 = 6 2/3, v 4 = 2 1/4 7,..., v = 3.. E. Heitsch, Tech 12

14 Metrics on spaces L 1 -norm Minkowski / rectilinear / Manhattan distance L 2 -norm Euclidean distance. L -norm hebyshev distance. E. Heitsch, Tech 13

15 cknowledgments Predicted RN foldings courtesy of Michael Zuker s mfold algorithm, available online through bioinfo.math.rpi.edu/ zukerm/. Taxicab geometry geometry.. E. Heitsch, Tech 14